
\section{Introduction}

\paragraph{\textbf{Background}}

Jones discovery of a new polynomial links invariant \cite{Jones85} and its interpretation by Witten in the context of Topological Quantum Field Theory \cite{Wi2}, initiated fruitful interactions between Von Neumann algebras, quantum groups and low dimensional topology. Though initially stated in the context of sub-factors, the construction of Jones polynomials were soon reformulated in the context of small representations of the quantum group $U_q\mathfrak{sl}_2$ \cite{Turaev_YB}. Drinfel'd $R$-matrix \cite{DrinfeldqRMatrix} permits to associate to any small representation a solution of the Yang-Baxter equation which, in turn, induces representations of the braid groups. Using an appropriate trace, these representations induce invariants of framed links (see \cite{OhtsukiBook, ChariPressley, Kassel} for surveys). In \cite{GeerPatureauTuraev_Trace, GeerPatureau_LinksInv, CGPInvariants}, Costantino,  Geer,  Patureau-Mirand and Turaev generalized this construction by defining framed links invariants from some non-small, projective indecomposable highest and lowest modules of  $U_q\mathfrak{sl}_2$ by modifying the trace and introducing a refined version of $U_q\mathfrak{sl}_2$, namely the unrolled quantum group. These invariants are related to the Akutsu-Deguchi-Ohtsuki invariants of \cite{ADO_inv}. Concerning the family of $U_q\mathfrak{sl}_2$- indecomposable modules which are not highest and lowest weight, namely the cyclic and semi-cyclic modules, an approach to define links invariants was started by Kashaev and Reshetikhin in \cite{KashaevReshetikhin04,KashaevReshetikhin05} and continued in \cite{GeerPatureau_GLinks} using semi-cyclic modules and further developed in  \cite{BGPR_Biquandle} to include cyclic modules as well.

\vspace{2mm}
\par In parallel to these quantum groups constructions, Kashaev initiated in \cite{Kashaev_6jSymbol} an original way to produce solutions of the Yang-Baxter equation from the cyclic quantum dilogarithms and the Pentagon identity they satisfy \cite{FadeevKashaevQDilog, KashaevVolkov}. This leads to a knots invariant \cite{KashaevLinkInv} which was conjectured to be related to the hyperbolic volume of the exterior of the knot \cite{Kashaev97}, when the latter is hyperbolic (see also \cite{Zagier_QModForms} for a generalization). Murakami and Murakami proved in \cite{MM01} that Kashaev's $R$-matrix is gauge equivalent to the Drinfel'd $R$-matrix giving rise to the $N$-th Jones polynomial at $N$-th root of unity, hence that these two framed links invariants are related. Kashaev's techniques were refined by Baseilhac and Benedetti to produce new families of links and $3$-manifolds invariants \cite{BaseilhacBenedettiInvariant, BaseilhacBenedettiLinkInv, BaseilhacBenedetti05, BaseilhacBenedetti15, BaseilhacBenedetti_NonAmbiguousStructures}. 

\vspace{2mm}
\par Kashaev \cite{Kashaev98} and Chekhov and Fock \cite{ChekhovFock} independently defined a family of algebras, the quantum Teichm\"uller spaces, associated to a triangulated punctured surface. The quantum dilogarithms and the Pentagon identity naturally appear in quantum Teichm\"uller theory by means of change of coordinates. Bonahon and Wong defined in \cite{BonahonWongqTrace} a refinement of the quantum Teichm\"uller spaces, namely the balanced Chekhov-Fock algebras, and defined an algebra morphism, the quantum trace, between the Kauffman-bracket skein algebra of a closed punctured surface and the balanced Chekhov-Fock algebra. A generalization of the Kauffman-bracket skein algebras (and of the quantum trace) for punctured surfaces $\mathbf{\Sigma}$ with boundary were defined by L\^e in \cite{LeStatedSkein}, following \cite{BonahonWongqTrace}, under the name \textit{stated skein algebras}. We will denote them by $\mathcal{S}_{\omega}(\mathbf{\Sigma})$.

\vspace{2mm}
\par The relation between Drinfel'd braidings, arising from quantum groups and the $R$-matrix, with the Kashaev braidings arising from the cyclic quantum dilogarithms and quantum Teichm\"uller theory, remains quite mysterious. Indeed, Murakami and Murakami theorem in \cite{MM01}, which relates two such braidings, is based on brute force computations which shed no light on this relation. An upshot of this paper is to provide a more conceptual explanation, based on skein algebras and the quantum trace, on how quantum groups naturally arise in quantum Teichm\"uller theory and how Drinfel'd $R$-matrix is related to the change of coordinates in quantum Teichm\"uller theory. Moreover, we'll see that Kashaev's braiding operators provide natural replacement for the $R$-matrix in the case of cyclic and semi-cyclic representations which satisfy some colored Yang-Baxter equation, compare them to the braidings in  \cite{KashaevReshetikhin04,KashaevReshetikhin05} and provide explicit formulas using quantum hyperbolic geometry. We'll eventually sketch the definition of new links invariants based on these braidings.

\vspace{2mm}
\paragraph{\textbf{Main results}}

Let $\mathbb{D}_n$ be a disc with two punctures on its boundary and $n$ inner punctures. Denote by $\mathcal{S}_{\omega}(\mathbb{D}_1)$ and $\mathcal{Z}_{\omega}(\mathbb{D}_1)$ the stated skein algebra and the balanced Chekhov-Fock algebra, respectively, of $\mathbb{D}_1$ and let $\operatorname{Tr} : \mathcal{S}_{\omega}(\mathbb{D}_1) \rightarrow 
\mathcal{Z}_{\omega}(\mathbb{D}_1)$ denote the quantum trace.
\vspace{2mm}
\par The main original idea of this paper, and the subject of the second section,  is the observation that $\mathcal{S}_{\omega}(\mathbb{D}_1)$ has a natural structure of Hopf algebra, closely related to Bigelow construction in \cite{BigelowQGroups}, and that its image $\mathcal{O}_q[D(B)]:= \operatorname{Tr} \left( \mathcal{S}_{\omega}(\mathbb{D}_1) \right) \subset \mathcal{Z}_{\omega}(\mathbb{D}_1)$, through the quantum trace map, is a Hopf algebra isomorphic to the simply laced quantum group $\widetilde{U}_q\mathfrak{gl}_2$ (using the notations in \cite{BrownGoodearl}). More precisely, we prove the

\begin{theorem}\label{theorem1}
The algebra $\mathcal{O}_q[D(B)]$ is presented by the generators $K^{\pm 1/2}, L^{\pm 1/2}, E$ and $F$ together with the following relations (where $q:=\omega^{-4}$):

\begin{align*}
&EK^{1/2}= q^{-1}K^{1/2}E; \quad EL^{1/2}= qL^{1/2} E; \quad  FK^{1/2}=qK^{1/2}F; \quad FL^{1/2}= q^{-1}L^{1/2}F; \\
&xy=yx \mbox{, for all }x,y\in \{K^{\pm 1/2}, L^{\pm 1/2} \}; \quad K^{1/2}K^{-1/2}=L^{1/2}L^{-1/2}= 1; \\
&EF-FE= \frac{K-L}{q-q^{-1}}. 
\end{align*}

The Hopf algebra structure is described by the following formulas:

\begin{align*}
& \Delta(E)= 1\otimes E + E\otimes K; \quad \Delta(F)= F\otimes 1 +L\otimes F; \\
& \Delta(x)=x\otimes x \mbox{, for all } x\in \{K^{\pm 1/2}, L^{\pm 1/2} \}; \\
& \epsilon(E)=\epsilon(F)=0; \quad \epsilon(K^{\pm 1/2})= \epsilon(L^{\pm 1/2})= 1; \\
& S(E)=-EK^{-1}; \quad S(F)= -L^{-1}F; \quad S(x)=x^{-1}\mbox{, for all } x\in \{K^{\pm 1/2}, L^{\pm 1/2} \}.
\end{align*}

\end{theorem}

In the third section we classify the weight indecomposable representations of   $\mathcal{O}_q[D(B)]$, in the case where $\omega$ is a root of unity of odd order $N>1$, and determine which ones extend to a $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-module. The motivation is that the braiding operators we will define next are linear morphisms between tensor products of pairs of such $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-modules. A $\mathcal{O}_q[D(B)]$-module is called a weight module if is semi-simple as a module over the center of $\mathcal{O}_q[D(B)]$. Weight modules form a tensor full subcategory $\mathcal{C}$ of $\mathcal{O}_q[D(B)]-\mathrm{Mod}$ and a projective object of $\mathcal{C}$ will be called a projective weight module. Beware that a projective weight modules are not necessary projective objects of  $\mathcal{O}_q[D(B)]-\mathrm{Mod}$.


 Let $D(B)\subset \operatorname{SL}_2(\mathbb{C})\times \operatorname{SL}_2(\mathbb{C})$ be the subgroup of pairs $(g_-, g_+)$ of matrices such that $g_-$ is lower triangular and $g_+$ is upper triangular, and $\varphi : D(B) \rightarrow \operatorname{SL}_2(\mathbb{C})$ be the map defined by $\varphi((g_-, g_+)):= g_-^{-1}g_+$. Let $\widetilde{D(B)}$ be the set of triples $(g,t,h_{\partial})$ where $g\in D(B)$,  $t\in \mathbb{C}$ is such that $T_N(t)=\operatorname{Tr} (\varphi(g))$, where $T_N(X)$ represents the $N$-th Chebyshev polynomial of first kind, and let $h_{\partial}\in \mathbb{C}^*$ be such that $h_{\partial}^{-N}$ is equal to the left upper coefficient of $g_- g_+$. Let $\mathcal{Z}$ be the center of $\mathcal{O}_q[D(B)]$ and $\mathrm{SpecMax}(\mathcal{Z})$ be the set of characters $\chi : \mathcal{Z} \rightarrow \mathbb{C}$.

\begin{theorem}\label{theorem2} Suppose that $\omega$ is a root of unity of odd order $N>1$.
\begin{enumerate}
\item The sets  $\mathrm{MaxSpec}(\mathcal{Z})$ and  $\widetilde{D(B)}$ are in canonical bijection. Moreover the map sending a indecomposable weight $\mathcal{O}_q[D(B)]$-module to the associated character in  $\mathrm{MaxSpec}(\mathcal{Z})$, induces a bijection between the set of isomorphism classes of projective  indecomposable weight  $\mathcal{O}_q[D(B)]$-modules and the set  $\widetilde{D(B)}$.
\item Every simple $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-module is a projective  indecomposable weight $\mathcal{O}_q[D(B)]$-module. Conversely, a projective indecomposable weight  $\mathcal{O}_q[D(B)]$-module, corresponding to a triple $(g,t,h_{\partial})\in \widetilde{D(B)}$,  extends to a  $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-module if and only one has
\begin{itemize}
\item either $\varphi(g)$ is not triangular (cyclic representations), 
\item or $\varphi(g)$ is triangular not diagonal (semi-cyclic representations) and the diagonal elements of $\varphi(g)$ are not equal to $\pm 1$, 
\item or $\varphi(g)$ is scalar (scalar representations).
\end{itemize}
\item For $(g,t,h_{\partial}) \in \widetilde{D(B)}$ fix a corresponding projective indecomposable representation $V(g,t,h_{\partial})$. Then we have the following Clebsch-Gordan type formula:
$$ V(g_1,t_1,h_{\partial}^{(1)}) \otimes V(g_2,t_2,h_{\partial}^{(2)}) \cong \oplus_{T_N(t)=\operatorname{Tr}(\varphi(g_1g_2))} m_t V(g_1g_2, t, h_{\partial}^{(1)}h_{\partial}^{(2)}),$$
where $m_t \in \{1,2\}$ is the multiplicity of $t$ as a zero of the polynomial $T_N(X)-\operatorname{Tr}(\varphi(g_1g_2))$.
\end{enumerate}
\end{theorem}

The fourth section is devoted to the study of Kashaev's braiding operators derived from the change of coordinates in the quantum Teichm\"uller spaces. We first review and slightly reformulate standard constructions and properties using the stated skein algebras. Let us sketch the construction. Giving two triangulations $\Delta$ and $\Delta'$ of a punctured surface $\mathbf{\Sigma}$, there exists an isomorphism $\Phi_{\Delta', \Delta}^{\mathbf{\Sigma}}: \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta) \rightarrow \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta')$ (change of coordinates) between the division algebras of the balanced Chekhov-Fock algebras. For a mapping class $\phi \in \operatorname{Mod}(\mathbf{\Sigma})$, whose restriction to the boundary is the identity, one associates an isomorphism $\phi^* : \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \phi(\Delta)) \rightarrow \mathcal{Z}_{\omega}(\mathbf{\Sigma},\Delta)$. We'll associate to a mapping class $\phi$ a linear operator $L(\phi)$ defined as an intertwiner for the automorphism $\phi^* \circ \Phi_{\phi(\Delta), \Delta}^{\mathbf{\Sigma}}\in \mathrm{Aut}(\widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta))$ between two irreducible representations of the balanced Chekhov-Fock algebra. In the case of the punctured disc $\mathbb{D}_n$, we show that these operators can be organized into a  projective representation of some braid groupoid. 
Let $X_n:=\mathrm{Specm}(\mathcal{Z}(\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)))$ be the maximal spectrum of the center of the reduced stated skein algebra associated to $\mathbb{D}_n$ (it depends on the odd order $N$ of the root of unity $\omega$). The braid group $B_n$ acts algebraically on $X_n$ (see Section \ref{sec_mcg_rep}). 

We define in Section \ref{sec_mcg_rep} a groupoid $\mathcal{B}^{\operatorname{SL}_2}_n$ whose objects are the points of $X_n$ and whose morphisms between $x$ and $y$ are (a subset of) those elements $b\in B_n$ such that $b\cdot x =y$. 

\begin{theorem}\label{theorem_functor}
The change of coordinates isomorphisms induce, for each root of unity $\omega$ of odd order $N>1$,  a functor $L_{\omega}^{\mathrm{SL}_2} : \mathcal{B}^{\mathrm{SL}_2}_n \rightarrow \mathrm{PVect}$.
\end{theorem}


The functor $L_{\omega}^{\mathrm{SL}_2}$ should be thought as arising from the holonomy of some projective flat bundle over $X_n$.  In the particular case where $n=2$ and $\tau$ represents the half-twist exchanging the two inner punctures in the clockwise order,  an intertwiner $L_{\omega}^{\mathrm{SL}_2}(\tau)$ will be called a \textit{Kashaev braiding}. Following Kashaev, we will see that these braidings satisfy a colored Yang-Baxter equation derived from the Pentagon identity. By definition, a Kashaev braiding is an intertwiner for the automorphism $\mathscr{R}:=  \tau^* \circ \Phi_{\tau(\Delta_2), \Delta_2}^{\mathbb{D}_2}\in \mathrm{Aut}(\widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \Delta_2))$. The relation between Kashaev braidings and the braidings defined by Drinfel'd $R$-matrix is summarized in the following:

\begin{theorem}\label{theorem3}
\begin{enumerate}
\item The braidings defined by Kashaev-Reshetikhin in \cite{KashaevReshetikhin04,KashaevReshetikhin05},  and used in \cite{BGPR_Biquandle} to produce link invariants, involving two pairs of $U_q\mathfrak{sl}_2$-modules which extend to $\mathcal{Z}_{\omega}(\mathbb{D}_1)$, are Kashaev braidings.
\item The Drinfel'd braiding associated to a pair of scalar representations of $U_q\mathfrak{sl}_2$ is a Kashaev braiding.
\item When $\omega$ is not a root of unity, one has the following decomposition:
$$\mathscr{R} = \sigma \circ \mathrm{Ad}(q^{-\frac{H\otimes G}{2}}) \circ \mathrm{Ad}\left(\exp_q((q-q^{-1})E\otimes F) \right),$$
where we refer to Section $4.2.3$ for the definition of the right-hand-side (outer) automorphism.
\end{enumerate}
\end{theorem}

In particular, the second assertion of Theorem \ref{theorem3} recovers Murakami and Murakami result of \cite{MM01} relating Kashaev's $R$-matrix to the braiding of the $N$-th Jones polynomial at $N$-th root of unity and generalizes it to the braidings of the ADO invariants as well.


  \par The main achievement of the present paper is an explicit formula for generic Kashaev braidings at odd roots of unity given in Section $\ref{sec_ClosedFormula}$. To a Kashaev braiding $R:V_1\otimes V_2\rightarrow V_3\otimes V_4$ we will associate some classical shape parameters $z_1,\ldots, z_4 \in \mathbb{CP}^1$ and the braiding is called $4T$-triangulable if they all belong to $\mathbb{C}\setminus \{0,1\}$ (so it is a generic condition). Following Baseilhac and Benedetti \cite{BaseilhacBenedettiLocRep}, we introduce in Section \ref{sec_ClosedFormula} the notion of q-shape parameters $\mathbf{w}_i$ lifting the classical ones. They consist in triples $\mathbf{w}_i=(w_i,w_i',w_i'')$ subject to some quantum Thurston equation. We also introduce the notion of flattening parameters $(\alpha, \beta)\in \mathbb{C}^2$ and prove the 
  
  \begin{theorem} 
  A $4T$-triangulable Kashaev braiding with q-shape parameters $\mathbf{w}_i$ and flattening parameters $(\alpha, \beta)$ is given by the formula 
  $$ R= \frac{1}{d} \tau \circ  D_{\alpha, \beta} \circ  \Phi_{w'_4}( -w_4[D^{-2}A] \otimes [A^{-1}D^2]) \Phi_{w''_3}(-w_3'[D^{-2}A] \otimes A^{-1})\Phi_{w''_2}(-w_2'A \otimes [A^{-1}D^2]) \Phi_{w'_1}( -w_1A \otimes A^{-1}),$$
  where $\frac{1}{d}$ is a scalar which ensures that $\operatorname{det}(R)=1$, $\tau(x\otimes y)=y\otimes x$ is the flip, $D_{\alpha, \beta}\in \operatorname{End}(\mathbb{C}^N \otimes \mathbb{C}^N)$ and  $A, D\in \operatorname{End}(\mathbb{C}^N)$ are linear maps defined in canonical bases $\{v_i\}_{i}$ by 
  $$ D_{\alpha, \beta}\cdot v_i\otimes v_j = q^{(\alpha-2i)(\beta-2j)/2} v_i \otimes v_j, \quad D\cdot v_i = q^i v_i, \quad A\cdot v_i=v_{i-1},$$
  and $\Phi_w(X)$ is a polynomial related to the cyclic dilogarithms. 
  \end{theorem}



 In Section $5$, we sketch how the results of this paper can lead to the construction of new links invariants, which seems related to the Baseilhac-Benedetti Quantum Hyperbolic Invariants in \cite{BaseilhacBenedettiInvariant, BaseilhacBenedetti05, BaseilhacBenedetti15} and to the Blanchet-Geer-Patureau Mirand-Reshetikhin invariants defined in \cite{BGPR_Biquandle}. We also formulate some open problems concerning these invariants which are the object of present investigations in collaboration with S.Baseilhac.
 
 
  We postpone to the appendices two technical proofs which are not quite enlightening and might be safely skipped at first reading.




\vspace{2mm}
\paragraph{\textbf{Acknowledgments.}} The author is thankful to S.Baseilhac, C.Blanchet, F.Bonahon, F.Costantino, L.Funar,  A.Quesney, P.Roche,  J.Toulisse and V.Verchinine for useful discussions and to the University of South California, the Federal University of S\~ao Carlos, the University of S\~ao Paulo, the University of Montpellier and the Waseda University for their kind hospitality during the development of this work.  He acknowledges  support from the Coordena\c{c}\~ao de Aperfei\c{c}oamento de Pessoal de n\'ivel Superior (CAPES), from the GEometric structures And Representation varieties (GEAR) network, from  the Japanese Society for Promotion of Science (JSPS) and from the Centre National de la Recherche Scientifique (CNRS).  

\begin{notations} All along this paper, $\mathcal{R}$ represents a commutative unital ring, $\omega \in \mathcal{R}^{\times}$ an invertible element and we will write $A:=\omega^{-2}$ and $q:= \omega^{-4}$. 
The element $q$ is said \textit{generic} if $q^n\neq 1$ for all $n\geq 0$, \textit{i.e.} if $q$ is not a root of unity.
\end{notations}


\section{A quantum group in quantum Teichm\"uller theory}

\subsection{Skein algebras, balanced Chekhov-Fock algebras and quantum trace}

In this subsection, we briefly recall the definitions and main properties of stated skein algebras and balanced Chekhov-Fock algebras.

\subsubsection{Kauffman-bracket (stated) skein algebras}\label{sec_def_skein}

\begin{definition}
A \textit{punctured surface} is a pair $\mathbf{\Sigma}=(\Sigma,\mathcal{P})$ where $\Sigma$ is a compact oriented surface and $\mathcal{P}$ is a finite subset of $\Sigma$ which intersects non-trivially each boundary component. A \textit{boundary arc} is a connected component of $\partial \Sigma \setminus \mathcal{P}$. We write $\Sigma_{\mathcal{P}}:= \Sigma \setminus \mathcal{P}$. A punctured surface $\mathbf{\Sigma}=(\Sigma,\mathcal{P})$ is said \textit{closed} if $\Sigma$ has no boundary components (in which case $\Sigma_{\mathcal{P}}$ might not be closed !) and is said \textit{open} otherwise.
\end{definition}

\textbf{Definition of stated skein algebras}
\\ Before stating precisely the definition of stated skein algebras, let us sketch it informally. Given a punctured surface $\mathbf{\Sigma}$, the stated skein algebra $\mathcal{S}_{\omega}(\mathbf{\Sigma})$ is the quotient of the $\mathcal{R}$-module freely spanned by isotopy classes of stated tangles in $\Sigma_{\mathcal{P}}\times (0,1)$  by some local skein relations. The left part of Figure \ref{fig_statedtangle} illustrates such a stated tangle: each point of $\partial T \subset \partial \Sigma_{\mathcal{P}}$ is equipped with a sign $+$ or $-$ (the state). Here the stated tangle is the union of three stated arcs and one closed curve. In order to work with two-dimensional pictures, we will consider the projection of tangles in $\Sigma_{\mathcal{P}}$ as in the right part of Figure \ref{fig_statedtangle}; such a projection will be referred to as a diagram.


\begin{figure}[!h] 
\centerline{\includegraphics[width=6cm]{StatedTangle.eps} }
\caption{On the left: a stated tangle. On the right: its associated diagram. The arrows represent the height orders. } 
\label{fig_statedtangle} 
\end{figure} 



\par   A \textit{tangle} in $ \Sigma_{\mathcal{P}}\times (0,1)$   is a  compact framed, properly embedded $1$-dimensional manifold $T\subset \Sigma_{\mathcal{P}}\times (0,1)$ such that for every point of $\partial T \subset \partial \Sigma_{\mathcal{P}}\times (0,1)$ the framing is parallel to the $(0,1)$ factor and points to the direction of $1$.  Here, by framing, we refer to a thickening of $T$ to an oriented surface. The \textit{height} of $(v,h)\in \Sigma_{\mathcal{P}}\times (0,1)$ is $h$.  If $b$ is a boundary arc and $T$ a tangle, we impose that no two points in $\partial_bT:= \partial T \cap b\times(0,1)$  have the same heights, hence the set $\partial_bT$ is totally ordered by the heights. Two tangles are isotopic if they are isotopic through the class of tangles that preserve the boundary height orders. By convention, the empty set is a tangle only isotopic to itself.
 
\vspace{2mm}


\par Let $\pi : \Sigma_{\mathcal{P}}\times (0,1)\rightarrow \Sigma_{\mathcal{P}}$ be the projection with $\pi(v,h)=v$. A tangle $T$ is in \textit{generic position} if for each of its points, the framing is parallel to the $(0,1)$ factor and points in the direction of $1$ and is such that $\pi_{\big| T} : T\rightarrow \Sigma_{\mathcal{P}}$ is an immersion with at most transversal double points in the interior of $\Sigma_{\mathcal{P}}$. Every tangle is isotopic to a tangle in generic position. We call \textit{diagram}  the image $D=\pi(T)$ of a tangle in generic position, together with the over/undercrossing information at each double point. An isotopy class of diagram $D$ together with a total order of $\partial_b D=\partial D\cap b$ for each boundary arc $b$, define uniquely an isotopy class of tangle. When choosing an orientation $\mathfrak{o}(b)$ of a boundary arc $b$ and a diagram $D$, the set $\partial_bD$ receives a natural order by setting that the points are increasing when going in the direction of $\mathfrak{o}(b)$. We will represent tangles by drawing a diagram and an orientation (an arrow) for each boundary arc. When a boundary arc $b$ is oriented we assume that $\partial_b D$ is ordered according to the orientation. A \textit{state} of a tangle is a map $s:\partial T \rightarrow \{-, +\}$. A pair $(T,s)$ is called a \textit{stated tangle}. We define a \textit{stated diagram} $(D,s)$ in a similar manner.

 \vspace{2mm}
\par  Let $\mathcal{R}$ be a commutative associative unital ring, $\omega\in \mathcal{R}^{\times}$ an invertible element and write $A:=\omega^{-2}$.

\begin{definition}\label{def_stated_skein}\cite{LeStatedSkein} 
  The \textit{stated skein algebra}  $\mathcal{S}_{\omega}(\mathbf{\Sigma})$ is the  free $\mathcal{R}$-module generated by isotopy classes of stated tangles in $\Sigma_{\mathcal{P}}\times (0, 1)$ modulo the following relations \eqref{eq: skein 1} and \eqref{eq: skein 2}, 
  	\begin{equation}\label{eq: skein 1} 
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]	
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2,-] (-0.4,-0.52) -- (.4,.53);
\draw[line width=1.2,-] (0.4,-0.52) -- (0.1,-0.12);
\draw[line width=1.2,-] (-0.1,0.12) -- (-.4,.53);
\end{tikzpicture}
=A
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth] 
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2] (-0.4,-0.52) ..controls +(.3,.5).. (-.4,.53);
\draw[line width=1.2] (0.4,-0.52) ..controls +(-.3,.5).. (.4,.53);
\end{tikzpicture}
+A^{-1}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90]	
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2] (-0.4,-0.52) ..controls +(.3,.5).. (-.4,.53);
\draw[line width=1.2] (0.4,-0.52) ..controls +(-.3,.5).. (.4,.53);
\end{tikzpicture}
\hspace{.5cm}
\text{ and }\hspace{.5cm}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90] 
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\draw[line width=1.2,black] (0,0)  circle (.4)   ;
\end{tikzpicture}
= -(A^2+A^{-2}) 
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,rotate=90] 
\draw [fill=gray!45,gray!45] (-.6,-.6)  rectangle (.6,.6)   ;
\end{tikzpicture}
;
\end{equation}

\begin{equation}\label{eq: skein 2} 
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.7,-.75)  rectangle (.4,.75)   ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (0,-.3);
\draw[line width=1.2] (0.4,0.3) to (0,.3);
\draw[line width=1.1] (0,0) ++(90:.3) arc (90:270:.3);
\draw (0.65,0.3) node {\scriptsize{$+$}}; 
\draw (0.65,-0.3) node {\scriptsize{$+$}}; 
\end{tikzpicture}
=
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.7,-.75)  rectangle (.4,.75)   ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (0,-.3);
\draw[line width=1.2] (0.4,0.3) to (0,.3);
\draw[line width=1.1] (0,0) ++(90:.3) arc (90:270:.3);
\draw (0.65,0.3) node {\scriptsize{$-$}}; 
\draw (0.65,-0.3) node {\scriptsize{$-$}}; 
\end{tikzpicture}
=0,
\hspace{.2cm}
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.7,-.75)  rectangle (.4,.75)   ;
\draw[->] (0.4,-0.75) to (.4,.75);
\draw[line width=1.2] (0.4,-0.3) to (0,-.3);
\draw[line width=1.2] (0.4,0.3) to (0,.3);
\draw[line width=1.1] (0,0) ++(90:.3) arc (90:270:.3);
\draw (0.65,0.3) node {\scriptsize{$+$}}; 
\draw (0.65,-0.3) node {\scriptsize{$-$}}; 
\end{tikzpicture}
=\omega
\begin{tikzpicture}[baseline=-0.4ex,scale=0.5,>=stealth]
\draw [fill=gray!45,gray!45] (-.7,-.75)  rectangle (.4,.75)   ;
\draw[-] (0.4,-0.75) to (.4,.75);
\end{tikzpicture}
\hspace{.1cm} \text{ and }
\hspace{.1cm}
\omega^{-1}
\heightexch{->}{-}{+}
- \omega^{-5}
\heightexch{->}{+}{-}
=
\heightcurve.
\end{equation}
The product of two classes of stated tangles $[T_1,s_1]$ and $[T_2,s_2]$ is defined by  isotoping $T_1$ and $T_2$  in $\Sigma_{\mathcal{P}}\times (1/2, 1) $ and $\Sigma_{\mathcal{P}}\times (0, 1/2)$ respectively and then setting $[T_1,s_1]\cdot [T_2,s_2]=[T_1\cup T_2, s_1\cup s_2]$. Figure \ref{fig_product} illustrates this product.
\end{definition}

\par  For a closed punctured surface, $\mathcal{S}_{\omega}(\mathbf{\Sigma})$ coincides with the classical (Turaev's) Kauffman-bracket skein algebra.

\begin{figure}[!h] 
\centerline{\includegraphics[width=8cm]{dessin.eps} }
\caption{An illustration of the product in stated skein algebras.} 
\label{fig_product} 
\end{figure} 

\textbf{The reduced stated skein algebras}
\\ 
\par A connected diagram is called a \textit{closed curve} if it is closed and an \textit{arc} if it is open. For $p \in \mathcal{P} \cap \partial \Sigma$, a \textit{corner arc} at $p$ is an arc 
$\alpha_p$ such that there exists an arc $\alpha' \subset \partial \Sigma$ with $\partial \alpha_p=\partial \alpha'$, and such that $\alpha_p\cup \alpha'$ bounds a disc in $\Sigma$ whose only intersection with $\mathcal{P}$ is $\{p\}$ (see Figure \ref{fig_bad_arc}). A stated corner arc is called a \textit{bad arc} if the state are $-$ followed by $+$ while we go along the arc counterclockwise around the puncture $p$ as in Figure \ref{fig_bad_arc}.

\begin{figure}[!h] 
\centerline{\includegraphics[width=4cm]{badarc.eps} }
\caption{A bad arc.} 
\label{fig_bad_arc} 
\end{figure} 

\begin{definition}\label{def_reduced_skein}\cite{CostantinoLe19} The \textit{reduced stated skein algebra} $\overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})$ is the quotient of the stated skein algebra $\mathcal{S}_{\omega}(\mathbf{\Sigma})$ by the ideal generated by bad arcs.
\end{definition}

\textbf{Bases for stated skein algebras}
\\  A closed component of a diagram $D$ is trivial if it bounds an embedded disc in $\Sigma_{\mathcal{P}}$. An open component of $D$ is trivial if it can be isotoped, relatively to its boundary, inside some boundary arc. A diagram is \textit{simple} if it has neither double point nor trivial component. By convention, the empty set is a simple diagram. Let $\mathfrak{o}$ be the orientation of the boundary arcs of $\mathbf{\Sigma}$ induced from the orientation of $\Sigma$ and denote by $\leq_{\mathfrak{o}}$ the total orders induced on each boundary arc. A state $s: \partial D \rightarrow \{ - , + \}$ is $\mathfrak{o}-$\textit{increasing} if for any boundary arc $b$ and any points $x,y \in \partial_bD$, then $x<_{\mathfrak{o}} y$ implies $s(x)< s(y)$. Here we choose the convention $- < +$. 

\begin{definition}\label{def_basis}
\begin{enumerate}
\item We denote by $\mathcal{B}\subset \mathcal{S}_{\omega}(\mathbf{\Sigma})$ the set of classes of stated diagrams $(D,s)$ such that $D$ is simple and $s$ is $\mathfrak{o}$-increasing. 
\item We denote by $\overline{\mathcal{B}} \subset \overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})$ the set of classes of stated diagrams $(D,s)\in \mathcal{B}$ which do not contain any bad arcs.
\end{enumerate}
\end{definition}

\begin{theorem} 
\begin{enumerate}
\item \cite[Theorem $2.11$]{LeStatedSkein} The set $\mathcal{B}$ is a  basis of $\mathcal{S}_{\omega}(\mathbf{\Sigma})$. 
\item \cite[Theorem $7.1$]{CostantinoLe19} The set $\overline{\mathcal{B}}$ is a basis of $\overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})$.
\end{enumerate}
\end{theorem}

\textbf{Gluing maps}
\\ Let $a$, $b$ be two boundary arcs of $\mathbf{\Sigma}$ and let $\mathbf{\Sigma}_{|a\#b}$ be the punctured surface obtained from $\mathbf{\Sigma}$ by gluing $a$ and $b$. Denote by $\pi : \Sigma_{\mathcal{P}} \rightarrow (\Sigma_{|a\#b})_{\mathcal{P}_{|a\#b}}$ the projection and $c:=\pi(a)=\pi(b)$. Let $(T_0, s_0)$ be a stated framed tangle of ${\Sigma_{|a\#b}}_{\mathcal{P}_{|a\#b}} \times (0,1)$ transversed to $c\times (0,1)$ and such that the heights of the points of $T_0 \cap c\times (0,1)$ are pairwise distinct and the framing of the points of $T_0 \cap c\times (0,1)$ is vertical. Let $T\subset \Sigma_{\mathcal{P}}\times (0,1)$ be the framed tangle obtained by cutting $T_0$ along $c$. 
Any two states $s_a : \partial_a T \rightarrow \{-,+\}$ and $s_b : \partial_b T \rightarrow \{-,+\}$ give rise to a state $(s_a, s, s_b)$ on $T$. 
Both the sets $\partial_a T$ and $\partial_b T$ are in canonical bijection with the set $T_0\cap c$ by the map $\pi$. Hence the two sets of states $s_a$ and $s_b$ are both in canonical bijection with the set $\mathrm{St}(c):=\{ s: c\cap T_0 \rightarrow \{-,+\} \}$. 

\begin{definition}\label{def_gluing_map}
Let $i_{|a\#b}: \mathcal{S}_{\omega}(\mathbf{\Sigma}_{|a\#b}) \rightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma})$ be the linear map given, for any $(T_0, s_0)$ as above, by: 
$$ i_{|a\#b} \left( [T_0,s_0] \right) := \sum_{s \in \mathrm{St}(c)} [T, (s, s_0 , s) ].$$
\end{definition}

\begin{figure}[!h] 
\centerline{\includegraphics[width=8cm]{gluing_map.eps} }
\caption{An illustration of the gluing map $i_{a\#b}$.} 
\label{fig_gluingmap} 
\end{figure} 

\begin{theorem}\label{theorem_gluing}
\begin{enumerate}
\item (\cite[Theorem $3.1$]{LeStatedSkein}) The linear map $i_{|a\#b}: \mathcal{S}_{\omega}(\mathbf{\Sigma}_{|a\#b}) \rightarrow \mathcal{S}_{\omega}(\mathbf{\Sigma})$ is an injective morphism of algebras. Moreover the gluing operation is coassociative in the sense that if $a,b,c,d$ are four distinct boundary arcs, then we have $i_{|a\#b} \circ i_{|c\#d} = i_{|c\#d} \circ i_{|a\#b}$.
\item (\cite[Theorem $7.6$]{CostantinoLe19} ) The morphism $i_{|a\#b}$ induces, by passing to the quotient, an injective algebra morphism (still denoted by the same letter) $i_{|a\#b}: \overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma}_{|a\#b}) \rightarrow \overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})$.
\end{enumerate}
\end{theorem}

The image of the gluing map is characterized as follows. The \textit{bigon} $\mathbb{B}$ is the punctured surface made of a disc with two punctures on its boundary. It has two boundary arcs $a$ and $b$ and is generated by the stated arcs $\alpha_{\varepsilon \varepsilon}, \varepsilon, \varepsilon'=\pm$ made of an arc $\alpha$ linking $a$ to $b$ with state $\varepsilon$ on $\alpha \cap a$ and $\varepsilon'$ on $\alpha \cap b$. Consider a disjoint union $\mathbb{B} \bigsqcup \mathbb{B}$ of two bigons; by gluing together the boundary arc $b_1$ of the first bigon with the boundary arc $a_2$ of the second, one obtains a morphism $\Delta:=i_{b_1\# a_2} : \mathcal{S}_{\omega}(\mathbb{B}) \rightarrow \mathcal{S}_{\omega}(\mathbb{B})^{\otimes 2}$ which endows $\mathcal{S}_{\omega}(\mathbb{B})$ with a structure of Hopf algebra where $\Delta$ is the coproduct. It was proved independently in \cite{CostantinoLe19, KojuQuesneyClassicalShadows} that the resulting Hopf algebra is canonically isomorphic to $\mathcal{O}_q[\operatorname{SL}_2]$ (recall that $q:= \omega^{-4}$). 


Now consider a punctured surface $\mathbf{\Sigma}$ and a boundary arc $c$. By gluing a bigon $\mathbb{B}$ along $\mathbf{\Sigma}$ while gluing $b$ with $c$, one obtains a punctured surface isomorphic to $\mathbf{\Sigma}$, hence a map $\Delta_a^L:= i_{b\#c} :   \mathcal{S}_{\omega}(\mathbf{\Sigma}) \rightarrow  \mathcal{S}_{\omega}(\mathbb{B}) \otimes \mathcal{S}_{\omega}(\mathbf{\Sigma})$ which endows $ \mathcal{S}_{\omega}(\mathbf{\Sigma})$ with a structure of left $\mathcal{S}_{\omega}(\mathbb{B})$ comodule. Similarly, gluing $c$ with $a$ induces a right comodule morphism $\Delta_a^R := i_{c\#a} :  \mathcal{S}_{\omega}(\mathbf{\Sigma}) \rightarrow  \mathcal{S}_{\omega}(\mathbf{\Sigma})\otimes  \mathcal{S}_{\omega}(\mathbb{B}) $.
The following theorem characterizes the image of the gluing map and  was proved independently in \cite{CostantinoLe19} and \cite{KojuQuesneyClassicalShadows}.


\begin{theorem}\label{theorem_exactsequence}(\cite[Theorem $4.7$]{CostantinoLe19}, \cite[Theorem $1.1$]{KojuQuesneyClassicalShadows})
Let $\mathbf{\Sigma}$ be a punctured surface and $a,b$ two boundary arcs. The following sequence is exact: 
$$ 0 \to \mathcal{S}_{\omega}(\mathbf{\Sigma}_{|a\#b}) \xrightarrow{i_{a\#b}} \mathcal{S}_{\omega}(\mathbf{\Sigma}) \xrightarrow{\Delta^L_a - \sigma \circ \Delta^R_b} \mathcal{S}_{\omega}(\mathbb{B}) \otimes \mathcal{S}_{\omega}(\mathbf{\Sigma}), $$
where $\sigma(x\otimes y) := y\otimes x$. 
\end{theorem}
For further use, let us reformulate  Theorem \ref{theorem_exactsequence} in the case where $\omega=+1$. Let $X:= \mathrm{Specm}(\mathcal{S}_{+1}(\mathbf{\Sigma})), X_{a\#b}:=\mathrm{Specm}(\mathcal{S}_{+1}(\mathbf{\Sigma}_{|a\#b}))$ and $\pi_{a\#b}: X\rightarrow X_{a\#b}$ the regular map induced by $i_{a\#b}$. Identifying $\mathcal{S}_{+1}(\mathbb{B})$ with $\mathbb{C}[\operatorname{SL}_2]$, the Hopf comodule maps $\Delta_a^L$ and $\Delta_b^R$ define on $X$ a structure of $\operatorname{SL}_2(\mathbb{C})$ bimodule. For $x\in X$ and $g\in \operatorname{SL}_2(\mathbb{C})$, denote by $g\cdot x$ and $x\cdot g$ the left and right group actions. Define a structure of left $\operatorname{SL}_2(\mathbb{C})$ module on $X$ by the formula $g\bullet x := g\cdot x \cdot g^{-1}$ and denote by $\Delta_{a\#b} : \mathbb{C}[X] \rightarrow \mathbb{C}[\operatorname{SL}_2]\otimes \mathbb{C}[X]$ the associated coaction. Then Theorem \ref{theorem_exactsequence} identifies $\mathbb{C}[X_{a\#b}] \subset \mathbb{C}[X]$ with the subset of coinvariant vectors for $\Delta_{a\#b}$, therefore it implies the: 

\begin{corollary}\label{coro_GIT_skein}
The variety $X_{a\#b}$ is canonically isomorphic to the GIT quotient $X\mathbin{/\mkern-6mu/} \operatorname{SL}_2(\mathbb{C})$ and under this isomorphism, the map $\pi_{a\#b} : X\rightarrow X_{a\#b}$ is identified with the projection map $X\rightarrow X \mathbin{/\mkern-6mu/} \operatorname{SL}_2(\mathbb{C})$. In particular $\pi_{a\#b}$ is surjective and $\pi_{a\#b}(x)=\pi_{a\#b}(y)$ if and only if there exists $g\in \operatorname{SL}_2(\mathbb{C})$ such that $y=g\bullet x$.
\end{corollary}



\textbf{Triangulations}


\begin{definition}\label{def_triangulation}
\begin{itemize}
\item The \textit{triangle} $\mathbb{T}$ is a disc with three punctures on its boundary. By abuse of notations, we also denote by $\mathbb{T}$ the underlying surface made of a disc with the three punctures removed.
\item A punctured surface $\mathbf{\Sigma}$ is said \textit{triangulable} if is can be obtained from a disjoint union $\bigsqcup_{\mathbb{T}\in F(\Delta)} \mathbb{T}$ of triangles by gluing some pairs of boundary arcs together. An  \textit{unordered triangulation} is the data of the disjoint union $\mathbf{\Sigma}_{\Delta}:= \bigsqcup_{\mathbb{T}\in F(\Delta)} \mathbb{T}$ together with the pairs of boundary arcs that are glued together to get $\mathbf{\Sigma}$. The connected component of $\mathbf{\Sigma}_{\Delta}$ are called \textit{faces} and their set is denoted by $F(\Delta)$. The image of a boundary arc of $\mathbf{\Sigma}_{\Delta}$ by the gluing map $\pi : \bigsqcup_{\mathbb{T}\in F(\Delta)} \mathbb{T} \rightarrow \Sigma_{\mathcal{P}}$ is called an \textit{edge} of $\Delta$ and the set of edges is denoted by $\mathcal{E}(\Delta)$.
\item A \textit{topological triangulation} $\Delta$ of $\mathbf{\Sigma}$ is the data of an unordered triangulation together with a bijection $I_{\Delta} : \mathcal{E}(\Delta) \xrightarrow{\cong} \{1, 2,  \ldots, |\mathcal{E}(\Delta)| \}$ named \textit{indexing map}.
 A pair $(\mathbf{\Sigma}, \Delta)$ is called a \textit{triangulated punctured surface}. 
\end{itemize}
\end{definition}

\par The only connected punctured surfaces which are not triangulable are: the punctured surfaces without punctures, the sphere with one or two punctures; the disc with one or two punctures on its boundary.   
 Given a triangulated punctured surface, by composing the associated gluing maps, one obtains injective morphisms of algebras: 
$$i^{\Delta} : \mathcal{S}_{\omega}(\mathbf{\Sigma}) \hookrightarrow \otimes_{\mathbb{T} \in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T}) \quad, \mbox{ and }
i^{\Delta} : \overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma}) \hookrightarrow \otimes_{\mathbb{T} \in F(\Delta)} \overline{\mathcal{S}}_{\omega}(\mathbb{T}) . $$

\vspace{2mm}
\par \textbf{Chebyshev morphisms and the center of reduced stated skein algebras}

\begin{definition}
\begin{enumerate}
\item Given $p\in \mathcal{P}\cap \mathring{\Sigma}$ an inner puncture, the \textit{peripheral curve} $\gamma_p\in \mathcal{S}_{\omega}(\mathbf{\Sigma})$ is the class of a simple closed curve which bounds a disc whose intersection with $\mathcal{P}$ is $\{p\}$. 
\item Given $\partial$ a connected component of $\partial \Sigma$, the \textit{boundary elements} $\alpha_{\partial}^{\pm 1} \in \overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})$ are the classes in $\overline{\mathcal{B}}$ of the stated diagrams $(\alpha_{\partial}, s_{\pm})$ where $\alpha_{\partial}$ is the union of all the corner arcs associated to punctures in $\mathcal{P}\cap \partial$ and the state $s_{\pm}$ sends every elements of $\partial \alpha_{\partial}$ to $\pm$.
\end{enumerate}
\end{definition}

The class of a peripheral curve belongs to the center of the stated skein algebra $\mathcal{S}_{\omega}(\mathbf{\Sigma})$. The class of a boundary element belongs to the center of the reduced stated skein algebra $\overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})$ (but is not central in $\mathcal{S}_{\omega}(\mathbf{\Sigma})$). Moreover in $\overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})$, one has $\alpha_{\partial} \alpha_{\partial}^{-1} = 1$. We refer to \cite[Lemma $6.2$]{KojuAzumayaSkein} for proofs.
\vspace{2mm}
\par 
When $\omega$ is a root of unity, the center of $\mathcal{S}_{\omega}(\mathbf{\Sigma})$ contains more elements.

\begin{definition}  The $N$-th Chebyshev polynomial of first kind is the polynomial  $T_N(X) \in \mathbb{Z}[X]$ defined by the recursive formulas $T_0(X)=2$, $T_1(X)=X$ and $T_{n+2}(X)=XT_{n+1}(X) -T_n(X)$ for $n\geq 0$.
\end{definition}

\begin{theorem}\label{theorem_center_skein}[\cite{BonahonWong1} when $\partial \Sigma=\emptyset$, \cite[Theorem $1.2$]{KojuQuesneyClassicalShadows} in general]
	Suppose that $\omega$ is a root of unity of odd order $N>1$. There exists an embedding 
	\begin{equation*}
	j_{\mathbf{\Sigma}} : \mathcal{S}_{+1}(\mathbf{\Sigma}) \hookrightarrow \mathcal{Z}\left( \mathcal{S}_{\omega}(\mathbf{\Sigma}) \right)
	\end{equation*}
	of the (commutative) stated skein algebra with parameter $+1$ into the center of the stated skein algebra with parameter $\omega$. Moreover, the morphism $j_{\mathbf{\Sigma}}$  is characterized by the property that it sends a closed curve $\gamma$ to $T_N(\gamma)$ and a stated arc $\alpha_{\varepsilon \varepsilon'}$ to the class $\alpha^{(N)}_{\varepsilon \varepsilon'}$ of a stated tangle made of $N$ parallel copies of $\alpha_{\varepsilon \varepsilon'}$ pushed along the framing in a small neighborhood.
\end{theorem}
Note that when $\alpha$ has its two endpoints in distinct boundary arcs, one has $\alpha^{(N)}_{\varepsilon \varepsilon'}= (\alpha_{\varepsilon \varepsilon'})^N$ (at the power $N$).
We also refer to \cite{BloomquistLe20} for a generalization of Theorem \ref{theorem_center_skein} including root of unity of even orders and stated skein modules of general $3$-manifolds.
The Chebyshev morphism passes to the quotient in the reduced stated skein algebra and induces a map (still denoted by the same symbol) $j_{\mathbf{\Sigma}} : \overline{\mathcal{S}}_{+1}(\mathbf{\Sigma}) \hookrightarrow \mathcal{Z}\left( \overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma}) \right)$.

\begin{theorem}\label{theorem_center_sskein}[\cite[Theorem $6.3$]{KojuAzumayaSkein}]
For $\omega$ a root of unity of odd order $N>1$, the center of the reduced stated skein algebra $\overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})$ is generated by the image of the Chebyshev morphism together with the classes of the peripheral curves $\gamma_p$ associated to inner punctures and the boundary elements $\alpha_{\partial}^{\pm 1}$. More precisely, we have an isomorphism
$$ \mathcal{Z}\left( \overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma}) \right) \cong \quotient{ j_{\mathbf{\Sigma}}(\overline{\mathcal{S}_{+1}})[\gamma_{p_i}, \alpha_{\partial_j}^{\pm 1}] } {\left( T_N(\gamma_{p_i})=j_{\mathbf{\Sigma}}(\gamma_{p_i}), \alpha_{\partial_j}^{\pm N} = j_{\mathbf{\Sigma}}(\alpha_{\partial_j}^{\pm 1}), \alpha_{\partial_j}\alpha_{\partial_j}^{-1}=1 \right)}.$$
\end{theorem}

 Let $X:= \mathrm{Specm}(\mathcal{Z}(\overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})))$ denote the maximal spectrum of the center of the reduced stated skein algebra and let $X^0:= \mathrm{Specm}(\overline{\mathcal{S}}_{+1}(\mathbf{\Sigma}))$ be the maximal spectrum of the reduced stated skein algebra at $+1$ (see \cite{KojuTriangularCharVar, KojuQuesneyClassicalShadows} for a geometric interpretation of $X^0$). The Chebyshev morphism induces a regular map $p : X \rightarrow X^0$ and Theorem \ref{theorem_center_sskein} shows that $p$ is a finite (branched) covering. More precisely, a closed point $\chi$ in $X$ (which we see as a character over the center of $\overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})$) is described the data $\{\chi_0, t_p, h_{\partial} | p \in \mathcal{P}\cap \mathring{\Sigma}, \partial \in \pi_0(\partial \Sigma)\}$ where $\chi_0:=p(\chi) \in X^0$, $t_p:= \chi(\gamma_p)$ is such that $T_N(t_p)=\chi_0(\gamma_p)$ and $h_{\partial}:= \chi(\alpha_{\partial})$ is such that $h_{\partial}^N = \chi_0(\alpha_{\partial})$.

\subsubsection{Balanced Chekhov-Fock algebras}

\textbf{Definition of the balanced Chekhov-Fock algebras}
\\
Let $(\mathbf{\Sigma}, \Delta)$ be a triangulated punctured surface. For  $e$ and $e'$ two edges in $\mathcal{E}(\Delta)$, denote by $a_{e,e'}$ the number of faces $\mathbb{T}\in F(\Delta)$ such that $e$ and $e'$ lift to edges $\widetilde{e}, \widetilde{e}'$ of $\mathbb{T}$ and such that we pass from $\widetilde{e}$ to $\widetilde{e}'$ in the counter-clockwise direction in $\mathbb{T}$. The \textit{Weil-Petersson}  form $\left<\cdot, \cdot \right>: \mathcal{E}(\Delta)\times \mathcal{E}(\Delta)\rightarrow \mathbb{Z}$ is the skew-symmetric form defined  by $\left< e, e'\right>:= a_{e,e'}-a_{e',e}$.  Fix $\mathcal{R}$ a unital commutative ring and $\omega$ an invertible element. The quantum torus $\mathcal{T}_{\omega}(\mathbf{\Sigma}, \Delta)$ is the polynomial $\mathcal{R}$-module $\mathcal{R}[Z_e^{\pm1}, e\in \mathcal{E}(\Delta)]$ with product:
  $$ Z_e Z_{e'} = \omega^{2\left<e, e'\right>}Z_{e'}Z_e \mbox{, for }e,e'\in \mathcal{E}(\Delta).$$
  
  \par For a monomial $Z_{e_1}^{k_1}\ldots Z_{e_n}^{k_n}$, its \textit{Weyl ordering} is the element $[Z_{e_1}^{k_1}\ldots Z_{e_n}^{k_n}]:= \omega^{-\sum_{i<j} \left<e_i, e_j\right>k_i k_j} Z_{e_1}^{k_1}\ldots Z_{e_n}^{k_n}$. This element is invariant under permutations in the sense that if $\pi \in \mathbb{S}_n$, then $[Z_{e_{i_1}}\ldots Z_{e_{i_p}}]= [Z_{e_{\pi(i_1)}}\ldots Z_{e_{\pi(i_p)}}]$. For $\mathbf{k}\in \mathcal{E}(\Delta)^{\mathbb{Z}}$, we write $Z^{\mathbf{k}}:= [\prod_e Z_e^{\mathbf{k}(e)}]$.  A monomial   $Z^{\mathbf{k}}$ is \textit{balanced} if for each face of $\Delta$ with edges $a, b, c$ then $\mathbf{k}(a)+\mathbf{k}(b)+\mathbf{k}(c)$ is even. Note that it is equivalent to the requirement that there exists a multi-curve $\mathcal{C}\subset \Sigma_{\mathcal{P}}$, with $\partial \mathcal{C} \subset \partial \Sigma_{\mathcal{P}}$, whose intersection with an edge $e$ has the same parity than $k_e$. If $I_{\Delta} : \mathcal{E}(\Delta) \xrightarrow{\cong} \{1, 2,  \ldots, |\mathcal{E}(\Delta)| \}$ is the indexing map of $\Delta$, for $1\leq n \leq |\mathcal{E}(\Delta)|$, we write $Z_n:= Z_{I_{\Delta}^{-1}(n)}$.
  
  \begin{definition}\label{def_CF_algebra}\cite{BonahonWongqTrace}
  The \textit{balanced Chekhov-Fock algebra} $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ is the sub-algebra of $\mathcal{T}_{\omega}(\mathbf{\Sigma}, \Delta)$ spanned by balanced monomials. 
\end{definition}  

\textbf{Representation theory of the balanced Chekhov-Fock algebras}
\\ The balanced Chekhov-Fock algebras are  particular cases of a family of algebras named quantum tori. Let $\mathcal{Z}$ denote the center of $\mathcal{Z}_{\omega}(\mathbf{\Sigma})$ and $R$ be the rank of $\mathcal{Z}_{\omega}(\mathbf{\Sigma})$ as a $\mathcal{Z}$-module.
By a classical theorem (\cite[Proposition $7.2$]{DeConciniProcesiBook}) of DeConcini and Procesi,  quantum tori at roots of unity are Azumaya of constant rank. 
Therefore the algebra $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ are semi-simple and the character map $\mathrm{Irrep}\left(\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \right) \rightarrow \mathrm{MaxSpec}(\mathcal{Z})$, sending an isomorphism class of irreducible representation to its associated character on $\mathcal{Z}$, is a bijection. Moreover, every irreducible representations of $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ have dimension $\sqrt{R}$. Let us describe $\mathcal{Z}$ and give a formula for $R$.


\begin{definition}\label{def_central_elements}
\begin{enumerate}
\item  Let $p\in \mathcal{P}\cap \mathring{\Sigma}$ be an inner puncture. For each edge $e\in \mathcal{E}(\Delta)$, denote by $\mathbf{k}_p(e)\in \{0,1,2\}$ the number of endpoints of $e$ equal to $p$. The \textit{central inner puncture element} is  $H_p:=Z^{\mathbf{k}_p}\in \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$.
\item  Let  $\partial$ a connected component of $\partial \Sigma$. For each edge $e$, denote by $\mathbf{k}_{\partial}(e)\in \{0, 1, 2\}$ the number of endpoints of $e$ lying in $\partial$. The \textit{central boundary element} is $H_{\partial}:=Z^{\mathbf{k}_{\partial}}\in \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$. 
\item Suppose that $\omega$ is a root of unity of odd order $N>1$. The  linear map $j_{(\mathbf{\Sigma}, \Delta)} : \mathcal{Z}_{+1}(\mathbf{\Sigma}, \Delta) \rightarrow \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ sending a balanced monomial $Z^{\mathbf{k}}$ to $Z^{N\mathbf{k}}$, is an injective morphism of algebras whose image lies in the center $\mathcal{Z}$ of $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$. We call it the \textit{Frobenius morphism}.
\end{enumerate}
\end{definition}


\begin{theorem}\label{theorem_CFClassification}[\cite{BonahonWong2} when $\partial \Sigma=\emptyset$, \cite[Theorem $1.2$]{KojuQuesneyQNonAb} in general]
Suppose that $\omega$ is a root of unity of odd order $N>1$.
\begin{enumerate}
\item The center $\mathcal{Z}$ of $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ is generated by the image of the Frobenius morphism together with the central inner puncture and boundary elements and their inverses. 
\item For a punctured surface $\mathbf{\Sigma}=(\Sigma, \mathcal{P})$ with $\Sigma$ connected of genus $g$ and $n_{\partial}$ boundary components and $s:=|\mathcal{P}|$ punctures, the simple  $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$-modules have dimension $N^{3g-3+s+n_{\partial}}$.
\end{enumerate}
\end{theorem}

 Let $Y:= \mathrm{Specm}(\mathcal{Z}(\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)))$ denote the maximal spectrum of the center of the balanced Chekhov-Fock algebra and let $Y^0:= \mathrm{Specm}(\mathcal{Z}_{+1}(\mathbf{\Sigma}, \Delta))$ be the maximal spectrum of the balanced Chekhov-Fock algebra at $+1$ (see \cite{KojuQuesneyQNonAb} for a geometric interpretation of $Y^0$ related to spectral network). The Frobenius morphism induces a regular map $p : Y \rightarrow Y^0$ and Theorem \ref{theorem_CFClassification} shows that $p$ is a finite (branched) covering. More precisely, a closed point $\chi$ in $Y$  is described the data $\{\chi_0, h_p, h_{\partial} | p \in \mathcal{P}\cap \mathring{\Sigma}, \partial \in \pi_0(\partial \Sigma)\}$ where $\chi_0:=p(\chi) \in Y^0$, $h_p:= \chi(H_p)$ is such that $h_p^N=\chi_0(H_p)$ and $h_{\partial}:= \chi(H_{\partial})$ is such that $h_{\partial}^N = \chi_0(H_{\partial})$.



\vspace{2mm}
\par 
\textbf{Gluing maps}
\\ 
Let $(\mathbf{\Sigma}_{|a\#b}, \Delta_{|a\#b})$ be the triangulated punctured surface obtained from $(\mathbf{\Sigma}, \Delta)$ by gluing two boundary arcs together and call $c$ the edge of $ \Delta_{|a\#b}$ corresponding to $a$ and $b$.

\begin{definition}\label{def_gluing_CF} The injective algebra morphism $i_{a\#b} : \mathcal{Z}_{\omega}(\mathbf{\Sigma}_{|a\#b}, \Delta_{|a\#b}) \hookrightarrow \mathcal{Z}_{\omega}(\mathbf{\Sigma},\Delta)$ is defined by $i_{a\#b}(Z^{\mathbf{k}}) := Z^{\mathbf{k'}}$, where $\mathbf{k'}(a)=\mathbf{k'}(b):=\mathbf{k}(c)$ and $\mathbf{k'}(e)=\mathbf{k}(e)$, if $e\neq a,b$.
\end{definition}

Note that the gluing is coassociative in the sense that if $a,b,c,d$ are four boundary arcs, one has  $i_{|a\#b} \circ i_{|c\#d} = i_{|c\#d} \circ i_{|a\#b}$.
Since the triangle $\mathbb{T}$ has only one triangulation (up to reindexing), we simply denote by $\mathcal{Z}_{\omega}(\mathbb{T})$ its balanced Chekhov Fock algebra. 
Given a topological triangulation $\Delta$, the punctured surface is obtained from the disjoint union $\bigsqcup_{\mathbb{T}\in F(\Delta)} \mathbb{T}$ of triangles by gluing the triangles along the boundary arcs corresponding to the edges of the triangulation.  By composing the associated gluing maps, one obtains an injective morphism of algebras:
$$i^{\Delta} : \mathcal{Z}_{\omega}(\mathbf{\Sigma},\Delta) \hookrightarrow \otimes_{\mathbb{T} \in F(\Delta)} \mathcal{Z}_{\omega}(\mathbb{T}).$$

\par The images of the gluing maps for balanced Chekhov-Fock algebras are characterized in a similar manner than for the stated skein algebras. For $a$ a boundary arc of $\mathbf{\Sigma}$, define some left and right comodule maps $\Delta_a^L : \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \rightarrow \mathcal{R}[X^{\pm 1}] \otimes \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ and $\Delta_a^R : \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \rightarrow  \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \otimes \mathcal{R}[X^{\pm 1}] $ by the formulas $\Delta_a^L (Z^{\mathbf{k}}):= X^{\mathbf{k}(a)} \otimes Z^{\mathbf{k}}$ and $\Delta_a^R(Z^{\mathbf{k}}):= Z^{\mathbf{k}}\otimes X^{\mathbf{k}(a)}$. 
\begin{lemma}\label{lemma_exactseq_CF} The following sequence is exact:
$$ 0 \to \mathcal{Z}_{\omega}(\mathbf{\Sigma}_{|a\#b}, \Delta_{|a\#b}) \xrightarrow{i_{a\#b}} \mathcal{Z}_{\omega}(\mathbf{\Sigma},\Delta) \xrightarrow{\Delta_a^L - \sigma \circ \Delta_b^R} \mathcal{R}[X^{\pm 1}] \otimes \mathcal{Z}_{\omega}(\mathbf{\Sigma},\Delta).$$
\end{lemma}
\begin{proof}
It is a straightforward consequence of the definitions.
\end{proof}


Now, in the case where $\mathcal{R}=\mathbb{C}$ and $\omega=+1$, write $Y:= \mathrm{Specm}(\mathcal{Z}_{+1}(\mathbf{\Sigma}, \Delta))$,  $Y_{a\#b}:= \mathrm{Specm}(\mathcal{Z}_{+1}(\mathbf{\Sigma}_{|a\#b}, \Delta_{|a\#b}))$ and $\pi_{a\#b}: Y \rightarrow Y_{a\#b}$ the regular map associated to $i_{a\#b}$. The comodule maps $\Delta_a^L$ and $\Delta_b^R$ induce a structure of $\mathbb{C}^*$ bimodule on $Y$. They induce, in turn, a structure of left $\mathbb{C}^*$-module by the formula $z\bullet y:= z\cdot y \cdot z^{-1}$ for $z\in \mathbb{C}^*, y\in Y$ and the exact sequence of Lemma \ref{lemma_exactseq_CF} identifies $\mathbb{C}[Y_{a\#b}]$ with the subset of $\mathbb{C}[Y]$ of coinvariant vectors for the associated coaction, therefore one has the: 

\begin{corollary}\label{coro_GIT_CF}
The variety $Y_{a\#b}$ is canonically isomorphic to the GIT quotient $Y\mathbin{/\mkern-6mu/} \mathbb{C}^*$ and under this isomorphism, the map $\pi_{a\#b} : Y\rightarrow Y_{a\#b}$ is identified with the projection map $Y\rightarrow Y \mathbin{/\mkern-6mu/} \mathbb{C}^*$. In particular $\pi_{a\#b}$ is surjective and $\pi_{a\#b}(x)=\pi_{a\#b}(y)$ if and only if there exists $z\in \mathbb{C}^*$ such that $y=z\bullet x$.

\end{corollary}





\subsubsection{Quantum Trace map}

\textbf{Definition of the quantum trace}
\\ We now define an algebra morphism $\operatorname{Tr}_{\omega}^{\Delta}: \mathcal{S}_{\omega}(\mathbf{\Sigma}) \rightarrow \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ following \cite{BonahonWongqTrace, LeStatedSkein}. Consider the three arcs $\alpha, \beta, \gamma$ in $\mathbb{T}$ drawn in Figure \ref{fig_triangle} and denote $a$, $b$, $c$ the three boundary arcs of $\mathbb{T}$ as in Figure \ref{fig_triangle}. The skein algebra $\mathcal{S}_{\omega}(\mathbb{T})$ is generated by the stated arcs $\alpha_{\varepsilon \varepsilon'}, \beta_{\varepsilon \varepsilon'}, \gamma_{\varepsilon \varepsilon'}$ for $\varepsilon, \varepsilon \in \{-,+\}$. By \cite[Proposition $5.1$]{LeStatedSkein}, there exists a unique morphism of algebras $\operatorname{Tr}_{\omega}^{\mathbb{T}} : \mathcal{S}_{\omega}(\mathbb{T}) \rightarrow \mathcal{Z}_{\omega}(\mathbb{T})$ satisfying:
\begin{eqnarray*}
 \operatorname{Tr}_{\omega}^{\mathbb{T}}(\alpha_{\varepsilon \varepsilon'})= \left\{ 
 \begin{array}{ll} 
 0 & \mbox{, if }(\varepsilon, \varepsilon')=(-,+); \\
 \left[Z_c^{\varepsilon} Z_b^{\varepsilon'}\right] & \mbox{, else.}
 \end{array}
 \right.
 \\
  \operatorname{Tr}_{\omega}^{\mathbb{T}}(\beta_{\varepsilon \varepsilon'})= \left\{ 
 \begin{array}{ll} 
 0 & \mbox{, if }(\varepsilon, \varepsilon')=(-,+); \\
 \left[Z_a^{\varepsilon} Z_c^{\varepsilon'}\right] & \mbox{, else.}
 \end{array}
 \right.
 \\
  \operatorname{Tr}_{\omega}^{\mathbb{T}}(\gamma_{\varepsilon \varepsilon'})= \left\{ 
 \begin{array}{ll} 
 0 & \mbox{, if }(\varepsilon, \varepsilon')=(-,+); \\
 \left[Z_b^{\varepsilon} Z_a^{\varepsilon'}\right] & \mbox{, else.}
 \end{array}
 \right.
 \end{eqnarray*}

 \begin{figure}[!h] 
\centerline{\includegraphics[width=6cm]{triangle_generateurs.eps} }
\caption{The triangle, the arcs $\alpha, \beta, \gamma$ and the stated arc $\alpha_{\varepsilon \varepsilon'}$.} 
\label{fig_triangle} 
\end{figure} 

\begin{definition}\label{def_trace_map}
The \textit{quantum trace map} $\operatorname{Tr}_{\omega}^{\Delta}: \mathcal{S}_{\omega}(\mathbf{\Sigma}) \rightarrow \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ is the unique algebra morphism making commuting the following diagram:
$$\begin{tikzcd}
\mathcal{S}_{\omega}(\mathbf{\Sigma}) \arrow[r, hook, "i^{\Delta}"] \arrow[d, dotted, "\operatorname{Tr}_{\omega}^{\Delta}"] 
& \otimes_{\mathbb{T}\in F(\Delta)} \mathcal{S}_{\omega}(\mathbb{T}) \arrow[d, "\otimes_{\mathbb{T}}\operatorname{Tr}_{\omega}^{\mathbb{T}}"] \\
\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \arrow[r, hook, "i^{\Delta}"] 
& \otimes_{\mathbb{T}\in F(\Delta)}\mathcal{Z}_{\omega}(\mathbb{T})
\end{tikzcd}$$
\end{definition}


If $\omega$ is a root of unity of odd order $N>1$, the following diagram is commutative:

\begin{equation}\label{diagram_shadows}
\begin{tikzcd}
\mathcal{S}_{+1}(\mathbf{\Sigma}) \arrow[r, hook, "j_{\mathbf{\Sigma}}"] \arrow[d, "\operatorname{Tr}_{+1}^{\Delta}"] &
\mathcal{S}_{\omega}(\mathbf{\Sigma}) \arrow[d, "\operatorname{Tr}_{\omega}^{\Delta}"] \\
\mathcal{Z}_{+1}(\mathbf{\Sigma}, \Delta) \arrow[r, hook, "j_{(\mathbf{\Sigma}, \Delta)}"] &
\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)
\end{tikzcd}
\end{equation}

When $\partial \Sigma=\emptyset$, the quantum trace is injective (\cite[Proposition $29$]{BonahonWongqTrace}). For open punctured surfaces, one has the:

\begin{theorem}\label{theorem_kernel_qtrace}[\cite{CostantinoLe19}]
The kernel of the quantum trace is generated, as an ideal, by the classes of bad arcs. Therefore the quantum trace induces an injective algebra morphism $\operatorname{Tr}_{\omega}^{\Delta} : \overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma}) \hookrightarrow \mathcal{Z}_{\omega}(\mathbf{\Sigma})$.
\end{theorem}
Theorem \ref{theorem_kernel_qtrace} justifies the definition of the reduced stated skein algebra.


\vspace{2mm}
\par \textbf{Explicit formulas for the quantum trace:}
We now give a formula for the image of a closed curve and a stated arc by the quantum trace.
 For each face $\mathbb{T}$ of $\Delta$, consider the clockwise cyclic order on its edges and denote by $\overline{\mathbb{T}}$ its image (possibly self-folded) in $\Sigma_{\mathcal{P}}$. 
	Let $\alpha$ be either a closed curve or an arc between two boundary components; suppose it is placed in minimal and transversal position with the edges of $\Delta$. Denote by $Q_{\alpha}$ the set $\alpha\cap \mathcal{E}(\Delta)$.  
	For any two points $p$ and $q$ in $Q_{\alpha}$, we write $p\to q$ if: 
	\begin{itemize}
		\item 	$p$ and $q$ respectively belong to edges $e$ and $e'$ of a common face $\overline{\mathbb{T}}$; 
		\item   $e$ is the immediate predecessor of $e'$ for the cyclic order; and, 
		\item $p$ and $q$ are on a same connected component of $\alpha\cap \overline{\mathbb{T}}$.   
	\end{itemize}
	Let $\operatorname{St}(\alpha)$ be the set of maps $s:Q_{\alpha} \rightarrow \{-,+\}$ and $\operatorname{St}^a(\alpha)$ be the set of maps $s: Q_{\alpha} \rightarrow \{-,+\}$ such that $(s(p),s(q))\neq (-,+)$ whenever $p\to q$. 
	For each state $s$, let $\mathbf{k}_s: \mathcal{E}(\Delta)\to \mathbb{Z}$ be given by $\mathbf{k}_s(e)=\sum_{q\in e} s(q)$ for $e\in  \mathcal{E}(\Delta)$, where we identify $-$ with $-1$ and $+$ with $+1$.  If $\alpha$ is an arc with state  $s_0:\partial \alpha \to \{-,+\}$ let $\operatorname{St}^a(\alpha, s_0)$ be the subset of $\operatorname{St}^a(\alpha)$ of those states that agree with $s_0$. 
	
	\begin{proposition}\label{prop_QTrace}
	\begin{enumerate}
	\item If $\alpha$ is a closed curve, one has
	\begin{equation}\label{eq_trace_curve}
 \operatorname{Tr}_{\omega}^{\Delta} (\alpha) = \sum_{s\in \operatorname{St}^a(\alpha)} \omega^{n_s}Z^{\mathbf{k}_s}, 
 \end{equation}
 for some integers $n_s\in \mathbb{Z}$.
 \item If $(\alpha,s_0)$  is a bad arc, then $\operatorname{Tr}_{\omega}^{\Delta} (\alpha, s_0) =0$. If $(\alpha, s_0)$ is a stated arc which is not a bad arc, one has
 \begin{equation}\label{eq_trace_arc}
	\operatorname{Tr}_{\omega}^{\Delta} (\alpha,s_0) = \sum_{s\in \operatorname{St}^a(\alpha,s_0)} \omega^{n_s}Z^{\mathbf{k}_s}, 
	\end{equation}
 for some integers $n_s\in \mathbb{Z}$.
 \item Moreover, if $\alpha$ intersects each edge of triangulation at most once, one has $n_s=0$ in Equations \ref{eq_trace_curve}, \ref{eq_trace_arc}.
 \end{enumerate}
 \end{proposition}

\begin{lemma}\label{lemma_formula}
Let $(D,s)$ be a simple diagram in the triangle $\mathbb{T}$ which does not contain any bad arcs in its connected components. Let $\mathfrak{o}_{\mathbb{T}}$ be an orientation of the boundary arcs (= edges) $a,b,c$ of $\mathbb{T}$ such that two edges point towards the same puncture. Write $[D,s]_{\mathfrak{o}_{\mathbb{T}}} \in \mathcal{S}_{\omega}(\mathbb{T})$ the corresponding element, where the height orders are given by $\mathfrak{o}_{\mathbb{T}}$. Let $\mathbf{k}_{(D,s)} : \mathcal{E}(\Delta) \rightarrow \mathbb{Z}$ be defined by $\mathbf{k}_{(D,s)}(d):= \sum_{v\in D \cap d}s(v)$, for $d\in \{a,b,c\}$. Then there exists $n\in \mathbb{Z}$ such that:

\begin{equation}\label{eq_triangle}
\operatorname{Tr}_{\omega}^{\mathbb{T}} ([D,s]_{\mathfrak{o}_{\mathbb{T}}}) 
= \omega^n Z^{\mathbf{k}_{(D,s)}}.
\end{equation}
Moreover if $D$ is connected, then $n=0$.
\end{lemma}

\begin{proof}
First, if $D$ is connected, then Equation \eqref{eq_triangle} with $n=0$ is just a reformulation of the definition of $\operatorname{Tr}_{\omega}^{\mathbb{T}}$. Next consider $(D,s), (D_1,s_1), (D_2,s_2)$  three simple stated diagrams without bad arcs, such that $(D,s) = (D_1,s_1) \cup (D_2,s_2)$ and such that the underlying stated tangle of $[D_1,s_1]_{\mathfrak{o}_{\mathbb{T}}}$ lies at the top (in the height direction) of the underlying stated tangle of $[D_2,s_2]_{\mathfrak{o}_{\mathbb{T}}}$ so that $[D,s]_{\mathfrak{o}_{\mathbb{T}}} = [D_1,s_1]_{\mathfrak{o}_{\mathbb{T}}} [D_2,s_2]_{\mathfrak{o}_{\mathbb{T}}}$. Suppose that Equation \eqref{eq_triangle} holds for both $(D_1,s_1)$ and $(D_2,s_2)$ for some $n_1,n_2\in \mathbb{T}$. Then, since $\mathbf{k}_{(D,s)}= \mathbf{k}_{(D_1,s_1)}+\mathbf{k}_{(D_2,s_2)}$, one has $Z^{\mathbf{k}_{(D,s)}}=\omega^{m} Z^{\mathbf{k}_{(D_1,s_1)}}Z^{\mathbf{k}_{(D_2,s_2)}}$ for some $m\in \mathbb{Z}$, so Equation \eqref{eq_triangle} holds for $(D,s)$ with $n=n_1+n_2+m$. 

 Call \textit{ordered} a simple diagram $(D,s)$, with no bad arcs, such that if $(D,s)=\beta_1 \cup \ldots \cup \beta_k$ with $\beta_i$ stated arcs, then up to reindexing, the underlying tangle of $[D,s]_{\mathfrak{o}_{\mathbb{T}}}$ is such that $\beta_i$ lies at the top (in the height direction) of $\beta_{i+1}$ so that $[D,s]_{\mathfrak{o}_{\mathbb{T}}}=[\beta_1] \ldots [\beta_k]$. By the preceding discussion, and by induction on $k$, Equation \eqref{eq_triangle} holds for simple diagrams.
 
 Since we choose the orientations $\mathfrak{o}_{\mathbb{T}}$ such that two edges points towards the same puncture, one can label the three punctures of $\mathbb{T}$ by $p_1, p_2, p_3$ in such a manner that a boundary with endpoints $p_i, p_j$ with $i<j$ is oriented from $p_i$ towards $p_j$. Let $\delta \in \{ \alpha, \beta, \gamma \}$ the corner arc surrounding $p_2$. If $(D,s)$ is such that no connected component of $D$ is parallel to $\delta$, then $(D,s)$ is ordered so Equation \eqref{eq_triangle} holds. Else, we prove the result by decreasing induction on the number $n_{\delta}$ of connected component of $D$ parallel to $\delta$. If $n_{\delta}\geq 1$, choose the innermost component $\delta_0$ of $D$ parallel to $\delta$ and apply to $\delta_0$ one of the two skein relations 
\begin{equation*}
\heightcurve =
\omega^{-1}
\heightexch{->}{-}{+}
- \omega^{-5}
\heightexch{->}{+}{-}
, \hspace{.2cm} or \hspace{.2cm}
\heightcurveright
= \omega
\heightexchright{->}{+}{-}
-\omega^5 
\heightexchright{->}{-}{+}.
\end{equation*}
This gives an equation of the form $[D,s]_{\mathfrak{o}_{\mathbb{T}}}= \omega^{\pm 1} [D_1,s_1]_{\mathfrak{o}_{\mathbb{T}}} - \omega^{\pm 5} [D_2,s_2]_{\mathfrak{o}_{\mathbb{T}}}$ in $\mathcal{S}_{\omega}(\mathbb{T})$ illustrated in Figure \ref{fig_proof}. A case-by-case analysis, depending on the states on $\partial \delta_0$, shows that $(D_2,s_2)$ contains a bad arc and is such that $\operatorname{Tr}_{\omega}^{\mathbb{T}}([D_2,s_2]_{\mathfrak{o}_{\mathbb{T}}})=0$, whereas one has $\mathbf{k}_{(D_1,s_1)}=\mathbf{k}_{(D,s)}$. Therefore, using the induction hypothesis on $(D_1,s_1)$, one has $\operatorname{Tr}_{\omega}^{\mathbb{T}}([D,s]_{\mathfrak{o}_{\mathbb{T}}}) = \omega^{\pm 1}\operatorname{Tr}_{\omega}^{\mathbb{T}}([D_1,s_1]_{\mathfrak{o}_{\mathbb{T}}})=\omega^n Z^{\mathbf{k}_{(D,s)}}$ for some $n\in \mathbb{Z}$. We conclude by decreasing induction on $n_{\delta}$.


 \begin{figure}[!h] 
\centerline{\includegraphics[width=9cm]{relation_triangle.eps} }
\caption{An illustration of the skein relation $[D,s]_{\mathfrak{o}_{\mathbb{T}}}= \omega [D_1,s_1]_{\mathfrak{o}_{\mathbb{T}}} - \omega^{5} [D_2,s_2]_{\mathfrak{o}_{\mathbb{T}}}$. The diagram $(D_2,s_2)$ contains a bad arc colored in red.} 
\label{fig_proof} 
\end{figure} 


\end{proof}



\begin{proof} [Proof of Proposition \ref{prop_QTrace}]

Fix a labelling $p_1, \ldots, p_s$ of the punctures in $\mathcal{P}$ and consider the branching $\mathfrak{o}_{\Delta}$ of $\Delta$ where each edge $e\in \mathcal{E}(\Delta)$ with endpoints $p_i, p_j$ is oriented from $p_i$ towards $p_j$ when $i<j$. In the particular case where $\alpha$ is an arc with endpoints $v,w$ in the same boundary arc $a$ and such that $h(v)<h(w)$, we further impose that $a$ is oriented from $v$ towards $w$. The branching induces on each face $\mathbb{T}$ an orientation $\mathfrak{o}_{\mathbb{T}}$ of its edges as in Lemma \ref{lemma_formula}. Choose a tangle $T_{\alpha} \subset \Sigma_{\mathcal{P}}\times (0,1)$ with vertical framing, whose projection on $\Sigma_{\mathcal{P}}\times \{\frac{1}{2}\}$ is $\alpha$ and such that for each edge $e\in \mathcal{E}(\Delta)$ the heights of the elements of $T_{\alpha}\cap e$ are pairwise distinct and ordered according to the orientation of $e$ given by the branching $\mathfrak{o}_{\Delta}$. For a face $\mathbb{T} \in F(\Delta)$, the restriction of $\alpha$ to $\overline{\mathbb{T}}$ lift to a simple diagram $\alpha^{(\mathbb{T})}$ in $\mathbb{T}$. Let $s$ be either in $\operatorname{St}^a(\alpha, s_0)$, if $\alpha$ is an arc, or in $\operatorname{St}^a(\alpha)$ if $\alpha$ is a closed curve. The restriction of $s$ to $\partial \overline{\mathbb{T}}$ lifts to a state $s_{| \partial \mathbb{T}}$ on $\alpha^{(\mathbb{T})}$. By Definition \ref{def_gluing_map}, one has: 

\begin{align*}
& i^{\Delta}(\alpha, s_0) = \sum_{s \in \operatorname{St}(\alpha, s_0)} \otimes_{\mathbb{T}} [\alpha^{(\mathbb{T})}, s_{|\partial \mathbb{T}}]_{\mathfrak{o}_{\mathbb{T}}} \quad \mbox{, if }\alpha\mbox{ is an arc;} \\
& i^{\Delta}(\alpha) = \sum_{s \in \operatorname{St}(\alpha)} \otimes_{\mathbb{T}} [\alpha^{(\mathbb{T})}, s_{|\partial \mathbb{T}}]_{\mathfrak{o}_{\mathbb{T}}} \quad \mbox{, if }\alpha\mbox{ is a closed curve.}
\end{align*}

By Lemma \ref{lemma_formula}, for each face $\mathbb{T}$, there exists $n_{\mathbb{T}, s}\in \mathbb{Z}$ such that $\operatorname{Tr}_{\omega}^{\mathbb{T}}([\alpha^{(\mathbb{T})}, s_{| \partial \mathbb{T}}])= \omega^{n_{\mathbb{T}, s}} Z^{\mathbf{k}_s^{(\mathbb{T})}}$, where $k_s^{(\mathbb{T})}: \mathcal{E}(\mathbb{T})\rightarrow \mathbb{Z}$ is the balanced map defined by $k_s^{(\mathbb{T})}(e)=\sum_{v\in e\cap \alpha}s(v)$. 
Therefore, setting $n_s:= \sum_{\mathbb{T}} n_s^{(\mathbb{T})}$, one finds
 $$  \left( \otimes_{\mathbb{T}}\operatorname{Tr}_{\omega}^{\mathbb{T}} \right)\circ i^{\Delta} (\alpha, s_0) = \sum_{s \in \operatorname{St}^a(\alpha, s_0)} \omega^{n_s} \otimes_{\mathbb{T}} Z^{k_s^{(\mathbb{T})}} = i^{\Delta}\left( \sum_{s\in \operatorname{St}^a(\alpha, s_0)} \omega^{n_s}Z^{k_s}\right).$$
 The formula $\operatorname{Tr}_{\omega}^{\Delta} (\alpha, s_0) = \sum_{s\in \operatorname{St}^a(\alpha, s_0)} \omega^{n_s}Z^{\mathbf{k}_s}$ thus follows from Definition \ref{def_trace_map} and the case where $\alpha$ is a closed curve is handled similarly. Moreover, if $\alpha$ intersects each edge of the triangulation at most once, the integers $n_s^{(\mathbb{T})}$ are all null since each diagram $\alpha^{(\mathbb{T})}$ is connected, so $n_s=0$.

\end{proof}	

\par \textbf{Graduations: }
We will need one last property of the quantum trace: both the algebras $\mathcal{S}_{\omega}(\mathbf{\Sigma})$ and $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ are $\mathrm{H}_1(\Sigma_{\mathcal{P}}, \partial \Sigma_{\mathcal{P}}; \mathbb{Z}/2\mathbb{Z})$-graded and $\operatorname{Tr}_{\omega}^{\Delta}$ is graded of degree $0$. We now define these gradings. Fix an orientation $\mathfrak{o}$ of the boundary arcs of $\mathbf{\Sigma}$. A simple diagram $D$ induces a relative homology class $[D]\in \mathrm{H}_1(\Sigma_{\mathcal{P}}, \partial \Sigma_{\mathcal{P}}; \mathbb{Z}/2\mathbb{Z})$. 

\begin{definition}
For $\chi \in \mathrm{H}_1(\Sigma_{\mathcal{P}}, \partial \Sigma_{\mathcal{P}}; \mathbb{Z}/2\mathbb{Z})$, we define the subspace:
$$ \mathcal{S}_{\omega}^{(\chi)}(\mathbf{\Sigma}):= \operatorname{Span} \left( [D,s]\in \mathcal{B}^{\mathfrak{o}} | [D]=\chi \right) \subset \mathcal{S}_{\omega}(\mathbf{\Sigma}).$$
\end{definition}
It follows from the defining skein relations \eqref{eq: skein 1} and \eqref{eq: skein 2} that $ \mathcal{S}_{\omega}^{(\chi)}(\mathbf{\Sigma})$ does not depend on the choice of $\mathfrak{o}$ and that $\mathcal{S}_{\omega}(\mathbf{\Sigma})= \oplus_{\chi \in \mathrm{H}_1(\Sigma_{\mathcal{P}}, \partial \Sigma_{\mathcal{P}}; \mathbb{Z}/2\mathbb{Z})}  \mathcal{S}_{\omega}^{(\chi)}(\mathbf{\Sigma})$ is a graded algebra, \textit{i.e.} that $ \mathcal{S}_{\omega}^{(\chi_1)}(\mathbf{\Sigma})\cdot  \mathcal{S}_{\omega}^{(\chi_2)}(\mathbf{\Sigma})\subset  \mathcal{S}_{\omega}^{(\chi_1+\chi_2)}(\mathbf{\Sigma})$.
\vspace{2mm}
\par An edge $e\in \mathcal{E}(\Delta)$ defines a Borel-Moore homology class $[e]\in \mathrm{H}_1^c(\Sigma_{\mathcal{P}}; \mathbb{Z}/2\mathbb{Z})$. Given a balanced monomial $Z^{\mathbf{k}}\in \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$, there exists a unique class $[\mathbf{k}] \in \mathrm{H}_1(\Sigma_{\mathcal{P}}, \partial \Sigma_{\mathcal{P}}; \mathbb{Z}/2\mathbb{Z})$ whose algebraic intersection with $[e]$ is $\mathbf{k}(e)$ modulo $2$. 

\begin{definition}
For $\chi \in \mathrm{H}_1(\Sigma_{\mathcal{P}}, \partial \Sigma_{\mathcal{P}}; \mathbb{Z}/2\mathbb{Z})$, we define the subspace:
$$ \mathcal{Z}_{\omega}^{(\chi)}(\mathbf{\Sigma}, \Delta) := \operatorname{Span} \left( Z^{\mathbf{k}} | [\mathbf{k}]=\chi \right)\subset \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta). $$
\end{definition}

It follows from definitions that $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)=  \oplus_{\chi \in \mathrm{H}_1(\Sigma_{\mathcal{P}}, \partial \Sigma_{\mathcal{P}}; \mathbb{Z}/2\mathbb{Z})}  \mathcal{Z}_{\omega}^{(\chi)}(\mathbf{\Sigma}, \Delta)$ is a graded algebra. Note that the $0$-graded part $ \mathcal{Z}_{\omega}^{(0)}(\mathbf{\Sigma}, \Delta)$ is generated, as an algebra, by the elements $X_e:= Z_e^2$ for $e\in \mathcal{E}(\Delta)$.

\begin{lemma}\label{lemma_grading}
The quantum trace map is a $\mathrm{H}_1(\Sigma_{\mathcal{P}}, \partial \Sigma_{\mathcal{P}}; \mathbb{Z}/2\mathbb{Z})$ graded morphism of degree $0$.
\end{lemma}

\begin{proof}
Let $\alpha_{\varepsilon \varepsilon'} \in \mathcal{S}_{\omega}(\mathbf{\Sigma})$ be a stated arc. Using Equation \eqref{eq_trace_arc}, one has $\operatorname{Tr}_{\omega}^{\Delta}(\alpha_{\varepsilon \varepsilon'}) = \sum_{s\in \operatorname{St}^a(\alpha_{\varepsilon \varepsilon'})} \omega^{n_s}Z^{\mathbf{k}_s}$ where each relative homology class $[\mathbf{k}_s]$ is equal to the class $[\alpha]$. Similarly, for a closed curve $\gamma  \in \mathcal{S}_{\omega}(\mathbf{\Sigma})$, by Equation \eqref{eq_trace_curve} one has $\operatorname{Tr}_{\omega}^{\Delta}(\gamma) = \sum_{s\in \operatorname{St}^a(\gamma)} \omega^{n_s} Z^{\mathbf{k}_s}$ where each relative homology class $[\mathbf{k}_s]$ is equal to the class $[\gamma]$. We conclude using the fact that $\mathcal{S}_{\omega}(\mathbf{\Sigma}, \Delta)$ is generated by stated arcs and closed curves.

\end{proof}

\begin{lemma}\label{lemma_grading_generator}
For each class $\chi \in \mathrm{H}_1(\Sigma_{\mathcal{P}}, \partial \Sigma_{\mathcal{P}}; \mathbb{Z}/2\mathbb{Z})$, fix a balanced monomial $Z^{\mathbf{k}_{\chi}}$ such that $[\mathbf{k}_{\chi}]=\chi$. As an algebra, $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ is generated by the elements $X_e=Z_e^2$ for $ e\in \mathcal{E}(\Delta)$, together with the monomials $Z^{\mathbf{k}_{\chi}}$.
\end{lemma}

\begin{proof} This follows from the facts that the $0$-th graded part $\mathcal{Z}^{(0)}_{\omega}(\mathbf{\Sigma}, \Delta)$ is generated by the elements $X_e$ and
 that the multiplication by $Z^{\mathbf{k}_{\chi}}$ induces an isomorphism between  $\mathcal{Z}^{(0)}_{\omega}(\mathbf{\Sigma}, \Delta)$ and  $\mathcal{Z}^{(\chi)}_{\omega}(\mathbf{\Sigma}, \Delta)$, whose inverse is the multiplication by $Z^{-\mathbf{k}_{\chi}}$.
\end{proof}

\begin{remark} The $0$-th graded part  $\mathcal{Z}^{(0)}_{\omega}(\mathbf{\Sigma}, \Delta)$ is an exponential version of the algebra defined by Chekhov and Fock in \cite{ChekhovFock}.  \end{remark}

\subsection{The Hopf algebra $\mathcal{S}_{\omega}(\mathbb{D}_1)$}

\begin{definition}
For $n\geq 0$, the punctured surface $\mathbb{D}_n=(D^2, \mathcal{P}_n)$ is a disc  with two punctures $p_N$ and $p_S$ on its boundary and $n$ inner punctures $p_1,\ldots, p_n$. \end{definition}

 Inspired by Bigelow's construction in \cite{BigelowQGroups}, we now endow the algebra $\mathcal{S}_{\omega}(\mathbb{D}_1)$ with a Hopf algebra structure. First define an algebra morphism $\mathrm{db}: \mathcal{S}_{\omega}(\mathbb{D}_1) \rightarrow \mathcal{S}_{\omega}(\mathbb{D}_2)$ as follows. Let $B\subset D^2$ be an open disc whose intersection with $\mathcal{P}_1$ is $\{p_1\}$ and whose intersection with $\mathcal{P}_2$ is $\{p_1, p_2\}$. Let $(T,s)$ be a stated tangle in $(D^2\setminus{\mathcal{P}_1})\times (0,1)$ isotoped such that $T$ does not intersect $B\times (0,1)$. Define $\mathrm{db}([T,s])$ to be the class of the same stated tangle embedded in $(D^2\setminus{\mathcal{P}_2})\times (0,1)$. Said differently the (doubling) morphism $\mathrm{db}$ consists in replacing the inner puncture of $\mathbb{D}_1$ by two punctures. Let $b_L$ and $b_R$ be the two boundary arcs of $\mathbb{D}_1$. Consider the disjoint union $\mathbb{D}_1\bigsqcup \mathbb{D}_1'$ of two copies of $\mathbb{D}_1$. Once gluing the boundary arcs $b_R$ and $b_L'$ of $\mathbb{D}_1\bigsqcup \mathbb{D}_1'$, one get the punctured surface $\mathbb{D}_2$. Let $i_{b_R \#b_L'} : \mathcal{S}_{\omega}(\mathbb{D}_2) \hookrightarrow \mathcal{S}_{\omega}(\mathbb{D}_1)^{\otimes 2}$  denote the gluing map of Definition \ref{def_gluing_map}. Define the morphism $\Delta : \mathcal{S}_{\omega}(\mathbb{D}_1) \rightarrow \mathcal{S}_{\omega}(\mathbb{D}_1)^{\otimes 2}$ as the composition:
$$ \Delta : \mathcal{S}_{\omega}(\mathbb{D}_1) \xrightarrow{\mathrm{db}} \mathcal{S}_{\omega}(\mathbb{D}_2) \xrightarrow{i_{b_R\# b_L'}} \mathcal{S}_{\omega}(\mathbb{D}_1)^{\otimes 2}.$$

 \begin{figure}[!h] 
\centerline{\includegraphics[width=9cm]{Disc_coproduct.eps} }
\caption{The arcs $\alpha$ and $\beta$ in $\mathbb{D}_1$ and an illustration of the equality $\Delta(\alpha_{\varepsilon \varepsilon'})= \alpha_{\varepsilon +}\otimes \alpha_{+ \varepsilon'} + \alpha_{\varepsilon -}\otimes \alpha_{- \varepsilon'}$.} 
\label{fig_skein_D1} 
\end{figure} 

It follows from the coassociativity of the gluing maps (Theorem \ref{theorem_gluing}) that $\Delta$ is coassociative. The algebra $\mathcal{S}_{\omega}(\mathbb{D}_1)$ is generated by the stated arcs $\alpha_{\varepsilon \varepsilon'}$ and $\beta_{\varepsilon \varepsilon'}$ drawn in Figure \ref{fig_skein_D1} and, for $\delta\in \{\alpha, \beta\}$, one has:



$$
 \begin{pmatrix} \Delta (\delta_{++}) & \Delta (\delta_{+-}) \\ \Delta(\delta_{-+}) & \Delta(\delta_{--}) \end{pmatrix} 
 = 
 \begin{pmatrix} \delta_{++} & \delta_{+-} \\ \delta_{-+} & \delta_{--} \end{pmatrix} 
 \otimes 
 \begin{pmatrix} \delta_{++} & \delta_{+-} \\ \delta_{-+} & \delta_{--} \end{pmatrix}. 
$$
 Further define the morphisms $\epsilon : \mathcal{S}_{\omega}(\mathbb{D}_1) \rightarrow \mathbb{C}$ and $S: \mathcal{S}_{\omega}(\mathbb{D}_1) \rightarrow \mathcal{S}_{\omega}(\mathbb{D}_1)$ by the formulas for $\delta\in \{\alpha, \beta\}$:
 
$$
 \begin{pmatrix} \epsilon(\delta_{++}) & \epsilon(\delta_{+-}) \\ \epsilon(\delta_{-+}) & \epsilon(\delta_{--}) \end{pmatrix} =
\begin{pmatrix} 1 &0 \\ 0& 1 \end{pmatrix}  
\text{ and }
\begin{pmatrix} S(\delta_{++}) & S(\delta_{+-}) \\ 	S(\delta_{-+}) & S(\delta_{--}) \end{pmatrix} 
	= 
	\begin{pmatrix} \delta_{--} & -q\delta_{+-} \\ -q^{-1}\delta_{-+} & \delta_{++} \end{pmatrix} .
$$

\begin{lemma} The coproduct $\Delta$, the counit $\epsilon$ and the antipode $S$ endow $\mathcal{S}_{\omega}(\mathbb{D}_1)$ with an Hopf algebra structure. 
\end{lemma}

\begin{proof} This follows from straightforward computations. \end{proof}



\begin{proposition}\label{prop_pres_skein}
The algebra $\mathcal{S}_{\omega}(\mathbb{D}_1)$ is presented by the generators  $\alpha_{\varepsilon \varepsilon'}, \beta_{\varepsilon \varepsilon'}$, for $\varepsilon , \varepsilon'\in \{-,+\}$, together with the following relations, where $\varepsilon, \varepsilon' \in \{-,+\}$:


\begin{align}
\left.
\begin{array}{llll}
x_{++}x_{+-} &= q^{-1}x_{+-}x_{++} & x_{++}x_{-+}&=q^{-1}x_{-+}x_{++} 
\\ x_{--}x_{+-} &= q x_{+-}x_{--} & x_{--}x_{-+}&=q x_{-+}x_{--} 
\\ x_{++}x_{--}&=1+q^{-1}x_{+-}x_{-+} &  x_{--}x_{++}&=1 + q x_{+-}x_{-+} 
\\ x_{-+}x_{+-}&=x_{+-}x_{-+} & &
\end{array}
\right\} \mbox{, for }x \in \{\alpha, \beta\}; \label{eq5} \\
\alpha_{\varepsilon \varepsilon'} \beta_{\varepsilon \varepsilon'}=\beta_{\varepsilon \varepsilon'} \alpha_{\varepsilon \varepsilon'}; \quad 
\beta_{+-}\alpha_{-+}=\alpha_{-+}\beta_{+-}; \label{eq6} \\
\beta_{\varepsilon -}\alpha_{\varepsilon +}=q\alpha_{\varepsilon +}\beta_{\varepsilon -}; \quad
\alpha_{- \varepsilon}\beta_{+\varepsilon}= q \beta_{+\varepsilon} \alpha_{- \varepsilon}; \label{eq7} \\
q^{-1}\beta_{\varepsilon +}\alpha_{\varepsilon -} - \alpha_{\varepsilon -}\beta_{\varepsilon +}= - (q-q^{-1})\alpha_{\varepsilon +}\beta_{\varepsilon -};\label{eq8} \\
q^{-1}\alpha_{+ \varepsilon}\beta_{-\varepsilon} - \beta_{-\varepsilon}\alpha_{+ \varepsilon}= -(q-q^{-1})\beta_{+ \varepsilon}\alpha_{-\varepsilon}; \label{eq9}\\
\beta_{--}\alpha_{++}-\alpha_{++}\beta_{--} = (q-q^{-1})\beta_{+-} \alpha_{-+}; \label{eq10}\\
\alpha_{--}\beta_{++} - \beta_{++}\alpha_{--} = (q-q^{-1})\alpha_{-+}\beta_{+-}; \label{eq11}\\
\beta_{-+}\alpha_{+-}-\alpha_{+-}\beta_{-+}= -(q-q^{-1})(\alpha_{++}\beta_{--} - \beta_{++}\alpha_{--}).\label{eq12}
\end{align}


\end{proposition}
 
 The proof of Proposition \ref{prop_pres_skein} is postponed to Appendix A.


\begin{remark} 
The Hopf algebra $\mathcal{S}_{\omega}(\mathbb{D}_1)$ is very similar to the Hopf algebra defined by Bigelow in \cite{BigelowQGroups} where our parameter $q$ is replaced by $-q$ in \cite{BigelowQGroups} and our generators $\alpha_{++}$, $\alpha_{+-}$, $\alpha_{-+}$, $\alpha_{--}$, $\beta_{++}$, $\beta_{+-}$, $\beta_{-+}$, $\beta_{--}$ are replaced by $k', e, e_0, k, l, f_0, f, l'$ respectively in \cite{BigelowQGroups}. Using this correspondence, the reader will find in \cite{BigelowQGroups} a nice topological interpretation of both the antipode and  the equality $\mu \circ (S\otimes id)\circ \Delta = \eta\circ \epsilon$.
\end{remark}









\subsection{The Hopf algebra $\mathcal{O}_q[D(B)]$}

Here and in all the rest of the paper, we suppose that the element $q-q^{-1}$ is invertible in $\mathcal{R}$.

\begin{notations}
Let $\Delta_1$ denote the topological triangulation of $\mathbb{D}_1$ drawn in Figure \ref{fig_triangulation_disc}. For simplicity, we will write $\mathcal{Z}_{\omega}(\mathbb{D}_1):= \mathcal{Z}_{\omega}(\mathbb{D}_1, \Delta_1)$ and $\operatorname{Tr}:= \operatorname{Tr}_{\omega}^{\Delta_1}: \mathcal{S}_{\omega}(\mathbb{D}_1) \rightarrow \mathcal{Z}_{\omega}(\mathbb{D}_1)$.
\end{notations}

 \begin{figure}[!h] 
\centerline{\includegraphics[width=2.5cm]{triangulation_disc.eps} }
\caption{A topological triangulation of $\mathbb{D}_1$.} 
\label{fig_triangulation_disc} 
\end{figure} 


\begin{definition} We denote by $\mathcal{O}_q[D(B)]$ the Hopf algebra $\mathcal{O}_q[D(B)]:= \operatorname{Tr}\left( \mathcal{S}_{\omega}(\mathbb{D}_1) \right) \subset \mathcal{Z}_{\omega}(\mathbb{D}_1)$. We also define the following elements of $\mathcal{O}_q[D(B)]$ (recall that $A=\omega^{-2}$, $q=A^2$ and $X_i=Z_i^2$):
\begin{align*}
&K^{1/2}:= \operatorname{Tr}(\alpha_{--})= [Z_1Z_3Z_4]^{-1}, K^{-1/2}:= \operatorname{Tr}(\alpha_{++}) = [Z_1Z_3Z_4], K^{\pm 1}:= (K^{\pm 1/2})^2, \\
&L^{1/2}:=  \operatorname{Tr}(\beta_{--}) = [Z_1Z_2Z_4]^{-1}, L^{-1/2}:= \operatorname{Tr}(\beta_{++}) = [Z_1Z_2Z_4], L^{\pm 1}:= (L^{\pm 1/2})^2, \\
&E:= \frac{-A}{q-q^{-1}} \operatorname{Tr}(\alpha_{+-}\alpha_{--}) = \frac{-1}{q-q^{-1}} (X_4^{-1} + [X_3X_4]^{-1}), \\
&F:= \frac{A^{-1}}{q-q^{-1}} \operatorname{Tr}(\beta_{--} \beta_{-+}) = \frac{1}{q-q^{-1}} (X_1^{-1} + [X_1X_2]^{-1}).
\end{align*}
\end{definition}




\begin{proof}[Proof of Theorem \ref{theorem1}]
Note that $\mathcal{O}_q[D(B)]$ is isomorphic to the reduced stated skein algebra $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_1)$. The fact that the elements $E,F,K^{\pm 1/2}, L^{\pm 1/2}$ generate the algebra $\mathcal{O}_q[D(B)]$ follows from the facts that the elements $\alpha_{\varepsilon \varepsilon'}, \beta_{\varepsilon \varepsilon'}$ generate $\mathcal{S}_{\omega}(\mathbb{D}_1)$ together with the equalities $\operatorname{Tr}(\alpha_{-+})=0$ and $\operatorname{Tr}(\beta_{+-})=0$. While passing the set of relations of $\mathcal{S}_{\omega}(\mathbb{D}_1)$ in Proposition \ref{prop_pres_skein} to the quotient by the ideal generated by the bad arcs $\alpha_{-+}$ and $\beta_{+-}$, one obtains the above set of relations for $\mathcal{O}_q[D(B)]$, hence they induce a presentation of  $\mathcal{O}_q[D(B)]$ by Theorem \ref{theorem_kernel_qtrace}. The formulas for the coproduct, counit and antipode follow from straightforward computations. For instance, one has:
\begin{eqnarray*}
\Delta(E) &=&  \frac{-A}{q-q^{-1}}\operatorname{Tr} \left(\Delta(\alpha_{+-})\Delta(\alpha_{--})\right) \\
&=&  \frac{-A}{q-q^{-1}}\operatorname{Tr} \left( \alpha_{+-}\otimes \alpha_{--} + \alpha_{++}\otimes \alpha_{+-} \right)\operatorname{Tr}(\alpha_{--})^{\otimes 2} \\
 &=& \frac{-A}{q-q^{-1}} \left( \operatorname{Tr}(\alpha_{+-}\alpha_{--})\otimes \operatorname{Tr}(\alpha_{--})^{ 2} + \operatorname{Tr}(\alpha_{++}\alpha_{--})\otimes \operatorname{Tr}(\alpha_{+-}\alpha_{--}) \right) \\
 &=& E\otimes K +1\otimes E. \\
 S(E) &=& \frac{-A}{q-q^{-1}} \operatorname{Tr}( S(\alpha_{--}) S(\alpha_{+-}) ) =  \frac{-A}{q-q^{-1}} \operatorname{Tr}(\alpha_{++}) (-q\operatorname{Tr}(\alpha_{+-})) \\
 &=& \frac{-A}{q-q^{-1}}  \operatorname{Tr}(\alpha_{++})^2 (-q^2 \operatorname{Tr}(\alpha_{+-}\alpha_{--}))= -q^2K^{-1}E= -EK^{-1}
 \end{eqnarray*}
 The other computations are similar and left to the reader.
 

\end{proof}

\begin{remark}
\begin{enumerate}
\item When $\mathcal{R}=\mathbb{C}[[\hbar]]$ and $q=\exp(\hbar)$, the sub Hopf algebra of $\mathcal{O}_q[D(B)]$ generated by $E,F, K^{\pm 1}$ and $L^{\pm 1}$ is the Drinfel'd double of the quantized Borel algebra (see \cite{DrinfeldqRMatrix, Kassel, ChariPressley}). In particular, this Hopf algebra has a canonical element ($R$-matrix) that will be related to the change of coordinates maps of quantum Teichm\"uller theory in Section $4.3$. The quotient of the latter algebra by the ideal generated by the central, group-like element $KL-1$ is  the Hopf algebra $U_q\mathfrak{sl}_2$ whose algebra structure was first defined by Kulish and Reshetikhin in \cite{KulishReshetikhin} and Hopf algebra structure by Sklyanin in \cite{Sklyanin}.
\item The fact that $U_q\mathfrak{sl}_2$ (and more generally $U_q\mathfrak{sl}_N$) can be embedded in some quantum torus was already found by Kashaev and Volkov in \cite{KashaevVolkov} (see also \cite{FadeevModularDouble} and \cite{SchraderShapiro}).
\end{enumerate}
\end{remark}



\begin{definition}\label{def_Cartan_involution}
Let $\theta_1 \in \mathrm{Aut}(\mathcal{S}_{\omega}(\mathbb{D}_1))$ be the involutive algebra automorphism sending a stated diagram to its image through the central symmetry of the disc (rotation of angle $\pi$). One has $\theta_1(\alpha_{\varepsilon \varepsilon'})=\beta_{\varepsilon' \varepsilon}$ and $\theta_1(\beta_{\varepsilon \varepsilon'})=\alpha_{\varepsilon' \varepsilon}$. By Theorem \ref{theorem_kernel_qtrace}, the kernel of the quantum trace is preserved by $\theta_1$, hence it induces an involutive algebra automorphism $\Theta_1 \in \mathrm{Aut}(\mathcal{O}_q[D(B)])$ defined by $\Theta_1(\operatorname{Tr} (x)) = \operatorname{Tr}(\theta_1(x))$ for all $x\in \mathcal{S}_{\omega}(\mathbb{D}_1)$. We call $\Theta_1$ the \textit{Cartan involution}. \end{definition}

Note that $\Theta_1(K^{\pm 1/2})= L^{\pm 1/2}$ and that $\Theta_1(E)=F$.






\section{Weight representations of $\mathcal{O}_q[D(B)]$ at odd roots of unity}

In this section, we suppose that $\mathcal{R}=\mathbb{C}$ and that $\omega$ is a root of unity of odd order $N>1$.

\subsection{The center of $\mathcal{O}_q[D(B)]$}

Let $\gamma_p \in \mathcal{S}_{\omega}(\mathbb{D}_1)$ be the peripheral curve around the inner puncture of $\mathbb{D}_1$. Since $\gamma_p$ is central in $\mathcal{S}_{\omega}(\mathbb{D}_1)$, its image $T:= \operatorname{Tr}(\gamma_p)$ is central in $\mathcal{O}_q[D(B)]$. In fact, since $T=H_p + H_p^{-1}$, where $H_p=[Z_2Z_3]$ is the central inner puncture element of Definition \ref{def_central_elements}, the element $T$ is central in the whole algebra $\mathcal{Z}_{\omega}(\mathbb{D}_1)$. So does the boundary puncture element $H_{\partial}= K^{-1/2}L^{-1/2}= [X_1Z_2Z_3X_4]$ and its inverse. It follows from Theorems \ref{theorem_center_skein}, \ref{theorem_CFClassification} and the commutativity of the diagram \eqref{diagram_shadows}, that the elements $E^N, F^N, K^{\pm N/2}, L^{\pm N/2}$ are central in $\mathcal{Z}_{\omega}(\mathbb{D}_1)$. 


A simple use of the skein relations shows the equality
$$ \gamma_p = -q \alpha_{++}\beta_{--}-q^{-1}\alpha_{--}\beta_{++}+\alpha_{+-}\beta_{-+}+\alpha_{-+}\beta_{+-},$$
in $\mathcal{S}_{\omega}(\mathbb{D}_1)$, and composing with the trace map, one finds the equality
\begin{equation}\label{tracetruc} T= - H_{\partial}(q^{-1}K +qL) -(q-q^{-1})^2 EFH_{\partial}. \end{equation}

By Theorem \ref{theorem_center_sskein}, one has the equality $T_N(\gamma_p)=j_{\mathbb{D}_1}(\gamma_p)$ which gives, after composing by the quantum trace, the equality
\begin{equation}\label{eq_center_rel}
T_N(T) +K^{N/2}L^{-N/2} + L^{N/2}K^{-N/2} + (q-q^{-1})^{2N}E^N F^N H_{\partial}^N = 0.
\end{equation}

\begin{remark} Equation \eqref{eq_center_rel} could have been proved directly, without using Theorem \ref{theorem_center_sskein}, by first showing by induction on $k\geq 1$ the equality
\begin{equation}\label{meuh}
 \prod_{i=0}^{k-1} \left( T+H_{\partial}(q^{-2i-1}K + q^{2i+1}L) \right) = (-1)^k (q-q^{-1})^{2k} E^k F^k H_{\partial}^k.
 \end{equation}
Next, remark that for any invertible elements $X$ and $Y$, one has
$$ \prod_{i=0}^{N-1} \left( X+X^{-1} +Yq^{2i} + Y^{-1}q^{-2i} \right) =X^N +X^{-N} + Y^N + Y^{-N}.$$
Applying the above equation with $X= H_p$ and $Y= qH_{\partial}L$ and combining with Equation \eqref{meuh} taken for $k=N$, one finds
\begin{align*}
 H_p^N +H_p^{-N} + H_{\partial}^NL^N + H_{\partial}^{-N}L^{-N} &=& \prod_{i=0}^{N-1} \left( T+H_{\partial}(q^{-2i-1}K + q^{2i+1}L) \right) \\
 &=& - (q-q^{-1})^{2N} E^N F^N H_{\partial}^N.
 \end{align*}
One obtains the equality \eqref{eq_center_rel} after noting that $T_N(T)= T_N(H_p + H_p^{-1}) = H_p^N + H_p^{-N}$.
\end{remark}

Theorem \ref{theorem_center_sskein} applied in the special case of $\mathbb{D}_1$ writes:

\begin{proposition}\label{prop_center}
The center of $\mathcal{O}_q[D(B)]$ is the quotient of the polynomial algebra $\mathbb{C}[T, H_{\partial}^{\pm 1}, E^N, F^N, K^{\pm N/2}, L^{\pm N/2}]$ by the ideal generated by the elements $T_N(T) +K^{N/2}L^{-N/2} + L^{N/2}K^{-N/2} + (q-q^{-1})^{2N}E^N F^N H_{\partial}^N$, $H_{\partial}^N - K^{-N/2}L^{-N/2}$ and $XX^{-1}-1$ for $X\in \{H_{\partial}, K^{N/2}, L^{N/2} \}$.
\end{proposition}

For latter use, we introduce the

\begin{definition}
The \textit{Casimir element} $C\in \mathcal{O}_q[D(B)]$ is the central element 
$$C:= - \frac{T H_{\partial}^{-1}}{(q-q^{-1})^2}= -\frac{1}{(q-q^{-1})^2}([X_1X_4]^{-1} +[X_1X_2X_3X_4]^{-1}).$$
\end{definition}
By Equation \eqref{tracetruc}, one has 
\begin{equation}\label{eq_Casimir} C= EF+\frac{qL +q^{-1}K}{(q-q^{-1})^2} = FE + \frac{qK+q^{-1}L}{(q-q^{-1})^2}. \end{equation}


\subsection{Weight indecomposable $\mathcal{O}_q[D(B)]$-modules}

A  $\mathcal{O}_q[D(B)]$-module $V$ is called a \textit{weight} module if $V$ is semi-simple as a module over the center of  $\mathcal{O}_q[D(B)]$.
In this subsection, we classify the indecomposable weight $\mathcal{O}_q[D(B)]$-modules at odd roots of unity. This classification extends the classification of $U_q\mathfrak{sl}_2$ simple modules made independently by De Concini-Kac in \cite{DeConciniKacRepQGroups} and Arnaudon-Roche in \cite{ArnaudonRoche} (see also \cite[Chapter $1$ Section $3.3$]{KlimykSchmudgenQGroups}). 

\begin{definition}\label{def_representations} We define three families of $\mathcal{O}_q[D(B)]$-modules as follows. For $n\in \mathbb{Z}$, we write $[n]=\frac{q^n-q^{-n}}{q-q^{-1}}$. As before, we write $A:= \omega^{-2}$ and $q:= \omega^{-4}$.
\begin{enumerate}
\item For $\mu \in \mathbb{C}^*$, $\varepsilon \in \{-1,+1\}$, $0\leq n \leq N-1$, the module $V_{\mu, \varepsilon, n}$ has canonical basis $(e_0, \ldots, e_n)$ and module structure given by: 
\begin{align*}
&K^{1/2}e_i = \varepsilon \mu A^{n-2i} e_i; \quad L^{1/2}e_i= \mu A^{2i-n}e_i \mbox{, for }i\in \{0, \ldots, n\}; \\
&Fe_n=0; \quad Fe_i = e_{i+1} \mbox{, for }i\in \{0,\ldots, n-1\}; \\
&Ee_0=0; \quad Ee_i= \mu^2 [i][n-i+1]e_{i-1}\mbox{, for }i\in \{1,\ldots, n\}.
\end{align*}
\item For $\lambda, \mu\in \mathbb{C}^*$ and $a,b\in \mathbb{C}$, the module $V(\lambda, \mu, a,b)$ has canonical basis $(v_0, \ldots, v_{N-1})$ and module structure given by:
\begin{align*}
& K^{1/2}v_i = \lambda q^{-i}v_i; \quad L^{1/2}v_i = \mu q^i v_i \mbox{, for }i\in \{0, \ldots, N-1\}; \\
& Fv_{N-1}=bv_0; \quad Fv_i=v_{i+1} \mbox{, for }i \in \{0, \ldots, N-2\}; \\
& Ev_0=a v_{N-1}; \quad Ev_{i}= \left( \frac{q^{-i+1}\lambda^2 - q^{i-1}\mu^2}{q-q^{-1}} [i] +ab \right) v_{i-1}\mbox{, for }i\in \{1, \ldots, N-1\}.
\end{align*}

\item For $\lambda, \mu \in \mathbb{C}^*$, $c\in \mathbb{C}$, the module $\widetilde{V}(\lambda, \mu, c)$ has canonical basis $(w_0, \ldots, w_{N-1})$  and module structure given by: 
\begin{align*}
& K^{1/2}w_i= \lambda q^{i}w_i; \quad L^{1/2}w_i= \mu q^{-i}w_i \mbox{, for }i \in \{0, \ldots, N-1\}; \\
& Ew_{N-1}= cw_0; \quad Ew_i= w_{i+1} \mbox{, for }i\in \{0, \ldots, N-2\}; \\
& F w_0= 0; \quad Fw_{i+1} = \left( \frac{ \mu^2q^{-1-i} - \lambda^2 q^{i+1} }{q-q^{-1}}[i] \right) w_i \mbox{, for }i\in \{1, \ldots, N-1\}.
\end{align*}

\end{enumerate}
\end{definition}

\begin{proposition}\label{prop_representations}
\begin{enumerate}
\item The weight modules $V_{\mu, \varepsilon, n}, V(\lambda, \mu, a,b), \widetilde{V}(\lambda, \mu, c)$ are indecomposable and any indecomposable weight $\mathcal{O}_q[D(B)]$-module is isomorphic to one of them.
\item The modules $V(\lambda, \mu, a, b), \widetilde{V}(\lambda, \mu, c)$ are projective objects in the category $\mathcal{C}$ of weight $\mathcal{O}_q[D(B)]$-modules. The module $V_{\mu, \varepsilon, n}$ is a projective object in $\mathcal{C}$  if and only if $n=N-1$.
\item The modules $V_{\mu, \varepsilon, n}$ are simple. The module $\widetilde{V}(\lambda, \mu, c)$ is simple if and only if either $c\neq 0$ or $\lambda \mu^{-1} \neq \pm q^{n-1}$ for all $n\in \{1, \ldots, N-1\}$. The module $V(\lambda, \mu, a, b)$ is simple if and only if either $\prod_{i\in \mathbb{Z}/N\mathbb{Z}} (ab+\frac{q^{1-i}\lambda^2 - q^{i-1}\mu^2}{q-q^{-1}}[i])\neq 0$ or $\lambda \mu^{-1} \neq \pm q^{n-1}$ for all $n\in \{1, \ldots, N-1\}$. Any simple $\mathcal{O}_q[D(B)]$-module is isomorphic to one of these modules.
\item One has an exact sequence:
$$ 0 \rightarrow V_{\mu, \varepsilon, N-n-2} \xrightarrow{i} V(\varepsilon \mu A^n, \mu A^{-n}, 0, 0) \xrightarrow{p} V_{\mu, \varepsilon, n} \rightarrow 0$$
where $i$ and $p$ are equivariant.
\item  \begin{itemize}
\item One has $\widetilde{V}(\lambda, \mu, c) \cong \widetilde{V}(\lambda', \mu', c')$ if and only if $c'=c$, $\lambda'=\lambda q^{-n}$, $\mu'=\mu q^{n}$ for $n\in \mathbb{Z}/N\mathbb{Z}$.
\item One has $V_{\mu, \varepsilon, N-1} \cong V(\varepsilon \mu A^{-1}, \mu A, 0, 0)$. 
\item One has $V(\lambda, \mu, a, b) \cong V(\lambda', \mu', a',b')$ if and only if $b'=b$, $\lambda'=\lambda q^{-n}$, $\mu'=\mu q^n$, $a'b=ab+\frac{q^{1-n}\lambda^2 - q^{n-1}\mu^2}{q-q^{-1}}[n]$ for $n\in \mathbb{Z}/N\mathbb{Z}$, with the additional assumption that $b\neq 0$ when $n\neq 0$.
\item If $\lambda \mu^{-1} \neq \pm q^{n-1}$, for all $n\in \{1, \ldots, N-1\}$, then $\widetilde{V}(\lambda, \mu, c) \cong V(\lambda, \mu, a, 0)$ for $c=a \prod_{k=1}^{N-1}$ $\frac{q^{-1-k}\mu^2 - q^{1+k}\lambda^2}{q-q^{-1}}[k+1]$.
\item A pair of isomorphic representations of the form $V_{\mu, \varepsilon, n}, V(\lambda, \mu, a,b)$ or $\widetilde{V}(\lambda, \mu, c)$ can be related by a sequence of isomorphisms of the previous type.
\end{itemize}
\end{enumerate}
\end{proposition}

The proof of Proposition \ref{prop_representations} is very similar to the classification of simple $U_q\mathfrak{sl}_2$-modules, and is postponed to Appendix B. 

\begin{notations} We will call projective weight module a projective object of the category of weight modules. Beware that projective weight modules might not be projective object in the whole category of $\mathcal{O}_q[D(B)]$-modules.
\end{notations}

 We now give a more simple classification of the  $\mathcal{O}_q[D(B)]$ projective indecomposable weight modules. 
 
 \begin{definition}
 The group $D(B)$ is the subgroup of $\operatorname{SL}_2(\mathbb{C})\times \operatorname{SL}_2(\mathbb{C})$ of pairs $g=(g_-, g_+)$ with $g_-$ lower triangular and $g_+$ upper triangular. The \textit{big cell}  $\operatorname{SL}^0_2(\mathbb{C})\subset \operatorname{SL}_2(\mathbb{C})$ is the subset of matrices $\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in  \operatorname{SL}_2(\mathbb{C})$ such that $a\neq 0$. Define a surjective map $\varphi : D(B) \rightarrow \operatorname{SL}_2^0(\mathbb{C})$ by $\varphi((g_-, g_+)):= (g_-)^{-1}g_+$.
 \end{definition}
 
 Note that $\varphi$ is not a group morphism. Let $W$ be an indecomposable weight $\mathcal{O}_q[D(B)]$-module. Since $W$ is indecomposable, it induces a character $\chi_{W}$ on the center of $\mathcal{Z}_{\omega}(\mathbb{D}_1)$.
 
 \begin{definition}
 The \textit{classical shadow} of $W$ is the element $g= (g_-, g_+)\in D(B)$ defined by:
 \begin{eqnarray*}
 g_- :=  \begin{pmatrix} \chi_W\circ \operatorname{Tr} (\beta_{++}^N) & \chi_W\circ \operatorname{Tr} (\beta_{+-}^N)  \\ \chi_W\circ \operatorname{Tr} (\beta_{-+}^N)  & \chi_W\circ \operatorname{Tr} (\beta_{--}^N)  \end{pmatrix}, 
 &
 g_+ :=  \begin{pmatrix} \chi_W\circ \operatorname{Tr} (\alpha_{++}^N) & \chi_W\circ \operatorname{Tr} (\alpha_{+-}^N)  \\ \chi_W\circ \operatorname{Tr} (\alpha_{-+}^N)  & \chi_W\circ \operatorname{Tr} (\alpha_{--}^N)  \end{pmatrix}.  
\end{eqnarray*}
The puncture invariant of $W$ is $t:= \chi_W(T)$ and the boundary invariant of $W$ is $h_{\partial}:=\chi_W(H_{\partial})$.
\end{definition}

Let $\mathcal{Z}$ denote the center of  $\mathcal{O}_q[D(B)]$. By Proposition \ref{prop_center}, the set $\mathrm{MaxSpec}(\mathcal{Z})$ is in bijection with the set $\widetilde{D(B)}$ of triples $(g, t, h_{\partial})$, where $g \in D(B)$, $t\in \mathbb{C}$ is such that $T_N(t)=\operatorname{Tr} (\varphi(g))$ and $h_{\partial}\in \mathbb{C}^*$ is such that $h_{\partial}^{-N}$ is equal to the left upper coefficient of $g_-\cdot g_+$. Note that, as an affine variety, the variety $\widetilde{D(B)}$ is a finite covering of $D(B)$ of degree $N^2$ ramified over those $g\in D(B)$ such that $\operatorname{Tr}( \varphi(g)) = \pm 2$.


\begin{lemma}\label{lemma_character} The character map $\chi$, sending an indecomposable weight $\mathcal{O}_q[D(B)]$-module to a character in $\mathrm{MaxSpec}(\mathcal{Z})\cong \widetilde{D(B)}$, induces a bijection between the set of isomorphism classes of projective indecomposable weight $\mathcal{O}_q[D(B)]$-modules and   $\widetilde{D(B)}$.
\end{lemma}

\begin{proof} This is an immediate consequence of Proposition \ref{prop_representations}.

\end{proof}

\begin{definition} An indecomposable weight $\mathcal{O}_q[D(B)]$-module $W$ with classical shadow $g$  is said: 
\begin{enumerate}
\item \textit{cyclic} if $\varphi(g)$ is not triangular (\textit{i.e.} if $\chi_W(E^NF^N)\neq 0$).
\item \textit{semi-cyclic} if $\varphi(g)$ is triangular not diagonal (\textit{i.e. } if either $(\chi_W(E^N)=0, \chi_W(F^N)\neq 0)$ or  $(\chi_W(E^N)\neq0, \chi_W(F^N)= 0)$).
\item \textit{diagonal} if $\varphi(g)$ is diagonal (\textit{i.e.} if $\chi_W(E^N)=\chi_W(F^N)=0$).
\item \textit{scalar} if $\varphi(g)$ is scalar.
\end{enumerate}
\end{definition}


\begin{remark} Hence $V_{\mu, \varepsilon, n}$ is diagonal and is scalar if and only if $\mu^{2N}=1$. The module $\widetilde{V}(\lambda, \mu, c)$ is semi-cyclic, is diagonal if and only if $c=0$ and is scalar if and only if $c=0$ and $(\lambda\mu^{-1})^{2N}=1$. The module $V(\lambda, \mu, a, b)$ is semi-cyclic if and only if either $b=0$ or $\widetilde{a}:= a\prod_{i=1}^{N-1} \left( \frac{q^{-i+1}\lambda^2 - q^{i-1}\mu^2}{q-q^{-1}} [i] +ab \right)=0$, is diagonal if and only if $b=\widetilde{a}=0$ and is scalar if and only if $b=\widetilde{a}=0$ and $(\lambda\mu^{-1})^{2N}=1$.
\par Moreover, by Proposition \ref{prop_representations}, a projective indecomposable weight $\mathcal{O}_q[D(B)]$-module is not simple if and only if it is scalar and $t\neq \pm 2$.
\end{remark}

\begin{remark}\label{remark_representations}
When $\mu=\lambda^{-1}$, the representations of Definition \ref{def_representations} induce representations of $U_q\mathfrak{sl}_2$ and any indecomposable weight $U_q\mathfrak{sl}_2$-module arises that way, so Proposition \ref{prop_representations} induces an obvious $U_q\mathfrak{sl}_2$ analogue which can be compared to the (very abondant) existing literature on the subject.
\begin{enumerate}
\item The classification of $U_q\mathfrak{sl}_2$ simple modules is well known since \cite{DeConciniKacRepQGroups, ArnaudonRoche}, however the classification of indecomposable $U_q\mathfrak{sl}_2$ modules is still unknown. Proposition \ref{prop_representations} gives a classification of the weight indecomposable modules, which seems new,  however there exists indecomposable modules which are not weight modules and which do not belong to our classification.
\item The fact that the representations $V(\lambda, \mu ,a ,b)$ and $\widetilde{V}(\lambda, \mu, c)$ are projective objects in the category of weight modules seems to be new as well. Let us emphasize that we do not claim that they are projective representations. 


\item It is a common mistake in quantum group literature (made for instance in \cite[Chapter $1$ Section $3.3$]{KlimykSchmudgenQGroups}) to believe that any simple $U_q\mathfrak{sl}_2$ module is isomorphic to one of the $V(\lambda, \lambda^{-1},a ,b)$ or $V(\mu,+1, n)$. As pointed out by Kassel in \cite{Kassel}, Proposition \ref{prop_representations} shows that we need to add the representations $\widetilde{V}(\lambda, \mu ,c)$ as well. More precisely, any character $\chi$ on the center of $\mathcal{O}_q[D(B)]$ such that either $\chi(F^N)\neq 0$ or $\chi(E^N)=0$ is induced by a representation of the form $V(\lambda, \mu, a ,b)$. However, for $g=(g_-,g_+)\in D(B)$ with $\varphi(g)$ lower triangular non-diagonal (\textit{i.e.} with $\chi(F^N)=0$ and $\chi(E^N)\neq 0$) and $h_{\partial}$ such that $h_{\partial}^{-N}$ is the upper left coefficient of $g_-g_+$, there exists a unique $t$ such that $T_N(t)=\operatorname{Tr}(\varphi(g))$ and such that $(g,t,h_{\partial})$ is the central character associated to a representation of the form $V(\lambda, \mu, a, b)$. The characters $(g,t',h_{\partial})$ for the other $t'\neq t$ are only obtained by representations of the form $\widetilde{V}(\lambda, \mu, c)$.
\end{enumerate}

\end{remark}

\subsection{Simple $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-modules}

Since $\mathcal{O}_q[D(B)]$ is a sub-algebra of  $\mathcal{Z}_{\omega}(\mathbb{D}_1)$, any module on $\mathcal{Z}_{\omega}(\mathbb{D}_1)$ is a module on $\mathcal{O}_q[D(B)]$. In this subsection we classify which $\mathcal{O}_q[D(B)]$-modules arise this way. 

\begin{proposition}\label{prop_classification_modules}
Let $W$ be a simple $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-module. As a $\mathcal{O}_q[D(B)]$-module, $W$ is projective weight indecomposable. Let $g$ denote its classical shadow. Then $W$ is:
\begin{itemize}
\item either cyclic, 
\item or semi-cyclic with the diagonal elements of $\varphi(g)$ not equal to $\pm 1$,
\item or scalar.
\end{itemize}
Moreover, each of the above $\mathcal{O}_q[D(B)]$-module extends (non-canonically) to a simple $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-module.
\end{proposition}

\begin{lemma}\label{Lemma1}
The algebra $\mathcal{Z}_{\omega}(\mathbb{D}_1)$ is presented by the generators $H_{\partial}^{\pm 1}, H_p^{\pm 1}, [Z_1Z_3Z_4]^{\pm 1}$ and  $X_1^{\pm 1}$ with relations $[Z_1Z_3Z_4]^{-1}X_1= q X_1[Z_1Z_3Z_4]^{-1}$ and $[H_{\partial}, x]=[H_p, x]=0$ for all generators $x$.
\end{lemma}

\begin{proof}
The fact that  $H_{\partial}^{\pm 1}, H_p^{\pm 1}, [Z_1Z_3Z_4]^{\pm 1}$ and  $X_1^{\pm 1}$ generate $\mathcal{Z}_{\omega}(\mathbb{D}_1)$  follows from Lemma \ref{lemma_grading_generator} and the fact that each $X_i^{\pm 1}$ for $i\in \{1,2,3,4\}$, is a product of these generators. The relations follow from definition.

\end{proof}

\begin{definition}
Let $(x_1, \lambda, h_p, h_{\partial})\in (\mathbb{C}^*)^4$. The $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-module $W(x_1, \lambda, h_p, h_{\partial})$ has canonical basis $(e_0, \ldots, e_{N-1})$ and module structure given by:
\begin{align*}
& H_p e_i = h_p e_i; \quad H_{\partial} e_i = h_{\partial} e_i; \quad  [Z_1Z_3Z_4]^{-1} e_i= \lambda q^{-i} e_i  \mbox{, for i} \in \{0, \ldots, N-1\}; \\
&X_1 e_0= x_1 e_{N-1}; \quad X_1 e_i = e_{i-1} \mbox{, for }i\in \{1, \ldots N-1\}.
\end{align*}
\end{definition}

\begin{lemma}\label{Lemma_2}
Any simple $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-module is isomorphic to a module $W(x_1, \lambda, h_p, h_{\partial})$. Moreover, $W(x_1, \lambda, h_p, h_{\partial})\cong W(x'_1, \lambda', h'_p, h'_{\partial})$ if and only if $h'_p=h_p$, $h'_{\partial}=h_{\partial}$, $x'_1=x_1$ and $\lambda'=\lambda q^n$ for some $n\in \mathbb{Z}/N\mathbb{Z}$.
\end{lemma}

\begin{proof} This is an immediate consequence of Theorem \ref{theorem_CFClassification}.
\end{proof}

\begin{lemma}\label{Lemma_3}
Let $(x_1, \lambda, h_p, h_{\partial})\in (\mathbb{C}^*)^4$.
\begin{enumerate}
\item If $(h_{\partial}h_p\lambda^2)^N \neq -1$, set $\mu := \lambda^{-1} h_{\partial}^{-1}$ and 
\begin{align*}
& a:= -(q-q^{-1})^{N-2}x_1\frac{h_{\partial}^{-1}(h_p+h_p^{-1}) +q \lambda^2 +q^{-1}\mu^2}{1+(h_{\partial}h_p\lambda^2)^{-N}}; \\
& b:= \frac{1}{(q-q^{-1})^N} x_1^{-1}(1+ (h_{\partial}h_p\lambda^2)^{-N}).
\end{align*}
One has $W(x_1, \lambda, h_p, h_{\partial})\cong V(\lambda, \mu, a, b)$ as a  $\mathcal{O}_q[D(B)]$-module.
\item If $(h_{\partial}h_p\lambda^2)^N = -1$, set $\mu := \lambda^{-1} h_{\partial}^{-1}$ and
$$ a:= \frac{1}{N}(q-q^{-1})^{N-2}x_1h_{\partial}^{-1}(h_p^{-1}-h_p).$$
One has  $W(x_1, \lambda, h_p, h_{\partial})\cong V(\lambda, \mu, a, 0)$ as a  $\mathcal{O}_q[D(B)]$-module.
\end{enumerate}
\end{lemma}


\begin{proof}
We denote by $\chi : \mathcal{Z}\rightarrow \mathbb{C}$ the character on the center $\mathcal{Z}$ of $\mathcal{O}_q[D(B)]$ induced by $W(x_1, \lambda, h_p, h_{\partial})$ and write $\mu := \lambda^{-1} h_{\partial}^{-1}$. First one has $\chi(K^{N/2})=\lambda^N$ and $\chi(L^{N/2})= h_{\partial}^{-1}\lambda^{-N}$. Since $X_4= H_{\partial} H_p [Z_1Z_3Z_4]^{-2}$, one has $\chi(X_2^N)=(h_{\partial} h_p \lambda^2)^N$. Using the equality $F^N= \frac{1}{q-q^{-1}} X_1^{-N}(1+X_2^{-N})$, one finds $\chi(F^N)=  \frac{1}{(q-q^{-1})^N} x_1^{-1}(1+ (h_{\partial}h_p\lambda^2)^{-N})=:b$. Next, we deduce from the equality $\chi(T)=h_p+h_p^{-1}$ that  $\chi(C)=-\frac{(h_p+h_p^{-1})h_{\partial}}{(q-q^{-1})^2}$.

\vspace{2mm}
\par First suppose that $(h_{\partial}h_p\lambda^2)^N \neq -1$, \textit{i.e.} that $\chi(F^N)\neq 0$. 
For $i\in \{0, \ldots, N-1\}$, define $v_i:=F^ie_0 \in W(x_1, \lambda, h_p, h_{\partial})$. It follows from the relations $FK^{1/2}=qK^{1/2}F$ and $FL^{1/2}=q^{-1}L^{1/2}F$ that one has $K^{1/2}v_i= \lambda q^{-i}v_i$ and $L^{1/2}v_i= \mu q^i v_i$ for all  $i\in \{0, \ldots, N-1\}$. Since $\chi(F^N)=b$, one has $Fv_{N-1}=bv_0$.  Consider the scalar $a$ defined in the first statement of the lemma and remark that one has the equality $\chi(C)=ab + \frac{q\lambda^2 +q^{-1}\mu^2}{(q-q^{-1})^2}$. Using Equation \eqref{eq_Casimir}, for any $i\in \{0, \ldots, N-2\}$ one has
$$ Ev_{i+1}= EFv_i= \left(C- \frac{qL+q^{-1}K}{(q-q^{-1})^2} \right)v_i = \left(ab + \frac{q^{-i}\lambda^2 - q^i\mu^2}{q-q^{-1}}[i+1] \right)v_i.$$
Similarly, one finds
$$ Ev_0 = b^{-1}EFv_{N-1} = b^{1}\left( C- \frac{qL+q^{-1}K}{(q-q^{-1})^2} \right)v_{N-1} = a v_{N-1}.$$
We have thus proved that  $W(x_1, \lambda, h_p, h_{\partial})\cong V(\lambda, \mu, a, b)$ as a  $\mathcal{O}_q[D(B)]$-module.

\vspace{2mm}
\par Next suppose that  $(h_{\partial}h_p\lambda^2)^N = -1$, \textit{i.e.} that $\chi(F^N)= 0$. Then there exists $i_0\in \{0, \ldots, N-1\}$ such that $h_{\partial}h_p\lambda^2=-q^{-2i_0-1}$. Using that $F=\frac{1}{q-q^{-1}}X_1^{-1}\left(1+q H_{\partial}^{-1}H_p^{-1}[Z_1Z_3Z_4]^2 \right)$, one finds that 
\begin{equation}\label{eqreloute}
 F e_i = \left\{ \begin{array}{ll}
\frac{1+q^{2i+1} h_{\partial}^{-1}h_p^{-1}\lambda^{-2} }{q-q^{-1}}e_{i+1} & \mbox{, if }i\neq N-1; \\
x_1^{-1}\frac{1+q^{-1} h_{\partial}^{-1}h_p^{-1}\lambda^{-2} }{q-q^{-1}}e_{0} & \mbox{, if }i=N-1.
\end{array} \right. 
\end{equation}
In particular $Fe_{i_0}=0$. For $i\in \{0, \ldots, N_1\}$, define $v_i:= F^i e_{i_0+1}$. Set  $\lambda':= \lambda q^{-(i_0+1)}$ and $\mu' := \lambda^{-1} h_{\partial}^{-1}q^{i_0+1}$. It follows from the relations $FK^{1/2}=qK^{1/2}F$ and $FL^{1/2}=q^{-1}L^{1/2}F$ that one has $K^{1/2}v_i= \lambda' q^{-i}v_i$ and $L^{1/2}v_i= \mu' q^i v_i$ for all  $i\in \{0, \ldots, N-1\}$. Next it follows from the equality $h_{\partial}h_p\lambda^2=-q^{-2i_0-1}$ that one has $\chi(C)= \frac{q \lambda'^2 + q^{-1}\mu'^2}{(q-q^{-1})^2}$.  Using Equation \eqref{eq_Casimir}, for any $i\{0, \ldots, N-2\}$ one has
$$ Ev_{i+1}= EFv_i= \left(C- \frac{qL+q^{-1}K}{(q-q^{-1})^2} \right)v_i = \left(\frac{q^{-i}\lambda'^2 - q^i\mu'^2}{q-q^{-1}}[i+1] \right)v_i.$$
It remains to compute $Ev_0$.
Since  $X_3^{-1}=H_{\partial} H_p^{-1}[Z_1Z_3Z_4]^2$ and $X_4^{-1}=H_{\partial}^{-1}H_pX_1$, one finds  $E= \frac{-1}{q-q^{-1}}H_{\partial}^{-1}H_pX_1\left( 1+qH_{\partial} H_p^{-1}[Z_1Z_3Z_4]^2 \right)$ from which we deduce that
$$ E e_i= \left\{ \begin{array}{ll}
- \frac{h_{\partial}^{-1}h_p + \lambda^2 q^{-2i+1}}{q-q^{-1}} e_{i-1} & \mbox{, if }i\neq 0; \\
-x_1 \frac{h_{\partial}^{-1}h_p + \lambda^2 q}{q-q^{-1}} e_{N-1} & \mbox{, if }i= 0.
\end{array} \right. $$
In particular, one has $Ev_0= \left\{ \begin{array}{ll} - \frac{h_{\partial}^{-1}h_p + \lambda^2 q^{-1-2i_0}}{q-q^{-1}} e_{i_0} & \mbox{, if }i_0\neq N-1; \\
-x_1 \frac{ h_{\partial}^{-1}h_p + \lambda^2 q}{q-q^{-1}} e_{N-1} & \mbox{, if }i_0=N-1. \end{array} \right.$
To express the relation between $e_{i_0}$ and $v_{N-1}$, we use Equation \eqref{eqreloute} and the equality $h_{\partial}h_p\lambda^2=-q^{-2i_0-1}$ and deduce that

\begin{eqnarray*}
v_{N-1} &=& F^{N-1}e_{i_0+1} = \left\{ \begin{array}{ll}
\frac{x_1^{-1} }{(q-q^{-1})^{N-1}} \prod_{k=i_0+1}^{N+i_0-1} \left( 1-q^{-(2i_0+1)}q^{2k+1} \right)e_{i_0} & \mbox{, if }i_0 \neq N-1; \\
\frac{1 }{(q-q^{-1})^{N-1}} \prod_{k=0}^{N-2} \left( 1-q^{2k+1} \right)e_{N-1} & \mbox{, if }i_0 = N-1. 
\end{array} \right. \\
&=&  \left\{ \begin{array}{ll}
\frac{x_1^{-1} }{(q-q^{-1})^{N-1}} P_N(1)e_{i_0} & \mbox{, if }i_0 \neq N-1; \\
\frac{1 }{(q-q^{-1})^{N-1}}P_N(1)e_{N-1} & \mbox{, if }i_0 = N-1. 
\end{array} \right.
\end{eqnarray*}

where the polynomial $P_N(X)$ is defined by  $P_N(X):= \prod_{i=1}^{N-1} (X-q^i)$. Since $P_N(X)(X-1)= X^N -1$, one has $P_N(X)= \sum_{k=0}^{N-1} X^k$, hence $P_N(1)=N$. Eventually we finds that 
$$ E v_0= \frac{1}{N} (q-q^{-1})^{N-2} x_1 h_{\partial}^{-1}(h_p^{-1}-h_p)v_{N-1}=av_{N-1}.$$
Hence $W(x_1, \lambda, h_p, h_{\partial})\cong V(\lambda', \mu', a, 0)$ as a  $\mathcal{O}_q[D(B)]$-module. By the last assertion of Proposition \ref{prop_representations}, this module is isomorphic to $V(\lambda, \mu, a ,0)$.
This concludes the proof.

\end{proof}


\begin{lemma}\label{Lemma_4}
Let $\lambda, \mu \in \mathbb{C}^*$ and $a,b\in \mathbb{C}$ and consider the following system with variable $h_p$:
\begin{equation}\label{system}
\left\{ \begin{array}{lll}
ab &=& -\frac{\lambda\mu (h_p+h_p^{-1}) + q\lambda^2 + q^{-1}\mu^2}{(q-q^{-1})^2} \\
h_p^N &\neq& - (\lambda \mu^{-1})^N
\end{array}\right.
\end{equation}
The system \eqref{system} admits a solution $h_p\in \mathbb{C}^*$ if and only if
\begin{enumerate}
\item either $S:= \prod_{i=0}^{N-1} \left( ab + [i]\frac{\lambda^2 q^{1-i} -\mu^2 q^{i-1}}{q-q^{-1}} \right)$ is non null, 
\item or $S=0$ and $(\lambda \mu^{-1})^{2N} \neq 1$.
\end{enumerate}
\end{lemma}

\begin{proof}
Consider the polynomial $$P(X)=X^2 +(\lambda \mu)^{-1}\left( ab(q-q^{-1})^2 + q\lambda^2 + q^{-1}\mu^2 \right) X +1.$$ Since its constant term is $+1$, there exists $h_p \in \mathbb{C}^*$ such that $P(X)=(X-h_p)(X-h_p^{-1})$. First, if $h_p^N = - (\lambda \mu^{-1})^N$, one deduce from the equality $P(h_p)=0$ that $S=0$. Hence if $S\neq 0$, the system \eqref{system} admits a solution. Next if $S=0$, there exists $n\in \{0, \ldots, N-1\}$ such that $ab+[n]\frac{\lambda^2q^{1-n}-\mu^2 q^{n-1}}{q-q^{-1}}=0$. It follows that $P(X)=(X+\lambda^{-1}\mu q^{2n-1})(X+ \lambda \mu^{-1} q^{1-2n})$, thus either $h_p$ or $h_p^{-1}$ is equal to $-\lambda^{-1}\mu q^{2n-1}$. It follows that the system \eqref{system} admits a solution such that $S=0$ if and only if  $(\lambda \mu^{-1})^{2N} \neq 1$.

\end{proof}

\begin{proof}[Proof of Proposition \ref{prop_classification_modules}]
By Lemmas \ref{Lemma_2} and \ref{Lemma_3}, any simple $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-module is isomorphic, as $\mathcal{O}_q[D(B)]$-module, to a module $V(\lambda, \mu, a, b)$. Conversely, consider such a module $V=V(\lambda, \mu, a, b)$, denote by $\chi: \mathcal{Z}\rightarrow \mathbb{C}$ its central character and write  $S:=  \prod_{i=0}^{N-1} \left( ab + [i]\frac{\lambda^2 q^{1-i} -\mu^2 q^{i-1}}{q-q^{-1}} \right)$. Note that $S=\chi(E^NF^N)$. 
\par First, if $V$ is cyclic then $S\neq 0$ and by Lemma \ref{Lemma_4} one can find $h_p\in \mathbb{C}^*$ solution of the system \eqref{system}. Set $x_1:= \frac{1+ \lambda^{2N}h_p^N (\lambda \mu)^N}{(q-q^{-1})^N}$. By Lemma \ref{Lemma_3}, one has $V\cong W(\lambda, x_1, h_p, (\lambda \mu)^{-1})$, hence $V$ extends to a $\mathcal{Z}_{\omega}(\mathbb{D}_1)$ simple module.
\par  Next suppose that $\chi(F^N)\neq 0$ and that $\chi(E^N)=0$, thus $S=0$. In this case, by Lemma \ref{Lemma_3}, $V$ extends to a $\mathcal{Z}_{\omega}(\mathbb{D}_1)$ module if and only if the system \eqref{system} admits a solution, hence if and only if $(\lambda \mu^{-1})^{2N} \neq 1$ by Lemma \ref{Lemma_4}.
\par  Now suppose that $\chi(F^N)=0$ and that $\chi(E^N)\neq 0$. The hypothesis $\chi(E^N)\neq 0$ implies that $a\neq 0$ and that for all $i\in \{1, \ldots, N-1\}$ one has $\lambda^2\mu^{-2} \neq q^{2(i-1)}$. Thus, if $(\lambda \mu^{-1})^{2N}=1$ then $\lambda \mu^{-1}= \pm q^{-1}$. Since $\chi(T) = -q\lambda \mu^{-1} -q^{-1}\lambda^{-1}\mu$, one has $\chi(T)= \pm 2$. Hence, if $V$ would extend to a simple $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-module, its puncture invariant would be $h_p=\pm 1$ and Lemma \ref{Lemma_3} would implies that $a=0$ which contradicts the hypothesis $\chi(E^N)\neq 0$. Conversely, if $(\lambda \mu^{-1})^{2N}\neq 1$, set $h_p:= -q\lambda \mu^{-1}$ and  $x_1:= \frac{aN}{(q-q^{-1})^{N-2}\lambda \mu (h_p - h_p^{-1})}$.  Note that $a\neq 0$ and,  since $(\lambda \mu^{-1})^{2N}\neq 1$, then $h_p-h_p^{-1}\neq 0$, hence $x_1\in \mathbb{C}^*$ is well defined. By Lemma \ref{Lemma_3} $V$ is isomorphic to $W(\lambda, x_1, h_p, (\lambda\mu)^{-1})$. 
\par Eventually, suppose that $\chi(E^N)=\chi(F^N)=0$. Since $b=0$ and $\chi(E^N)=0$, one has either $a=0$, or there exists $i \in \{1, \ldots, N-1\}$ such that $(\lambda\mu^{-1})^2=q^{2(i-1)}$. The latter case implies $(\lambda \mu^{-1})^{2N}=1$. If $a=0$ and $V$ extends to a simple $\mathcal{O}_q[D(B)]$-module, then by Lemma \ref{Lemma_3}, one has $h_p=\pm 1$, hence $\chi(T)=\pm 2=-q\lambda \mu^{-1} -q^{-1}\lambda^{-1}\mu$ and one has again  $(\lambda \mu^{-1})^{2N}=1$.  Conversely, if $(\lambda \mu^{-1})^{2N}=1$, then one can write $\lambda^2 \mu^{-2}=q^{2(i-1)}$ for some $i \in \{0, \ldots, N-1\}$. If $i\neq 0$, then $a\neq 0$. In this case we define $h_p:= -q\lambda \mu^{-1}$ (different from $\pm 1$ by hypothesis) and $x_1:= \frac{aN}{(q-q^{-1})^{N-2}\lambda \mu (h_p - h_p^{-1})}$. By Lemma \ref{Lemma_3},  $V$ is isomorphic to $W(\lambda, x_1, h_p, (\lambda\mu)^{-1})$. Finally, if $\lambda^2\mu^{-2}=q^{-2}$, then the hypothesis $\chi(E^N)=0$ implies $a=0$. We choose $x_1\in \mathbb{C}^*$ and $h_p=\pm 1$ arbitrarily  and  Lemma \ref{Lemma_3} implies that  $V$ is isomorphic to $W(\lambda, x_1, h_p, (\lambda\mu)^{-1})$.





\end{proof}

















\subsection{Factorizable representations of $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)$ and gauge group action}\label{sec_factorizable_rep}

In this subsection, we prove some properties concerning $\overline{\mathcal{S}}_{\omega} (\mathbb{D}_n)$-modules which will be crucial in order to define the representation functor $L_{\omega}^{\mathrm{SL}_2}$ of Theorem \ref{theorem_functor}. As before, we suppose that $\omega$ is a root of unity of odd order $N>1$.
By gluing $n$ copies of $\mathbb{D}_1$ along boundary arcs, one obtains the punctured surface $\mathbb{D}_n$. Let $i_n : \overline{\mathcal{S}}_{\omega}(\mathbb{D}_n) \hookrightarrow  \overline{\mathcal{S}}_{\omega}(\mathbb{D}_1)^{\otimes n} \cong \mathcal{O}_q[D(B)]^{\otimes n}$ denote the gluing map. For $V_1, \ldots, V_n$ some projective indecomposable $\mathcal{O}_q[D(B)]$-modules, the map $i_n$ induces on $V_1\otimes \ldots \otimes V_n$ a structure of $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)$-module.


\begin{definition}
A  $\overline{\mathcal{S}}_{\omega} (\mathbb{D}_n)$-module is said \textit{factorizable} if it is isomorphic to a module of the form $V_1\otimes \ldots \otimes V_n$, for $V_i$ some projective indecomposable $\mathcal{O}_q[D(B)]$-modules. 
\end{definition}

Let $X_n:= \mathrm{Specm}(\mathcal{Z}(\overline{\mathcal{S}}_{\omega} (\mathbb{D}_n)))$ denote the maximal spectrum of the center of $\overline{\mathcal{S}}_{\omega} (\mathbb{D}_n)$. A factorizable $\overline{\mathcal{S}}_{\omega} (\mathbb{D}_n)$-module $V$ induces a character $\chi(V)\in X_n$. Denote by $\Delta_n$ the topological triangulation of $\mathbb{D}_n$ obtained by gluing $n$ copies of $\Delta_1$ together, as depicted in Figure \ref{fig_triangulation_Dn}. 



\begin{theorem}\label{theorem_factorization}
\begin{enumerate}
\item The character map $\chi$ induces a bijection between the set of isomorphism classes of factorizable $\overline{\mathcal{S}}_{\omega} (\mathbb{D}_n)$-modules and the set of  closed points of $X_n$. 
\item Every simple $\mathcal{Z}_{\omega}(\mathbb{D}_n, \Delta_n)$-module induces, via the quantum trace, a factorizable $\overline{\mathcal{S}}_{\omega} (\mathbb{D}_n)$-module.
\item If $n\geq 2$, every factorizable $\overline{\mathcal{S}}_{\omega} (\mathbb{D}_n)$-module is isomorphic to a module of the form $V_1\otimes \ldots \otimes V_n$, where $V_i$ is a projective indecomposable $\mathcal{O}_q[D(B)]$-module that extends to a $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-module.
\end{enumerate}
\end{theorem}


\begin{notations}
\begin{itemize}
\item
Let $X_n^{0}:= \mathrm{Specm}(\overline{\mathcal{S}}_{+1}(\mathbb{D}_n))$ and $p_1 : X_n \rightarrow X_n^{0}$ the finite branched cover induced by the Chebyshev morphism $j_{\mathbb{D}_n} : \overline{\mathcal{S}}_{+1}(\mathbb{D}_n) \hookrightarrow \mathcal{Z}(\overline{\mathcal{S}}_{\omega} (\mathbb{D}_n))$. 
\item Let $\pi_1 : (X_1)^n \rightarrow X_n$ and $\pi_1^0 : (X_1^0)^n \rightarrow X_n^0$ denote the regular maps induced by  the gluing map $i_n$.
\item Set $\widehat{X}_n^0 := \mathrm{Specm}({\mathcal{S}}_{+1}(\mathbb{D}_n))$, denote by $\widehat{\pi}_1^0 : (\widehat{X}_1^0)^n \rightarrow \widehat{X}_n^0$ the morphism induced by the gluing map. Note that since the reduced stated skein algebra is, by definition, a quotient of its non-reduced version, $X_n^0$ is a subvariety of $\widehat{X}_n^0$ and $\pi_1^0$ is the restriction of $\widehat{\pi}_1^0$.
\item Write $Y_n:= \mathrm{Specm}(\mathcal{Z}(\mathcal{Z}_{\omega}(\mathbb{D}_n, \Delta_n)))$ and $Y_n^0:= \mathrm{Specm}(\mathcal{Z}_{+1}(\mathbb{D}_n, \Delta_n))$ and let $p_2: Y_n \rightarrow Y_n^0$ be the finite branched cover induced by the Frobenius morphism. Write $\pi_2: (Y_1)^n \rightarrow Y_n$ and $\pi_2^0: (Y_1^0)^n \rightarrow Y_n^0$ the regular morphisms induced by the gluing map $i': \mathcal{Z}_{\omega}(\mathbb{D}_n, \Delta_n) \hookrightarrow  \mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes n}$.
\item Denote by $\theta_n: Y_n \rightarrow X_n$ the regular map induced by the restriction to the center of the  quantum trace and $\theta_n^0 : Y_n^0 \rightarrow X_n^0$ the regular map induced by the quantum trace at $\omega=+1$.
\end{itemize}
\end{notations}

Note that the following diagram commutes: 
\begin{equation*}
\begin{tikzcd}
(Y_1)^n \arrow[r, "\pi_2"] \arrow[d, "(\theta_1)^n"] & Y_n  \arrow[d, "\theta_n"] \\
(X_1)^n \arrow[r, "\pi_1"] & X_n
\end{tikzcd}
\end{equation*}

Let us describe the maps $\pi_i$ and $p_i$ explicitly. By Theorem \ref{theorem_CFClassification},  an element $\hat{\chi}\in Y_n$ is an element $\hat{\chi}=(\chi, h_{p_1}, \ldots, h_{p_n}, h_{\partial})$, where $\chi \in Y_n^0$ and $h_{p_i}:=\hat{\chi}(H_{p_i})$ is such that $h_{p_i}^N=\chi(H_{p_i})$ and $h_{\partial}:= \hat{\chi}(H_{\partial})$ is such that $h_{\partial}^N=\chi(H_{\partial})$. The projection $p_2 : Y_n \rightarrow Y_n^0$ sends $(\chi, h_{p_i}, h_{\partial})$ to $\chi$. 


By Theorem \ref{theorem_center_skein}, an element $\hat{\rho}\in X_n$ is an element $\hat{\rho}=(\rho, t_1, \ldots, t_n, h_{\partial})$ where $\rho \in X_n^0$ and $t_i:= \hat{\rho}(\gamma_{p_i})$ is such that $T_N(t_i) = \rho(\gamma_{p_i})$ and $h_{\partial}:=\hat{\rho}(\alpha_{\partial})$ is such that $h_{\partial}^N = \rho(\alpha_{\partial})$. The projection $p_1 : X_n \rightarrow X_n^0$ sends $(\rho, t_i, h_{\partial})$ to $\rho$. The map $\theta_n$ sends $(\chi, h_{p_1}, \ldots, h_{p_n}, h_{\partial})$ to $(\theta_n^0(\chi), h_{p_1}+h_{p_1}^{-1}, \ldots, h_{p_n}+h_{p_n}^{-1}, h_{\partial})$. Note also that it follows from Proposition \ref{prop_pres_skein}, that 
 $\widehat{X}_1^0 = \operatorname{SL}_2(\mathbb{C})\times \operatorname{SL}_2(\mathbb{C})$ and that 
$X_1=\widetilde{D(B)}$ and $X_1^0=D(B)\subset  \operatorname{SL}_2(\mathbb{C})\times \operatorname{SL}_2(\mathbb{C})$ is the subset of pairs $(g_-, g_+)$ where $g_-$ is lower triangular and $g_+$ is upper triangular. 

\begin{definition}\textit{(Gauge group actions)}
\begin{enumerate}
\item We endow $Y_1^0$ with an algebraic structure of  $\mathbb{C}^*$-bimodule via the formula 
$$ (z\cdot \chi \cdot z') ([Z_1^{k_1}Z_2^{k_2}Z_3^{k_3}Z_4^{k_4}]) := z^{k_1} z'^{k_4} \chi ([Z_1^{k_1}Z_2^{k_2}Z_3^{k_3}Z_4^{k_4}]), $$
 for all $ \chi \in Y_1^0, z\in \mathbb{C}^*$.
\item We endow the variety $(Y^0_1)^n$  with the structure of left $(\mathbb{C}^*)^{n-1}$ module defined for  $\mathbf{z}=(z_1, \ldots, z_{n-1}) \in (\mathbb{C}^*)^{n-1}$ and $\mathbf{\chi}=(\chi_1, \ldots, \chi_n) \in (Y^0_1)^n$ by the formula
$$ \mathbf{z} \bullet \mathbf{\chi} := (\chi_1 \cdot z_1^{-1}, z_1 \cdot \chi_2 \cdot z_2^{-1}, \ldots, z_i \cdot \chi_{i+1} \cdot z_{i+1}^{-1}, \ldots, z_{n-1}^{-1}\cdot \chi_n).$$
 \item We endow $\widehat{X}_1^0=  \operatorname{SL}_2(\mathbb{C})\times \operatorname{SL}_2(\mathbb{C})$ with an algebraic structure of $\operatorname{SL}_2(\mathbb{C})$ bimodule by setting for $(g_-, g_+) \in \widehat{X}_1^0$ and $h, h' \in \operatorname{SL}_2(\mathbb{C})$ the formula $h\cdot (g_-, g_+) \cdot h' := (hg_- h', hg_+ h')$.
 \item We endow the variety $(\widehat{X}^0_1)^n$ with the structure of left $(\operatorname{SL}_2(\mathbb{C}))^{n-1}$ module defined for $\mathbf{h}=(h_1, \ldots, h_{n_1}) \in (\operatorname{SL}_2(\mathbb{C}))^{n-1}$ and $\mathbf{g}= (g_1, \ldots, g_n) \in (\widehat{X_1^0})^n$ by the formula
 $$ \mathbf{h} \bullet \mathbf{g}:= (g_1\cdot h_1^{-1}, h_1 \cdot g_2 \cdot h_2^{-1}, \ldots, h_i \cdot  g_{i+1} \cdot h_{i+1}^{-1}, \ldots, h_{n-1} \cdot g_n).$$
 \item Let $\iota : (\mathbb{C}^*)^{n-1} \hookrightarrow (\operatorname{SL}_2(\mathbb{C}))^{n-1}$ denote the group-morphism obtained by assigning to $z\in \mathbb{C}^*$ the matrix $\begin{pmatrix} z & 0 \\ 0 & z^{-1} \end{pmatrix}$. The action of $(\operatorname{SL}_2(\mathbb{C}))^{n-1}$ on $(\widehat{X}^0_1)^n$ induces, via $\iota$, an action of $(\mathbb{C}^*)^{n-1}$ which preserves the subvariety $(X_1^0)^n \subset (\widehat{X}_1^0)^n$ (because the multiplication by a diagonal matrix preserves both the sets of upper and lower matrices). We still denote by $\bullet$ the action of  $(\mathbb{C}^*)^{n-1}$ on $(X_1^0)^n$.
\end{enumerate}
\end{definition}
Note that the map $(\theta_1^0)^n : (Y_1^0) ^n \rightarrow (X_1^0)^n $ is $(\mathbb{C}^*)^{n-1}$ equivariant.

\begin{lemma}\label{lemma_factorization1}
\begin{enumerate}
\item The maps $\pi_1$ and $\pi_2$ are both surjective.
\item  Let $\hat{\mathbf{\chi}}=(\hat{\chi}_1, \ldots, \hat{\chi}_n) \in (Y_1)^n $, with $\hat{\chi}_i=(\chi_i, h_{p}^{(i)},  h_{\partial}^{(i)})$ and $\hat{\mathbf{\chi}}'=(\hat{\chi}'_1, \ldots, \hat{\chi}'_n) \in (Y_1)^n $ , with $\hat{\chi}'_i=(\chi'_i, h_{p}^{(i)'}, h_{\partial}^{(i)'})$. Write $\mathbf{\chi}:=p_2(\mathbf{\chi})= (\chi_1, \ldots, \chi_n)$ and $\mathbf{\chi}':=p_2(\mathbf{\chi})$. 
\par Then one has $\pi_2(\hat{\mathbf{\chi}})=\pi_2(\hat{\mathbf{\chi}}')$ if and only if $(i)$ there exists $\mathbf{z}\in (\mathbb{C}^*)^{n-1}$ such that $\mathbf{z}\bullet \mathbf{\chi} = \mathbf{\chi}'$ and $(ii)$ one has $h_p^{(i)}=h_p^{(i)'}$ for all $1\leq i \leq n$ and $\prod_{i=1}^n h_{\partial}^{(i)}= \prod_{i=1}^n h_{\partial}^{(i)'}$.
\item Let $\hat{\mathbf{g}}= (\hat{g}_1, \ldots, \hat{g}_n) \in (X_1)^n $, with $\hat{g}_i= (g_i, t_p^{(i)}, h^{(i)}_{\partial})$, and $\hat{\mathbf{g}}'= (\hat{g}'_1, \ldots, \hat{g}'_n) \in (X_1)^n $, with $\hat{g}'_i= (g'_i, t_p^{(i)'}, h^{(i)'}_{\partial})$. Write $\mathbf{g}=p_1(\hat{\mathbf{g}})$ and $\mathbf{g}'=p_1(\hat{\mathbf{g}}')$.
\par Then one has $\pi_1(\hat{\mathbf{g}})= \pi_1(\hat{\mathbf{g}}')$ if and only if $(i)$ there exists $\mathbf{z}\in (\mathbb{C}^*)^{n-1}$ such that $\mathbf{z}\bullet \mathbf{g} = \mathbf{g}'$ and $(ii)$ one has $t_p^{(i)}=t_p^{(i)'}$ for all $1\leq i \leq n$ and $\prod_{i=1}^n h_{\partial}^{(i)}= \prod_{i=1}^n h_{\partial}^{(i)'}$.
\end{enumerate}
\end{lemma}

\begin{proof}
First, by Corollary \ref{coro_GIT_CF}, the variety $Y_n^0$ is isomorphic to the GIT quotient $(Y_1^0)^n \mathbin{/\mkern-6mu/} (\mathbb{C}^*)^{n-1}$ and, using this isomorphism, the map $\pi_2^0 : (Y_1^0)^n \rightarrow Y_n^0$ is identified with the quotient map $(Y_1^0)^n \rightarrow (Y_1^0)^n \mathbin{/\mkern-6mu/} (\mathbb{C}^*)^{n-1}$. So $\pi_2^0$ is surjective and $\pi_2^0(\mathbf{\chi})= \pi_2^0 (\mathbf{\chi}')$ if and only if there exists $\mathbf{z}\in (\mathbb{C}^*)^{n-1}$ such that $\mathbf{z}\bullet \mathbf{\chi} = \mathbf{\chi}'$. This implies the surjectivity of $\pi_2$ and the second assertion of the lemma.


 Next, by Corollary \ref{coro_GIT_skein}, the variety $\widehat{X}_n^0$ is isomorphic to the GIT quotient $(\widehat{X}_1^0)^n \mathbin{/\mkern-6mu/} (\operatorname{SL}_2(\mathbb{C}))^{n-1}$ and, using this isomorphism, the map $\pi_1^0 : (X_1^0)^n \rightarrow X_n^0$ is identified with the quotient map. So $\widehat{\pi}_1^0$ is surjective and $\widehat{\pi}_1^0(\mathbf{g})= \widehat{\pi}_1^0(\mathbf{g}')$ if and only if there exists $\mathbf{h} \in (\operatorname{SL}_2(\mathbb{C}))^{n-1}$ such that $\mathbf{h}\bullet \mathbf{g}= \mathbf{g'}$. 
 
  In particular, if both $\mathbf{g}$ and $\mathbf{g}'$ belong to $(X_1^0)^n \subset (\widehat{X}_1^0)^n$ and $\mathbf{g}'=\iota(\mathbf{z}) \bullet \mathbf{g}$ for some $\mathbf{z}\in (\mathbb{C}^*)^{n-1}$, then $\pi_1^0(\mathbf{g})=\pi_1^0(\mathbf{g}')$. Conversely,  if both $\mathbf{g}$ and $\mathbf{g}'$ belong to $(X_1^0)^n \subset (\widehat{X}_1^0)^n$ are such that $\pi_1^0(\mathbf{g})=\pi_1^0(\mathbf{g}')$, there exists $\mathbf{h} \in (\operatorname{SL}_2(\mathbb{C}))^{n-1}$ such that $\mathbf{h}\bullet \mathbf{g}= \mathbf{g'}$. Write  $\mathbf{g}= (g_1, \ldots, g_n)$, with $g_i=(g_{i-} g_{i+})$, and $\mathbf{h}=(h_1, \ldots, h_{n-1})$, and let us prove by induction on  $i\in \{1, \ldots, n-1\}$ that $h_i$ is a diagonal matrix. First since both $g_{1-}$ and $g_{1-}h_1^{-1}$ are lower triangular and both $g_{1+}$ and $g_{1+} h_1^{-1}$ are upper triangular, then $h_1$ is diagonal. Next, supposing that $h_{i-1}$ is diagonal, then both $h_{i-1} g_{i-}$ and $h_{i-1} g_{i-}h_{i}$ are lower triangular and both $h_{i-1} g_{i+}$ and $h_{i-1}g_{i+}h_{i}$ are upper triangular so $h_{i}$ is diagonal. By induction on $i$, this proves that there exists $\mathbf{z}\in (\mathbb{C}^*)^{n-1}$ such that $\mathbf{h}= \iota(\mathbf{z})$. So we have proved that $\pi_1^0(\mathbf{g})=\pi_1^0(\mathbf{g}')$ if and only if there exists $\mathbf{z}\in (\mathbb{C}^*)^{n-1}$ such that $\mathbf{g}'=\mathbf{z}\bullet \mathbf{g}$. This implies the third assertion of the lemma.
  
  
  It remains to prove the surjectivity of $\pi_1^0$, which will imply the surjectivity of $\pi_1$ and will conclude the proof. Let $x\in X_n^0 \subset \widehat{X}_n^0$ and let us find an element $\mathbf{g}' \in (X_1^0)^n$ such that $\pi_1^0(\mathbf{g})=x$. By surjectivity of $\widehat{\pi}_1^0$, there exists $\mathbf{g} \in (\widehat{X}_1^0)^n$ such that $\widehat{\pi}_1^0(\mathbf{g})=x$. Let us define an element $\mathbf{h}=(h_1, \ldots, h_{n-1}) \in (\operatorname{SL}_2(\mathbf{C}))^{n-1}$ such that $\mathbf{g}':=\mathbf{h}\bullet \mathbf{g}$ belongs to $(X_1^0)^n \subset (\widehat{X}_1^0)^n$. This will imply that $\pi_1^0(\mathbf{g}')=x$ and conclude the proof. We define the $h_i$ recursively using the following key remark: for any two matrices $a,b\in \operatorname{SL}_2(\mathbb{C})$ there exists $h\in \operatorname{SL}_2(\mathbb{C})$ such that $ah$ is upper triangular and $bh$ is lower triangular. 
  Write $\mathbf{g}=(g_1, \ldots, g_n)$ with $g_i=(g_{i-}, g_{i+})$.
  We first choose $h_1$ such that $g_{1-}h_1^{-1}$ is lower triangular and $g_{1+}h_1^{-1}$ is upper triangular. For $i\in \{2, \ldots, n-1\}$, assuming we have defined $h_{i-1}$, we choose $h_{i}$ such that $(h_{i-1}g_{i -}) h_{i}^{-1}$ is lower triangular and $(h_{i-1} g_{i +})h_i^{-1}$ is upper triangular. Set $\mathbf{g}':= \mathbf{h} \bullet \mathbf{g} = (g_1', \ldots, g_n')$. By construction, for $i\in \{1, \ldots, n-1\}$ one has $g_i'\in D(B)$. Moreover, since $\pi_1^0(\mathbf{g}')=\pi_1^0(\mathbf{g})=x \in X_n^0$, the product $g_{1-}'\ldots g_{n-}'$ is lower triangular and $g_{1+}'\ldots g_{n+}'$ is upper triangular. This implies that $g_n \in D(B)$ as well, so $\mathbf{g}'\in D(B)^n= (X_1^0)^n$ and the proof is completed.
 


\end{proof}


\begin{lemma}\label{lemma_factorization2}
When $n\geq 2$, the map $\pi_1 \circ (\theta_1)^n : (Y_1)^n \rightarrow X_n$ is surjective.
\end{lemma}

\begin{proof}
Let $x\in X_n$. By Lemma \ref{lemma_factorization1}, $\pi_1$ is surjective, so there exists $\widehat{\mathbf{g}} \in (X_1)^n $ such that $\pi_1(\widehat{\mathbf{g}})=x$. Write $\mathbf{g}:= p_1(\widehat{\mathbf{g}}) = (g_1, \ldots, g_n)\in D(B)^n$. By Proposition \ref{prop_classification_modules}, $\widehat{\mathbf{g}}$ lies in the image of $(\theta_1)^n $ if and only if for every $i \in \{1, \ldots, n\}$, $\varphi(g_i)\in \operatorname{SL}_2^0$ is neither a non-scalar diagonal matrix nor a non-diagonal triangular matrix with diagonal elements equal to $\pm 1$. For any pair $(g,g') \in D(B)$, one can find $z\in \mathbb{C}^*$ such that nor $g \cdot \begin{pmatrix} z^{-1} & 0 \\ 0 & z \end{pmatrix}$ nor $\begin{pmatrix} z & 0 \\ 0 & z^{-1} \end{pmatrix}\cdot g'$ is non-scalar diagonal or non-diagonal triangular with diagonal entries equal to $\pm 1$. Using this remark, we construct by induction on $i$ some $z_i \in \mathbb{C}^*$ such that for $\mathbf{z}=(z_1, \ldots, z_{n-1})$, the element  $\mathbf{g}':=\mathbf{z} \bullet \mathbf{g}= (g_1', \ldots, g_n')$ lies in the image of $(\theta_1^0)^n$. Lift $\mathbf{g}'$ to an element $\widehat{\mathbf{g}}' \in (X_1)^n$ by imposing that if $\hat{g}_i = (g_i, t_i, h_{\partial}^{(i)})$ then $\widehat{\mathbf{g}'} = (\widehat{g}'_1, \ldots, \widehat{g}'_n)$ with $\widehat{g}_i'=(g_i', t_i, h_{\partial}^{(i)'})$ where the $h_{\partial}^{(i)'}$ are chosen arbitrarily such that $(h_{\partial}^{(i)'})^{-N}$ is equal to the left upper coefficient of $g'_{i-}g'_{i+}$ and such that $\prod_{i=1}^n h_{\partial}^{(i)'} = \prod_{i=1}^n h_{\partial}^{(i)}$. Then  $\widehat{\mathbf{g}}'= (\theta_1)^n (y)$ for some $y\in (Y_1)^n$ and $\pi_1(\widehat{\mathbf{g}'})= \pi_1(\widehat{\mathbf{g}})=x$ so 
$\pi_1 \circ (\theta_1)^n (y) =  x$. This proves the surjectivity of $\pi_1 \circ (\theta_1)^n : (Y_1)^n \rightarrow X_n$.

\end{proof}

\begin{lemma}\label{lemma_factorization3}
The image of $\overline{\mathcal{S}_{\omega}}(\mathbb{D}_2)$ in $\mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}$ by the composition of the gluing maps and quantum trace is included in the subalgebra generated by 
$$G:=\{ K^{\pm 1/2}\otimes K^{\pm 1/2}, \operatorname{Tr}(\gamma_{p_1}), \operatorname{Tr}(\gamma_{p_2}), H_{\partial}^{\pm 1}\otimes H_{\partial}^{\pm 1}, EH_{\partial}\otimes 1, K^{1/2}L^{-1/2}\otimes 1, F\otimes 1, L\otimes F, E\otimes K, 1\otimes E \}.$$
\end{lemma}

\begin{proof}
Consider the three arcs $\alpha^{(2)}, \beta^{(2)}$ and $\delta$ of $\mathbb{D}_2$ drawn in Figure \ref{fig_arcs_D2}, denote by $\kappa:= \operatorname{Tr}^{\otimes 2} \circ i_2 : \overline{\mathcal{S}_{\omega}}(\mathbb{D}_2) \hookrightarrow \mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}$ the composition of the gluing map and the quantum trace
and let $\mathcal{A} \subset \mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}$ be the algebra generated by $G$.


\begin{figure}[!h] 
\centerline{\includegraphics[width=8cm]{arcs_D2.eps} }
\caption{The stated arcs $\delta_{\varepsilon \varepsilon'}, \alpha^{(2)}_{\varepsilon \varepsilon'}$ and $ \beta^{(2)}_{\varepsilon \varepsilon'}$ in $\mathbb{D}_2$.} 
\label{fig_arcs_D2} 
\end{figure} 


 By \cite[Proposition $3.4$]{KojuAzumayaSkein}, $\overline{\mathcal{S}_{\omega}}(\mathbb{D}_2)$ is generated, as an algebra, by the stated arcs $\alpha^{(2)}_{\varepsilon \varepsilon'}, \beta^{(2)}_{\varepsilon \varepsilon'}, \delta_{\varepsilon \varepsilon'}$ for $\varepsilon, \varepsilon' \in \{-, +\}$ so it is sufficient to prove that the image by $\kappa$ of each of these $12$ generators lies in $\mathcal{A}$. First note that $\kappa (\alpha^{(2)}_{\varepsilon \varepsilon'})= \Delta( \operatorname{Tr}(\alpha_{\varepsilon \varepsilon'}))$ and  $\kappa (\beta^{(2)}_{\varepsilon \varepsilon'})= \Delta( \operatorname{Tr}(\beta_{\varepsilon \varepsilon'}))$. So one has 
 \begin{align*}
& \kappa(\alpha^{(2)}_{++}) = K^{-1/2}\otimes K^{-1/2} \in \mathcal{A}, \quad  \kappa(\alpha^{(2)}_{--})= K^{1/2}\otimes K^{1/2} \in \mathcal{A}, \quad  \kappa(\alpha^{(2)}_{-+})=\kappa(\beta^{(2)}_{+-})=0, \\
 & \kappa(\beta^{(2)}_{++})= L^{-1/2}\otimes L^{-1/2}=(H_{\partial})^{\otimes 2} (K^{1/2})^{\otimes 2} \in \mathcal{A}, \quad 
  \kappa(\beta^{(2)}_{--})= L^{1/2}\otimes L^{1/2}=(H_{\partial}^{-1})^{\otimes 2} (K^{-1/2})^{\otimes 2} \in \mathcal{A}, \\
 & \kappa(\alpha^{(2)}_{+-}) = -A^{-1}(q-q^{-1}) \Delta(EK^{-1/2}) = -A^{-1}(q-q^{-1})(E\otimes K + 1 \otimes E) (K^{-1/2} \otimes K^{-1/2}) \in \mathcal{A}, \\
 & \kappa(\beta^{(2)}_{-+}) = -A(q-q^{-1}) \Delta(L^{-1/2}F) = -A(q-q^{-1}) H_{\partial}(K^{1/2}\otimes K^{1/2})(F\otimes 1 + L\otimes F) \in \mathcal{A}.
 \end{align*}
 
 To compute $\kappa(\delta_{\varepsilon \varepsilon'})$, we use the formula
 $$ i_2(\delta_{\varepsilon \varepsilon'}) = \omega^{-1} (\alpha_{\varepsilon -} \beta_{\varepsilon' +} \otimes 1) - \omega^{-5}(\alpha_{\varepsilon +} \beta_{\varepsilon' -} \otimes 1),$$
 from which we deduce that 
 \begin{align*}
 & \kappa(\delta_{++}) = -\omega(q-q^{-1}) (EH_{\partial}\otimes 1) \in \mathcal{A}, \quad \kappa(\delta_{-+})= \omega^{-1}(K^{1/2}L^{-1/2}\otimes 1) \in \mathcal{A}, \\
 & \kappa(\delta_{--})= \omega^{-3}(q-q^{-1})(K^{1/2}L^{-1/2}\otimes 1)(F\otimes 1) \in \mathcal{A}.
 \end{align*}
 Eventually, using the skein relation $\gamma_{p_1}=\omega \delta_{+-} - \omega^5 \delta_{-+}$, one finds
 $$ \kappa(\delta_{+-}) = \omega^{-1}\operatorname{Tr}(\gamma_{p_1}) + \omega^{-5} \kappa(\delta_{-+}) \in \mathcal{A}.$$
 This concludes the proof.
 

\end{proof}

\begin{lemma}\label{lemma_factorization4}
Let $V_1, V_2, V_1', V_2'$ be some projective indecomposable representations of $\mathcal{O}_q[D(B)]$ such that $V_1\otimes V_2$ and $V_1'\otimes V_2'$ induce the same character over the center of $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_2)$. Then $V_1\otimes V_2$ and $V_1'\otimes V_2'$ are isomorphic as $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_2)$-modules.
\end{lemma}

\begin{proof}
By Proposition \ref{prop_representations}, one can suppose that each module $V_1,V_2, V_1', V_2'$ is either of the form $V(\lambda, \mu, a,b)$ or of the form $\widetilde{V}(\lambda, \mu, c)$. We detail the proof in the particular case where all these modules have the form $V(\lambda, \mu, a,b)$, the other cases are very similar and left to the reader. So suppose that for $i=1,2$, $V_i = V(\lambda_i, \mu_i, a_i, b_i)$ and $V_i' = V(\lambda'_i, \mu'_i, a'_i, b'_i)$ and let $\chi \in X_2$ denote the character over the center of $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_2)$ induced by both $V_1\otimes V_2$ and $V_1'\otimes V'_2$. For $i=1,2$, we set $h_{\partial i}:= (\lambda_i \mu_i)^{-1}$ and $h'_{\partial i}:= (\lambda'_i \mu'_i)^{-1}$.


Note that
\begin{align*}
& (\lambda_1\lambda_2)^N = (\lambda'_1 \lambda'_2)^N = \chi(\alpha_{--}^{(N)}), \quad (\lambda_1 \mu_1^{-1})^N = (\lambda_1'\mu_1'^{-1})^N = \chi(\delta_{-+}^{(N)}), \quad \lambda_1\lambda_2\mu_1\mu_2 = \lambda_1'\lambda_2'\mu_1'\mu_2' = \chi(H_{\partial}^{-1}), \\
& \lambda_i \mu_i^{-1}q + \mu_i \lambda_i^{-1}q^{-1} + h_{\partial_i} a_ib_i(q-q^{-1})^2 = \lambda'_i \mu_i^{'-1}q + \mu'_i \lambda_i^{'-1}q^{-1} + h'_{\partial_i} a'_ib'_i(q-q^{-1})^2 = -\chi(H_{p_i}), \mbox{ for }i=1,2.
\end{align*}
Recall from Proposition \ref{prop_representations} that $V(\lambda, \mu, a, b) \cong V(\lambda q^{-n}, \mu q^n, a', b)$ when $a'b=ab + \frac{q^{1-n}\lambda^2 - q^{n-1}\mu^2}{q-q^{-1}}[n]$. So, replacing $V_1'$ and $V_2'$ by some isomorphic $\mathcal{O}_q[D(B)]$-modules if necessary, one can suppose that 
\begin{equation}\label{eq_relations_coefficients}
\lambda_1\lambda_2 = \lambda_1'\lambda_2', \quad \lambda_1 \mu_1^{-1} = \lambda_1' \mu_1^{'-1},\quad \lambda_1\lambda_2\mu_1\mu_2 =  \lambda_1'\lambda_2'\mu_1'\mu_2', \quad h_{\partial i}a_i b_i = h_{\partial i}' a_i'b_i'  \quad \mbox{for}\quad i=1,2.
\end{equation}

Let $\{ c_{i,j} \}_{ 0 \leq i, j \leq N-1}\in (\mathbb{C}^*)^{N^2}$ be a family of non-vanishing complex numbers satisfying the following system of equations:

\begin{align}
\label{K1} 
& c_{i+1, j} = c_{i,j}, \mbox{ for }i\neq N-1; \quad b_1c_{N-1, j}= b_1' c_{0,j} \mbox{ and } h'_{\partial 1}a'_1 c_{N-1, j}= h_{\partial 1} a_1 c_{0,j}, \mbox{ for all }j; \\
\label{K2} & c_{i,j+1} = \frac{h'_{\partial 2}}{h_{\partial 2}} c_{i,j}, \mbox{ for }j \neq N-1; \quad a'_2 c_{i,0} = a_2 c_{i,N-1} \mbox{ and }b_2h_{\partial 2} c_{i,0} = b_2'h'_{\partial 2} c_{i, N-1}, \mbox{ for all }i.
\end{align}

Note that the equations $ h_{\partial i}a_i b_i = h_{\partial i}' a_i'b_i', i=1,2$ in Equation \eqref{eq_relations_coefficients} ensures that the system \{\eqref{K1}, \eqref{K2}\} admits a solution. Let $\Psi : V_1 \otimes V_2 \xrightarrow{\cong} V_1'\otimes V_2'$ be the isomorphism defined by $\Psi(v_i\otimes v_j):= c_{i,j} v'_i \otimes v'_j$ for all $0 \leq i,j \leq N-1$ and let us prove that $\Psi$ is $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_2)$-equivariant. By Lemma \ref{lemma_factorization3}, it is sufficient to show that 
\begin{equation}\label{K3}
\Psi(x \cdot v_i\otimes v_j) = x\cdot \Psi(v_i\otimes v_j), \quad \mbox{ for all }x\in G, 0 \leq i, j \leq N-1.
\end{equation}

Replacing $x$ by each of the $12$ elements of $G$ and using the formulas of Definition \ref{def_representations} and Equations \eqref{eq_relations_coefficients}, straightforward computations left to the reader show that the system of equations \eqref{K3} is equivalent to the system of equations \{\eqref{K1}, \eqref{K2}\}, so $\Psi$ is equivariant and the proof is completed.

\end{proof}

\begin{proof}[Proof of Theorem \ref{theorem_factorization}]
By Lemma \ref{lemma_character}, a family $(V_1, \ldots, V_n)$ of $n$ projective indecomposable $\mathcal{O}_q[D(B)]$-modules is characterized, up to isomorphism, by a point $\widehat{\mathbf{g}}\in (X_1)^n$. The corresponding factorizable module $V_1\otimes \ldots \otimes V_n$ induces the character $\pi_1(\widehat{\mathbf{g}})$ on the center of $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)$. The algebra $\mathcal{Z}_{\omega}(\mathbb{D}_n, \Delta_n)$, like any quantum torus at root of unity, is Azumaya of constant rank, so $Y_n$ is in bijection with the set of isomorphism classes of simple $\mathcal{Z}_{\omega}(\mathbb{D}_n, \Delta_n)$-modules. 


 The fact that any simple $\mathcal{Z}_{\omega}(\mathbb{D}_n, \Delta_n)$-module induces a factorizable $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)$-module follows from the surjectivity of $\pi_2$ proved in Lemma \ref{lemma_factorization1}. 
 
 
  Let $\chi$ denote the map sending an isomorphism class of factorizable $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)$-module to its induced character in $X_n$ and let us prove that $\chi$ is a bijection. The surjectivity of $\chi$ follows from the surjectivity of $\pi_1$ proved in Lemma \ref{lemma_factorization1}. To prove that $\chi$ is injective, we need to show that if $V_1\otimes \ldots \otimes V_n$ and $V'_1\otimes \ldots \otimes V'_n$ are two factorizable $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)$-modules which induce the same character $x\in X_n$, then they are isomorphic as $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)$-modules. Let $\widehat{\mathbf{g}}, \widehat{\mathbf{g}}' \in (X_1)^n$ be the characters induced by the families $(V_1, \ldots, V_n)$ and $(V'_1, \ldots, V'_n)$ respectively and write $\mathbf{g}=p_1(\widehat{\mathbf{g}})$ and $\mathbf{g}'=p_1(\widehat{\mathbf{g}}')$. Since $\pi_1(\widehat{\mathbf{g}})= \pi_1(\widehat{\mathbf{g}}')=x$, by Lemma \ref{lemma_factorization1}, there exists $\mathbf{z}\in (\mathbb{C}^*)^{n-1}$ such that $\mathbf{g}'= \mathbf{z}\bullet \mathbf{g}$ and such that $\widehat{\mathbf{g}}$ and $ \widehat{\mathbf{g}}' $ have the same inner punctures and boundary invariant. Since the group $(\mathbb{C}^*)^{n-1}$ is generated by elements of form $(1, \ldots, 1, z, 1, \ldots, 1)$, where $z\in \mathbb{C}^*$ is in $i$-th position, it is sufficient to make the proof when  $\mathbf{z}= (1, \ldots, 1, z, 1, \ldots, 1)$, the general case will follow using an induction on the number of such generators in a decomposition. For this particular $\mathbf{z}$, replacing the family $(V_1', \ldots, V_n')$ by a family of isomorphic $\mathcal{O}_q[D(B)]$ if necessary, one can suppose that $V_j = V_j'$ for $j\notin \{ i, i+1\}$. Now $V_i\otimes V_{i+1}$ and $V'_i \otimes V'_{i+1}$ induce the same character in $X_2$, so by Lemma \ref{lemma_factorization4}, there exists an isomorphism $\Psi : V_i\otimes V_{i+1} \xrightarrow{\cong} V'_i \otimes V'_{i+1}$ of $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_2)$-modules. The isomorphism $id^{\otimes i-1} \otimes \Psi \otimes id^{\otimes n-i-1} : V_1 \otimes \ldots \otimes V_n \xrightarrow{\cong} V'_1\otimes \ldots \otimes V'_n$ is $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)$-equivariant so we have proved that $\chi$ is a bijection.
  
  
  It remains to prove that, for $n\geq 2$,  any factorizable $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)$-module is isomorphic to a module $V_1\otimes \ldots \otimes V_n$, where $V_i$ extends to a $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-module, but this follows from Lemma \ref{lemma_factorization2} and the injectivity of $\chi$.

\end{proof}













\subsection{Clebsch-Gordan type formula}

For each $\hat{g} \in \widetilde{D(B)}$, fix an arbitrary projective indecomposable $\mathcal{O}_q[D(B)]$-module $V(\hat{g})$ whose induced character is $\hat{g}$. By Lemma \ref{lemma_character}, the isomorphism class of $V(\hat{g})$ only depends on $\hat{g}$. Recall that for $\hat{g}=(g,t,h_{\partial})$, the element $t$ is a zero of the polynomial $T_N(X)-\operatorname{Tr}(\varphi(g))$ and denote by $m_t$ its multiplicity. If $c=c_0^N+c_0^{-N}$, one has the factorization $T_N(X)-c = \prod_{i\in \mathbb{Z}/N{\mathbb{Z}}} (X-q^ic_0 - q^{-i}c_0^{-1})$, therefore 
$m_t = \left\{ \begin{array}{ll} 2 &, \mbox{ if }\operatorname{Tr}(\varphi(g))=\pm 2\mbox{ and } t\neq \pm 2; \\ 1 &, \mbox{ else.} \end{array}\right.$


The goal of this subsection is to prove the following 
\begin{theorem}\label{theorem_CG}
For any $(g_1,t_1, h_{\partial}^{(1)})$ and $(g_2,t_2, h_{\partial}^{(2)})$ in $\widetilde{D(B)}$, one has an isomorphism of $\mathcal{O}_q[D(B)]$-modules:
\begin{equation}\label{eq_CB}
V(g_1,t_1,h_{\partial}^{(1)}) \otimes V(g_2, t_2, h_{\partial}^{(2)}) \cong \oplus_{T_N(t)=\operatorname{Tr}(\varphi(g_1g_2))} m_t V(g_1g_2, t, h_{\partial}^{(1)}h_{\partial}^{(2)}).
\end{equation}

\end{theorem}


\begin{lemma}\label{lemma_CG1}
There exists some integers $n_t \in \{0, \ldots N\}$ such that $\sum_t n_t =N$ and 
\begin{equation}\label{eq_CG2}
V(g_1,t_1,h_{\partial}^{(1)}) \otimes V(g_2, t_2, h_{\partial}^{(2)}) \cong \oplus_{T_N(t)=\operatorname{Tr}(\varphi(g_1g_2))} n_t V(g_1g_2, t, h_{\partial}^{(1)}h_{\partial}^{(2)}).
\end{equation}
\end{lemma}

\begin{proof}
For $i=1,2$, write $V_i:= V(g_i,t_i,h_{\partial}^{(i)})$. Since $V_1\otimes V_2$ is finite dimensional, it is totally decomposable, i.e. it is a direct sum of indecomposable $\mathcal{O}_q[D(B)]$-modules. Since $V_1$ and $V_2$ are projective objects of the category of weight modules, so is $V_1\otimes V_2$, thus each indecomposable summand of $V_1\otimes V_2$ is a projective object of the category of weight modules, so is of the form $V(\hat{g})$ by Lemma \ref{lemma_character}. Let $V(g,t,h_{\partial})$ be such an indecomposable summand of $V_1\otimes V_2$. First, since $H_{\partial}$ is group-like in $\mathcal{O}_q[D(B)]$, one has $h_{\partial}=h_{\partial}^{(1)}h_{\partial}^{(2)}$. Next, denoting by $\Delta_0 : \mathbb{C}[D(B)] \rightarrow \mathbb{C}[D(B)]^{\otimes 2}$ the coproduct defining the group product in the algebraic group $D(B)$ and identifying $\mathbb{C}[D(B)]$ with $\overline{\mathcal{S}}_{+1}(\mathbb{D}_1)$ and $\mathcal{O}_q[D(B)]$ with $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_1)$, one has the following commuting diagram: 
$$
 \begin{tikzcd}
\mathbb{C}[D(B)] 
\arrow[r, "\Delta_0"] \arrow[d, hook, "j_{\mathbb{D}_1}"]
 &
  \mathbb{C}[D(B)]^{\otimes 2}
   \arrow[d, hook, "j_{\mathbb{D}_1}^{\otimes 2}"] 
   \\
\mathcal{O}_q[D(B)]
 \arrow[r, "\Delta"] 
 & 
 \mathcal{O}_q[D(B)]^{\otimes 2}
\end{tikzcd}
$$
where $j_{\mathbb{D}_1}$ represents the Chebyshev morphism. It follows that $g=g_1g_2$, therefore $V_1\otimes V_2$ is isomorphic to a direct sum of modules $ V(g_1g_2, t, h_{\partial}^{(1)}h_{\partial}^{(2)})$. The formula $\sum_t n_t =N$ is proved by identifying the dimension of both sizes of Equation \eqref{eq_CG2}.

\end{proof}

The strategy for the proof of the fact that $n_t=m_t$ was suggested to the author by S.Baseilhac and is somehow implicit in \cite[Proposition $5.8$]{Baseilhac_QCoadjointOrbit}. First, we suppose that for $i=1,2$, $V_i$ has the form $V_i=V(\lambda_i,\mu_i,a_i,b_i)$ of Definition \ref{def_representations}. The Borel subalgebra $B_q \subset \mathcal{O}_q[D(B)]$ is the Hopf subalgebra generated by $K^{\pm 1/2}$ and $F$. In \cite{FadeevKashaevQDilog, Kashaev_6jSymbol} (see also \cite{BaseilhacThesis}), Fadeev and Kashaev defined explicit Clebsch-Gordan operators for the tensor product of two cyclic $B_q$-modules. The main idea is to use those Fadeev-Kashaev operators to define Clebsch-Gordan type operators for $V_1\otimes V_2$, in the case where the parameters $(\lambda_i,\mu_i,a_i,b_i )$ are generic, to obtain the decomposition \eqref{eq_CB} explicitly. The non-generic cases will be obtained by analytic continuation in the parameters $(\lambda_i,\mu_i,a_i,b_i )$. We first introduce some notations.

\begin{notations}
\begin{enumerate}
\item
Let $x,y,z \in \mathbb{C}^*$ such that $x^N+y^N=z^N$. For $k\geq 0$, we set 
$$ w(x,y,z | k):= \prod_{i=0}^k \frac{y}{z-xq^{2i}}.$$
Note that $w(x,y,z| k+N)=w(x,y,z |k)$, so we can (and do) define $w(x,y,z|k)$ for $k\in \mathbb{Z}/N\mathbb{Z}$. It is designed to satisfy the following equation:
\begin{equation}\label{eq_w}
z w(x,y,z| k) = y w(x,y,z|k-1) + xq^{2k} w(x,y,z|k).
\end{equation}
\item
For $k\in \mathbb{Z}/N\mathbb{Z}$, we denote by $\lceil k \rfloor\in \{ 0, \ldots, N-1\}$ the integer whose class is $k$ modulo $N$. 
\item
We write $x^{\delta_{i,j}}$ the element whose value is $x$ if $i=j$ and $1$ else.
\item Suppose that  $b_1, b_2$ and $b:= b_1+b_2\mu_1^{2N}$ are non-zero and fix $\widetilde{b_1}, \widetilde{b_2}, \widetilde{b}$ such that $\widetilde{b_1}^N=b_1,\widetilde{b_2}^N=b_2,\widetilde{b}^N=b$. For $\alpha \in \mathbb{Z}/N\mathbb{Z}$, we set $\widetilde{b_{\alpha}}:= \widetilde{b}q^{2\alpha}$. We define
$$ x_{\alpha}:= \widetilde{b_2}\mu_1^2q^{2\alpha}, \quad y:= \widetilde{b_1}, \quad z_{\alpha}:=\widetilde{b_{\alpha}}.$$
Note that $x_{\alpha}^N +y_{\alpha}^N = z_{\alpha}^N$. Let

$$ (K_{\alpha})_k^{a,b}:= \frac{ \widetilde{b_{\alpha}}^{\lceil k \rfloor}} { \widetilde{b_1}^{\lceil a \rfloor } \widetilde{b_2}^{\lceil b \rfloor}} w(x_{\alpha}, y , z_{\alpha} |a-\alpha).$$
Using Equation \eqref{eq_w}, we deduce the following formula:

\begin{equation}\label{eq_w2}
b^{\delta_{k, N-1}} (K_{\alpha})_{k+1}^{a, b} = b_1^{\delta_{a, N-1}} (K_{\alpha})_k^{a-1, b} + b_2^{\delta_{b,N-1}} \mu_1^2 q^{2a} (K_{\alpha})_k^{a,b-1}.
\end{equation}

\item We eventually set
$$ a_{\alpha} := \frac{\widetilde{b} q^{2\alpha}a_2}{b\widetilde{b_2}} + \frac{\widetilde{b} \lambda_2^2 q^{2\alpha}}{ \widetilde{b}q^{2\alpha-2} - \widetilde{b_2}\mu_1^2}\left( \frac{\lambda_1^2 - \mu_1^2}{q -q^{-1}} +a_1b_1\right).$$
\end{enumerate}
\end{notations}

Let $C:= (\mathbb{C}^*)^2 \times \mathbb{C}^2$ be the space of parameters $(\lambda, \mu, a, b)$.

\begin{lemma}\label{lemma_CG2}
There exists an open Zariski subset $U \subset C\times C$ such that if $V_i= V(\lambda_i, \mu_i, a_i, b_i)$, for $i=1,2$, where $\left( (\lambda_1,\mu_1,a_1,b_1), (\lambda_2,\mu_2,a_2,b_2)\right) \in U$, then the decomposition \eqref{eq_CB} holds, where each multiplicity is $m_t=1$.
\end{lemma}

\begin{proof}
\textbf{Step 1} Suppose that  $b_1, b_2$ and $b:= b_1+b_2\mu_1^{2N}$ are non-zero. For $\alpha \in \mathbb{Z}/N\mathbb{Z}$, we define 
$$ K_{\alpha} : V(\lambda_1\lambda_2, \mu_1\mu_2, a_{\alpha}, b_{\alpha}) \hookrightarrow V_1\otimes V_2$$
by
$$ K_{\alpha} v_k := \sum_{i+j=k} (K_{\alpha})_k^{i, j} v_i \otimes v_j.$$
Let us first prove that $K_{\alpha}$ is $\mathcal{O}_q[D(B)]$-equivariant. First, the fact that it sends $v_k$ to the subspace spanned by vectors $v_i\otimes v_j$ where $i+j=k$, implies that $K_{\alpha}$ commutes with the actions of $K^{\pm 1/2}$ and $L^{\pm 1/2}$. We next show that it commutes with the action of $F$. On the one hand
$$ K_{\alpha} F v_k = b^{\delta_{k, N-1}} K_{\alpha} v_{k+1} = b^{\delta_{k, N-1}} \sum_{a+b=k+1} (K_{\alpha})_{k+1}^{a,b} v_a \otimes v_b.$$
On the other hand, since $\Delta(F)=F\otimes 1 + L\otimes F$, one has:
\begin{eqnarray*}
 F K_{\alpha} v_k &=&  \sum_{i+j=k} (F\otimes 1 +L\otimes F) (K_{\alpha})_k^{i,j} v_i \otimes v_j \\
  &=& \sum_{i+j=k} (K_{\alpha})_k^{i,j} \left( b_1^{\delta_{i, N-1}} v_{i+1} \otimes v_j + b_2^{\delta_{j, N-1}} \mu_1^2 q^{2i} v_i \otimes v_{j+1} \right) \\
  &=& \sum_{a+b=k+1} \left( b_1^{\delta_{a, N-1}} (K_{\alpha})_k^{a-1, b} + b_2^{\delta_{b, N-1}} \mu_1^2 q^{2a} (K_{\alpha})_k^{a, b-1} \right) v_a\otimes v_b
  \end{eqnarray*}
  So the equality $F K_{\alpha} = K_{\alpha} F$ follows from Equation \eqref{eq_w2}. Let $V_{\alpha}\subset V_1\otimes V_2$ be the image of $K_{\alpha}$. Since the eigenvalues of the operator $K^{1/2}$ acting on $V_{\alpha}$ are simple and $K_{\alpha}$ is $K^{1/2}$-equivariant, the vectors $\{K_{\alpha} v_k, k\in \mathbb{Z}/N\mathbb{Z}\}$ form a basis of $V_{\alpha}$ and $K_{\alpha}$ is injective. Let 
  $$t_{\alpha} := -(q\lambda_1\mu_1^{-1}\lambda_2\mu_2^{-1} + q^{-1} \mu_1\lambda_1^{-1} \mu_2\lambda_2^{-1}) - (q -q^{-1})^2 (\lambda_1\mu_1\lambda_2\mu_2)^{-1}ba_{\alpha}.$$
  A direct computation, left to the reader, shows that $T \cdot K_{\alpha}v_0 = t_{\alpha}v_0$. Since $T$ commutes with $F$, and since $v_k=F^k v_0$, one has $T\cdot K_{\alpha} v_k = t_{\alpha}v_k$ for all $k\in \mathbb{Z}/N\mathbb{Z}$. So $K_{\alpha}$ commutes with $T$. Now using the facts that the action of $F$ on $V_{\alpha}$ is an invertible operator (since $b\neq 0$) and using the formula 
  $$ T= -H_{\partial}(q^{-1}K+qL) - (q- q^{-1})^2 EF H_{\partial}, $$
  we see that the facts that $K_{\alpha}$ commutes with the actions of $T, F, K, L$ and $H_{\partial}$, imply that $K_{\alpha}$ commutes with $E$. Therefore $K_{\alpha}$ is $\mathcal{O}_q[D(B)]$-equivariant.
  
 \vspace{2mm}
 \par \textbf{Step 2} We now prove that if the parameters $(\lambda_i, \mu_i, a_i, b_i)$ are chosen generically, then $\alpha\neq \beta$ implies that $t_{\alpha}\neq t_{\beta}$. This will imply that $V_{\alpha}\cap V_{\beta}=\{0\}$, so that $V_1\otimes V_2 = \oplus_{\alpha \in \mathbb{Z}/N\mathbb{Z}} V_{\alpha}$, where each multiplicity is $n_{t_{\alpha}}=1$, so it will conclude the proof. Let us fix $\alpha\neq \beta$. Clearly, to prove $t_{\alpha}\neq t_{\beta}$ it is sufficient to prove that $a_{\alpha}\neq a_{\beta}$. Write
 $$ u:= \frac{\widetilde{b}a_2}{b\widetilde{b_2}}, \quad v:= \lambda_2^2 q^2\left( \frac{\lambda_1^2-\mu_1^2}{q-q^{-1}} +a_1b_1 \right), \quad w:= \frac{\widetilde{b_2}\mu_1^2}{\widetilde{b}}.$$
 So one has $a_{\alpha} = uq^{2\alpha} +\frac{bq^{2\alpha}}{q^{2\alpha}-w}$. A straightforward computation shows that
 $$a_{\alpha} = a_{\beta} \Leftrightarrow (q^{2\alpha}-w)(q^{2\beta}-w) = \frac{vw}{u}.$$
 It is clear from this expression that for generic parameters $(\lambda_i, \mu_i, a_i, b_i)$, the equation $a_{\alpha}=a_{\beta}$ has no solution with $\alpha\neq \beta$. This concludes the proof.
  

\end{proof}



\begin{lemma}\label{lemma_CG3}
The decomposition \eqref{eq_CB} holds whenever $V(g_i,t_i,h_{\partial}^{(i)}) \cong V(\lambda_i,\mu_i,a_i,b_i)$, for $i=1,2$ and some arbitrary parameters $(\lambda_i,\mu_i,a_i,b_i)\in C$.
\end{lemma}

\begin{proof}
Write $c_i=(\lambda_i, \mu_i,a_i,b_i)\in C$. If $(c_1,c_2) \in U$, then the proof follows from Lemma \ref{lemma_CG2}. Else, since $U$ is dense in $C^2$, one can choose a sequence of parameters $(c_1^{(k)}, c_2^{(k)}) \in U$ such that $(c_1^{(k)} , c_2^{(k)}) \xrightarrow{k\to \infty} (c_1,c_2)$ in the analytic topology. Let $g_1^{(k)}, g_2^{(k)} \in D(B)$ be the elements associated to $V(c_1^{(k)})$ and $V(c_2^{(k)})$ respectively. Write $\mathbf{T}\in \operatorname{End}(V(c_1)\otimes V(c_2))$ and $\mathbf{T}^{(k)} \in \operatorname{End}(V(c_1^{(k)})\otimes V(c_2^{(k)}))$ the operators associated to the central element $T$. Since $(c_1^{(k)}, c_2^{(k)}) \in U$, all eigenvalues of $\mathbf{T}^{(k)}$ have multiplicity one and the characteristic polynomial of $\mathbf{T}^{(k)}$ is $\chi_{\mathbf{T}^{(k)}}(X):= \operatorname{det}( \mathbf{T}^{(k)} - X id ) = -( T_N(X) - \operatorname{Tr}(\varphi(g_1^{(k)}g_2^{(k)}) ) )$. In the basis $\{v_i\otimes v_j\}_{i,j \in \mathbb{Z}/N\mathbb{Z}}$, the matrix coefficients of $\mathbf{T}^{(k)}$ depend analytically on the parameters $(c_1^{(k)}, c_2^{(k)})$, so does the characteristic polynomial $\chi_{\mathbf{T}^{(k)}}(X)$. Therefore, by passing to the limit, one finds that $\chi_{\mathbf{T}}(X) =  -( T_N(X) - \operatorname{Tr}(\varphi(g_1g_2) ) )$. Together with Lemma \ref{lemma_CG1}, this implies  \eqref{eq_CB}.


\end{proof}



\begin{proof}[Proof of Theorem \ref{theorem_CG}]
 When both $V(g_1,t_1,h_{\partial}^{(1)})$ and $V(g_2,t_2,h_{\partial}^{(2)})$ are isomorphic to some modules $V(\lambda_i,\mu_i,a_i,b_i)$, Lemma \ref{lemma_CG3} concludes. Recall from Remark \ref{remark_representations} that even though not every weight indecomposable projective modules $V(g,t,h_{\partial})$ is isomorphic to such a $V(\lambda, \mu ,a ,b)$ (one might need to use the modules $\widetilde{V}(\lambda, \mu, c)$), for each $(g,\partial) \in \widetilde{D(B)}$ there exists a $t'$, with $T_N(t')=\operatorname{Tr} (\varphi(g))$, such that $V(g,t',h_{\partial})$ is isomorphic to a module $V(\lambda, \mu ,a ,b)$. Therefore to prove Theorem \ref{theorem_CG} in the general case, it is sufficient to prove that for any other $t_i'$ such that $T_N(t'_i)= \operatorname{Tr}(\varphi(g_i))$, then 
 $$ V(g_1,t_1,h_{\partial}^{(1)})\otimes V(g_2,t_2,h_{\partial}^{(2)}) \cong  V(g_1,t'_1,h_{\partial}^{(1)})\otimes V(g_2,t'_2,h_{\partial}^{(2)}), $$
 as $\mathcal{O}_q[D(B)]$-modules. This is an immediate consequence of Lemma \ref{lemma_factorization4} which proves that they are isomorphic as $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_2)$-modules. 



\end{proof}


\begin{remark}
If $V(g_i,t_i,h_{\partial}^{(i)}) = \widetilde{V}(\lambda_i,\mu_i,c_i)$, for $i=1,2$, where $c_1, c_2$ and $c:= c_2 +c_1 \lambda_2^2$ are non zero, one can define Clebsch-Gordan operators explicitly using some $N$ roots $\widetilde{c_1}, \widetilde{c_2}, \widetilde{c}$ of $c_1,c_2$ and $c$ respectively. A similar argument to the one in the proof of Lemma \ref{lemma_CG2}, shows that, for $\alpha \in \mathbb{Z}/N\mathbb{Z}$, the linear map 
$$K_{\alpha} : \widetilde{V}(\lambda_1\lambda_2q^{\alpha}, \mu_1\mu_2q^{-\alpha}, c)\hookrightarrow  \widetilde{V}(\lambda_1,\mu_1,c_1)\otimes  \widetilde{V}(\lambda_2,\mu_2,c_2),$$
defined by 
$$ K_{\alpha} w_k = \sum_{a+b=k} \frac{\widetilde{c}^{\lceil k \rfloor}}{ \widetilde{c_1}^{\lceil b \rfloor} \widetilde{c_2}^{\lceil a \rfloor}} 
w(\widetilde{c_1}\lambda_2^2q^{2\alpha}, \widetilde{c_2}, \widetilde{c} | b-\alpha) w_a\otimes w_b $$
is $\mathcal{O}_q[D(B)]$-equivariant. Moreover, the restriction to the image of $K_{\alpha}$ of the central element $T$ is a scalar matrix with value
$$ t_{\alpha}= -q^{2\alpha - 1}\lambda_1\lambda_2(\mu_1\mu_2)^{-1} - q^{1-2\alpha}\mu_1\mu_2 (\lambda_1\lambda_2)^{-1}, $$
so $\alpha\neq \beta$ implies $t_{\alpha}\neq t_{\beta}$ and we obtain an explicit decomposition 
$$ \widetilde{V}(\lambda_1,\mu_1,c_1)\otimes  \widetilde{V}(\lambda_2,\mu_2,c_2) \cong  \oplus_{\alpha}  \widetilde{V}(\lambda_1\lambda_2q^{\alpha}, \mu_1\mu_2q^{-\alpha}, c).
$$
However, our strategy does not generalise to pairs of representations of the form $V(\lambda_1,\mu_1, a_1,b_1) \otimes \widetilde{V}(\lambda_2,\mu_2, c_2)$ or $\widetilde{V}(\lambda_1, \mu_1,c_1) \otimes V(\lambda_2, \mu_2, a_2, b_2)$. In those cases, finding explicit Clebsch-Gordan operators seems to be a difficult problem and the author did not find a way to avoid the use of Lemma \ref{lemma_factorization4}.
\end{remark}











\section{Braid groupoid representations and braiding operators}

\subsection{Change of coordinates in quantum Teichm\"uller theory}


 The principal motivation for the authors of \cite{ChekhovFock, Kashaev98} to introduce the quantum Teichm\"uller spaces lies in the change of coordinates isomorphisms related the  quantum dilogarithm. Since the existing literature on the subject predates the construction of the stated skein algebras, so only deals with closed punctured surfaces,  
 in this subsection, we briefly revisit and generalize the definitions and main properties of change of coordinates. The techniques and results are  similar to \cite{LiuQTeichNC, BonahonWongqTrace, Hiatt}, so no essential originality is claimed here. However the use of the gluing morphisms $i_{a\#b}$ significantly simplifies the presentation, even for closed punctured surfaces.

\subsubsection{Definition of the change of coordinates isomorphism}

\begin{notations} Since the balanced Chekhov-Fock algebra $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$  is Noetherian, its multiplicative subset of non-zero elements satisfies the Ore condition and the balanced Chekhov-Fock algebra admits a division algebra that will be denoted $\widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta)$ (see \textit{e.g.} \cite{Kassel, Cohn_SkewFields} for details).
\end{notations}

In this subsection, for any two triangulations $\Delta$ and $\Delta'$, we will define an isomorphism $\phi_{\Delta, \Delta'}^{\mathbf{\Sigma}}: \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta') \xrightarrow{\cong} \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta)$ such that $\Phi_{\Delta, \Delta''}^{\mathbf{\Sigma}}= \Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}} \circ \Phi_{\Delta', \Delta''}^{\mathbf{\Sigma}}$ for any three topological triangulations $\Delta, \Delta', \Delta''$.
 
 \begin{definition}\label{def_reindexing_map}
 \begin{enumerate}
\item  The triangulation $\Delta$ is obtained from $\Delta'$ by \textit{reindexing its edges} if $\mathcal{E}(\Delta)= \mathcal{E}(\Delta')$ and there exists  a permutation  $\pi$  of $ \{1, 2,  \ldots, |\mathcal{E}(\Delta)| \}$ such that $I_{\Delta}=\pi \circ I_{\Delta'}$. We write $\Delta'= \pi_* \Delta$. The permutation $\pi$  induces an isomorphism $\phi_{\pi_* \Delta, \Delta}^{\mathbf{\Sigma}}: \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \xrightarrow{\cong} \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \pi_*\Delta)$ which sends a balanced monomial $Z^{\mathbf{k}}$ to $Z^{\pi \circ \mathbf{k}}$.
 \item Let $e$ be an inner edge of $\Delta$ bounding two distinct faces $\mathbb{T}_1$ and $\mathbb{T}_2$.  The union of these two faces forms a square $Q$ for which $e$ is one diagonal. The topological triangulation $\Delta'$ is obtained from $\Delta$ by a \textit{flip along} $e$, if the set $\mathcal{E}(\Delta')$ is obtained from $\mathcal{E}(\Delta)$ by removing $e$ and adding the other diagonal $e'$ of $Q$. The indexing $I_{\Delta'}$ is required to satisfy $I_{\Delta'}(a)=I_{\Delta}(a)$ for $a\in \mathcal{E}(\Delta)\setminus \{e\}$ and $I_{\Delta'}(e')=I_{\Delta}(e)$. We write $\Delta'= F_e \Delta$.
 \end{enumerate}
 \end{definition}
 
  Consider the pentagon $\mathbb{P}$, \textit{i.e.} a disc with five punctures on its boundary, and let $\Delta_1, \ldots, \Delta_5$ be the five topological triangulations of $\mathbb{P}$ drawn in Figure \ref{fig_pentagone}.
 
 
 \begin{figure}[!h] 
\centerline{\includegraphics[width=8cm]{pentagone.eps} }
\caption{Some arcs and five topological triangulations of $\mathbb{P}$ illustrating the Pentagon relation.} 
\label{fig_pentagone} 
\end{figure} 
 
\begin{theorem}[Penner \cite{Penner87}]\label{theorem_Penner} Any two topological triangulations can be related by a finite sequence of flips and reindexings, and any two sequences of flips and reindexings relating the same two topological triangulations are related by a finite number of the following elementary moves, together with their inverses:
 \begin{enumerate}
 \item Replace a reindexing $(\pi_1 \circ \pi_2)_*$ by a subsequence $\left( (\pi_1)_*,  (\pi_2)_*\right)$.
 \item Remove or add a subsequence $(F_{e'},  F_e)$,  of two flips of the diagonals of the same square.
 \item Replace a subsequence $(F_e, \pi_*)$ by $(\pi_*, F_{\pi(e)})$.
 \item Replace a subsequence $(F_{e_1}, F_{e_2})$ by $(F_{e_2}, F_{e_1})$, if $e_1$ and $e_2$ are inner edges which do not belong to the same face.
 \item (Pentagon identity) Given a Pentagon $\mathbb{P}$ inside $\mathbf{\Sigma}$ obtained by gluing three faces and denoting by $e_1, e_2, e_3$ the three edges drawn in Figure \ref{fig_pentagone}, then replace a subsequence $ (F_{e_2}, F_{e_1})$ by $(F_{e_1}, F_{e_2}, F_{e_3})$.
 \end{enumerate}
 \end{theorem}
 To define the isomorphism $\phi_{\Delta, \Delta'}^{\mathbf{\Sigma}}$, we thus need to define it when $\Delta$ and $\Delta'$ differs either by a reindexing (this was done in Definition \ref{def_reindexing_map}) or by a flip and then verify that the five moves of Theorem \ref{theorem_Penner} are satisfied.

 
\vspace{2mm}
\par Let $Q$ denote the square, \textit{i.e.} a disc with four punctures on its boundary,  let $\Delta, \Delta'$ be its two topological triangulations drawn in Figure \ref{fig_square}, let $\beta^{(1)}, \ldots, \beta^{(4)}$ be the four arcs of $Q$ drawn in Figure \ref{fig_square}. By Lemma \ref{lemma_grading_generator}, the algebra $\mathcal{Z}_{\omega}(Q, \Delta)$ (resp. the algebra $\mathcal{Z}_{\omega}(Q, \Delta')$)  is generated by the elements $\operatorname{Tr}_{\omega}^{\Delta} (\beta^{(i)}_{++})$,  (resp. by the elements $\operatorname{Tr}_{\omega}^{\Delta'} (\beta^{(i)}_{++})$),  for $i\in \{1, 2, 3, 4\}$ , together with the elements $X_1, \ldots, X_5$ (resp. the elements $X_1', \ldots, X_5'$).

 \begin{figure}[!h] 
\centerline{\includegraphics[width=9cm]{square_triangulation2.eps} }
\caption{The square $Q$, some arcs and two topological triangulations.} 
\label{fig_square} 
\end{figure} 


\begin{definition}\label{def_square_changecoordinates}
The isomorphism $\Phi_{\Delta, \Delta'}^{Q} : \widehat{\mathcal{Z}}_{\omega}(Q, \Delta') \xrightarrow{\cong} \widehat{\mathcal{Z}}_{\omega}(Q, \Delta)$ is defined by $\Phi^Q_{\Delta, \Delta'}\left(\operatorname{Tr}_{\omega}^{\Delta'}(\beta^{(i)}_{++})\right)= \operatorname{Tr}_{\omega}^{\Delta}(\beta^{(i)}_{++})$, for $i\in \{1, 2, 3, 4\}$, and by:
\begin{align*}
&  \Phi^Q_{\Delta, \Delta'}(X_1')=X_1+[X_1X_5]; &\Phi^Q_{\Delta, \Delta'}(X_3')=X_3+[X_3X_5]; \\
&  \Phi^Q_{\Delta, \Delta'}(X_2')=(X_2^{-1}+[X_2X_5]^{-1})^{-1}; & \Phi^Q_{\Delta, \Delta'}(X_4')=(X_4^{-1}+[X_4X_5]^{-1})^{-1}; \\
& \Phi^Q_{\Delta, \Delta'}(X_5')=X_5^{-1}. &
\end{align*}
\end{definition}

\begin{lemma}\label{lemma_skein_square}
The following diagram commutes:
 
 $$\begin{tikzcd}
  {} &{} &  \widehat{\mathcal{Z}}_{\omega}(Q, \Delta') \arrow[dd, "\cong"', "\Phi_{\Delta, \Delta'}^{Q}"] \\
  \mathcal{S}_{\omega}(Q) \arrow[urr, hook, "\operatorname{Tr}_{\omega}^{\Delta'}"] \arrow[drr, hook, "\operatorname{Tr}_{\omega}^{\Delta}"] & {} & {} \\
 {} & {} & \widehat{\mathcal{Z}}_{\omega}(Q, \Delta)
  \end{tikzcd}
 $$

\end{lemma}

\begin{proof}
The algebra $\mathcal{S}_{\omega}(Q)$ is generated by the stated arcs $(\beta_i)_{\varepsilon \varepsilon'}$, for $i\in \{1,2,3,4\}$ and $\varepsilon, \varepsilon'= \pm$. We need to show that the images of each these generators through the two maps $\Phi_{\Delta, \Delta'}^{Q} \circ \operatorname{Tr}_{\omega}^{\Delta'}$ and $\operatorname{Tr}_{\omega}^{\Delta}$ are equal. For the generators $(\beta_i)_{++}$, this is true by definition. Since $\operatorname{Tr}_{\omega}^{\Delta}( (\beta_i)_{--})= \operatorname{Tr}_{\omega}^{\Delta}( (\beta_i)_{++})^{-1}$ and $\operatorname{Tr}_{\omega}^{\Delta'}( (\beta_i)_{--})= \operatorname{Tr}_{\omega}^{\Delta'}( (\beta_i)_{++})^{-1}$, the equality also holds for the generators $(\beta_i)_{--}$. The image through both quantum traces of the elements $(\beta_i)_{-+}$ is null, hence we simply need to verify the equality for the generators $(\beta_i)_{+-}$. For $i=1$, one has:
$$
\operatorname{Tr}_{\omega}^{\Delta'} \left( (\beta_1)_{+-} \right) = [Z_{4'}Z_{5'}Z_{1'}^{-1}] + [Z_{4'}Z_{5'}^{-1}Z_{1'}^{-1}] = q\operatorname{Tr}_{\omega}^{\Delta'} \left( (\beta_1)_{++} \right)X_{1'}^{-1}(1+q^2 X_{5'}^{-1}).
$$
We deduce the equalities:
\begin{eqnarray*}
\Phi_{\Delta, \Delta'}^{Q} \circ \operatorname{Tr}_{\omega}^{\Delta'} \left( (\beta_1)_{+-} \right) &=& q \operatorname{Tr}_{\omega}^{\Delta} \left( (\beta_1)_{++} \right)(X_1 + [X_1X_5])^{-1}(1+q^2 X_5) \\
&=& q[Z_4Z_1]X_1^{-1}= [Z_4Z_1^{-1}]= \operatorname{Tr}_{\omega}^{\Delta} \left( (\beta_1)_{+-} \right).
\end{eqnarray*}

The verifications for the generators  $(\beta_i)_{+-}$ with $i=2,3,4$ are similar and left to the reader.

\end{proof}

Suppose that $\Delta'=F_e\Delta$ is obtained from $\Delta$ by flipping an inner edge $e$ inside a square $Q$ as in Definition \ref{def_reindexing_map}. The punctured surface $\mathbf{\Sigma}$ 
 is obtained by gluing $Q \bigsqcup \mathbf{\Sigma}\setminus Q$ along some pairs of boundary arcs. The gluing map $i: \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \hookrightarrow \mathcal{Z}_{\omega}(Q, \Delta_Q) \otimes \mathcal{Z}_{\omega}(\mathbf{\Sigma}\setminus Q, \Delta_{\mathbf{\Sigma}\setminus Q})$ (Definition \ref{def_gluing_CF}) induces an injective morphism $\widehat{i} : \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta) \hookrightarrow \widehat{\mathcal{Z}}_{\omega}(Q, \Delta_Q) \otimes \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}\setminus Q, \Delta_{\mathbf{\Sigma}\setminus Q})$. 
  Here $\mathbf{\Sigma}\setminus Q$ might be empty in which case we set $\mathcal{S}_{\omega}(\mathbf{\Sigma}\setminus Q):= \mathcal{R}$ (think for instance to the once punctured torus). Similarly, denote by $\widehat{i}' : \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta') \hookrightarrow \widehat{\mathcal{Z}}_{\omega}(Q, \Delta'_Q) \otimes \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}\setminus Q, \Delta_{\mathbf{\Sigma}\setminus Q})$ the corresponding gluing morphism.


\begin{definition}\label{def_flip_changecoordinates}
The isomorphism $\Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}} : \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta') \xrightarrow{\cong} \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta)$ is the unique morphism making the following diagram commuting:
$$ \begin{tikzcd}
\widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta')  \arrow[r, hook, "\widehat{i}"] \arrow[d, dotted, "\Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}}"] &
 \widehat{\mathcal{Z}}_{\omega}(Q, \Delta_Q) \otimes \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}\setminus Q, \Delta_{\mathbf{\Sigma}\setminus Q}) \arrow[d, "\cong"', "\Phi_{\Delta, \Delta'}^{Q}\otimes id"] \\
 \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta) \arrow[r, hook, "\widehat{i}'"] &
 \widehat{\mathcal{Z}}_{\omega}(Q, \Delta'_Q) \otimes \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}\setminus Q, \Delta_{\mathbf{\Sigma}\setminus Q})
 \end{tikzcd}$$
 \end{definition}
 By a straightforward verification, the isomorphism $\Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}}$ of Definition \ref{def_flip_changecoordinates} for a closed punctured surface, coincides with the isomorphism defined by Hiatt in \cite{Hiatt}.
 The fact that the operators  $\Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}} $, that we associated to a reindexing and a flip, satisfy the first four elementary moves of Theorem \ref{theorem_Penner}, is a straightforward consequence of the definitions. The non trivial verification is the Pentagon identity. 
 

 
 
 \begin{lemma}\label{lemma_pentagone}
 The flip and reindexing operators of Definitions \ref{def_reindexing_map} and \ref{def_square_changecoordinates} satisfy the Pentagon identity.
 \end{lemma}
 
 \begin{proof}
 We first suppose that $\mathbf{\Sigma}=\mathbb{P}$.
 By Lemma \ref{lemma_grading_generator}, the algebra $\mathcal{Z}_{\omega}(\mathbb{P}, \Delta_1)$ is generated by the elements $X_e$, for $e \in \mathcal{E}(\Delta_1)$, and by the images through the quantum trace of the stated arcs $(\beta_i)_{++}$, for $i\in \{1, \ldots, 5\}$, drawn in Figure \ref{fig_pentagone}. We need to show that for each of these generator $X$, one has:
 \begin{equation}\label{eq_pentagon}
 \Phi_{\Delta_5, \Delta_4}^{\mathbb{P}} \circ \Phi_{\Delta_4, \Delta_1}^{\mathbb{P}} (X) = \Phi_{\Delta_5, \Delta_3}^{\mathbb{P}} \circ \Phi_{\Delta_3, \Delta_2}^{\mathbb{P}}\circ \Phi_{\Delta_2, \Delta_1}^{\mathbb{P}}(X). 
 \end{equation}
 The equality \eqref{eq_pentagon} for $X=X_e$, where $e\in \mathcal{E}(\Delta_1)$, was proved (by a straightforward computation) by Chekhov and Fock in \cite{ChekhovFock} (see also \cite[Proposition $9$]{LiuQTeichNC}). When $X= \operatorname{Tr}_{\omega}^{\Delta_1} \left( (\beta_i)_{++} \right)$, it is a straightforward consequence of Lemma \ref{lemma_skein_square} that both the left-hand-side and the right-hand-side of \eqref{eq_pentagon} are equal to $\operatorname{Tr}_{\omega}^{\Delta_5} \left( (\beta_i)_{++} \right)$, hence the proof is complete in this case.
 
 \vspace{2mm}
 \par For a general punctured surface $\mathbf{\Sigma}$ with five topological triangulations $\Delta_1, \ldots, \Delta_5$ related by the Pentagon identity as in Figure \ref{fig_pentagone} for a Pentagon $\mathbb{P}\subset \mathbf{\Sigma}$ made of three adjacent faces, and for $k=1,5$, denote by $i_k : \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta_k) \hookrightarrow \widehat{\mathcal{Z}}_{\omega}(\mathbb{P}, \Delta^{\mathbb{P}}_k) \otimes \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}\setminus \mathbb{P}, \Delta_k^{\mathbf{\Sigma} \setminus \mathbb{P}})$ the gluing morphism arising from Definition \ref{def_gluing_CF}. Consider the following diagram:
 $$
 \begin{tikzcd}
 \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta_1) 
 \arrow[r, hook, "i_1"] \arrow[d, dotted] &
 \widehat{\mathcal{Z}}_{\omega}(\mathbb{P}, \Delta^{\mathbb{P}}_1) \otimes \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}\setminus \mathbb{P}, \Delta_1^{\mathbf{\Sigma} \setminus \mathbb{P}})
 \arrow[d, "\cong"', "( \Phi_{\Delta_5, \Delta_4}^{\mathbb{P}} \circ \Phi_{\Delta_4, \Delta_1}^{\mathbb{P}})\otimes id = (\Phi_{\Delta_5, \Delta_3}^{\mathbb{P}} \circ \Phi_{\Delta_3, \Delta_2}^{\mathbb{P}}\circ \Phi_{\Delta_2, \Delta_1}^{\mathbb{P}}) \otimes id"]
 \\
  \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta_5)
  \arrow[r, hook, "i_5"] &
    \widehat{\mathcal{Z}}_{\omega}(\mathbb{P}, \Delta^{\mathbb{P}}_5) \otimes \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}\setminus \mathbb{P}, \Delta_5^{\mathbf{\Sigma} \setminus \mathbb{P}})
    \end{tikzcd}
 $$
 By the preceding case, the above diagram commutes once we replace the dotted arrow both by  $ \Phi_{\Delta_5, \Delta_4}^{\mathbf{\Sigma}} \circ \Phi_{\Delta_4, \Delta_1}^{\mathbf{\Sigma}}$ and by  $\Phi_{\Delta_5, \Delta_3}^{\mathbf{\Sigma}} \circ \Phi_{\Delta_3, \Delta_2}^{\mathbf{\Sigma}}\circ \Phi_{\Delta_2, \Delta_1}^{\mathbf{\Sigma}}$, hence these two operators are equal and the proof is complete.
    
    
 \end{proof}

 
 
 \begin{definition} Let $\Delta$ and $\Delta'$ be two topological triangulations of $\mathbf{\Sigma}$. Let $\Delta=\Delta_0, \Delta_1, \ldots, \Delta_n=\Delta'$ be a sequence of topological triangulations of $\mathbf{\Sigma}$ such that $\Delta_{i+1}$ is obtained from $\Delta_i$ by either a flip or a reindexing. The \textit{change of coordinates} isomorphism  $\Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}} : \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta') \xrightarrow{\cong} \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta)$ is the composition:
 $$ \Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}} := \Phi_{\Delta, \Delta_1}^{\mathbf{\Sigma}}\circ \Phi_{\Delta_1, \Delta_2}^{\mathbf{\Sigma}}\circ \ldots \circ \Phi_{\Delta_{n-1}, \Delta'}.$$
 \end{definition}
 
 It follows from Theorem \ref{theorem_Penner} and Lemma \ref{lemma_pentagone} that the isomorphism $\Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}}$ does not depend on the choice of the sequence of flips and reindexings used to define it. 
 
 \begin{proposition}\label{prop_change_coordinates}
 The followings hold: 
 \begin{enumerate}
 \item One has $\Phi_{\Delta, \Delta''}^{\mathbf{\Sigma}}= \Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}} \circ \Phi_{\Delta', \Delta''}^{\mathbf{\Sigma}}$ for any three topological triangulations $\Delta, \Delta', \Delta''$.
 \item For $\Delta, \Delta'$ two topological triangulations of $\mathbf{\Sigma}$, the following diagram commutes:
 
 $$\begin{tikzcd}
  {} &{} &  \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta') \arrow[dd, "\cong"', "\Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}}"] \\
  \mathcal{S}_{\omega}(\mathbf{\Sigma}) \arrow[urr, hook, "\operatorname{Tr}_{\omega}^{\Delta'}"] \arrow[drr, hook, "\operatorname{Tr}_{\omega}^{\Delta}"] & {} & {} \\
 {} & {} & \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta)
  \end{tikzcd}
 $$

\end{enumerate}
\end{proposition}

\begin{proof}
The first assertion is an immediate consequence of the definitions. The second assertion is also immediate when $\Delta$ and $\Delta'$ differ from a reindexing. It remains to prove the commutativity of the diagram when $\Delta$ and $\Delta'$ differ from a flip. In the case where $\mathbf{\Sigma}=Q$, this was proved in Lemma \ref{lemma_skein_square}. In the general case, consider the injective morphisms $i : \widehat{\mathcal{Z}}_{\omega} (\mathbf{\Sigma}, \Delta) \hookrightarrow \widehat{\mathcal{Z}}_{\omega} (Q, \Delta^Q)\otimes \widehat{\mathcal{Z}}_{\omega} (\mathbf{\Sigma}\setminus Q, \Delta^{\mathbf{\Sigma}\setminus Q})$ and   $i' : \widehat{\mathcal{Z}}_{\omega} (\mathbf{\Sigma}, \Delta') \hookrightarrow \widehat{\mathcal{Z}}_{\omega} (Q, \Delta'^Q)\otimes \widehat{\mathcal{Z}}_{\omega} (\mathbf{\Sigma}\setminus Q, \Delta'^{\mathbf{\Sigma}\setminus Q})$ arising from Definition \ref{def_gluing_CF} and $j: \mathcal{S}_{\omega}(\mathbf{\Sigma}) \hookrightarrow \mathcal{S}_{\omega}(Q)\otimes \mathcal{S}_{\omega}(\mathbf{\Sigma}\setminus Q)$ the gluing morphism from Definition \ref{def_gluing_map}. Consider the following diagram:

$$
\begin{tikzcd}
{} &
 \mathcal{S}_{\omega}(Q)\otimes \mathcal{S}_{\omega}(\mathbf{\Sigma}\setminus Q) 
 \arrow[rd, "\operatorname{Tr}_{\omega}^{\Delta'^Q} \otimes \operatorname{Tr}_{\omega}^{\Delta'^{\mathbf{\Sigma}\setminus Q}}"] 
 & {} \\
{} & 
\widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta')
 \arrow[r, hook, "i'"] \arrow[dd, "\cong"', "\Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}}"] 
&
\widehat{\mathcal{Z}}_{\omega}(Q, \Delta'^Q)\otimes \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}\setminus Q, \Delta'^{\mathbf{\Sigma}\setminus Q}) 
\arrow[dd, "\cong"', "\Phi_{\Delta^Q, \Delta'^Q}^Q \otimes id"] 
\\
\mathcal{S}_{\omega}(\mathbf{\Sigma})
 \arrow[ruu, hook, bend left,  "j"] 
 \arrow[ru, "\operatorname{Tr}_{\omega}^{\Delta'}"] \arrow[rd, "\operatorname{Tr}_{\omega}^{\Delta}"] \arrow[rdd, hook, bend right, "j"'] 
&{}&{} \\
{} &
 \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta) 
 \arrow[r, hook, "i"] 
 & 
 \widehat{\mathcal{Z}}_{\omega}(Q, \Delta^Q) \otimes \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta^{\mathbf{\Sigma}\setminus Q})
 \\ {} & 
 \mathcal{S}_{\omega}(Q)\otimes \mathcal{S}_{\omega}(\mathbf{\Sigma}\setminus Q) 
 \arrow[ru, "\operatorname{Tr}_{\omega}^{\Delta^Q} \otimes \operatorname{Tr}_{\omega}^{\Delta^{\mathbf{\Sigma}\setminus Q}}"']
  & {} 
\end{tikzcd}
$$
In this diagram, the commutativity of the inner square is Definition \ref{def_flip_changecoordinates}, the commutativity of the outer pentagon follows from Lemma \ref{lemma_skein_square}, the commutativity of the  squares on the top and bottom of the diagram follows from Definition \ref{def_trace_map}. These facts, together with the injectivity of $j$, $i$ and $i'$, imply  that the inner triangle in the diagram commutes, which is the second statement of the lemma.


\end{proof}

 
 \subsubsection{The Fock-Goncharov decomposition and the quantum exponential}
 
 Following Fock and Goncharov in \cite[Section $1.3$]{FockGoncharovClusterVariety}, we now show that the flip isomorphisms $\Phi^{\mathbf{\Sigma}}_{F_e\Delta, \Delta}$ of Definition \ref{def_flip_changecoordinates}, can be decomposed as   $\Phi_{F_e\Delta, \Delta}^{\mathbf{\Sigma}} = \nu^{\mathbf{\Sigma}}_{F_e\Delta, \Delta} \circ \mathrm{Ad}(s_q(X_e^{-1}))$, where $ \nu^{\mathbf{\Sigma}}_{F_e\Delta, \Delta}$ is an automorphism of $\mathcal{Z}_{\omega}(\Sigma,\Delta)$ (not only of its division algebra) and the automorphism $\mathrm{Ad}(s_q(X_e^{-1}))$ is inner in a suitable extension of $\widehat{\mathcal{Z}}_{\omega}(\Sigma,\Delta)$.   Recall that we defined $[n]:= \frac{q^n - q^{-n}}{q-q^{-1}}$ and write $[n]! = [n][n-1]\ldots [1]$ with the convention $[0]!=1$. We will use the notation $\mathrm{Ad}(x)y:=xyx^{-1}$.
 
 \begin{definition}\label{def_quantum_exponential}
  The  \textit{quantum exponential} is the power series  defined as follows.
 \begin{enumerate}
 \item If $q\in \mathcal{R}^*$ is generic (\textit{i.e.} if $q$ is not a root of unity) and $q-q^{-1}$ is invertible, we set:
 $$ \exp_q(x) := \sum_{n=0}^{\infty} \frac{q^{n(n-1)/2}}{[n]!} x^n\in \mathcal{R}[[x]].$$
 \item If $q\in \mathcal{R}^*$ is a root of unity of odd order $N>1$, we set:
 $$ \exp^{<N}_q(x):= \sum_{n=0}^{N-1} \frac{q^{n(n-1)/2}}{[n]!} x^n\in \mathcal{R}[x].$$
 \end{enumerate}
 
 We also write $s_q(x):= \exp_q\left( -\frac{x}{q-q^{-1}} \right)$ and $s_q^{<N}(x):= \exp^{<N}_q\left( -\frac{x}{q-q^{-1}} \right)$.
 \end{definition}
 
 \begin{lemma}\label{lemma_expq} Consider an $\mathcal{R}$-algebra $\mathcal{A}$ with two elements $u,v$ such that $uv=q^2vu$.
  \begin{enumerate}
 \item If $q\in \mathcal{R}^*$ is generic and the elements $\exp_q(u)^{\pm 1}$ and $\exp_q(v)^{\pm 1}$ are well-defined, that is if $\mathcal{A}$ is a topological algebra with topological embeddings $\mathcal{R}[[u]]\hookrightarrow \mathcal{A}$ and $\mathcal{R}[[v]]\hookrightarrow \mathcal{A}$, 
  then the following results hold.
 \begin{enumerate}
 \item One has $s_q(u+v)=s_q(u)s_q(v)$.
 \item One has $s_q(q^2u)s_q(u)^{-1}=(1+qu)^{-1}$ and $s_q(q^{-2}u)s_q(u)^{-1}=1+q^{-1}u$. Hence $\mathrm{Ad}(s_q(u))v=v(1+qu)^{-1}$ and $\mathrm{Ad}(s_q(v))u=u(1+q^{-1}v)$. 
 \end{enumerate}
 \item  If $q\in \mathcal{R}^*$ is a root of unity of odd order $N>1$, then the following results hold.
 \begin{enumerate}
 \item If  $u^nv^{N-n}=0$ for all $n\in \{0, \ldots, N\}$, then $s_q^{<N}(u+v)=s_q^{<N}(u)s_q^{<N}(v)$.
 \item If $u^N=0$ then  $s_q^{<N}(q^2u)s_q^{<N}(u)^{-1}=(1+qu)^{-1}$ and $s^{<N}_q(q^{-2}u)s^{<N}_q(u)^{-1}=1+q^{-1}u$. 
  \end{enumerate}
 \end{enumerate}
 \end{lemma}
 
 \begin{proof}
 Formula $(1)(a)$ is \cite[Lemma A.$3$]{OhtsukiBook}. The formulas $(1)(b)$ are immediate consequence of the following factorization, which is proved for instance in \cite[Chapter II.D, Proposition $2$]{ZagierDilogarithm}:
 $$s_q(x)=\prod_{n=0}^{\infty}(1+q^{2n+1}x).$$
 Formula $(2)(a)$ is \cite[Lemma A.$12$]{OhtsukiBook}. The formulas $(2)(b)$ follow from straightforward computations.
 \end{proof}
 
 
  \begin{remark}The q-Pochhammer symbol  $(x;q)_{\infty}:=\prod_{i=0}^{\infty}(1-q^ix)$ is related to the quantum dilogarithm $\mathrm{Li}_2(x;q):=\sum_{i=1}^{\infty}\frac{x^n}{n(1-q^n)}$ by the relation $(x;q)_{\infty}=\exp(-\mathrm{Li}_2(x;q))$ (see \cite[Proposition $2$]{ZagierDilogarithm}). Therefore,  the quantum exponential is related to the quantum dilogarithm by
   $$\exp_q\left( -\frac{x}{q-q^{-1}} \right)=s_q(x)=(-qx;q^2)_{\infty}=\exp(-\mathrm{Li}_2(-qx;q^2)).$$
    \end{remark}
 
 
 
 
 \begin{definition}\label{def_nu}
 \begin{enumerate}
 \item \textit{(The case of the square)}
 The isomorphism \\ $\nu^Q_{\Delta, \Delta'} : \mathcal{Z}_{\omega}(Q, \Delta') \xrightarrow{\cong} \mathcal{Z}_{\omega}(Q, \Delta)$ is defined by: 
 \begin{align*}
 & \nu^Q_{\Delta, \Delta'} \left(\operatorname{Tr}_{\omega}^{\Delta'} (\beta^{(i)}_{++})\right)= \operatorname{Tr}_{\omega}^{\Delta} (\beta^{(i)}_{++}) \mbox{, for }i\in \{1,2,3,4\}; \\
 & \nu^Q_{\Delta, \Delta'} (X'_1) = X_1; \quad \nu^Q_{\Delta, \Delta'}(X'_3) = X_3;  \\
 & \nu^Q_{\Delta, \Delta'} (X'_2)= [X_2X_5]; \quad \nu^Q_{\Delta, \Delta}(X'_4)= [X_4X_5]; \\
 & \nu^Q_{\Delta, \Delta'} (X'_5)= X_5^{-1}.
 \end{align*}
 \item \textit{(The general case)} Let $\Delta$ and $\Delta'=F_{e}\Delta$ be two topological triangulations of $\mathbf{\Sigma}$ which differ from a flip along an inner edge $e\in\mathcal{E}(\Delta')$ and denote by $Q\subset \mathbf{\Sigma}$ the square whose $e$ is a diagonal. The isomorphism $\nu^{\mathbf{\Sigma}}_{\Delta, \Delta'} : \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta') \xrightarrow{\cong} \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ is the unique isomorphism making the following diagram commuting: 
 $$ \begin{tikzcd}
\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta')  \arrow[r, hook, "\widehat{i}"] \arrow[d, dotted, "\nu_{\Delta, \Delta'}^{\mathbf{\Sigma}}"] &
 \mathcal{Z}_{\omega}(Q, \Delta_Q) \otimes \mathcal{Z}_{\omega}(\mathbf{\Sigma}\setminus Q, \Delta_{\mathbf{\Sigma}\setminus Q}) \arrow[d, "\cong"', "\nu_{\Delta, \Delta'}^{Q}\otimes id"] \\
 \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \arrow[r, hook, "\widehat{i}'"] &
 \mathcal{Z}_{\omega}(Q, \Delta'_Q) \otimes \mathcal{Z}_{\omega}(\mathbf{\Sigma}\setminus Q, \Delta_{\mathbf{\Sigma}\setminus Q})
 \end{tikzcd}$$
 
 \end{enumerate}
 
 \end{definition}
To formulate the Fock-Goncharov decomposition, one needs to consider the element $s_q(X_e^{-1})$ for $X_e\in  \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta')$ and one would like to be able to consider an automorphism $\mathrm{Ad}(s_q(X_e^{-1}))$. Since the quantum exponential is a power series, let us introduce the following:

\begin{definition}\label{def_series} Let $(\mathbf{\Sigma}, \Delta)$ be a triangulated punctured surface. The \textit{completed balanced Chekhov-Fock module} is the topological $\mathcal{R}$-module $\overline{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta)$ whose elements are the power series $\sum_{\mathbf{k}} c_{\mathbf{k}} Z^{\mathbf{k}}$, associated to balanced monomials $Z^{\mathbf{k}}$. It has a product only partially defined  induced by $Z^{\mathbf{k}}Z^{\mathbf{k'}} = \omega^{\left<\mathbf{k}, \mathbf{k}'\right>^{WP}}Z^{\mathbf{k} + \mathbf{k}'}$, where $\left<\cdot, \cdot \right>^{WP}$ represents the $\mathbb{Z}$-bilinear extension of the Weil-Petersson form. 
\end{definition}

 The product between  two elements of $\overline{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta)$ is not always defined. For instance, for $x:=\sum_{\mathbf{k}}  Z^{\mathbf{k}}$, the product $x^2$ does not make sense. However, one has a natural inclusion $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \subset \overline{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta)$ and the product of an element of $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ by an element of $\overline{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta)$ is well defined. For $x\in \overline{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta)$, we will say that the automorphism $\mathrm{Ad}(x) \in \mathrm{Aut} (\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta))$ is well defined if for any $y \in \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$, there exists $z\in \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ such that $xy=zx$. In this case, $\mathrm{Ad}(x)\cdot y := z$ is uniquely determined and $\mathrm{Ad}(x)$ is indeed an automorphism of $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$.


The following lemma is stated in a different form in \cite[Section $1.3$]{FockGoncharovClusterVariety}, where it appears as a definition. Since the authors of \cite{FockGoncharovClusterVariety} only considered the $0$-th graded part of the balanced Chekhov-Fock algebra, we prove that their decomposition generalizes to the full algebra as-well.
 
 \begin{lemma}\label{lemma_FockGoncharov}[Fock-Goncharov decomposition]
Let $\Delta'$ and $\Delta=F_{e}\Delta'$ be two topological triangulations of $\mathbf{\Sigma}$ which differ from a flip along an inner edge $e\in\mathcal{E}(\Delta')$.
 If $q\in \mathcal{R}^*$ is generic, then the automorphism $\mathrm{Ad}(s_q(X_e^{-1})) \in \mathrm{Aut}(\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta'))$  is well defined and 
 one has the equality $\Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}} = \nu^{\mathbf{\Sigma}}_{\Delta, \Delta'} \circ \mathrm{Ad}(s_q(X_e^{-1}))$.
 \end{lemma}
 
 \begin{proof} In view of Definitions \ref{def_flip_changecoordinates} and \ref{def_nu}, it is sufficient to make the proof in the case where $\mathbf{\Sigma}=Q$.
 
 A straightforward computation shows that $\mathrm{Ad}(s_q(X_5'^{-1})) \cdot \operatorname{Tr}_{\omega}^{\Delta'} (\beta^{(i)}_{++}) = \operatorname{Tr}_{\omega}^{\Delta'} (\beta^{(i)}_{++})$ for every $i\in\{1,2,3,4\}$, hence one has $ \nu^Q_{\Delta, \Delta'} \circ \mathrm{Ad}(s_q(X_5'^{-1})) \left(   \operatorname{Tr}_{\omega}^{\Delta'} (\beta^{(i)}_{++})\right) = \Phi_{\Delta, \Delta'}^Q \left( \operatorname{Tr}_{\omega}^{\Delta'} (\beta^{(i)}_{++})\right) = \operatorname{Tr}_{\omega}^{\Delta} (\beta^{(i)}_{++})$. The equalities $\nu_{\Delta, \Delta'} \circ \mathrm{Ad}(s_q(X_5'^{-1})) (X'_i)= \Phi_{\Delta, \Delta'}^Q(X'_i)$, for $1\leq i \leq 5$, follow from Lemma \ref{lemma_expq}.
 \end{proof}
 
 
 
 \subsubsection{Intertwiners and mapping class representations}
 
 In this subsection, we suppose that $\omega$ is a root of unity of odd order $N>1$. Let $\mathbf{\Sigma}$ a punctured surface and $\Delta, \Delta'$ two topological triangulations. Given $r: \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \rightarrow \operatorname{End} (W)$ an irreducible representation, we would like to consider the composition $r':= r\circ \Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}} : \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta') \rightarrow \operatorname{End} (W)$ to obtain an irreducible representation of $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta')$. Since the change of coordinates $\Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}}$ takes values in the division algebra of the balanced Chekhov-Fock algebra, it is not clear that this composition is well-defined.
 
 \begin{definition}\label{def_subtil} We say that the irreducible representation $r: \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \rightarrow \operatorname{End} (W)$ is \textit{compatible} with $\Delta\to \Delta'$ if for any $X\in \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta')$, writing $\Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}}(X)= A B^{-1}=C^{-1}D$ with $A,B, C, D \in \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$, then the operators $r(B), r(C) \in \operatorname{End}(W)$ are invertible. In this case, we define the representation $r':= r\circ \Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}} : \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta') \rightarrow \operatorname{End} (W)$ by $r'(X):= r(A) r(B)^{-1}=r(C)^{-1}r(D)$.
 \end{definition}

 \begin{remark}
 \begin{enumerate}
\item By Theorem \ref{theorem_CFClassification}, since the balanced Chekhov-Fock algebra is semi-simple and its simple modules all have the same dimension which only depends on $\mathbf{\Sigma}$, the representation $r'=r\circ \Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}} $ is irreducible as well. 
\item Write $Y:= \mathrm{Specm}\left(\mathcal{Z}(\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)) \right)$. The property of being compatible with $\Delta\to \Delta'$ obviously only depends on the isomorphism class of $r$, so only depends on the associated character $\chi \in Y$ on the center of $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$.
 Moreover, to verify that $r$ is compatible with $\Delta\to \Delta'$, it is sufficient to verify the invertibility conditions of Definition \ref{def_subtil} for $X$ lying in a finite set of generators of $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$. For an element $B \in \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$, the set of characters in $Y$ whose associated irreducible representation $r$ is such that $r(B)$ is invertible, is a Zariski dense open subset set. Therefore, the subset of 
$Y$ whose associated representation is compatible with $\Delta \to \Delta'$ is a Zariski dense open subset as well. Said differently, a "generic" irreducible representation of $Y$ is compatible with $\Delta\to \Delta'$.
\item Eventually, note  that if $r: \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \rightarrow \operatorname{End} (W)$ is compatible with $\Delta\to\Delta'$ and $r':=r\circ \Phi_{\Delta, \Delta'}^{\mathbf{\Sigma}} $ is compatible with $\Delta'\to \Delta''$ then $r$ is compatible with $\Delta \to \Delta''$.
\end{enumerate}
\end{remark}

\begin{definition}\begin{enumerate}
\item Let $\operatorname{Homeo}^+(\mathbf{\Sigma})$ be the topological group of preserving orientation homeomorphisms of $\Sigma$ that preserve the set $\mathcal{P}$ and whose restriction to the boundary is the identity map. The \textit{mapping class group} is the group $\operatorname{Mod}(\mathbf{\Sigma}):= \pi_{0} \left( \operatorname{Homeo}^+(\mathbf{\Sigma}) \right)$.
\item For $\phi \in \operatorname{Mod}(\mathbf{\Sigma})$ and $\Delta$ a topological triangulation of $\mathbf{\Sigma}$, the topological triangulation $\phi(\Delta)$ has edges $\mathcal{E}(\phi(\Delta)):= \{ \phi(e), e\in \mathcal{E}(\Delta) \}$ and indexing map $I_{\phi(\Delta)}:=\phi \circ I_{\Delta}$.
\item The isomorphism $\phi^* : \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \phi(\Delta)) \xrightarrow{\cong} \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ is defined by $\phi^*(Z^{\mathbf{k}}):= Z^{\mathbf{k}\circ \phi^{-1}}$.
\item An irreducible representation $r: \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \rightarrow \operatorname{End} (W)$ is \textit{compatible} with $\phi \in \operatorname{Mod}(\mathbf{\Sigma})$ if it is compatible with $\Delta\to \phi(\Delta)$.
\end{enumerate}
\end{definition}
For $\phi \in \operatorname{Mod}(\mathbf{\Sigma})$ and $r: \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \rightarrow \operatorname{End} (W)$ an irreducible representation compatible with $\phi$, define $\phi\cdot r := r\circ \phi^* \circ \Phi^{\mathbf{\Sigma}}_{\phi(\Delta), \Delta}$. For $\psi \in \operatorname{Mod}(\mathbf{\Sigma})$, then  $\phi\cdot r$ is compatible with $\psi$ if and only if $r$ is compatible with $\psi \circ \phi$ and in this case one has $\psi \cdot (\phi \cdot r) = (\psi \circ \phi)\cdot r$, so one has a partially defined left action of $\operatorname{Mod}(\mathbf{\Sigma})$ on the set of isomorphisms classes of simple $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$-modules. The corresponding partially defined left action of $\operatorname{Mod}(\mathbf{\Sigma})$ on $Y:= \mathrm{Specm}\left(\mathcal{Z}(\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)) \right)$ is characterized by $\phi\cdot \chi := \chi\circ \phi^{-1}\circ \Phi^{\mathbf{\Sigma}}_{\phi(\Delta), \Delta}$, whenever the irreducible representation associated to $\chi$ is compatible with $\phi$. In this case, we say that $\chi$ is compatible with $\phi$.



\begin{definition}\label{def_intertwiner}
\begin{enumerate}
\item Let $\phi \in \operatorname{Mod}(\mathbf{\Sigma})$ and consider $r: \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \rightarrow \operatorname{End}(V(\chi))$ and $r': \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \rightarrow \operatorname{End}(V(\phi\cdot \chi))$ two irreducible representations with central characters $\chi$ and $\phi\cdot \chi$ respectively such that $\chi$ is compatible with $\phi$. Since $r'$ and $r\circ \phi^* \circ \Phi^{\mathbf{\Sigma}}_{\phi(\Delta), \Delta}$ are isomorphic, as simple $ \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) $-modules, there exists an isomorphism $L(\phi) : V(\chi)\xrightarrow{\cong} V(\phi\cdot\chi) $, unique up to multiplication by a scalar, such that:
$$ r'(X) = L(\phi) \left(r\circ \phi^* \circ \Phi^{\mathbf{\Sigma}}_{\phi(\Delta), \Delta}(X)\right) L(\phi)^{-1} \mbox{, for all }X \in  \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta).$$
The isomorphism $L(\phi)$ is called the  \textit{intertwining operator} associated to $\phi$.
\item Let $\mathcal{B}(\mathbf{\Sigma}, \Delta)$ be the groupoid whose objects are the characters of $ \widetilde{\mathcal{X}}_{\mathbb{C}^*}(\hat{\mathbf{\Sigma}}(\Delta))$ and whose morphisms are:
$$ \operatorname{Hom}_{\mathcal{B}(\mathbf{\Sigma}, \Delta)}(\chi_1, \chi_2) := \{ \phi \in \operatorname{Mod}(\mathbf{\Sigma}) | \phi \mbox{ is compatible with }\chi_1 \mbox{ and } \phi\cdot \chi_1 = \chi_2 \}.$$
Let $\mathrm{PVect}$ be the category whose objects are finite dimensional vector spaces and morphisms are projective maps. For each $\chi \in \widetilde{\mathcal{X}}_{\mathbb{C}^*}(\hat{\mathbf{\Sigma}}(\Delta))$, fix an arbitrary simple $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) $-module $V(\chi)$ with central character $\chi$.
The \textit{intertwining functor} $L : \mathcal{B}(\mathbf{\Sigma}, \Delta)\rightarrow \mathrm{PVect}$ is the functor sending a character $\chi$ to the vector space $V(\chi)$ and sending a morphism $\phi \in  \operatorname{Hom}_{\mathcal{B}(\mathbf{\Sigma}, \Delta)}(\chi_1, \chi_2) $ to the projective class of an intertwiner $L(\phi): V(\chi_1)\xrightarrow{\cong} V(\chi_2)$.
\end{enumerate}
\end{definition}


\begin{definition}\label{def_mcg_skein} The mapping class group acts on the stated skein algebra by the action $s: \operatorname{Mod}(\mathbf{\Sigma}) \rightarrow \mathrm{Aut}(\mathcal{S}_{\omega}(\mathbf{\Sigma}))$ defined by $s(\phi)\cdot [D,s]:= [\phi(D), s\circ \phi^{-1}]$, where $\phi\in \operatorname{Mod}(\mathbf{\Sigma})$ and $[D,s]\in \mathcal{S}_{\omega}(\mathbf{\Sigma})$ is a stated diagram.
\end{definition}

\begin{lemma}\label{lemma_mcg_action} For $\phi \in \operatorname{Mod}(\mathbf{\Sigma})$, the following diagram commutes:
$$
\begin{tikzcd}
\mathcal{S}_{\omega}(\mathbf{\Sigma})  \arrow[rr, "\cong"', "s(\phi)^{-1}"] \arrow[d, "\operatorname{Tr}_{\omega}^{\Delta}"] &{}&
\mathcal{S}_{\omega}(\mathbf{\Sigma}) \arrow[d, "\operatorname{Tr}_{\omega}^{\Delta}"] \\
\widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta) \arrow[rr, "\phi^* \circ \Phi_{\phi(\Delta), \Delta}^{\mathbf{\Sigma}}", "\cong"'] &{}&
\widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta)
\end{tikzcd}
$$
\end{lemma}

\begin{proof}
Consider the following diagram:

$$
\begin{tikzcd}
\mathcal{S}_{\omega}(\mathbf{\Sigma}) \arrow[r, "\operatorname{Tr}_{\omega}^{\Delta}"] \arrow[dd, "\cong", "s(\phi)^{-1}"'] \arrow[rd, "\operatorname{Tr}_{\omega}^{\phi(\Delta)}"'] &
\widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta) \arrow[d, "\cong"', "\Phi_{\phi(\Delta), \Delta}"] \\
{}& \widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \phi(\Delta)) \arrow[d, "\cong"', "\phi^*"] \\
\mathcal{S}_{\omega}(\mathbf{\Sigma}) \arrow[r, "\operatorname{Tr}_{\omega}^{\Delta}"] &
\widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta) 
\end{tikzcd}
$$
In the above diagram, the upper triangle commutes by Proposition \ref{prop_change_coordinates} and the commutativity of the lower square follows from Equations \eqref{eq_trace_curve} and \eqref{eq_trace_arc}. This proves the lemma.


\end{proof}
















\subsection{Mapping class groupoid projective representations}\label{sec_mcg_rep}

\subsubsection{The general case}

\begin{notations}
In this section, we fix $\omega\in \mathbb{C}$ a root of unity of odd order $N>1$ and $(\mathbf{\Sigma}, \Delta)$ a triangulated punctured surface. Let $X:= \mathrm{Specm}\left(  \mathcal{Z}\left(\overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})\right) \right)$, $X^0:= \mathrm{Specm} \left( \mathcal{S}_{+1}(\mathbf{\Sigma}) \right)$ and $Y:= \mathrm{Specm} \left(  \mathcal{Z}( \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)) \right)$. Write $p:X\rightarrow X^0$ the projection map induced by the Chebyshev morphism and $\pi: Y\rightarrow X$ the morphism induced by the quantum trace. We also consider the action of the mapping class group $\operatorname{Mod}(\mathbf{\Sigma})$ on $X$ and $X^0$  defined by $\phi\cdot \chi := \chi \circ s(\phi)^{-1}$  and the partially defined action of $\operatorname{Mod}(\mathbf{\Sigma})$ on $Y$ defined by $\phi\cdot \chi := \chi \circ \phi^{-1} \circ \Phi^{\mathbf{\Sigma}}_{\phi(\Delta), \Delta}$ whenever $\chi$ is $\phi$-compatible.
\end{notations}

Note that, by injectivity of $\operatorname{Tr}_{\omega}^{\Delta}: \mathcal{Z}\left(\overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})\right) \hookrightarrow \mathcal{Z}\left( \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \right)$ (and because the image of a dominant map between two irreducible varieties is Zariski dense, see \textit{e.g.} \cite[Proposition $3.34$]{Milne_AG}), the image $p(Y)\subset X$  is Zariski dense. Note also that by Lemma \ref{lemma_mcg_action}, the map $\pi$ is $\operatorname{Mod}(\mathbf{\Sigma})$-equivariant. The equivariance of $p$ is an immediate consequence of the definition of the Chebyshev morphism.

\vspace{2mm}\par 
 If $Y$ would admit a finite orbit $\chi_1, \ldots, \chi_n$ through the action of the mapping class group, then the intertwining functor would induce a finite dimensional projective representation of the mapping class group on the vector space $\oplus_i V(\chi_i)$. The author does not expect such a finite orbit to exist in general, however we formulate the following:

\begin{conjecture}\label{conjecture} Suppose that $\chi_1, \chi_2 \in Y$ are  such that $\pi(\chi_1)=\pi(\chi_2)$, then the $\overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})$-modules $V(\chi_1)$ and $V(\chi_2)$ are isomorphic.
\end{conjecture}

\begin{definition}
Let  $\mathcal{B}^{\operatorname{SL}_2}(\mathbf{\Sigma})$ be the groupoid 
\begin{itemize}
\item
whose objects are characters on the center of $\overline{\mathcal{S}}_{\omega}(\mathbf{\Sigma})$ which extend to characters on the center of $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$, \textit{i.e.} are the points of $\pi(Y)\subset X $
\item and whose morphisms between two characters $\chi_1$ and $\chi_2$ are those elements $\phi \in \operatorname{Mod}(\mathbf{\Sigma})$ such that $\chi_2 = \chi_1 \circ s(\phi)^{-1}$ and such that $\chi_1$ is a character associated to a simple $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ which is $\phi$-compatible.
\end{itemize}
\end{definition}

\vspace{2mm}
\par 
 If Conjecture \ref{conjecture} is true, by Lemma \ref{lemma_mcg_action},  the intertwining functor $L: \mathcal{B}(\mathbf{\Sigma}, \Delta) \rightarrow \mathrm{PVect}$ would induce a functor $L_{\omega}^{\operatorname{SL}_2} : \mathcal{B}^{\operatorname{SL}_2}(\mathbf{\Sigma}) \rightarrow \mathrm{PVect}$. 
 \vspace{2mm}
\par 
 \textbf{Claim:}  unlike for $Y$, it is easy to see that the action of the mapping class group on $X$ admits
 finite orbits from which we could extract finite dimensional mapping class group projective representations assuming that Conjecture \ref{conjecture} is true and that the orbit lies in the dense subset $\mathrm{obj}(\mathcal{B}^{\operatorname{SL}_2}(\mathbf{\Sigma})) \subset X$.
 
 
  Indeed, using the explicit description of the finite branched covering $p:X\rightarrow X_0$ given at the end of Section \ref{sec_def_skein}, any finite $\operatorname{Mod}(\mathbf{\Sigma})$-orbit in $X^0$ lifts to (possibly many) finite orbits in $X$ by choosing the boundary and inner punctures invariants. 
  
   \vspace{2mm}
\par 
    When $\mathbf{\Sigma}$ is closed, it is well known (\cite{Bullock, PS00,Barett}) that $X^0$ is (non-canonically) isomorphic to the character variety $ \mathcal{X}_{\operatorname{SL}_2}(\Sigma_{\mathcal{P}})$ through an isomorphism which is $\operatorname{Mod}(\mathbf{\Sigma})$-equivariant and depends on the choice of a spin $S$ structure on $\Sigma_{\mathcal{P}}$. Denote by $\Psi^S :   \mathcal{X}_{\operatorname{SL}_2}(\Sigma_{\mathcal{P}})\xrightarrow{\cong}  X^0$ such an isomorphism.
    
    For any finite subgroup $G\subset \operatorname{SL}_2(\mathbb{C})$, one has an inclusion $\mathcal{X}_{G}(\Sigma_{\mathcal{P}}) \subset \mathcal{X}_{\operatorname{SL}_2}(\Sigma_{\mathcal{P}})$ and the subset $\mathcal{X}_{G}(\Sigma_{\mathcal{P}}) $ is a finite orbit for $\operatorname{Mod}(\mathbf{\Sigma})$. It is not clear whether such an orbit lies in the image of $\pi$, however we can use the following trick. The inclusion $\Sigma_{\mathcal{P}} \subset \Sigma$ induces an surjective morphism $\pi_1(\Sigma_{\mathcal{P}}, v_0) \rightarrow \pi_1(\Sigma, v_0)$, on the fundamental groups, which in turn induces an embedding $e: \mathcal{X}_{\operatorname{SL}_2} (\Sigma) \rightarrow \mathcal{X}_{\operatorname{SL}_2}(\Sigma_{\mathcal{P}})$; its image is the subset of classes of representations $\rho : \pi_1(\Sigma_{\mathcal{P}}, v_0) \rightarrow \operatorname{SL}_2(\mathbb{C})$ which extend to $\pi_1(\Sigma, v_0)$.  If $\Delta$ is a combinatoric triangulation, \textit{i.e.} if each edge has two distinct endpoints, it follows from \cite[Lemma $7$]{BonahonWong3} that any element in the image $e\left(  \mathcal{X}_{\operatorname{SL}_2} (\Sigma)  \right) \subset  \mathcal{X}_{\operatorname{SL}_2}(\Sigma_{\mathcal{P}})$ is sent by $\Psi^S$ to an element in the image of $\pi : Y \rightarrow X$. Therefore, when $\Delta$ is combinatoric and $G\subset \operatorname{SL}_2(\mathbf{\Sigma})$ is a finite group, the set $X^0_G:= \Psi^S \circ e( \mathcal{X}_G(\Sigma)) \subset X^0$ is a finite $\operatorname{Mod}(\Sigma)$ orbit which lies in the image of $\pi$, so $X_G := p^{-1}(X_G^0) \subset X$ is a finite orbit, and the functor $L^s$ induces a finite dimensional projective representation of $\operatorname{Mod}(\Sigma)$ on $V_G:= \oplus_{\chi \in X_G} V(\chi)$. 
    
    \vspace{2mm}
\par 
    When $\mathbf{\Sigma}$ is open, there is a natural generalization of the notion of character variety, namely the \textit{relative character variety} $ \mathcal{X}_{\operatorname{SL}_2}(\Sigma_{\mathcal{P}})$  defined and studied in \cite{KojuTriangularCharVar} and it is proved in \cite[Theorem $1.6$]{KojuQuesneyClassicalShadows} that any relative spin structure $S$ induces an isomorphism $\Psi^S :  \mathcal{X}_{\operatorname{SL}_2}(\Sigma_{\mathcal{P}})\xrightarrow{\cong}  \mathrm{Specm}(\mathcal{S}_{+1}(\mathbf{\Sigma}))$ (see also \cite[Theorem $8.12$]{CostantinoLe19} for a related construction). Roughly speaking, a point in the relative character variety is the gauge equivalence class $[\rho]$ of a functor $\rho : \Pi_1(\Sigma_{\mathcal{P}}) \rightarrow \operatorname{SL}_2(\mathbb{C})$ whose domain is the fundamental groupoid of $\Sigma_{\mathcal{P}}$; we refer to \cite{KojuTriangularCharVar} for details. The natural action of the mapping class group on the fundamental groupoid induces an action of $\operatorname{Mod}(\mathbf{\Sigma})$ on $\mathcal{X}_{\operatorname{SL}_2}(\mathbf{\Sigma})$ such that $\Psi^S$ is equivariant.
    Under the isomorphism $(\Psi^S)^{-1}$, the subvariety $X^0 \subset \mathrm{Specm}(\mathcal{S}_{+1}(\mathbf{\Sigma}))$ is identified with a subvariety $\overline{\mathcal{X}}_{\operatorname{SL}_2}(\Sigma_{\mathcal{P}})$ whose points are gauge equivalence classes of functors $[\rho]$ sending certain paths in $\Pi_1(\Sigma_{\mathcal{P}}) $ (related to the corner arcs surrounding the boundary punctures in $\mathcal{P}\cap \partial \Sigma$) to elements of the Borel subgroup $B\subset \operatorname{SL}_2(\mathbb{C})$ of upper triangular matrices. As in the closed case, any finite subgroup $G\subset B$ induces a finite $\operatorname{Mod}(\mathbf{\Sigma})$ orbit $\overline{\mathcal{X}}_{G}(\Sigma_{\mathcal{P}})\subset \overline{\mathcal{X}}_{\operatorname{SL}_2}(\Sigma_{\mathcal{P}})$ which is sent by $\Psi^S$ to a finite orbit of $X^0$. However, unlike the closed case, there is no guarantee that such an orbit can be chosen to lye in the image of $\pi$, so Conjecture \ref{conjecture} is not sufficient to provide finite dimensional projective representations of $\operatorname{Mod}(\mathbf{\Sigma})$ for open punctured surfaces.     
    
    
    \vspace{2mm}
    \par  To motivate Conjecture \ref{conjecture}, note that it follows from (\cite[Theorem $2$]{FrohmanKaniaLe_UnicityRep}, \cite[Corollary $1.3$]{KojuAzumayaSkein}) that it is \textit{generically} true  in the sense that there exists a Zariski open subset $\mathcal{O} \subset X^0$ (named the Azumaya locus) such that Conjecture \ref{conjecture} is true for any pair of characters $\chi_1, \chi_2$ in $\mathcal{O}$. Unfortunately, for closed punctured surfaces, the trivial representation $[\rho^0]$ does not belong to $\mathcal{O}$ (see \cite{BonahonWong4}).

    
   
 \subsubsection{Braid groupoid representations}
 
 We now specialize the preceding discussion to the particular case where $(\mathbf{\Sigma}, \Delta) = (\mathbb{D}_n, \Delta_n)$ for some $n\geq 2$. Using the notations of Section \ref{sec_factorizable_rep}, we write $X_n$,  $Y_n$ and $\pi_1$ instead of $X$,  $Y$ and $\pi$. The braid group is $B_n := \operatorname{Mod}(\mathbb{D}_n)$ and we write $\mathcal{B}^{\operatorname{SL}_2}_n:= \mathcal{B}^{\operatorname{SL}_2}(\mathbb{D}_n)$.
 Thanks to our study of factorizable representations in Section \ref{sec_factorizable_rep}, we deduce the following: 
 
 \begin{theorem}\label{theorem_braidgroupoid}
 Conjecture \ref{conjecture} is true for $(\mathbf{\Sigma}, \Delta)=(\mathbb{D}_n, \Delta_n)$. Moreover, if $n\geq 2$, the set $\pi_1(Y_n)$ of objects of $\mathcal{B}^{\operatorname{SL}_2}_n$ is equal to the whole $X_n$. 
 \end{theorem} 
 
 \begin{proof}
 By the second assertion of Theorem \ref{theorem_factorization}, every simple $\mathcal{Z}_{\omega}(\mathbb{D}_n, \Delta_n)$-module is a factorizable $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)$ and by the injectivity of the map $\chi$ in the first assertion of Theorem \ref{theorem_factorization}, any two factorizable $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)$-modules which induce the same character on the center of $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)$ are isomorphic, so Conjecture \ref{conjecture} is verified for $(\mathbb{D}_n, \Delta_n)$. When $n\geq 2$, the equality $\pi_1(Y_n)=X_n$ follows from the third assertion of Theorem \ref{theorem_factorization}. 

 
 \end{proof}

By Theorem \ref{theorem_braidgroupoid}, we have a functor $L^{\operatorname{SL}_2}_{\omega} : \mathcal{B}^{\operatorname{SL}_2}_n \rightarrow \mathrm{PVect}$ that can be refined as follows. Suppose $n\geq 2$ and that the image of any object by $L^{\operatorname{SL}_2}_{\omega}$ is a factorizable representation of the form $V_1\otimes \ldots \otimes V_n$, where $V_i$ extends to a simple $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-modules. The braid groupoid $ \mathcal{B}^{\operatorname{SL}_2}_n$ is generated by the half-twists exchanging two consecutive inner punctures of $\mathbb{D}_n$. The image of these generators by $L^{\operatorname{SL}_2}_{\omega}$ will be called Kashaev's braidings in the next subsection and will be explicitly described. Suppose that $\tau_k: \chi_1 \to \chi_2$ is a half-twist exchanging the punctures $p_k$ and $p_{k+1}$. Then the image $L_{\omega}^{\operatorname{SL}_2}(\tau_k)$ is the composition of isomorphisms of the form $id^{\otimes k-1}\otimes R \otimes id^{n-k-2}$, where $R: V_k\otimes V_{k+1} \to V_k'\otimes V_{k+1}'$ is an intertwiner for the half-twist in $\mathbb{D}_2$, and of isomorphisms of the form $id^{\otimes l-1} \otimes \phi \otimes id^{\otimes n-l-2}$, where $\phi:V_l\otimes V_{l+1} \xrightarrow{\cong} V_{l}' \otimes V_{l+1}'$ is an isomorphism of $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_2)$ modules. 
 By imposing such operators $R$ and $\phi$ to have determinant equal to $1$, one reduces the projective ambiguity of $L_{\omega}^{\operatorname{SL}_2}$ to the multiplication by a $N^2$-th root of unity.


\begin{notations}
Let $\mathcal{C}$ denote the category of weight $\mathcal{O}_q[D(B)]$-modules, $\mathrm{Proj}(\mathcal{C})$ its full subcategory of projective objects and $\mathrm{Proj}(\mathcal{C})^{(N^2)}$ the category with the same objects that $\mathrm{Proj}(\mathcal{C})$ but where morphisms are only considered up to multiplication by a $N^2$-th root of unity.
\end{notations}

We obtain the following refinement of Theorem \ref{theorem_functor}:


\begin{theorem}\label{theorem_functor2}
The change of coordinates isomorphisms induce a functor $L_{\omega}^{\mathrm{SL}_2} : \mathcal{B}^{\mathrm{SL}_2}_n \rightarrow \mathrm{Proj}(\mathcal{C})^{(N^2)}$.
\end{theorem}

The functor of Theorem \ref{theorem_functor2} is the key ingredient to define framed links invariant using the renormalized trace (see Section \ref{sec_LinksInv}).


 Let $e \in D(B)$ be the neutral element  $e:=(\mathds{1}_2, \mathds{1}_2) \in D(B)$ and let $\rho_0 \in X_n^0$ be the image through $p_1 : (D(B))^n =(X_1^0)^n \rightarrow X_n^0$ of the element $(e, \ldots, e)$. The point $\rho_0 \in X_n^0$ is fixed by the braid group $B_n$ so we obtain finite dimensional projective representations of the braid group by lifting $\rho_0$ in $X_n$. In the next subsection, we will identified those representations of $B_n$ with the ones defining the Kashaev invariant and the ADO invariants using Drinfel'd $R$-matrices.
 




















 
 














\subsection{Kashaev's braidings and colored Yang-Baxter equation}
\subsubsection{Definition of Kashaev's braidings}

\begin{notations}
In this subsection, we suppose that $\omega$ is a root of unity of odd order $N>1$. Let $p_1, \ldots, p_n$ be the inner punctures of $\mathbb{D}_n$.
For $1\leq i \leq n-1$, we  denote by $\sigma_i\in B_n$ the mapping class of a half-twist exchanging $p_i$ and $p_{i+1}$ in the clockwise order and leaving fixed the other inner punctures. When $n=2$, we just write $\tau:= \tau_1$. 
\end{notations}


  \begin{figure}[!h] 
\centerline{\includegraphics[width=4cm]{triangulation_Dn.eps} }
\caption{A topological triangulation of $\mathbb{D}_n$.} 
\label{fig_triangulation_Dn} 
\end{figure} 
 


For $n\geq 2$, let $\Delta_n$ denote the topological triangulation of $\mathbb{D}_n$, drawn in Figure \ref{fig_triangulation_Dn}, which is obtained by gluing $n$ copies of $\Delta_1$. 
Recall from Theorem \ref{theorem_factorization} that any simple $\mathcal{Z}_{\omega}(\mathbb{D}_n, \Delta_n)$-module is isomorphic as a $\overline{\mathcal{S}}_{\omega}(\mathbb{D}_n)$-module, to a module of the form  $V:=V_1\otimes \ldots \otimes V_n$ where each $V_i$ is a simple $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-module.



\begin{definition}\label{def_Kashaev_braiding}
\begin{enumerate}
\item Consider two pairs $(V_1, V_2)$ and $(V_3,V_4)$ of simple $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-modules. Let $\chi, \chi' \in X_2$ be the central characters of $V_1\otimes V_2$ and $V_3\otimes V_4$ respectively, seen as simple $\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta_2)$-modules. The two pairs  $(V_1, V_2)$ and $(V_3,V_4)$ are said $\tau$\textit{-compatible} if  $\chi$ is compatible with $\tau$ and $\chi'= \tau \cdot \chi$.
\item If two pairs $(V_1, V_2)$ and $(V_3,V_4)$ are $\tau$-compatible, a \textit{Kashaev braiding} is an intertwining operator $L(\tau) : V_1\otimes V_2 \xrightarrow{\cong} V_3\otimes V_4$ whose determinant is $1$. 
\end{enumerate}
\end{definition}

Note that a Kashaev braiding is uniquely defined, up to multiplication by a $N^2$-th root of unity. Moreover, since each $V_i$ is a $\mathcal{O}_q[D(B)]$-module, the vector spaces $V_1\otimes V_2$ and $V_3\otimes V_4$ are also $\mathcal{O}_q[D(B)]$-modules and a Kashaev braiding $R: V_1\otimes V_2 \xrightarrow{\cong} V_3\otimes V_4$ is a morphism of $\mathcal{O}_q[D(B)]$-modules.

\begin{proposition}\label{prop_YB}
Let $V_1, \ldots, V_{12}$ be some simple $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-modules such that there exists some Kashaev braidings $R_1: V_1\otimes V_2\rightarrow V_7\otimes V_8$, $R_2: V_8\otimes V_3 \rightarrow V_9\otimes V_6$, $R_3: V_7\otimes V_9 \rightarrow V_4\otimes V_5$, $R_4: V_2\otimes V_3 \rightarrow V_{10}\otimes V_{11}$, $R_5: V_1\otimes V_{10} \rightarrow V_4\otimes V_{12}$ and $R_6: V_{12}\otimes V_{11} \rightarrow V_5\otimes V_6$, that is such that each involved pairs are $\tau$-compatible. Then there exists a $N^2$-th root of unity $c$ such that the following colored Yang-Baxter equation holds:
$$ (R_3\otimes id)\circ (id \otimes R_2)\circ (R_1\otimes id) = c(id\otimes R_6)\circ (R_5\otimes id) \circ (id \otimes R_4). $$
\end{proposition}

Figure \ref{fig_YB} illustrates this colored Yang-Baxter equation.

  \begin{figure}[!h] 
\centerline{\includegraphics[width=4cm]{YangBaxter.eps} }
\caption{An illustration of the colored Yang-Baxter equation. Here the morphisms are composed from bottom to top.} 
\label{fig_YB} 
\end{figure} 
 
\begin{proof}
Both the vector spaces $V_1\otimes V_2\otimes V_3$ and $V_4\otimes V_5\otimes V_6$ are simple $\mathcal{Z}_{\omega}(\mathbb{D}_3, \Delta_3)$-modules and, by functoriality of the intertwining functor of Definition \ref{def_intertwiner}, the composition $(R_3\otimes id)\circ (id \otimes R_2)\circ (R_1\otimes id): V_1\otimes V_2\otimes V_3\rightarrow V_4\otimes V_5\otimes V_6$ is an intertwiner $L(\tau_1\circ \tau_2\circ \tau_1)$ whereas the composition $(id\otimes R_6)\circ (R_5\otimes id) \circ (id \otimes R_4) : V_1\otimes V_2\otimes V_3\rightarrow V_4\otimes V_5\otimes V_6$ is an intertwiner $L(\tau_2\circ \tau_1\circ \tau_2)$. Hence the  proposition follows from the braid relation $\tau_1\circ \tau_2\circ \tau_1 = \tau_2\circ \tau_1\circ \tau_2$ together with the unicity, up to multiplication by a $N^2$-th root of unity, of intertwiners.
\end{proof}



\subsubsection{Relation with the Kashaev-Reshetikhin braidings}

\begin{notations} We denote by $\mathscr{R} \in \mathrm{Aut}\left( \widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \Delta_2)\right)$ the automorphism $\mathscr{R}:=\tau^* \circ \Phi_{\tau(\Delta_2), \Delta_2}^{\mathbb{D}_2}$. Let $i_1 : \mathcal{S}_{\omega}(\mathbb{D}_2)\hookrightarrow \mathcal{S}_{\omega}(\mathbb{D}_1)^{\otimes 2}$ and $i_2 : \widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \Delta_2) \hookrightarrow \widehat{\mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}}$ the gluing morphisms from Definitions \ref{def_gluing_map} and \ref{def_gluing_CF} respectively. By abuse of notation, using $i_1$ and $i_2$,  we will consider  $\mathcal{S}_{\omega}(\mathbb{D}_2)$ as a sub-algebra of $ \mathcal{S}_{\omega}(\mathbb{D}_1)^{\otimes 2}$ and consider $\widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \Delta_2)$ as a subalgebra of $ \widehat{\mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}}$. \end{notations}

By definition, a Kashaev's braiding is an intertwiner for the automorphism $\mathscr{R}$, hence the rest of the section will be devoted to the study of $\mathscr{R}$. 
\begin{definition}\label{def_rotation} Let $\theta_2 \in \mathrm{Aut}(\mathcal{S}_{\omega}(\mathbb{D}_2))$ be involutive automorphism sending a stated diagram to its image through the central symmetry of the disc exchanging the two boundary punctures and the two inner punctures. The automorphism $\theta_2$ induces an involutive automorphism $\Theta_2$ on the image of the quantum trace $\operatorname{Tr}_{\omega}^{\Delta_2}(\mathcal{S}_{\omega}(\mathbb{D}_2)) \subset \mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta_2)$.
\end{definition}
Note that $\Theta_2(x\otimes y)=\Theta_1(y)\otimes \Theta_1(x)$, where $\Theta_1$ is the Cartan involution of Definition \ref{def_Cartan_involution}.


By Lemma \ref{lemma_mcg_action}, one has the following equalities:
\begin{equation}\label{eq_twist}
\mathscr{R}(\operatorname{Tr}_{\omega}^{\Delta_2} (x)) = \operatorname{Tr}_{\omega}^{\Delta_2}( s(\tau^{-1})\cdot x) \quad \mbox{, for all }x \in \mathcal{S}_{\omega}(\mathbb{D}_2).
\end{equation}
Since the automorphism $\theta_2$ commutes with $s(\tau^{-1})$, one has:
\begin{equation}\label{eq_Cartan}
\mathscr{R} \circ \Theta_2 (X) = \Theta_2 \circ \mathscr{R} (X) \quad \mbox{, for all }X \in \operatorname{Tr}_{\omega}^{\Delta_2}(\mathcal{S}_{\omega}(\mathbb{D}_2)).
\end{equation}


We derive from Equation \eqref{eq_twist} the following:

\begin{proposition}\label{prop_KR}
The automorphism $\mathscr{R}$ satisfies the following equalities:
\begin{eqnarray}
\mathscr{R}(1\otimes K) &=&(K\otimes 1) \left( 1-q(q-q^{-1})^2 FL^{-1}\otimes K^{-1}E \right)^{-1}; \label{eq_KR1} \\
\mathscr{R}(1\otimes L^{-1}) &=& (L^{-1} \otimes 1)  \left( 1-q(q-q^{-1})^2 FL^{-1}\otimes K^{-1}E \right)^{-1}; \label{eq_KR2} \\
\mathscr{R}(E\otimes 1) &=& L^{-1} \otimes E; \label{eq_KR3} \\
\mathscr{R}(1 \otimes F) &=& F \otimes K^{-1}; \label{eq_KR4} \\
\mathscr{R}(\Delta(x)) &=& \Delta(x) \quad \mbox{, for all }x \in \mathcal{O}_q[D(B)]. \label{eq_KR5}
\end{eqnarray}
\end{proposition}

\begin{proof}
First, remember that the coproduct of $\mathcal{S}_{\omega}(\mathbb{D}_1)$ is defined by doubling the inner puncture, thus one has $s(\tau^{-1}) \cdot \Delta(X) = \Delta(X)$ for all $X\in \mathcal{S}_{\omega}(\mathbb{D}_1)$. Hence Equation \eqref{eq_KR5} follows from Equation \eqref{eq_twist}. Let $\delta_{\varepsilon \varepsilon'} \in \mathcal{S}_{\omega}(\mathbb{D}_1)$ be the class of the stated arc draw in Figure \ref{fig_twist_skein}. Using the defining relation \eqref{eq: skein 2}, one finds that 
$$\delta_{\varepsilon \varepsilon'} = \omega^{-1} \alpha_{\varepsilon - } \beta_{\varepsilon' +} - \omega^{-5} \alpha_{\varepsilon +} \beta_{\varepsilon' -}.$$
Composing with the quantum trace, one finds:
\begin{eqnarray*}
\operatorname{Tr}_{\omega}^{\Delta_2} (\delta_{++}\otimes 1) &=& -(q-q^{-1}) \omega E H_{\partial}\otimes 1;  \\
\operatorname{Tr}_{\omega}^{\Delta_2} (\delta_{-+}\otimes 1) &=& \omega^{-1} K H_{\partial}\otimes 1; \\
\operatorname{Tr}_{\omega}^{\Delta_2} (\delta_{--}\otimes 1) &=& \omega^{-3}(q-q^{-1})K H_{\partial} F\otimes 1.
\end{eqnarray*}

 \begin{figure}[!h] 
\centerline{\includegraphics[width=9cm]{Twist_skein.eps} }
\caption{The figure illustrates how the Kashaev-Reshetikhin automorphism can be derived from elementary skein manipulations.} 
\label{fig_twist_skein} 
\end{figure} 
 

By a simple skein computation drawn in Figure \ref{fig_twist_skein}, one finds:
\begin{equation}\label{eq_twist_truc}
 s(\tau^{-1}) \cdot (\delta_{\varepsilon \varepsilon'}\otimes 1) = \sum_{\mu, \mu' = \pm} \beta_{\varepsilon \mu} \beta_{\varepsilon' \mu'} \otimes \delta_{\mu \mu'}.
 \end{equation}
By composing Equation \eqref{eq_twist_truc} for $(\varepsilon, \varepsilon')=(+,+)$  with the quantum trace, one finds:
$$ \operatorname{Tr}_{\omega}^{\Delta_2} \left( s(\tau^{-1})\cdot (\delta_{++}\otimes 1) \right) = -(q-q^{-1})  L^{-1} \otimes EH_{\partial}.$$
Hence Equation \eqref{eq_KR3} follows from Equation \eqref{eq_twist} applied to $x= \delta_{++}\otimes 1$. By composing with the automorphism $\Theta_2$ and using Equation \eqref{eq_Cartan}, one finds Equation \eqref{eq_KR4}. Composing Equation \eqref{eq_twist_truc} for  $(\varepsilon, \varepsilon')= (-,+)$ with the quantum trace, one has:
$$  \operatorname{Tr}_{\omega}^{\Delta_2} \left( s(\tau^{-1})\cdot (\delta_{-+}\otimes 1) \right) = \omega^{-1} 1\otimes KH_{\partial} - \omega^{3}(q-q^{-1})^2 FL^{-1}\otimes EH_{\partial}.$$
Hence Equation \eqref{eq_twist} applied to $x= \delta_{-+}\otimes 1$ gives the relation:
\begin{equation}\label{eq_RRR}
\mathscr{R} (KH_{\partial} \otimes 1) = 1\otimes KH_{\partial} - q^{-1}(q-q^{-1})^2 FL^{-1}\otimes E H_{\partial}.
\end{equation}
Composing Equation \eqref{eq_twist_truc} for $(\varepsilon, \varepsilon')= (-,-)$ with the quantum trace, one has:
\begin{multline*}  \operatorname{Tr}_{\omega}^{\Delta_2} \left( s(\tau^{-1})\cdot (\delta_{--}\otimes 1) \right) =\omega^{-3}(q-q^{-1}) \left( -(q-q^{-1})^2 q^{-3} F^2L^{-1} \otimes EH_{\partial} \right.
\\ \left.+ L \otimes KFH_{\partial} -F\otimes (K-L-q^{-1}(q-q^{-1})EF)H_{\partial}\right).\end{multline*}
Hence Equation \eqref{eq_twist} applied to $x= \delta_{--}\otimes 1$ gives the relation:

\begin{multline}\label{eq_RR}
\mathscr{R}(KFH_{\partial} \otimes 1) =  -(q-q^{-1})^2 q^{-3} F^2L^{-1} \otimes EH_{\partial} \\ + L \otimes KFH_{\partial} -F\otimes (K-L-q^{-1}(q-q^{-1})EF)H_{\partial}.
\end{multline}
Combining Equations \eqref{eq_RRR} and \eqref{eq_RR}, one finds:
$$
\mathscr{R}(F\otimes 1) = F\otimes 1 +L\otimes F - \left( 1 -q^{-1}(q-q^{-1})^2 FL^{-1} \otimes K^{-1}E \right)^{-1} (F\otimes LK^{-1}).
$$
Using the fact that $\mathscr{R}(\Delta(F))= \Delta(F)$, we derive the equality:
$$\mathscr{R}(L\otimes F) =  \left( 1 -q^{-1}(q-q^{-1})^2 FL^{-1} \otimes K^{-1}E \right)^{-1} (F\otimes LK^{-1}).$$
Using the equality $\mathscr{R}(1\otimes F) = F \otimes K^{-1}$, one finds:
\begin{eqnarray*}
 \mathscr{R}(L\otimes 1) &=&  \left( 1 -q^{-1}(q-q^{-1})^2 FL^{-1} \otimes K^{-1}E \right)^{-1} (1\otimes L) \\
  &=& (1\otimes L)  \left( 1 -q(q-q^{-1})^2 FL^{-1} \otimes K^{-1}E \right)^{-1}.
  \end{eqnarray*}
 Composing the above equation with the automorphism $\Theta_2$ and using Equation \eqref{eq_Cartan}, one finds Equation \eqref{eq_KR1}, while combining the same equation with the equality $\mathscr{R}(L^{-1}\otimes L^{-1}) = L^{-1}\otimes L^{-1}$ one finds \eqref{eq_KR2}.

\end{proof}

In \cite{KashaevReshetikhin04,KashaevReshetikhin05}, Kashaev and Reshetikhin defined a Hopf algebra $\mathcal{U}$ generated by some elements $\mathbf{K}^{\pm 1}, \mathbf{L}^{\pm 1}, \mathbf{E}, \mathbf{F}$. It follows from Theorem \ref{theorem1} that one has an injective Hopf algebra morphism $\Psi : \mathcal{U} \hookrightarrow \mathcal{O}_q[D(B)]$ defined by $\Psi(\mathbf{K}^{\pm 1})= K^{\pm 1}, \Psi(\mathbf{L}^{\pm 1})= L^{\mp 1}, \Psi(\mathbf{E})= (q-q^{-1}) E$ and $\Psi(\mathbf{F})= (q-q^{-1})F$. Moreover the authors of  \cite{KashaevReshetikhin04,KashaevReshetikhin05} defined an automorphism $\mathscr{R}^{KR} \in \mathrm{Aut}\left( \widehat{\mathcal{U}^{\otimes 2} } \right)$ by the formulas: 
\begin{eqnarray*}
\mathscr{R}^{KR} (1\otimes \mathbf{K}) &=& (1\otimes \mathbf{K})\left( 1-q\mathbf{K}^{-1}\mathbf{E} \otimes \mathbf{F}\mathbf{L} \right)^{-1}; \\
\mathscr{R}^{KR} (1\otimes \mathbf{L}) &=& (1\otimes \mathbf{L})\left( 1-q\mathbf{K}^{-1}\mathbf{E} \otimes \mathbf{F}\mathbf{L} \right)^{-1}; \\
\mathscr{R}^{KR}(\mathbf{E}\otimes 1) &=& \mathbf{E}\otimes \mathbf{L}; \\
\mathscr{R}^{KR}(1\otimes \mathbf{F}) &=& \mathbf{K}^{-1} \otimes \mathbf{F}; \\
\mathscr{R}^{KR}( \Delta(x)) &=& \sigma \circ \Delta(x) \quad \mbox{, for all }x\in \mathcal{U}; 
\end{eqnarray*}
where $\sigma(a\otimes b):=b\otimes a$. Let $\Psi_2 : \widehat{\mathcal{U}^{\otimes 2} } \hookrightarrow \widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \Delta_2)$ the injective morphism induced by $\Psi$. We deduce from Proposition \ref{prop_KR} the following:

\begin{corollary}\label{coro_KR}
The following diagram commutes:
$$
\begin{tikzcd}
\widehat{\mathcal{U}^{\otimes 2} }  \arrow[r, "\cong"', "\sigma \circ \mathscr{R}^{KR}"] \arrow[d, hook, "\Psi_2"] &
\widehat{\mathcal{U}^{\otimes 2} }   \arrow[d, hook, "\Psi_2"] \\
 \widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \Delta_2) \arrow[r, "\cong"', "\mathscr{R}"] &
 \widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \Delta_2) 
 \end{tikzcd}
 $$
 Hence the Kashaev-Reshetikhin braidings in \cite{KashaevReshetikhin04,KashaevReshetikhin05, BGPR_Biquandle} associated to pairs of simple modules which extend to $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-simple modules, are Kashaev braidings.
 \end{corollary}





\subsubsection{Relation with Drinfel'd $R$-matrices}

Since the quantum trace $\operatorname{Tr}_{\omega}^{\Delta_2}$ is not surjective, Proposition \ref{prop_KR} is not sufficient to characterize the automorphism $\mathscr{R}$. To further study this automorphism, and compare with the Drinfel'd $R$-matrices,  we will follow the approach of Hikami and Inoue in \cite{Hikami_Inoue_braiding} who characterized the $0$-th graded part of $\mathscr{R}^{-1}$ when $\omega$ is generic. Note that in \cite{Hikami_Inoue_braiding}, the authors studied braidings associated to half-twist in the counter-clockwise direction, which is the opposite of our convention. First let us recall Drinfel'd construction of braiding operators when $\omega\in \mathbb{C}$ is a root of unity of odd order $N>1$. The main idea is to consider the following formula:
$$
R^{QG}:= q^{-\frac{H\otimes G}{2}} \exp_q^{<N}\left( (q-q^{-1})E\otimes F \right) 
$$
where the symbols $H$ and $G$ represent some operators such that $K^{1/2}=q^{\frac{H}{2}}$ and $L^{1/2}=q^{\frac{G}{2}}$. We will not try to give a sense to this formula but instead, we will make sense to its image through a pair of diagonal representations of $\mathcal{O}_q[D(B)]$.

\begin{definition}\label{def_Drinfeld_braiding}
Let $r_1 : \mathcal{O}_q[D(B)] \rightarrow \operatorname{End}(V_1)$ and $r_2 : \mathcal{O}_q[D(B)] \rightarrow \operatorname{End}(V_2)$ be two weight indecomposable representations.
\begin{enumerate}
\item Let $(e_i)_i$ be a basis of $V_1$ such that $r_1(K^{1/2})e_i= q^{\frac{h}{2}-i} e_i$ for some $h \in \mathbb{C}/N\mathbb{Z}$ (here $q^{1/2}:=\omega^{-2}$) and let $(f_i)_i$ be a basis of $V_2$ such that $r_2(L^{1/2})f_i = q^{\frac{g}{2}+i}e_i$ for some $g  \in \mathbb{C}/N\mathbb{Z}$. Define an operator $(r_1\otimes r_2) ( q^{-\frac{H\otimes G}{2}})\in \operatorname{End}(V_1\otimes V_2)$ by the formula:
$$ (r_1\otimes r_2) ( q^{-\frac{H\otimes G}{2}}) \cdot e_i \otimes f_j := q^{-\frac{(h-2i)(g+2j)}{2}} e_i\otimes f_j.$$
\item If $V_1$ and $V_2$ are diagonal, we call \textit{Drinfel'd braiding} the operator $R^D : V_1\otimes V_2 \rightarrow V_2 \otimes V_1$ defined by:
$$ R^D := \sigma \circ (r_1\otimes r_2 )( q^{-\frac{H\otimes G}{2}})\circ (r_1\otimes r_2) \left(  \exp_q^{<N}\left( (q-q^{-1})E\otimes F \right) \right), $$
where $\sigma(x\otimes y):= y\otimes x$.
\end{enumerate}
\end{definition}

\begin{remark} Unless the representations $V_1$ and $V_2$ are both scalars, the operator  $(r_1\otimes r_2) ( q^{-\frac{H\otimes G}{2}})$ is not uniquely defined since it depends on the choice of the $q$-logarithms $h$ and $g$. This is the main reason for the introduction of the unrolled quantum groups in \cite{GeerPatureau_LinksInv, CGP_unrolledQG}. Since we will focus on the cases where $V_1$ and $V_2$ are both scalars, for which the operator $(r_1\otimes r_2) ( q^{-\frac{H\otimes G}{2}})$ is unique up to sign, we will ignore this ambiguity.
\end{remark}

\begin{theorem}\label{theorem_Drinfeld} Let $V_1, V_2$ be two scalars projective indecomposable $\mathcal{O}_q[D(B)]$-modules and $R^D: V_1\otimes V_2 \rightarrow V_2\otimes V_1$ a Drinfel'd braiding as in Definition \ref{def_Drinfeld_braiding}. Then $R^D$ is a Kashaev braiding.

\end{theorem}

\begin{remark} In the particular case where $V_1=V_2=V(A^{-1}, A, 0,0)$, the associated Drinfel'd braiding $R^D$ is the braiding defining the $N$-th Jones polynomial at $N$-th root of unity. Hence Theorem \ref{theorem_Drinfeld} provides an alternative proof of Murakami and Murakami result in \cite{MM01} which relates this braiding to Kashaev's $R$-matrix of \cite{KashaevLinkInv}. Moreover, when $V_1=V_2=V(A^n, A^{-n}, 0,0)$, the associated Drinfel'd braiding is the braiding defining the ADO invariants, hence our theorem is more general.
\end{remark}



Our strategy to prove Theorem \ref{theorem_Drinfeld} goes as follows. We first characterize $\mathscr{R}$ when $q$ is generic using the Fock-Goncharov decomposition of Lemma \ref{lemma_FockGoncharov} and compare with Drinfel'd $R$-matrix. We will deduce from this study explicit formulas for the images $\mathscr{R}(X_i), 1\leq i \leq 7$. Next, we use the fact that these formulas does not depend whether $q$ is generic or not, hence they hold for $q$ a root of unity. We eventually deduce from these formulas that the Drinfel'd braidings at roots of unity intertwine $\mathscr{R}$, hence are Kashaev braiding in the sense of Definition \ref{def_Kashaev_braiding}.

\vspace{2mm}
\par \textbf{The case where $\omega$ is generic.} Consider the topological triangulations $\Delta^{(2)}, \ldots, \Delta^{(5)}$ of $\mathbb{D}_2$ drawn in Figure \ref{fig_flip_triangulations} and write $\Delta^{(1)}:= \Delta_2$. 

 
  \begin{figure}[!h] 
\centerline{\includegraphics[width=9cm]{Flip_triangulations.eps} }
\caption{Six triangulations of $\mathbb{D}_2$.} 
\label{fig_flip_triangulations} 
\end{figure} 

By definition, the automorphism $\mathscr{R}$ decomposes as:
$$\mathscr{R}= \tau^* \circ \Phi^{\mathbb{D}_2}_{\tau(\Delta_2), \Delta^{(5)}} \circ  \Phi^{\mathbb{D}_2}_{\Delta^{(5)}, \Delta^{(4)}} \circ  \Phi^{\mathbb{D}_2}_{\Delta^{(4)}, \Delta^{(3)}} \circ  \Phi^{\mathbb{D}_2}_{\Delta^{(3)}, \Delta^{(2)}}  \circ  \Phi^{\mathbb{D}_2}_{\Delta^{(2)}, \Delta^{(1)}}.$$
By Lemma \ref{lemma_FockGoncharov}, for $i\in \{1, \ldots, 4\}$, the flip automorphism  $ \Phi^{\mathbb{D}_2}_{\Delta^{(i+1)}, \Delta^{(i)}}$ decomposes as  $ \Phi^{\mathbb{D}_2}_{\Delta^{(i+1)}, \Delta^{(i)}}= \nu_i \circ \mathrm{Ad}(s_q(X_{n_i}^{-1}))$, where $\nu_i:= \nu^{\mathbb{D}_2}_{\Delta^{(i+1)}, \Delta^{(i)}}$ and $(n_1,n_2,n_3, n_4)= (4, 5,3,4)$. Set
$$ \mathcal{K}:= \tau^* \circ  \Phi^{\mathbb{D}_2}_{\tau(\Delta_2), \Delta^{(5)}} \circ \nu_4 \circ \nu_3 \circ \nu_2 \circ \nu_1 \in \mathrm{Aut}(\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta_2)).$$
One thus has the expression:
\begin{equation*}
\mathscr{R}= \mathcal{K} \circ \mathrm{Ad}(s_q\left( (\nu_3\circ \nu_2\circ\nu_1)^{-1}(X_4^{-1})\right)) \circ  \mathrm{Ad}(s_q\left(  (\nu_2\circ\nu_1)^{-1}(X_3^{-1})\right)) 
 \circ \mathrm{Ad}(s_q\left( \nu_1^{-1}(X_5^{-1})\right)) \circ  \mathrm{Ad}(s_q\left( X_4^{-1}\right)).\end{equation*}

By a straightforward computation using Definition \ref{def_nu} (see Table $1$ below), one has:
\begin{equation}\label{eq_Rexpression1}
\mathscr{R}= \mathcal{K} \circ \mathrm{Ad}(s_q\left( [X_3X_4X_5]^{-1} \right)) \circ \mathrm{Ad}(s_q\left( [X_3X_4]^{-1} \right))\circ \mathrm{Ad}(s_q\left( [X_4X_5]^{-1} \right))
\circ \mathrm{Ad}(s_q\left( X_4^{-1} \right)).
\end{equation}

\begin{definition}
The automorphism $\mathrm{Ad}(q^{-\frac{H\otimes G}{2}}) \in \mathrm{Aut}(\mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2})$ is defined by the formula:
$$ \mathrm{Ad}(q^{-\frac{H\otimes G}{2}}) \left( Z^{\mathbf{k}} \otimes Z^{\mathbf{k}'} \right):= (K^{1/2})^{\mathbf{k}'(e_1) - \mathbf{k}'(e_4)}Z^{\mathbf{k}} \otimes Z^{\mathbf{k}'} (L^{1/2})^{\mathbf{k}(e_4)- \mathbf{k}(e_1)}.$$
We still denote by the same symbol $\mathrm{Ad}(q^{-\frac{H\otimes G}{2}})$ its restriction to the subalgebras $\mathcal{Z}_{\omega}(\mathbb{D}_2)$ and $\mathcal{O}_q[D(B)]^{\otimes 2}$.
\end{definition}
That $\mathrm{Ad}(q^{-\frac{H\otimes G}{2}})$ is well defined and restricts to automorphisms of $\mathcal{Z}_{\omega}(\mathbb{D}_2)$ and $\mathcal{O}_q[D(B)]^{\otimes 2}$ follows from straightforward computations. The automorphism $\mathrm{Ad}(q^{-\frac{H\otimes G}{2}})$ is not inner, so the notation is abusive. However, it is justified by the following:

\begin{lemma}\label{lemma_qq}
Let $r_1 : \mathcal{O}_q[D(B)] \rightarrow \operatorname{End}(V_1)$ and $r_2 : \mathcal{O}_q[D(B)] \rightarrow \operatorname{End}(V_2)$ be two weight indecomposable representations and consider an operator 
$(r_1\otimes r_2) ( q^{-\frac{H\otimes G}{2}})\in \operatorname{End}(V_1\otimes V_2)$ as in Definition \ref{def_Drinfeld_braiding}. Then the following diagram commutes:
$$ 
\begin{tikzcd}
\mathcal{O}_q[D(B)]^{\otimes 2} \arrow[rrr, "\cong"', "\mathrm{Ad}(q^{-\frac{H\otimes G}{2}})"] \arrow[d, "r_1 \otimes r_2"] &{}&{}&
\mathcal{O}_q[D(B)]^{\otimes 2}  \arrow[d, "r_1 \otimes r_2"]  \\
\operatorname{End}(V_1\otimes V_2) \arrow[rrr, "\cong"', "\mathrm{Ad}\left((r_1\otimes r_2) \left( q^{-\frac{H\otimes G}{2}}\right)\right)"] &{}&{}&
\operatorname{End}(V_1\otimes V_2)
\end{tikzcd}
$$
\end{lemma}

\begin{proof}
The restriction of $\mathrm{Ad}(q^{-\frac{H\otimes G}{2}})$ to $\mathcal{O}_q[D(B)]^{\otimes 2}$ is characterized by the following formulas:
\begin{eqnarray*}
\mathrm{Ad}(q^{-\frac{H\otimes G}{2}})(x\otimes y) = x\otimes y \quad \mbox{, for all }x,y\in \{K^{\pm 1/2}, L^{\pm 1/2}\}; \\
\mathrm{Ad}(q^{-\frac{H\otimes G}{2}})(E\otimes 1) =E\otimes L^{-1}; \quad \mathrm{Ad}(q^{-\frac{H\otimes G}{2}})(F \otimes 1)= F\otimes L; \\
\mathrm{Ad}(q^{-\frac{H\otimes G}{2}})(1\otimes E)= K\otimes E; \quad \mathrm{Ad}(q^{-\frac{H\otimes G}{2}})(1\otimes F) = K^{-1}\otimes F.
\end{eqnarray*}
The proof then  follows from these formulas and the fact that the representations in Definition \ref{def_representations} have weight bases $(e_i)_i$ such that either $e_i=F^i e_0$ or $e_i=E^i e_0$.
\end{proof}


\begin{lemma}\label{lemma_kappa} The following diagram commutes:
$$
\begin{tikzcd}
\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta_2) \arrow[rr, "\cong"', "\mathcal{K}"] \arrow[d, hook] &{}&
\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta_2) \arrow[d, hook] \\
\mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2} \arrow[rr, "\cong"', "\sigma \circ \mathrm{Ad}(q^{-\frac{H\otimes G}{2}})"]   &{}&
\mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}   
\end{tikzcd}
$$
\end{lemma}

\begin{proof} By Lemma \ref{lemma_grading_generator}, the algebra $\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta_2)$ is generated by the elements $\Delta(K^{1/2})$, $\Delta(L^{1/2})$,  $H_{p_1}$, $H_{p_2}$ and the $X_i$ for $i\in \{1,\ldots, 7\}$, we must thus prove that the image of these generators by $\mathcal{K}$ and $\sigma \circ \mathrm{Ad}(q^{-\frac{H\otimes G}{2}})$ are equal. For the generators $\Delta(K^{1/2})$ and $\Delta(L^{1/2})$, this fact follows from Equation \ref{eq_Rexpression1} and Equation \ref{eq_KR5} of Proposition \ref{prop_KR}. For the other generators, this follows from a straightforward computation summarized in Table $1$.
\begin{table}[h!]
  \begin{center}
    \label{table}
    \caption{Computation of $\mathcal{K}$}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta^{(1)})$ & $\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta^{(2)})$ & $\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta^{(3)})$ & $\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta^{(4)})$ & $\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta^{(5)})$ & $\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta^{(1)})$ \\
\hline
$X_1$ & $X_1$ & $X_1$ & $[X_1X_3]$ & $[X_1X_3X_4]$ & $[X_1X_2X_4]$ \\
\hline
$X_2$ & $X_2$ & $X_2$ & $X_2$ & $X_2$ & $X_5$ \\
\hline
$X_3$ & $[X_3X_4]$ & $[X_3X_4X_5]$ & $[X_4X_5]$ & $X_5$ & $X_6$ \\
\hline
$X_4$ & $X_4^{-1}$ & $[X_4X_5]^{-1}$ & $[X_3X_4X_5]^{-1}$ & $[X_3X_4X_5]^{-1}$ & $[X_2X_4X_6]^{-1}$ \\
\hline
$X_5$ & $[X_4X_5]$ & $X_4$ & $[X_3X_4]$ & $X_3$ & $X_2$ \\
\hline
$X_6$ & $X_6$ &$X_6$ &$X_6$ &$X_6$ & $X_3$ \\
\hline
$X_7$ & $X_7$ & $[X_5X_7]$ & $[X_5X_7]$ & $[X_4X_5X_7]$ & $[X_4X_6X_7]$ \\
\hline
$H_{p_1}=[Z_2Z_3]$ & $[Z_2Z_3Z_4]$ & $[Z_2Z_3Z_4Z_5]$ & $[Z_2Z_4Z_5]$ & $[Z_2Z_5]$ & $H_{p_2}$ \\
\hline
$H_{p_2}=[Z_5Z_6]$ & $[Z_4Z_5Z_6]$ & $[Z_4Z_6]$ & $[Z_3Z_4Z_6]$ & $[Z_3Z_6]$ &$H_{p_1}$ \\
\hline

\end{tabular}
  \end{center}
\end{table}
In this table, for $1\leq i \leq 4$, one passes from the $i$-th column to the $i+1$-th column by composing with $\nu_i$ and one passes from the $5$-th column to the last one by composing with $ \tau^* \circ  \Phi^{\mathbb{D}_2}_{\tau(\Delta_2), \Delta^{(5)}}$. Thus one passes from the first column to the last one by composing with $\mathcal{K}$. We conclude by comparing the formulas in the last column with the image through $\sigma \circ \mathrm{Ad}(q^{-\frac{H\otimes G}{2}})$ of the corresponding generators.

\end{proof}

\begin{proposition} Suppose that  $\omega$ is generic. The automorphism \\ $\mathrm{Ad}\left(\exp_q((q-q^{-1})E\otimes F) \right) \in \mathrm{Aut}(\mathcal{Z}_{\omega}(\mathbb{D}_2,\Delta_2))$ is well defined. Moreover, one has:
\begin{equation*}
\mathscr{R} = \sigma \circ \mathrm{Ad}(q^{-\frac{H\otimes G}{2}}) \circ \mathrm{Ad}\left(\exp_q((q-q^{-1})E\otimes F) \right).
\end{equation*}
\end{proposition}

\begin{proof}
In $\overline{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \Delta_2)$, one has the equality (where all products are well defined):

\begin{align*}
& \exp_q\left( (q-q^{-1})E\otimes F\right) = s_q\left( (X_4^{-1} +[X_3X_4]^{-1})\otimes (X_1^{-1}+[X_1X_2]^{-1}) \right) \\
&= s_q\left( [X_3X_4]^{-1}\otimes (X_1^{-1} + [X_1X_2]^{-1}) \right)  s_q\left( X_4^{-1}\otimes (X_1^{-1} + [X_1X_2]^{-1}) \right) \\
&= s_q \left( [X_3X_4]^{-1}\otimes [X_1X_2]^{-1} \right) s_q\left([X_3X_4]^{-1}\otimes X_1^{-1}\right)s_q\left(X_4^{-1}\otimes [X_1X_2]^{-1}\right) s_q\left(X_4^{-1}\otimes X_1^{-1}\right) \\
&= s_q\left( [X_3X_4X_5]^{-1}\right)s_q\left([X_3X_4]^{-1}\right)s_q\left([X_4X_5]^{-1}\right)s_q\left(X_4^{-1}\right)
\end{align*}
Where we used formula $(1)(a)$ of Lemma \ref{lemma_expq} three times. We conclude using Lemma \ref{lemma_kappa} and Equation \ref{eq_Rexpression1}.


\end{proof}

\begin{proposition}\label{prop_calcul_R}
Suppose that $\omega$ is generic. The automorphism $\mathscr{R}$ is characterized by the following formulas:
\begin{align*}
&\mathscr{R}(\Delta(K^{1/2}))= \Delta(K^{1/2}); \mathscr{R}(\Delta(L^{1/2})) = \Delta(L^{1/2}); \quad\mathscr{R}(H_{p_1})=H_{p_2};  \mathscr{R}(H_{p_2})=H_{p_1}; \\
&\mathscr{R}(X_1) = \left(X_1^{-1}+  [X_1X_2]^{-1} +  [X_1X_2X_4]^{-1}  \right)^{-1}; \\
&\mathscr{R}(X_2) = X_4+X_5+[X_2X_4X_5] + [X_4X_5X_6]+[X_2X_4X_5X_6]; \\
&\mathscr{R}(X_3) = \left(X_6^{-1} +X_4+[X_2X_4] + [X_6^{-1}X_4] + [X_2X_4X_6^{-1}]\right)^{-1}; \\
&\mathscr{R}(X_5)= \left(X_2^{-1} + X_4+  [X_4X_6] + [X_2^{-1}X_4] + [X_2^{-1}X_4X_6] \right)^{-1}; \\
&\mathscr{R}(X_6) =  X_3+X_4+[X_3X_4X_6]+[X_2X_3X_4]+[X_2X_3X_4X_6]; \\
&\mathscr{R}(X_7) = \left( X_7^{-1} + [X_6X_7]^{-1}+[X_4X_6X_7]^{-1} \right)^{-1}.
\end{align*}
\end{proposition}

\begin{proof} The formulas for $\mathscr{R}(\Delta(K^{1/2}))$ and  $\mathscr{R}(\Delta(L^{1/2}))$ follow from Proposition \ref{prop_KR}. The formulas for  $\mathscr{R}(H_{p_1})$ and   $\mathscr{R}(H_{p_2})$ follow from the fact that both $H_{p_1}$ and $H_{p_2}$ commute with $E\otimes F$ (since those are central elements) together with the computation of $\mathcal{K}(H_{p_i})$ made in Table $1$. For simplicity, write $\mathcal{AD}:= \mathrm{Ad}\left(\exp_q\left( (q-q^{-1})E\otimes F\right)\right)$. Let us first compute $\mathscr{R}(X_1)$. Using the formula $(1)(a)$ in Lemma \ref{lemma_expq}, one has $\exp_q\left( (q-q^{-1}) E\otimes F\right) = s_q\left( (q-q^{-1})[X_3X_4]^{-1}\otimes F\right) s_q\left( (q-q^{-1})X_4^{-1}\otimes F\right)$. Since $X_1=X_1\otimes 1$ commutes with $X_4^{-1}\otimes F$ and $q^2$-commutes with $[X_3X_4]^{-1}\otimes F$, one has:
\begin{eqnarray*}
\mathcal{AD}(X_1) &=& X_1 s_q\left( q^2 \left( (q-q^{-1})[X_3X_4]^{-1}\otimes F \right) \right) s_q \left( (q-q^{-1})[X_3X_4]^{-1}\otimes F \right)^{-1} \\
&=& X_1\left( 1+q(q-q^{-1})[X_3X_4]^{-1}\otimes F\right)^{-1}
\end{eqnarray*}
where we used formula $(1)(b)$ of Lemma \ref{lemma_expq} to pass from the first to the second line. We obtain the expression for $\mathscr{R}(X_1)$ by composing the above formula with the automorphism $\mathcal{K}$ using Table $1$. Next, to compute $\mathscr{R}(X_2)$, note that $X_2=X_2\otimes 1$ $q^{-2}$-commutes with $E\otimes F$, thus one has:
\begin{eqnarray*}
\mathcal{AD}(X_2) &=& X_2 s_q\left( q^{-2} \left(-(q-q^{-1})^2 E\otimes F \right) \right) s_q \left(-(q-q^{-1})^2 E\otimes F \right)^{-1} \\
&=& X_2\left( 1-q^{-1}(q-q^{-1})^2 E\otimes  F\right)
\end{eqnarray*}
where we used formula $(1)(b)$ of Lemma \ref{lemma_expq}. We obtain the expression for $\mathscr{R}(X_2)$ by composing the above formula with the automorphism $\mathcal{K}$ using Table $1$. For $X_3$, note that $X_3=X_3\otimes 1$ $q^2$-commutes with $E\otimes F$, thus one has:
\begin{eqnarray*}
\mathcal{AD}(X_3) &=& X_3 s_q\left( q^{2} \left(-(q-q^{-1})^2 E\otimes F \right) \right) s_q \left(-(q-q^{-1})^2 E\otimes F \right)^{-1} \\
&=& X_3\left( 1-q(q-q^{-1})^2 E\otimes  F\right)^{-1}
\end{eqnarray*}
Again, we conclude by composing with $\mathcal{K}$ using Table $1$. The formulas for $\mathscr{R}(X_5)$, $\mathscr{R}(X_6)$, $\mathscr{R}(X_7)$ are obtained by replacing in the above computations the elements $X_1, X_2, X_3$, $X_4, X_5, X_6, X_7$ by the elements $X_7, X_6, X_5$, $X_4, X_3, X_2, X_1$ respectively.


\end{proof}

\begin{remark}\begin{itemize}
\item In \cite{Hikami_Inoue_braiding}, the authors computed explicitly the images $\mathscr{R}^{-1}(X_i)$ directly from the formulas of Definition \ref{def_square_changecoordinates}. The formulas they obtained are much more complicated than the formulas we obtained in Proposition \ref{prop_calcul_R} using the Fock-Goncharov decomposition.
\item The reader might ask why we did not provide a formula for $\mathscr{R}(X_4)$. First because we do not need it to characterize $\mathscr{R}$: indeed it can be recovered from the above formulas using the equality $X_4= [\Delta(K^{-1})X_1^{-1}X_3^{-1}X_6^{-1}X_7^{-1}]$. The actual reason is that we will argue in the proof of Theorem \ref{theorem_Drinfeld} that the above computations for $\mathscr{R}(X_i)$ hold when $\omega$ is a root of unity of odd order $N>1$ with the additional assumption that $E^N\otimes1=1\otimes F^N=0$ (through the image of appropriate representations). The author did not find a straightforward computation for $\mathscr{R}(X_4)$ that generalizes to this case.
\end{itemize}
\end{remark}


\textbf{The case where $\omega$ is a root of unity.}
\vspace{2mm}
\par We first begin to prove that the formulas in Proposition \ref{prop_calcul_R} also hold when $\omega$ is a root of unity. Consider two commutative unital rings $\mathcal{R}_1$ and $\mathcal{R}_2$ with invertible elements $\omega_1\in \mathcal{R}_1^{\times}$, $\omega_2\in \mathcal{R}_2^{\times}$, such that both $\omega_1^4-\omega_1^{-4}$ and $\omega_2^4-\omega_2^{-4}$ are invertible and suppose that we have a surjective ring morphism $\mu : \mathcal{R}_1 \rightarrow \mathcal{R}_2$ such that $\mu(\omega_1)=\omega_2$. By extension of scalars, the morphism $\mu$ induces a structure of $\mathcal{R}_2$-algebra on $\mathcal{R}_2\otimes_{\mathcal{R}_1} \widehat{\mathcal{Z}}_{\omega_1}(\mathbf{\Sigma}, \Delta)$ and it follows from Definition \ref{def_CF_algebra} that this algebra is isomorphic to $\widehat{\mathcal{Z}}_{\omega_2}(\mathbf{\Sigma}, \Delta)$. A straightforward analysis of Definitions \ref{def_reindexing_map}, \ref{def_square_changecoordinates}, \ref{def_flip_changecoordinates}, shows that the following diagram commutes:
\begin{equation}\label{eq_diaaa}
\begin{tikzcd}
\mathcal{R}_2\otimes_{\mathcal{R}_1} \widehat{\mathcal{Z}}_{\omega_1}(\mathbb{D}_2, \Delta_2) 
\arrow[r, "\cong"', "\mathcal{R}_2\otimes_{\mathcal{R}_1}\mathscr{R}"] \arrow[d, "\cong"] &
\mathcal{R}_2\otimes_{\mathcal{R}_1} \widehat{\mathcal{Z}}_{\omega_1}(\mathbb{D}_2, \Delta_2) 
\arrow[d, "\cong"] \\
\widehat{\mathcal{Z}}_{\omega_2}(\mathbb{D}_2, \Delta_2)
\arrow[r, "\cong"', "\mathscr{R}"] &
\widehat{\mathcal{Z}}_{\omega_2}(\mathbb{D}_2, \Delta_2)
\end{tikzcd}\end{equation}

\begin{lemma}\label{lemma_rootunity}
The formulas of Proposition \ref{prop_calcul_R} hold when $\omega_N \in \mathcal{R}^{\times}$ is a root of unity of order $N>1$.
\end{lemma}

\begin{proof}
Consider the ring $\mathcal{R}_1$ obtained by localizing the ring $\mathbb{Z}[\omega^{\pm 1}]$ with respect to $\omega^{-4}-\omega^{4}$ and denote by $\mu : \mathcal{R}_1 \rightarrow \mathcal{R}$ the unique surjective ring morphism sending $\omega$ to the root of unity $\omega_N\in \mathcal{R}$. Since $\omega\in \mathcal{R}_1$ is generic, one can apply Proposition \ref{prop_calcul_R} to $\widehat{\mathcal{Z}}_{\omega}(\mathbf{\Sigma}, \Delta)$ to compute $\mathscr{R}$. We conclude using the commutativity of the diagram \eqref{eq_diaaa}.
\end{proof}

\begin{proof}[Proof of Theorem \ref{theorem_Drinfeld}]
Write $\mathscr{A}:= \mathcal{K}^{-1}\circ \mathscr{R}$. By Lemma \ref{lemma_qq}, it sufficient to show that the following diagram commutes:
$$\begin{tikzcd}
\widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \Delta_2) 
\arrow[rrrr, "\cong"', "\mathscr{A}"] \arrow[d, "r_1\otimes r_2"'] &{}&{}&{}&
\widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \Delta_2) 
\arrow[d, "r_1\otimes r_2"] \\
\operatorname{End}(V_1\otimes V_2) 
\arrow[rrrr, "\cong"', "\mathrm{Ad}\left( (r_1\otimes r_2)(\exp_q^{<N}((q-q^{-1})E\otimes F))\right)"] &{}&{}&{}&
\operatorname{End}(V_1\otimes V_2) 
\end{tikzcd}$$
Said differently, we need to show that for each generator $X$ of $\widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \Delta_2) $, one has
\begin{equation*}
(r_1\otimes r_2)\circ \mathscr{A}(X) = \mathrm{Ad}\left( (r_1\otimes r_2)(\exp_q^{<N}((q-q^{-1})E\otimes F))\right) \circ (r_1\otimes r_2)(X).
\end{equation*}
When $X= \Delta(K^{1/2})$ and $X=\Delta(L^{1/2})$ the equality  follows from Proposition \ref{prop_KR}, Lemma \ref{lemma_rootunity} and the fact that both $\Delta(K^{1/2})$ and $\Delta(L^{1/2})$ commute with $E\otimes F$. For $X=H_{p_1}$ and $X=H_{p_2}$, this follows from Table $1$, Lemma \ref{lemma_rootunity} and the fact that $H_{p_i}$ commutes with $E\otimes F$. For $X\in \{X_1, X_2, X_3, X_5, X_6, X_7\}$, the computations made in the proof of Proposition \ref{prop_calcul_R} can be translated word-by-word by replacing $\exp_q$ and $s_q$ by $(r_1\otimes r_2)\circ \exp_q^{<N}$ and $(r_1\otimes r_2)\circ s_q^{<N}$ respectively, to prove the desired equality. Indeed, the hypothesis $r_1(E^N)=r_2(F^N)=0$ implies that we can replace each computation involving formulas $(1)(a)$ and $(1)(b)$ of Lemma \ref{lemma_expq}  by equivalent computation using formulas $(2)(a)$ and $(2)(b)$ respectively. We then conclude using Lemma \ref{lemma_rootunity}.



\end{proof}


\subsection{A closed formula for Kashaev's braidings}\label{sec_ClosedFormula}

We now generalize the techniques used in the preceding subsections to provide an explicit formula for generic Kashaev's braidings at odd roots of unity.
\begin{notations} For $\chi$ a character over the center of $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$, we denote by $\mathcal{I}_{\chi} \subset \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ the ideal generated by the kernel of $\chi$. Recall that two characters $\chi$ and $\chi'$ over the centers of $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)$ and $\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta')$ are said compatible if $\chi = \Phi_{\Delta \Delta'}^{\mathbf{\Sigma}} \circ \chi'$. In this case, we get an isomorphism
$$ \Phi_{(\Delta, \chi), (\Delta', \chi')}^{\mathbf{\Sigma}} : \quotient{\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta')}{\mathcal{I}_{\chi'}} \xrightarrow{\cong} \quotient{\mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta)}{\mathcal{I}_{\chi}}.$$

\end{notations}

 \textbf{(Quantum) shape parameters in (quantum) hyperbolic geometry}
\vspace{2mm}

\par The Kashaev braidings that we will soon define depend on some quantum shape parameters introduced by Baseilhac and Benedetti in \cite{BaseilhacBenedettiLocRep}, inspired by Thurston's classical shape parameters. To make clear the geometric meaning of these parameters, we briefly recall their origin following \cite{ThurstonLectNotes, NeumannZagier_VolHypMfd}. 

\begin{definition}
 A \textit{shape parameter} is an element of $\mathbb{C}^{**}:= \{ \mathbf{z}=(z,z',z'') | zz'z''=-1, zz'-z=-1\}$.
\end{definition}
Obviously, each projection $\mathbf{z}\mapsto z$, $\mathbf{z}\mapsto z'$ and $\mathbf{z}\mapsto z''$ induces a bijection between $\mathbb{C}^{**}$ and $\mathbb{C}\setminus \{0,1\}$. 
Let $M$ a compact oriented connected $3$-manifold such that $\partial M$ is a disjoint union of spheres and tori and let $\overline{M}$ be obtained by shrinking each connected component of $\partial M$ to a point which we call an ideal point. An ideal triangulation $\mathcal{T}$ is a triangulation of $\overline{M}$ whose set of vertices is the set of ideal points. So $\overline{M}$ is obtained from a disjoint union $M_T=\bigsqcup_{i\in I} T_i$ of tetrahedrons $T_i$ by gluing some pairs of faces while respecting the orientations. We denote by $\widehat{\mathcal{E}}(T)$ the set of edges of $M_T$, by $\mathcal{E}(T)$ the set of edges of $T$ and by $\pi : \widehat{\mathcal{E}}(T)\rightarrow \mathcal{E}(T)$ the surjective map sending an edge of a tetrahedron $T_i$ to the corresponding edge in $T$. 


\par A \textit{branching} $b_i$ of a tetrahedron $T_i$ with set of vertices  $V(T_i)$ is a bijection $b_i : V(T_i) \xrightarrow{\cong} \{0, 1, 2, 3\}$.  Denote by $[ab]$ the edge of $T_i$ whose endpoints are the vertices sent to $a$ and $b$ by $b_i$. The map $b_i$ induces an orientation of the edges of $T_i$ by orienting $[ab]$ from $a$ towards $b$. A \textit{branching} of $T$ is an orientation of the edges of $T$ such that the restriction to each $T_i$ is induced by a branching $b_i$.
A branching $b_i$ induces an orientation of $T_i$ and we write $\star_{b_i}=+1$ if this orientation agrees with the one of $M$ and $\star_{b_i}=-1$ else. 
\par For $(T,b)$ a branched triangulation and $\{\mathbf{z}_i\}_i \in (\mathbb{C}^{**})^I$ an assignment of a shape parameter $\mathbf{z}_i=(z_i,z'_i,z''_i)$ to each tetrahedron $T_i$, we get a map $\mathbf{z} : \widehat{\mathcal{E}}(T) \rightarrow \mathbb{C}\setminus \{0,1\}$ defined on each tetrahedron $T_i$ by sending the edges $[01], [23]$ to $z_i$, the edges $[02], [13]$ to $z'_i$ and the edges  $[03], [12]$ to $z''_i$. The \textit{gluing variety} $\mathcal{G}(T,b)$ is the subvariety of $(\mathbb{C}^{**})^I$ defined by the following \textit{Thurston's equations}
$$ \prod_{\widehat{e} \in \pi^{-1}(e)} \mathbf{z}(\widehat{e}) = 1, \quad \forall e\in \mathcal{E}(T).$$
An \textit{augmented representation} is a pair $(\rho, h)$ where $\rho : \pi_1(M) \to \operatorname{PSL}_2(\mathbb{C})$ is a representation such that $\rho([\gamma]) \in PB$ for all simple closed curves $\gamma \subset \partial M$ and $PB\subset \operatorname{PSL}_2(\mathbb{C})$ is the Borel subgroup of classes of upper triangular matrices,  and $h\in \mathrm{H}^1(\partial M; \mathbb{C}^*)$ is such that $\operatorname{Tr}(\rho(\gamma)) = h(\gamma) + h(\gamma)^{-1}$ for all $\gamma \subset \partial M$. The augmented representations form the closed points of an affine variety $\mathcal{R}_{\operatorname{PSL}_2}^{aug}(M)$ on which $\operatorname{PSL}_2(\mathbb{C})$ acts by conjugacy. The \textit{augmented character variety} is the GIT quotient

$$ \mathcal{X}_{\operatorname{PSL}_2}^{aug}(M) := \mathcal{R}_{\operatorname{PSL}_2}^{aug}(M) \mathbin{/\mkern-6mu/} \operatorname{PSL}_2(\mathbb{C}).$$

The main interest in the introduction of Thurston's shape parameters and the gluing variety lye in the existence of an algebraic morphism (see \textit{e.g.} \cite{BDRV, KimKimYoon} for definitions):

$$ \mathcal{D} : \mathcal{G}(\mathcal{T}, b) \rightarrow \mathcal{X}_{\operatorname{PSL}_2}^{aug}(M).$$

 Therefore, a set of shape parameters $\{\mathbf{z}_i\}_i$ satisfying the Thurston's equation should be thought as a combinatorial way to encode the data of an augmented representation (for instance, the monodromy representation associated to some hyperbolic structure). A big success of this approach is a combinatorial description, made by Dupont \cite{Dupont}, Neumann \cite{Neumann92, Neumann98, Neumann04} and Zickert \cite{Zickert}, of the Chern-Simons functional (see also \cite{DupontZickert}). The abelianization map $ab : \pi_1(\mathbb{C}^{**}) \cong \mathbb{F}_2 \to \mathbb{Z}^2$ defines an infinite cyclic abelian covering $\widetilde{\mathbb{C}^{**}} \rightarrow \mathbb{C}^{**}$ of the set of shape parameters. The Roger's dilogarithm $R$ is an analytic function on $\widetilde{\mathbb{C}^{**}}$.  By lifting Thurston's equations in $(\widetilde{\mathbb{C}^{**}})^{\mathcal{I}}$ in a suitable manner, one gets an infinite cover $\widetilde{\mathcal{G}}(\mathcal{T}, b)\to \mathcal{G}(\mathcal{T}, b)$ and the Roger's dilogarithm induces an analytic function $R^T : \widetilde{\mathcal{G}}(\mathcal{T}, b)\to \mathbb{C}$. A lift of a family $\{\mathbf{z}_i\}_i \in \mathcal{G}(\mathcal{T}, b)$ in $\widetilde{\mathcal{G}}(\mathcal{T}, b)$ is called a \textit{combinatorial flattening}. A \textit{flattening} of an enhanced representation $(\rho, h)$ is an element $l\in \mathrm{H}^1(\partial M; \mathbb{C})$ such that $\exp(l([\gamma]))= h(\gamma)$ for all $\gamma \subset \partial M$. Conjugacy classes of flattened enhanced representations form an analytic variety $\mathcal{X}_{\operatorname{PSL}_2}^{flat}(M)$, which is an infinite cover over  $\mathcal{X}_{\operatorname{PSL}_2}^{aug}(M)$. The morphism $\mathcal{D}$ lifts to a map
 $$ \widetilde{\mathcal{D}} : \widetilde{\mathcal{G}}(\mathcal{T}, b) \to \mathcal{X}_{\operatorname{PSL}_2}^{flat}(M).$$ 
 When $\widetilde{\mathcal{D}}( \{\mathbf{w}_i\}_i) = [\rho, l]$, the main upshot of the work in \cite{Dupont,Neumann92, Neumann98, Neumann04,Zickert} is the following formula
 $$ CS([\rho, l]) = R^T(\{\mathbf{w}_i\}_i)= \sum_{i\in I} \star_{b_i} R(\mathbf{w}_i, b_i), $$
 where $CS([\rho, l])$ is the $\operatorname{PSL}_2$ Chern-Simons invariant (containing the volume) and $R(\mathbf{w}_i, b_i)$ is an evaluation of the Roger's dilogarithm depending on the flattened shape parameter and the branching. We refer to March\'e \cite{Marche_SimplicialFormula} for a beautiful gauge theoretical interpretation of this formula and to \cite{Baseilhac_CSNotes} for a self-contained introduction to the formula.
 
 \vspace{2mm}
 \par The main purpose of the present paper is to define some new framed links invariants that would refine the Quantum Hyperbolic Invariants (QHI) and contains the non semi simple invariants of \cite{GeerPatureau_LinksInv} (including the ADO and Kashaev's invariant). The construction and study of the QHI was initiated by Kashaev in \cite{KashaevLinkInv} and further continued by Baseilhac-Benedetti in \cite{BaseilhacBenedettiInvariant, BaseilhacBenedetti05, BaseilhacBenedetti15, BaseilhacBenedetti_NonAmbiguousStructures}. Their definition shares many similarities with the above combinatorial formula for the (exponential of the) Chern-Simons invariant which is conjectured to be related to their semi-classical asymptotic. Since our invariants depend on parameters very similar to the ones used to define both the QHI and the non-semi simple ones, we briefly introduce them here. We fix $q$ a root of unity of odd order $N>1$.
 
 \begin{definition}
 A \textit{q-shape parameter} is an element of $$\mathbb{C}_q^{**}:= \{ \mathbf{w}=(w,w',w'') | (w^N,{w'}^N, {w''}^N)\in \mathbb{C}^{**}, ww'w''=-q^{-1}\}.$$
\end{definition}

So $\mathbb{C}_q^{**}$ is a $N^2$ branched cover of $\mathbb{C}^{**}$ and the choice of a lift $\mathbf{w}$ of a shape parameter $\mathbf{z}$ will be called a combinatorial $q$-flattening. The q-gluing variety $\mathcal{G}_q(T,b)\subset (\mathbb{C}_q^{**})^{\mathcal{I}}$ is the subvariety of  $(\mathbb{C}_q^{**})^{\mathcal{I}}$ defined by the following quantum Thurston's equations: 
$$ \prod_{\widehat{e} \in \pi^{-1}(e)} \mathbf{w}(\widehat{e}) = q^{-2}, \quad \forall e\in \mathcal{E}(T).$$
A preimage of $\mathcal{G}_q(T,b)\rightarrow \mathcal{G}(T,b)$ is called a combinatorial q-flattening. Such a q-flattening is equivalent to the data of a classical flattening together with a so-called \textit{integral charge} map $d : \widehat{\mathcal{E}}(T) \rightarrow \mathbb{Z}$ satisfying some condition (see \cite{BaseilhacBenedettiInvariant, BaseilhacBenedetti05} for the definition of integral charges and  \cite[Remark $3.4$]{BaseilhacBenedettiLocRep} for the relation with q-flattenings). The map $\mathcal{D}$ lifts to a regular morphism
$$ \mathcal{D}_q : \mathcal{G}_q(T,b) \rightarrow \mathcal{X}_{\operatorname{PSL}_2,q}(M), $$
where $\mathcal{X}_{\operatorname{PSL}_2, q}(M)=\mathcal{R}_{\operatorname{PSL}_2, q}(M) \mathbin{/\mkern-6mu/} \operatorname{PSL}_2(\mathbb{C})$ represents the classes of pairs $(\rho, h)\in \mathcal{R}_{\operatorname{PSL}_2, q}(M)$ with $\rho \in \mathcal{R}_{\operatorname{PSL}_2}(M)$ and $h\in \mathrm{H}^1(\partial M; \mathbb{C})$ is such that $q^{h([\gamma])} +q^{-h([\gamma])} = \operatorname{Tr}(\rho(\gamma)^2)$ for all simple closed curve $\gamma \subset \partial M$. Let $\mathcal{O} \subset \mathcal{X}_{\operatorname{PSL}_2,q}(M)$ be the union over all branched triangulations of the images of the maps $\mathcal{D}_q=\mathcal{D}_q(T,b)$. The QHI are maps $\mathcal{H}_q : \mathcal{O} \rightarrow \quotient{\mathbb{C}}{\mu_N}$ valued in complex numbers up to multiplication by a $N$-th root of unity. By assigning a $2-2$-tensor $\mathcal{L}_q(T_i,b_i,\mathbf{w}_i, \star_i)\in \operatorname{End}( \mathbb{C}^N \otimes \mathbb{C}^N)$ (the so-called cyclic dilogarithm or Kashaev $6j$-symbol related to flip operators \cite{Bai6jsymbols, BaseilhacBenedettiLocRep}) to each tetrahedron and then contracting the tensors along the pairs of glued faces, we get a regular map $\mathcal{H}^{red}_q : \mathcal{G}_q(T,b) \rightarrow \mathbb{C}$ named the reduced state-sum. By introducing a suitable symmetrization factor $\alpha_q  \mathcal{G}_q(T,b) \rightarrow \mathbb{C}$ and setting $\mathcal{H}_q = \alpha_q \times \mathcal{H}^{red}_q$, one gets a regular map such that, if $\mathcal{D}_q(\{\mathbf{w}_i\}_i)=(\rho, h)$, the value $\mathcal{H}_q(\{\mathbf{w}_i\}_i)$ only depends, up to multiplication by a $N$-th root of unity, on $(\rho,h)$. As we will point out in Section \ref{sec_LinksInv}, the role of the symmetrization factor seems very similar to the role of the renormalized trace in quantum groups theory.


\vspace{2mm}
\par Since we are interested in invariants of framed links $L\subset S^3$, we now specialize the preceding discussion to the case where $M=S^3\setminus (\mathring{N}(L) \cup B_N \cup B_S)$ is the complement of the interior of a tubular neighborhood of $L$ and of two disjoint open balls $B_N, B_S$ away from $N(L)$.  Fix a preserving orientation homeomorphism $S^3 \setminus (B_N\cup B_S) \cong S^2\times [0,1]$ and consider a planar projection diagram $D\subset S^2\times \{1/2\}$ of $L$. There is a canonical ideal triangulation $\mathcal{T}^{4T}(D)$ of $M$ associated to $D$ named the $4T$-triangulation. It was first considered by Kashaev in his construction of links invariants \cite{KashaevLinkInv} and underlines the heuristic argument behind his celebrated volume conjecture \cite{Kashaev97} (see \cite{Yokota_VolConj,Yokota_ComplexVol, ChoKimKim_OptimisticLimitKashaevInv} for a precise formulation and proofs of this heuristic using the notion of optimistic limit);  this triangulation was further studied by Weeks in his thesis and underlines the SnapPea software \cite{Weeks_SnapPea} (see also \cite{Thurston_JonesPoly}). We sketch the construction and refer to \cite{KimKimYoon} for a detailed construction.  First assign to each crossing of $D$ an octahedron whose top vertex is on the overpass and bottom vertex is on the underpass. Pull the two middle vertices lying on the two sides of the overpass up to $B_N$ and the two other middle vertices down to $B_S$. By gluing these twisted octahedra together, we get an octahedral decomposition of $S^3\setminus (B_N\cup B_S)$. The triangulation $\mathcal{T}^{4D}$ is then obtained by further subdividing each twisted octahedron into $4$ tetrahedra (see Figure \ref{fig_octahedron}). We fix the branching $b$ on $\mathcal{T}^{4D}$ described in Figure \ref{fig_decomposition_octahedre}; in particular one has $\star_{b_i}=+1$ for each $i=1,\ldots, 4$. A closed point of $\mathcal{G}_q(\mathcal{T}^{4T}(D), b)$ is a assignment of $4$ quantum shape parameters $\mathbf{w}_1^{(i)}, \ldots, \mathbf{w}_4^{(i)}$ to each crossing of $D$ that satisfy some quantum Thurston equations (which can be easily explicited from the formulas for the classical Thurston's equations in \cite{KimKimYoon} by changing the branching). In particular, the choice of branching is designed so that the quantum Thurston equation associated to the inner edge lying inside the octahedron is $w_1w_2w_3w_4=q^{-2}$ (see Figure \ref{fig_decomposition_octahedre}). 

   \begin{figure}[!h] 
\centerline{\includegraphics[width=9cm]{Octahedron_flip.eps} }
\caption{The octahedron associated to a Kashaev braiding.} 
\label{fig_octahedron} 
\end{figure} 


\vspace{2mm}
\par \textbf{Special polynomials}
\vspace{2mm}

 \par We first need a generalisation of the Fock-Goncharov decomposition for flip operators. For this, we need the following generalization of the q-exponential $\exp_q^{<N}$ polynomial, which appeared in the work of Fadeev-Kashaev \cite{FadeevKashaevQDilog} and Baseilhac-Benedetti \cite{BaseilhacBenedettiLocRep}:




\begin{definition}
For $w\in \mathbb{C}^*$, we set
$$ \Phi_w(x):= \sum_{n=0}^{N-1} \frac{(-wx)^nq^{n(n-1)/2}}{(q^nw-q^{-n})\ldots (qw-q^{-1})} \in \mathbb{C}[x].$$
\end{definition}

\begin{lemma}\label{lemma_special_poly}
Consider $\mathcal{A}$ a $\mathbb{C}$-algebra and $x\in \mathcal{A}$ such that $x^N\in \mathbb{C}\setminus \{-1\}$. Choose $w\in \mathbb{C}^*$ such that $w^N(1+x^N) = 1$. Then one has:
\begin{enumerate}
\item The summand $e_n(x):=\frac{(-wx)^nq^{n(n-1)/2}}{(q^nw-q^{-n})\ldots (qw-q^{-1})}$ only depends on $n$ modulo $N$, therefore
$$ \Phi_w(x):= \sum_{n\in \mathbb{Z}/N\mathbb{Z}}e_n(x) \in \mathbb{C}[x].$$
\item One has $\Phi_w(q^{-2}x)=w(1+q^{-1}x)\Phi_w (x)$ and $\Phi_w(q^2x) = w^{-1}(1+qx)^{-1}\Phi_w(x)$.
\item If $y \in \mathcal{A}$ is such that $xy=q^2yx$, then $\mathrm{Ad}(\Phi_w(x)) y = w^{-1} y(1+qx)^{-1}$. 
\\ If $y \in \mathcal{A}$ is such that $xy=q^{-2}yx$, then $\mathrm{Ad}(\Phi_w(x)) y = w y(1+q^{-1}x)$.
\item If $w=1$, $\Phi_{w=1}(x)=\exp_q^{<N}\left(\frac{-x}{q-q^{-1}}\right)$.
\end{enumerate}
\end{lemma}

\begin{proof}
First, from the expression $e_n(x)= \frac{(-q^{-1}x)^n}{(1-w^{-1}q^{-2n}) \ldots (1-w^{-1}q^{-2})}$, we see that 
$$e_{n+N}(x)=\left( \frac{-x^N}{(1-w^{-1}q^{-2(n+N)})\ldots (1-w^{-1}q^{-2(n+1)})}\right) e_n(x) = \left( \frac{-x^N}{1-w^{-N}}\right) e_n(x) = e_n(x).$$
So the first assertion is proved. Then, using the equality $e_{n+1}(x)= \frac{-q^{-1}x}{1-w^{-1}q^{-2(n+1)}} e_n(x)$, which holds for any $n\in \mathbb{Z}/N\mathbb{Z}$, one finds that 
$$ e_{n+1}(q^{-2}x) = we_{n+1}(x) +w q^{-1}x e_n(x), $$
and the equality $\Phi_w(q^{-2}x)=w(1+q^{-1}x)\Phi_w (x)$ follows by summing over $n\in \mathbb{Z}/N\mathbb{Z}$. The equality $\Phi_w(q^2x) = w^{-1}(1+qx)^{-1}\Phi_w(x)$ is deduced by replacing $x$ by $q^2x$. The third assertion is an immediate consequence of the second one and the last assertion is immediate.

\end{proof}


\vspace{2mm}
\par \textbf{Generalization of the Fock-Goncharov decomposition}
\vspace{2mm}
\par Recall the two triangulations $\Delta$ and $\Delta'$ of the square $Q$ depicted in Figure \ref{fig_square}.
We  fix $\chi$ and $\chi'$ two compatible characters on the centers of $\mathcal{Z}_{\omega}(Q,\Delta)$ and  $\mathcal{Z}_{\omega}(Q,\Delta')$ and suppose that $\chi(X_5^N)\neq -1$. Write $x_i:= \chi(X_i^N), x_i':= \chi({X'}_i^N)$ and fix $w_0\in \mathbb{C}^*$ such that $w_0^N(1+x_5)=1$.

\begin{definition}\label{def_nu}
 The isomorphism 
  $$\nu^Q_{w_0} : \quotient{\mathcal{Z}_{\omega}(Q, \Delta')}{I_{\chi'}} \xrightarrow{\cong} \quotient{\mathcal{Z}_{\omega}(Q, \Delta)}{I_{\chi}}$$ is defined by: 
 \begin{align*}
 & \nu^Q_{w_0} \left(T_{\omega}^{\Delta'} (\beta^{(i)}_{++})\right)= T_{\omega}^{\Delta} (\beta^{(i)}_{++}) \mbox{, for }i\in \{1,2,3,4\}; \\
 & \nu^Q_{w_0} (X'_1) = w_0^{-1}X_1; \quad \nu^Q_{w_0}(X'_3) = w_0^{-1}X_3;  \\
 & \nu^Q_{w_0} (X'_2)= w_0 [X_2X_5]; \quad \nu^Q_{w_0}(X'_4)= w_0 [X_4X_5]; \\
 & \nu^Q_{w_0} (X'_5)= X_5^{-1}.
 \end{align*}
\end{definition}

\begin{lemma}\label{lemma_decomp_flip}
The isomorphism $\Phi_{(\Delta,\chi),  (\Delta', \chi')}^{Q} : \quotient{\widehat{\mathcal{Z}}_{\omega}(Q, \Delta')}{I_{\chi'}} \xrightarrow{\cong} \quotient{\widehat{\mathcal{Z}}_{\omega}(Q, \Delta)}{I_{\chi}}$ decomposes as:
$$ \Phi_{(\Delta,\chi),  (\Delta', \chi')}^{Q} = \nu_{w_0} \circ \mathrm{Ad}(\Phi_{w_0}({X'}_5^{-1})).$$
\end{lemma}

\begin{proof}
Like in the proof of Lemma \ref{lemma_FockGoncharov}, we show the equality $\Phi_{(\Delta,\chi),  (\Delta', \chi')}^{Q} (x)= \nu_{w_0} \circ \mathrm{Ad}(\Phi_{w_0}({X'}_5^{-1}))(x)$ for $x$ either a $X'_i, i=1, \ldots, 5$, or a $\operatorname{Tr}^{\Delta'}_{\omega}(\beta^{(i)}_{++}), i=1, \ldots, 4$. When $x=\operatorname{Tr}^{\Delta'}_{\omega}(\beta^{(i)}_{++})$, since $x$ commutes with $X_5'$, it is invariant under $\mathrm{Ad}(\Phi_{w_0}({X'}_5^{-1}))$ and one has $\Phi_{(\Delta,\chi),  (\Delta', \chi')}^{Q} (x)= \operatorname{Tr}_{\omega}^{Q}(\beta^{(i)}_{++}) = \nu_{w_0}^Q (x)$ by definition. When $x=X_i$, the equality $\Phi_{(\Delta,\chi),  (\Delta', \chi')}^{Q} (x)= \nu_{w_0} \circ \mathrm{Ad}(\Phi_{w_0}({X'}_5^{-1}))(x)$ follows from the third assertion of Lemma \ref{lemma_special_poly}. 



\end{proof}


\begin{remark}
Consider the classical shape parameter $\mathbf{z}=(z,z',z''):= (-x_5, (1+x_5)^{-1}, 1+x_5^{-1})$. The existence of a character $\chi$ compatible with $\chi'$ is equivalent to the assumption that $\mathbf{z}$ is indeed a shape parameter, \textit{i.e.} that $z\in \mathbb{C} \setminus \{0,1\}$. By Definition \ref{def_square_changecoordinates}, the character $\chi$ is then related  to $\chi'$ and $\mathbf{z}$ via the expressions: 
$$ x_5=-z, \quad x_1=z'x_1',\quad x_2=z''x_2',\quad x_3=z'x_3',\quad x_4=z''x_4'.$$
Figure \ref{fig_square_shape_parameters} illustrates how these formula can be interpreted by the operation of gluing a branched tetrahedron with shape parameters $\mathbf{z}$. The presence of a minus sign in $x_5=-z$ is purely conventional: the $z$, attached to the tetrahedron,  represents a bicross product whereas the shear bend coordinate $x_i$, parametrizing a pleated surface, are minus a bicross product.

 \begin{figure}[!h] 
\centerline{\includegraphics[width=8cm]{square_shape_parameters.eps} }
\caption{The figure illustrates how the flip operator $\Phi_{\Delta, \Delta'}$ at $\omega=+1$ can be interpreted as the action of gluing a tetrahedron with shape parameters.} 
\label{fig_square_shape_parameters}
\end{figure} 

\end{remark}

 \textbf{Decomposition of $\mathscr{R}$}
\vspace{2mm}
\par
Recall the triangulations $\Delta_2=\Delta^{(1)}, \ldots, \Delta^{(5)}$ of $\mathbb{D}_2$ depicted in Figure \ref{fig_flip_triangulations}.
\begin{definition}
\begin{enumerate}
\item
We consider two pairs $(V_1,V_2)$ and $(V_3,V_4)$ of $\mathcal{Z}_{\omega}(\mathbb{D}_1)$ simple modules which are $\tau$-compatible and let $\chi_1$ denote the character over the center of $\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta_2)$ defined by $V_1\otimes V_2$ (so the character associated to $V_3\otimes V_4$ is $\tau\cdot \chi_1$). The pairs $(V_1,V_2)$ and $(V_3,V_4)$ (or the character $\chi_1$, or a Kashaev braiding $R:V_1\otimes V_2 \rightarrow V_3\otimes V_4$) are said $4T$-\textit{triangulable} if there exist some characters $\chi_2, \ldots, \chi_5$ over the centers of $\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta^{(2)}), \ldots, \mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta^{(5)})$ such that $\chi_i$ and $\chi_{i+1}$ are compatible for $i=1, \ldots, 4$. 
 \item For $i=1, \ldots, 7$, write $x_i:= \chi_1(X_i^N)$.The \textit{classical shape parameters} associated to $\chi_1$ (or to the pair $(V_1,V_2)$ or to a Kashaev braiding $R:V_1\otimes V_2 \rightarrow V_3\otimes V_4$) are the shape parameters $\mathbf{z}_1, \ldots, \mathbf{z}_4 \in \mathbb{C}^{**}$ defined by 
 \begin{align*}
 & z_1 := -\chi_2(X_4^N) = -x_4^{-1} ; \\
 & z_2':= -\chi_3(X_5^N) = -x_5^{-1}(1+x_4)^{-1}; \\
 & z_3':= -\chi_4(X_3^N) = -x_3^{-1}(1+x_4)^{-1}; \\
 & z_4:= -\chi_5(X_4^N) = - x_4(1+x_5(1+x_4))^{-1}(1+x_3(1+x_4))^{-1}.
 \end{align*}
 
 Note that $\chi_1$ is $4T$ triangulable if and only if $z_1,z_2',z_3'$ and $z_4$  are in $\mathbb{C}\setminus \{0, 1\}$, \textit{i.e.} if and only if they indeed define classical shape parameters $\mathbf{z}_i$. In particular, being $4T$ triangulable is a generic condition. Note also that they satisfy the Thurston's equation:
 $$ z_1z_2z_3z_4=1.$$ 
 Figure \ref{fig_decomposition_octahedre} illustrates how the parameters $\tau \cdot \chi_1(X_i^N)$ can be computed using the classical shape parameters and the $x_i$ (compare with the formulas obtained from Proposition \ref{prop_calcul_R}).
 \end{enumerate}
 \end{definition}
 
 \begin{figure}[!h] 
\centerline{\includegraphics[width=\textwidth]{Decomposition_octahedre.eps} }
\caption{The top figure illustrates how the parameters $\tau \cdot \chi_1(X_i^N)$ can be computed using the classical shape parameters and the $x_i$. The bottom figure shows the four tetrahedra associated to a crossing and their branching. The edges in red are glued together inside the twisted octahedron giving the Thurston's equation $z_1z_2z_3z_4=1$.} 
\label{fig_decomposition_octahedre} 
\end{figure} 

 Fix $\chi_1$, a character over the center of $\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta_2)$ corresponding to $V_1\otimes V_2$ and consider 
 $$\mathscr{R}:= \tau^* \circ \Phi_{(\tau(\lambda), \chi_1\circ \tau), (\lambda, \chi_1)}^{\mathbb{D}_2} :  \quotient{ \widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \lambda_2)}{I_{\chi_1}} \xrightarrow{\cong}   \quotient{ \widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \lambda_2)}{I_{\tau \cdot \chi_1}}.$$
 
  By definition, a Kashaev braiding is an isomorphism $R : V_1\otimes V_2 \xrightarrow{\cong} V_3\otimes V_4$ such that $\mathscr{R}=\mathrm{Ad}(R)$.
The condition for $\chi_1$ to be $4T$ triangulable is equivalent to the possibility of decomposing the automorphism $\mathscr{R}$ as:
$$\mathscr{R}= \tau^* \circ \Phi^{\mathbb{D}_2}_{(\tau(\lambda_2), \tau\cdot \chi_1), (\lambda^{(5)}, \chi_5)} \circ  \Phi^{\mathbb{D}_2}_{(\lambda^{(5)}, \chi_5), (\lambda^{(4)}, \chi_4)} \circ  \Phi^{\mathbb{D}_2}_{(\lambda^{(4),\chi_4)}, (\lambda^{(3)}, \chi_3)} \circ  \Phi^{\mathbb{D}_2}_{(\lambda^{(3), \chi_3)},( \lambda^{(2)}, \chi_2)}  \circ  \Phi^{\mathbb{D}_2}_{(\lambda^{(2)}, \chi_2), (\lambda^{(1)}, \chi_1)}.$$

Fix $w_1', w_2'', w_3'', w_4' \in \mathbb{C}^*$ some $N$-th roots of $z_1', z_2'',z_3''$ and $z_4'$ respectively. 
By Lemma \ref{lemma_decomp_flip}, for $i\in \{1, \ldots, 4\}$, one has $ \Phi^{\mathbb{D}_2}_{(\lambda^{(i+1)}, \chi_{i+1}), (\lambda^{(i)}, \chi_i)}= \nu_{w_i^{'(')}} \circ \mathrm{Ad}(\Phi_{w_i^{'(')}}(X_{n_i}^{-1}))$.   Set
$$ \mathcal{K}:= \tau^* \circ  \Phi^{\mathbb{D}_2}_{(\tau(\lambda_2), \tau\cdot \chi_1), (\lambda^{(5)}, \chi_5)} \circ \nu_{w'_4} \circ \nu_{w''_3} \circ \nu_{w''_2} \circ \nu_{w'_1} : \quotient{ \widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \lambda_2)}{I_{\chi_1}} \xrightarrow{\cong}   \quotient{ \widehat{\mathcal{Z}}_{\omega}(\mathbb{D}_2, \lambda_2)}{I_{\tau \cdot \chi_1}}.$$
One has:
\begin{multline*}
\mathscr{R}= \mathcal{K} \circ \mathrm{Ad}(\Phi_{w'_4}\left( (\nu_{w''_3}\circ \nu_{w''_2}\circ\nu_{w'_1})^{-1}(X_4^{-1})\right)) \circ  \mathrm{Ad}(\Phi_{w''_3}\left(  (\nu_{w''_2}\circ\nu_{w'_1})^{-1}(X_3^{-1})\right)) \\
 \circ \mathrm{Ad}(\Phi_{w''_2}\left( \nu_{w'_1}^{-1}(X_5^{-1})\right)) \circ  \mathrm{Ad}(\Phi_{w'_1}\left( X_4^{-1}\right)).
 \end{multline*}

By a straightforward computation using Definition \ref{def_nu} (see Table \ref{table_k} below), we find:
\begin{equation}\label{eq_Rexpression1}
\mathscr{R}= \mathcal{K} \circ \mathrm{Ad}(\Phi_{w'_4}\left( {w'_1}^{2} w''_2 w''_3 [X_3X_4X_5]^{-1} \right)) \circ \mathrm{Ad}(\Phi_{w''_3}\left( w'_1 [X_3X_4]^{-1} \right))\circ \mathrm{Ad}(\Phi_{w''_2}\left( w'_1[X_4X_5]^{-1} \right))
\circ \mathrm{Ad}(\Phi_{w'_1}\left( X_4^{-1} \right)).
\end{equation}

The isomorphism $\mathcal{K}$ is described by Table \ref{table_k}:
\begin{table}[h!]
   
     \centering
  \begin{adjustbox}{max width=\textwidth}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$\mathcal{Z}_{\omega}(\mathbb{D}_2, \lambda^{(1)})$ & $\mathcal{Z}_{\omega}(\mathbb{D}_2, \lambda^{(2)})$ & $\mathcal{Z}_{\omega}(\mathbb{D}_2, \lambda^{(3)})$ & $\mathcal{Z}_{\omega}(\mathbb{D}_2, \lambda^{(4)})$ & $\mathcal{Z}_{\omega}(\mathbb{D}_2, \lambda^{(5)})$ & $\mathcal{Z}_{\omega}(\mathbb{D}_2, \lambda^{(1)})$ \\
\hline
$X_1$ & $X_1$ & $X_1$ & $w''_3[X_1X_3]$ & $w''_3w'_4[X_1X_3X_4]$ & $w''_3w'_4[X_1X_2X_4]$ \\
\hline
$X_2$ & $(w'_1)^{-1}X_2$ & $(w'_1w''_2)^{-1}X_2$ & $(w'_1w''_2w''_3)^{-1}X_2$ & $(w'_1w''_2w''_3w'_4)^{-1}X_2$ & $(w'_1w''_2w''_3w'_4)^{-1}X_5$ \\
\hline
$X_3$ & $w'_1[X_3X_4]$ & $(w'_1w''_2)[X_3X_4X_5]$ & $(w'_1w''_2w''_3)[X_4X_5]$ & $(w'_1w''_2w''_3w'_4)X_5$ & $(w'_1w''_2w''_3w'_4)X_6$ \\
\hline
$X_4$ & $X_4^{-1}$ & $(w''_2)^{-1}[X_4X_5]^{-1}$ & $(w''_2w''_3)^{-1}[X_3X_4X_5]^{-1}$ & $(w''_2w''_3{w'_4}^2)^{-1}[X_3X_4X_5]^{-1}$ & $(w''_2w''_3{w'_4}^2)^{-1}[X_2X_4X_6]^{-1}$ \\
\hline
$X_5$ & $w'_1[X_4X_5]$ & $(w'_1w''_2)X_4$ & $(w'_1w''_2w''_3)[X_3X_4]$ & $(w'_1w''_2w''_3w'_4)X_3$ & $(w'_1w''_2w''_3w'_4)X_2$ \\
\hline
$X_6$ & $(w'_1)^{-1}X_6$ & $(w'_1w''_2)^{-1}X_6$ &$(w'_1w''_2w''_3)^{-1}X_6$ &$(w'_1w''_2w''_3w'_4)^{-1}X_6$ & $(w'_1w''_2w''_3w'_4)^{-1}X_3$ \\
\hline
$X_7$ & $X_7$ & $w''_2[X_5X_7]$ & $w''_2[X_5X_7]$ & $(w''_2w'_4)[X_4X_5X_7]$ & $(w''_2w'_4)[X_4X_6X_7]$ \\
\hline
$H_{p_1}=[Z_2Z_3]$ & $[Z_2Z_3Z_4]$ & $[Z_2Z_3Z_4Z_5]$ & $[Z_2Z_4Z_5]$ & $[Z_2Z_5]$ & $H_{p_2}$ \\
\hline
$H_{p_2}=[Z_5Z_6]$ & $[Z_4Z_5Z_6]$ & $[Z_4Z_6]$ & $[Z_3Z_4Z_6]$ & $[Z_3Z_6]$ &$H_{p_1}$ \\
\hline

\end{tabular}
\end{adjustbox}
 \label{table_k}
 \caption{Computation of $\mathcal{K}$}
\end{table}
 In this table, for $1\leq i \leq 4$, one passes from the $i$-th column to the $i+1$-th column by composing with $\nu_{w_i^{'(')}}$ and one passes from the $5$-th column to the last one by composing with $ \tau^* \circ  \Phi^{\mathbb{D}_2}_{(\tau(\lambda_2), \tau\cdot \chi_1), (\lambda^{(5)}, \chi_5)}$. Thus one passes from the first column to the last one by composing with $\mathcal{K}$. In order to reorder the parameters used to define Kashaev's braidings, we now introduce a class of simple $\mathcal{Z}_{\omega}(\mathbb{D}_1)$-modules named standard.
 
 \begin{definition}
 \begin{enumerate}
 \item Let $V=\mathbb{C}^N$ and $\{ v_i, i \in \mathbb{Z}/N\mathbb{Z} \}$ be its canonical basis. We define the operators $D, A \in \operatorname{End}{V}$ by the formulas:
 $$ D v_i= q^i v_i, \quad Av_i=v_{i-1}, \quad \forall i \in \mathbb{Z}/N\mathbb{Z}.$$
 They satisfy $AD=qDA$ and $A^N=D^N=id$. 
 \item  Let $(\widetilde{x_1}, \lambda, h_p, h_{\partial}) \in (\mathbb{C}^*)^4$ and define a simple $\mathcal{Z}_{\omega}(\mathbb{D}_1, \Delta_1)$-module $W(\widetilde{x_1}, \lambda, h_p, h_{\partial})$ by the formulas: 
$$ \rho(H_p)= h_p id, \quad \rho(H_{\partial})=h_{\partial} id, \quad \rho(X_1)= \widetilde{x_1} A, \quad \rho([Z_1Z_3Z_4])= \lambda^{-1}D.$$
We call \textit{standard} such a simple $\mathcal{Z}_{\omega}(\mathbb{D}_1, \Delta_1)$-module. By Lemma \ref{Lemma_2}, every simple $\mathcal{Z}_{\omega}(\mathbb{D}_1, \Delta_1)$-module is isomorphic to a standard one and two standards are isomorphic if and only if they have the same parameters except for $\widetilde{x_1}$ which can be multiplied by a $q^i$. 
\item Let $(V_1,V_2)$ be some $4T$ triangulable standard modules of the form $V_i=W(x_1^{(i)}, \lambda_i, h_p^{(i)}, h_{\partial}^{(i)})$. Write
$$ \widetilde{x_4}:= \frac{h_{\partial}^{(1)} \widetilde{x_1}^{(2)}}{h_p^{(1)} \widetilde{x_1}^{(1)}}, \quad \widetilde{x_3}:=\frac{h_p^{(1)}}{h_{\partial}^{(1)} \lambda_1^2}, \quad \widetilde{x_5}:= \lambda_2^2 h_p^{(2)}h_{\partial}^{(2)}.$$
Note that for $i=3,4,5$, one has $\widetilde{x_i}^N=x_i$. A \textit{system of quantum shape parameters for} $(V_1,V_2)$ is the data of quantum shape parameters $\mathbf{w}_1, \ldots, \mathbf{w}_4$ such that $w_1',w_2'',w_3'',w_4'$ are $N$-th roots of $z_1', z_2'',z_3'',z_4'$ and such that
\begin{align}
\label{E1} & w_1=-\widetilde{x_4}; \\
\label{E2} & w_2'= -w'_1\widetilde{x_4}^{-1} \widetilde{x_5}^{-1}; \\
\label{E3} & w_3' = -w'_1 \widetilde{x_4}^{-1} \widetilde{x_3}^{-1}; \\
\label{E4} &w_4 = -{w'_1}^2 w''_2 w''_3 \widetilde{x_3}^{-1}\widetilde{x_4}^{-1}\widetilde{x_5}^{-1}.
\end{align}
Note that $w_1,w_2',w_3',w_4$ are $N$-th roots of $z_1,z_2',z_3',z_4$ so the $\mathbf{w}_i$ define some q-flattenings of the $\mathbf{z}_i$. Note also that they satisfy the following quantum Thurston equation:
 $$ w_1w_2w_3w_4= q^{-2}.$$

\end{enumerate}
\end{definition}
 
 When $(V_1,V_2)$ are $4T$ triangulable standard representations with quantum shape parameters $\mathbf{w}_i$, Equation \eqref{eq_Rexpression1} takes the form:
 
 \begin{equation}\label{eq_Rexpression2}
\mathscr{R}= \mathcal{K} \circ \mathrm{Ad} \left[ \Phi_{w'_4}( -w_4[D^{-2}A] \otimes [A^{-1}D^2]) \Phi_{w''_3}(-w_3'[D^{-2}A] \otimes A^{-1})\Phi_{w''_2}(-w_2'A \otimes [A^{-1}D^2]) \Phi_{w'_1}( -w_1A \otimes A^{-1}) \right].
\end{equation}
 

\vspace{2mm}
 \textbf{Flattening parameters and intertwiners for $\mathcal{K}$ }
\vspace{2mm}
\par Fix $(V_1,V_2)$ and $(V_3, V_4)$ some $4T$ triangulable $\tau$ compatible pairs of standard modules and denote by $\pi_i$ the character induced by $V_i=W(\widetilde{x_1}^{(i)}, \lambda_i, h_p^{(i)}, h_{\partial}^{(i)})$ on the center of $\mathcal{Z}_{\omega}(\mathbb{D}_1)$.  In view of the decomposition \eqref{eq_Rexpression2} and in order to provide an explicit formula for a Kashaev's braiding $R:V_1\otimes V_2\to V_3 \otimes V_4$, it remains to find an intertwiner for $\mathcal{K}$.  We will make the (non restrictive) assumption that
\begin{equation}\label{eq_convention}
h_{\partial}^{(1)}=h_{\partial}^{(4)} \quad \mbox{and} \quad h_{\partial}^{(2)} = h_{\partial}^{(3)}.
\end{equation}
Note that the assumption of being $\tau$-compatible implies $h_p^{(2)}=h_p^{(3)}$ and $h_p^{(1)}=h_p^{(4)}$. We choose a system of quantum shape parameters $\mathbf{w}_i$ and fix a square root $(w'_1w''_2w''_3w'_4)^{1/2}$ of $w'_1w''_2w''_3w'_4$,  such that
 \begin{align}
 \label{E5}& w''_3w'_4=  \frac{\widetilde{x_1}^{(1)}}{\widetilde{x_1}^{(4)}} \lambda_3^{-2} {h_{\partial}^{(3)}}^{-1} \\
 \label{E6}& w''_2w'_4=  \frac{\widetilde{x_1}^{(3)}}{\widetilde{x_1}^{(2)}} \lambda_4^{-2}  \\
 \label{E7}& \frac{\lambda_4}{\lambda_1} = \frac{\lambda_2}{\lambda_3} = (w'_1w''_2w''_3w'_4)^{1/2}, 
 \end{align}
 
 \begin{remark}
By inspection of Equations \eqref{E5}, \eqref{E6}, \eqref{E7}, we see that once a $4T$-triangulable pair $(V_1,V_2)$ of standard representations is fixed, there is a $1$-to-$1$ correspondance between the set of $\tau$-compatible standard pairs $(V_3,V_4)$ satisfying \eqref{eq_convention} and the set of systems of quantum shape parameters $\mathbf{w}_1, \ldots, \mathbf{w}_4$, endowed with a square root $(w'_1w''_2w''_3w'_4)^{1/2}$, which satisfy Equations \eqref{E1}, \eqref{E2}, \eqref{E3}, \eqref{E4}, \eqref{E5}, \eqref{E6} and \eqref{E7}.
\end{remark}

The intertwiners for $\mathcal{K}$ will have a form similar to the factor $\tau \circ q^{-H\otimes G/2}$ in Drinfeld's braidings. In particular, they will depend on some additional parameters that will play a role similar to the information added by lifting a $U_q\mathfrak{sl}_2$ diagonal representation to a representation of the unrolled quantum group as in \cite{CGP_unrolledQG}. Their geometric role seems very similar to Neumann's notion of flattening in hyperbolic geometry.

\begin{definition}
\begin{enumerate}
\item A pair $(\alpha, \beta)\in \mathbb{C}^2$ is a pair of \textit{flattening parameters} if they verify
 \begin{equation}\label{E8}
 q^{\alpha} = \frac{\lambda_1^2}{w''_2w'_4}, \quad \mbox{ and }\quad q^{\beta} = \frac{w''_3w'_4}{\mu_2^2}.
 \end{equation}
 Here, if $q=\exp(2i\pi k /N)$, with $k$ prime to $2N$, we used the convention $q^{\alpha}:= \exp(2i\pi k \alpha/N)$.  A system of $4T$ \textit{octahedron parameters} for $(V_1, V_2)$ is the data of a system of quantum shape parameters $\mathbf{w_1}, \ldots, \mathbf{w_4}$, of a square root $(w'_1w''_2w''_3w'_4)^{1/2}$ and of a pair $(\alpha, \beta)$ of flattening parameters which satisfy Equations \eqref{E1}, \ldots, \eqref{E8}. 
 \item The associated diagonal operator $D_{\alpha, \beta} : V_1\otimes V_2 \rightarrow V_4\otimes V_3$ is defined by 
 $$ D_{\alpha, \beta} v_i\otimes v_j := q^{\frac{(\alpha - 2i)(\beta - 2j)}{2}} v_i \otimes v_j, \quad \forall i, j \in \mathbb{Z}/N\mathbb{Z}. $$
\end{enumerate}
\end{definition}


\begin{lemma}
Let $\widehat{\mathcal{K}} : \mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2} \xrightarrow{\cong} \mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}$ be the isomorphism defined by 
$$ \widehat{K} (Z^{\mathbf{k}} \otimes Z^{\mathbf{k}'}) := w^{(\mathbf{k}, \mathbf{k}')} \left( Z^{\mathbf{k}'} [Z_1Z_2Z_4]^{k_1-k_4} \otimes [Z_1Z_3Z_4]^{k_4' - k_1'} Z^{\mathbf{k}}\right), $$
where 
$$ w^{(\mathbf{k}, \mathbf{k}')} := (w'_1w''_2w''_3w'_4)^{(k_3-k_2+k'_2-k'_3)/2}(w''_3w'_4)^{(k_1-k_4)/2} (w''_2w'_4)^{(k_4'-k_1')/2}.$$
Note that, since $\mathbf{k}, \mathbf{k}'$ are balanced, the integers $k_1-k_4$ and $k'_4-k'_1$ are even. Then $\widehat{\mathcal{K}}$ is an algebra isomorphism which passes to the quotient to an isomorphism (still denoted by the same letter) 
$$\widehat{\mathcal{K}} : \quotient{\mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}}{(I_{\pi_1}\otimes 1 +1\otimes I_{\pi_2})} \xrightarrow{\cong} \quotient{\mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}}{(I_{\pi_3}\otimes 1 +1\otimes I_{\pi_4})}$$
 such that the following diagram commutes: 
$$ \begin{tikzcd}
\quotient{\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta_2)}{I_{\chi_1}} \arrow[r, "\cong"', "\mathcal{K}"] \arrow[d, "\cong"', "i"] &
\quotient{\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta_2)}{I_{\chi_2}}  \arrow[d, "\cong"', "i"] \\
\quotient{\mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}}{(I_{\pi_1}\otimes 1 +1\otimes I_{\pi_2})} \arrow[r, "\cong"', "\widehat{\mathcal{K}}"] &
 \quotient{\mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}}{(I_{\pi_3}\otimes 1 +1\otimes I_{\pi_4})}
 \end{tikzcd}
 $$
 where the isomorphisms $i$ are induced by the gluing morphism $\mathcal{Z}_{\omega}(\mathbb{D}_2, \Delta_2) \hookrightarrow \mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}$.


\end{lemma}

\begin{proof} 
The fact that  $\widehat{\mathcal{K}} : \mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2} \xrightarrow{\cong} \mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}$ is an algebra isomorphism is an immediate consequence of the definition and of the fact that $w^{(\mathbf{k}+\mathbf{h}, \mathbf{k}'+\mathbf{h}')}=w^{(\mathbf{k}, \mathbf{h})}w^{(\mathbf{k}',\mathbf{h}')}$. To prove the inclusion 
$$ \widehat{\mathcal{K}}( \mathcal{I}_{\pi_1}\otimes 1 + 1\otimes \mathcal{I}_{\pi_2}) \subset (\mathcal{I}_{\pi_3} \otimes 1 +1\otimes \mathcal{I}_{\pi_4}), $$
we need to show that 
$$ \pi_1(x) = (\pi_3 \otimes \pi_4)(\widehat{\mathcal{K}} (x\otimes 1)) \quad \mbox{and} \quad \pi_2(x)=(\pi_3\otimes \pi_4)(\widehat{\mathcal{K}}(1\otimes x)), $$
for each generator $x\in \{X_1^{\pm N}, K^{\pm N/2}, H_p^{\pm 1}, H_{\partial}^{\pm 1} \}$. For $x=H_p^{\pm 1}$, this follows from the equalities $h_p^{(1)}=h_p^{(4)}$ and $h_p^{(2)}=h_p^{(3)}$. For $x=H_p^{\pm 1}$, this follows from the convention  \eqref{eq_convention} that $h_{\partial}^{(1)}=h_{\partial}^{(4)}$ and $h_{\partial}^{(2)} = h_{\partial}^{(3)}$.  For $x= K^{\pm N/2}$, this follows from the equalities $\left( \frac{\lambda_4}{\lambda_1} \right)^N = \left( \frac{\lambda_2}{\lambda_3} \right)^N = (w'_1w''_2w''_3w'_4)^{N/2}$ given by Equation \eqref{E7}. For $x=X_1^{\pm N}$, this follows from Equations \eqref{E5}, \eqref{E6} by elevating both sides of the equations at the power $N$. 

\end{proof}


\begin{lemma}\label{lemma_D}
Let $\tau: V_4\otimes V_3 \to V_4\otimes V_3$ be the flip morphism $\tau(x\otimes y)=y\otimes x$. Then the operator $\tau \circ D_{\alpha, \beta} : V_1\otimes V_2 \rightarrow V_3\otimes V_4$ is an intertwiner for $\widehat{K}$, \textit{i.e.} $\mathrm{Ad}(\tau \circ D_{\alpha, \beta}) = \widehat{K}$.
\end{lemma}

\begin{proof} Let $\sigma : \quotient{\mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}}{(I_{\pi_3}\otimes 1 +1\otimes I_{\pi_4})} \xrightarrow{\cong} \quotient{\mathcal{Z}_{\omega}(\mathbb{D}_1)^{\otimes 2}}{(I_{\pi_4}\otimes 1 +1\otimes I_{\pi_3})}$ be the flip $\sigma(x\otimes y):= y\otimes x$, so that $\mathrm{Ad}(\tau)=\sigma$. We need to prove the equality 
\begin{equation}\label{eq_alacon}
 \sigma \circ \widehat{\mathcal{K}} (X) = \mathrm{Ad}(D_{\alpha, \beta}) (X), 
 \end{equation}
for $X$ a generator of the form $a\otimes 1$ or $1\otimes a$ and $a\in \{X_1^{\pm 1}, [Z_1Z_2Z_4]^{\pm 1}, H_p^{\pm 1}, H_{\partial}^{\pm 1}\}$. When $a=H_p$, this follows from the equalities $h_p^{(1)}=h_p^{(4)}$ and $h_p^{(2)}=h_p^{(3)}$. When $a=H_{\partial}$, this follows from the convention we made in Equation \eqref{eq_convention} that $h_{\partial}^{(1)}=h_{\partial}^{(4)}$ and $h_{\partial}^{(2)} = h_{\partial}^{(3)}$. When $a=K^{1/2}$, on the one hand, we have
$$ \sigma \circ \widehat{\mathcal{K}} (K^{1/2}\otimes 1)= (w'_1w''_2w''_3w'_4)^{-1/2} (K^{1/2}\otimes 1) \quad \mbox{and} \quad \sigma \circ \widehat{\mathcal{K}}(1\otimes K^{1/2})= (w'_1w''_2w''_3w'_4)^{1/2}(1\otimes K^{1/2}).$$
On the other hand
$$ \mathrm{Ad}(D_{\alpha, \beta}) (K^{1/2}\otimes 1)= \frac{\lambda_1}{\lambda_4} (K^{1/2}\otimes 1) \quad \mbox{and}\quad \mathrm{Ad}(D_{\alpha, \beta})(1\otimes K^{1/2}) = \frac{\lambda_2}{\lambda_3} (1\otimes K^{1/2}),$$
so Equation \eqref{eq_alacon} follows from Equation \eqref{E7} in that case. When $a=X_1$, write $d_{i,j}:=  q^{\frac{(\alpha - 2i)(\beta - 2j)}{2}}$, so that $D_{\alpha, \beta}\cdot v_i\otimes v_j = d_{i,j}v_i \otimes v_j$. On the one hand, we compute
$$ \sigma \circ \widehat{\mathcal{K}} (X_1\otimes 1) \cdot (v_i\otimes v_j) = w''_3w'_4 \widetilde{x}_1^{(1)} \mu_2^{-2} q^{-2j} (v_{i-1}\otimes v_j), \quad
\sigma \circ \widehat{\mathcal{K}} (1\otimes X_1) \cdot (v_i\otimes v_j) = (w''_2w'_4)^{-1} \widetilde{x}_1^{(2)} \lambda_1^{2} q^{-2i} (v_{i}\otimes v_{j-1}).$$
On the other hand
$$\mathrm{Ad}(D_{\alpha, \beta})(X_1\otimes 1) \cdot(v_i \otimes v_j) = \widetilde{x}_1^{(1)} \frac{d_{i-1, j}}{d_{i,j}} (v_{i-1}\otimes v_j), \quad
 \mathrm{Ad}(D_{\alpha, \beta})(1\otimes X_1) \cdot(v_i \otimes v_j) = \widetilde{x}_1^{(2)} \frac{d_{i, j-1}}{d_{i,j}} (v_{i}\otimes v_{j-1}).$$
 So Equation \eqref{eq_alacon} follows from Equations \eqref{E5} and \eqref{E6} in that case.






\end{proof}

\begin{theorem}\label{theorem_ClosedFormula}
Let $(V_1,V_2)$ be a pair of $4T$ triangulable standard representations and $(\mathbf{w}_1, \mathbf{w}_2, \mathbf{w}_3, \mathbf{w}_4,$ $ (w'_1w''_2w''_3w'_4)^{1/2}, \alpha, \beta)$ a system of  octahedron parameters with associated pair $(V_3,V_4)$. Then the isomorphism $R: V_1\otimes V_2 \to V_3\otimes V_4$ defined by
$$ R= \tau \circ D_{\alpha, \beta} \circ  \Phi_{w'_4}( -w_4[D^{-2}A] \otimes [A^{-1}D^2]) \Phi_{w''_3}(-w_3'[D^{-2}A] \otimes A^{-1})\Phi_{w''_2}(-w_2'A \otimes [A^{-1}D^2]) \Phi_{w'_1}( -w_1A \otimes A^{-1}) $$ 
is an intertwiner for $\mathscr{R}$ (satisfies $\mathrm{Ad}(R)=\mathscr{R}$), so if $d^{N^2}= \operatorname{det}(R)$, then $\frac{1}{d}R$ is a Kashaev braiding.
\end{theorem}

\begin{proof}
This follows from the decomposition \eqref{eq_Rexpression2} and Lemma \ref{lemma_D}.
\end{proof}




 \section{Concluding remarks concerning $\operatorname{SL}_2$-links invariants}\label{sec_LinksInv}
 
  The original purpose of this paper was to define and study some new links invariants using quantum Teichm\"uller theory. This goal will be achieved in a separate publication \cite{KojuBaseilhac_ToAppear}, in collaboration with S.Baseilhac. In this section, we briefly sketch the definition of these invariants, using the results of the present paper, and state some open questions/conjectures. The section serves as a motivation for the work done in this paper and no original result is claimed here. 
 During the writing of this paper, Blanchet, Geer, Patureau-Mirand and Reshetikhin independently defined in \cite{BGPR_Biquandle} a family of links invariants which are very similar to the ones described in this section. Though their construction is completely different and only relies on quantum groups techniques, the main algebraic inputs (the Kashaev-Reshetikhin braidings and the renormalized trace) are the same. However the authors of  \cite{BGPR_Biquandle} did not provided explicit formulas for the Kashaev-Reshetikhin braidings and, as a result, did not compute their invariants explicitly on non-trivial (non-abelian) exemples, whereas our formula in Theorem \ref{theorem_ClosedFormula} is quite explicit.
 
 \subsection{Sketch of construction of the $\operatorname{SL}_2$-links invariants}
 
Let $L\subset S^3$ be a framed link and consider the affine variety $\mathcal{X}_{\operatorname{SL}_2,q}(S^3\setminus L)$, defined in the same manner than $\mathcal{X}_{\operatorname{PSL}_2,q}(S^3\setminus L)$ at the beginning of Section \ref{sec_ClosedFormula} by replacing $\operatorname{PSL}_2$ by $\operatorname{SL}_2$ and considering pairs $(\rho, l)$ where $\operatorname{Tr}(\rho(\gamma))=q^{h([\gamma])}+q^{-h([\gamma])}$. Our purpose is to define invariants $\left< L, [\rho, h]\right>_{\omega} \in \quotient{\mathbb{C}}{\mu_{N^2}}$ for $[\rho,h]\in  \mathcal{X}_{\operatorname{SL}_2,q}(S^3\setminus L)$ and valued in the set of complex numbers up to multiplication by a $N^2$-th root of unity. Let $\mathcal{C}$ denote (a small squeleton of) the tensor category of weight $\mathcal{O}_q[D(B)]$-modules and $\mathrm{Proj}(\mathcal{C})$ is full subcategory of projective objects. We endow $\mathcal{C}$ with the structure of a pivotal category by writing $V^*=\operatorname{Hom}(V, \mathbb{C})$ for the dual of an object $V \in \mathcal{C}$ and setting: 
\begin{align*}
& \overleftarrow{coev}_V : \mathbb{C} \to V\otimes V^*,  1\mapsto \sum_i v_i\otimes v_i^*; \\
& \overleftarrow{ev}_V: V^*\otimes V \to \mathbb{C},  \phi\otimes v \mapsto \phi(v); \\
& \overrightarrow{coev}_V : \mathbb{C} \to V^* \otimes V,  1\mapsto \sum_i v_i^* \otimes (K^{1/2}L^{-1/2})^{N-1}v_i; \\
& \overrightarrow{ev}_V: V\otimes V^* \to \mathbb{C},  v\otimes \phi \mapsto \phi\left( (K^{1/2}L^{-1/2})^{N-1} v\right).
\end{align*}
 The $0$-th Hochschield-Mitchell homological group $\mathrm{HH}_0(\mathcal{D})$ of a (small) linear category $\mathcal{D}$ is the coend
 $$ \mathrm{HH}_0(\mathcal{D}) := \int^{x\in \mathcal{D}} \operatorname{Hom}_{\mathcal{D}}(x,x) =\quotient{ \oplus_{x \in \mathcal{C}} \operatorname{Hom}_{\mathcal{D}}(x,x)}{ (uv-vu)} \in \mathrm{Vect}.$$
 A trace on $\mathcal{D}$ is a linear map $\mathrm{HH}_0(\mathcal{D})\rightarrow \mathbb{C}$. When $\mathcal{C}$ is a pivotal tensor category and $\mathcal{D}=\mathrm{Proj}(\mathcal{C})$ its full subcategory of projective objects, there is a canonical trace $\operatorname{Tr}^{R} : \mathrm{HH}_0(\mathrm{Proj}(\mathcal{C}))\to \mathbb{C}$, named the \textit{renormalized trace} introduced by Geer, Patureau-Mirand and Turaev in \cite{GeerPatureauTuraev_Trace} and further generalized and studied in \cite{GeerPatureauVirelizier_Trace}. 
 \vspace{2mm}
 \par Let $\mathrm{Proj}(\mathcal{C})^{(N^2)}$ be the category obtained from $\mathrm{Proj}(\mathcal{C})$ by considering the morphisms only up to multiplication by a $N^2$ root of unity. By Theorem \ref{theorem_functor2}, for $n\geq 2$, we have a functor $L_{\omega}^{\operatorname{SL}_2} : \mathcal{B}^{\operatorname{SL}_2}_n \rightarrow \mathrm{Proj}(\mathcal{C})^{(N^2)}$. By passing to homology and post-composing with the renormalized trace, we get a map
 $$ \mathcal{J}_{\omega, n} : \mathrm{HH}_0(\mathcal{B}_n^{\operatorname{SL}_2}) \xrightarrow{ L_{\omega}^{\operatorname{SL}_2}} \mathrm{HH}_0(\mathrm{Proj}(\mathcal{C})^{(N^2)}) \xrightarrow{\operatorname{Tr}^R} \quotient{\mathbb{C}}{\mu_{N^2}}.$$ 
 Using the geometric description of \cite[Theorem $1.3$]{KojuQuesneyClassicalShadows} of a character over the center of the stated skein algebra, one can identify an object of $\mathcal{B}_n^{\operatorname{SL}_2}$ with a point in the relative $\operatorname{SL}_2$-character variety of $\mathbb{D}_n$, as defined in \cite{KojuTriangularCharVar}, together with punctures and boundary invariants $h_{p_i}, h_{\partial_i}$. An endomorphism in $\mathcal{B}_n^{\operatorname{SL}_2}$ can then be identified with a braid equipped with an $\operatorname{SL}_2$-representation of its fundamental group and the elements $h_{p_i}, h_{\partial_i}$. Using these remark, we obtain a map $\iota_n: \mathrm{HH}_0(\mathcal{B}_n^{\operatorname{SL}_2}) \rightarrow \mathcal{X}_{\operatorname{SL}_2,q}(S^3\setminus L)$ by closing the braid to a framed link and using the $h_{p_i}, h_{\partial_i}$ to define the q-flattening. The precise definition of $\iota_n$ will appear elsewhere \cite{KojuBaseilhac_ToAppear}. 
 For a triple $(L,[\rho, h])=\iota_n(\beta)$, we then define $\left<L,[\rho,h]\right>_{\omega}:= \mathcal{J}_{\omega, n}(\beta)$. To prove that it is independent on $\beta$, we still need to verify the invariance under some Markov stabilisation moves as in \cite{GeerPatureau_GLinks}. We leave this verification for latter work \cite{KojuBaseilhac_ToAppear}.
 
 
 
 \subsection{Some open problems regarding the links invariants}
 
 
 We list a few open problems concerning the links invariants sketched in the previous section. 
 
 \par $(1)$ Consider a character $\chi \in \mathcal{B}_n^{\operatorname{SL}_2}$ with $L_{\omega}^{\operatorname{SL}_2}(\beta) = V_1\otimes \ldots \otimes V_n$ and a braid $\beta \in \operatorname{End}_{\mathcal{B}_n^{\operatorname{SL}_2}}(\chi)$. Decompose the braid $\beta$ as a composition of half-twists operators 
 $$\beta : V_1\otimes \ldots \otimes V_n \xrightarrow{T_1} V_1^{(2)} \otimes \ldots \otimes V_n^{(2)} \xrightarrow{T_2} \ldots \xrightarrow{T_k} V_1^{(k+1)}\otimes \ldots \otimes V_n^{(k+1)} = V_1\otimes \ldots \otimes V_n, $$
 where $T_i= id^{\otimes n_i}\otimes R_i^{\epsilon_i} \otimes id^{\otimes n-n_i-2}$ with $R_i$ a Kashaev braiding with octahedra parameters $o_i=(\mathbf{w}_j^{(i)}, $
 \\  $({w_1'}^{(i)} {w_2''}^{(i)} {w_3''}^{(i)} {w_4'}^{(i)})^{1/2},  \alpha_i, \beta_i)$. 
 The above decomposition induces a sliced planar diagram $D_{\beta}$ representing $\beta$. Let $(L, [\rho, h])=\iota_n(\beta)$ be the decorated framed links obtained by closing the braid and $D$ the diagram of $L$ obtained from $D_{\beta}$. 
 
 
 The equalities $V_i^{(k+1)}=V_i$ for $i=1, \ldots, n$, induce a set of equations for the octahedron parameters. 
 \begin{problem}\label{problem1}
 Relate these equations to the quantum Thurston equations defining $\mathcal{G}_q(\mathcal{T}^{4T}(D))$.
 \end{problem}
 
 It is natural to expect that the quantum shape parameters $\{\mathbf{w}_j^{(i)}\}$ satisfy the quantum Thurston equations and thus define a point in $\mathcal{G}_q(\mathcal{T}^{4T}(D))$ whose image by $\mathcal{D}_q$ would be the $\operatorname{PSL}_2$ reduction of $[\rho,h]$. The role of the additional square roots $({w_1'}^{(i)} {w_2''}^{(i)} {w_3''}^{(i)} {w_4'}^{(i)})^{1/2}$
seems to encode the lift to a $\operatorname{SL}_2$ representation. As their name suggests, the flattening parameters $(\alpha_i,\beta_i)$ seems to encode a classical flattening, though the invariant  $\left<L,[\rho,h]\right>_{\omega}$ should not depend on this additional structure.

\vspace{2mm}
\par $(2)$ In the definition of Kashaev braidings, we have impose them to have determinant $1$, that is we have renormalized our braiding operators $R$ to $\frac{1}{d}R$ with $d^{N^2}=\operatorname{det}(R)$. This condition permitted to reduce the projective ambiguity in the definition of the braid groupoid representation to a multiplication by a $N^2$-th root of unity, and thus we obtained invariants in $\mathbb{C}$ which are only defined up to $N^2$-th root of unity. A similar strategy is used by Baseilhac and Benedetti to define the QHI. However, the condition to have determinant $1$ is purely technical and has no deep geometric origin. In the case of the non-semi simple links invariants of \cite{GeerPatureau_LinksInv}, which are supposed to be the particular case of our invariants when $\rho: \pi_1(M)\to \mathbb{C}^* \subset \operatorname{SL}_2(\mathbb{C})$ is a diagonal representation, the Drinfel'd braiding operators do not have determinant $1$ and the invariants are unambiguously defined in $\mathbb{C}$. The additional information used to define the braidings are the $h$ parameters encoding the lift of a $U_q\mathfrak{sl}_2$ representation to a representation of the unrolled quantum group. They are related to our flattening parameters by $(\alpha, \beta)=(h, -h)$ and the condition imposed to a Drinfel'd braiding $R : V_1\otimes V_2 \to V_2\otimes V_1$ is to switch the two $h$ parameters (to lift to an equivariant map for the unrolled quantum group). 

\begin{problem}\label{problem2}
Find a global system of equations for the flattening parameters $(\alpha_i, \beta_i)$ in such a way that the unrenormalized Kashaev braidings (the $R$ in Theorem \ref{theorem_ClosedFormula} and not the $\frac{1}{d}R$) induce a linear representation $L_{\omega}^{\operatorname{SL}_2}: \mathcal{B}_n^{\operatorname{SL}_2} \to \mathrm{Proj}(\mathcal{C})$ with no projective ambiguity.
\end{problem}
 
 We would get this way links invariants valued in $\mathbb{C}$ instead of $\quotient{\mathbb{C}}{\mu_{N^2}}$.

\vspace{2mm}
\par $(3)$ Our links invariants seems very close to the Baseilhac-Benedetti QHI. Given $\Delta$ a triangulation of $\mathbf{\Sigma}$, by fixing a simple $\mathcal{Z}_{\omega}(\mathbb{T})$ module $W_i$ for each face $\mathbb{T}_i \in F(\Delta)$, we get a representation
$$ r : \mathcal{Z}_{\omega}(\mathbf{\Sigma}, \Delta) \xrightarrow{i^{\Delta}} \otimes_{\mathbb{T}_i \in F(\Delta)}\mathcal{Z}_{\omega}(\mathbb{T}_i) \to \operatorname{End} \left( \otimes_{\mathbb{T}_i \in F(\Delta)} W_i\right) $$
named local representation. A local representation of $\mathcal{Z}_{\omega}(\mathbb{D}_n, \Delta_n)$ is characterized, up to isomorphism, by a character $\chi_0$ over $\mathcal{Z}_{+1}(\mathbb{D}_n, \Delta_n)$ and a central charge $h_C$ such that $h_C^N=\prod_{e \in \mathcal{E}(\Delta)}\chi_0(X_e^N)$, whereas an irreducible representation of $\mathcal{Z}_{\omega}(\mathbb{D}_n, \Delta_n)$ is characterized, up to isomorphism, by a similar character $\chi_0$, a boundary invariant $h_{\partial}$ and some puncture invariants $h_{p_1}, \ldots, h_{p_n}$ whose $N$-th power are given by $\chi_0$. By \cite[Corollary $1.3$]{KojuQuesneyQNonAb}, a local representation $W^{loc}(\chi_0, h_C)$ decomposes into irreducible ones as 
$$ W^{loc}(\chi_0, h_C) \cong \oplus_{h_{\partial}h_{p_1}\ldots h_{p_n}=h_C} W^{irr}(\chi_0, h_{p_i}, h_{\partial}), $$
so is the direct sum of $N^n$ irreducible representations. Write $W^{loc}:=  W^{loc}(\chi_0, h_C)$ and for $\mathbf{p}=(p_1, \ldots, p_n)$ denote by $W^{irr}(\mathbf{p})$ its corresponding irreducible summand such that $W^{loc}=\oplus_{\mathbf{p}}W^{irr}(\mathbf{p})$ and set $\chi_{\mathbf{p}} \in \mathcal{B}_n^{\operatorname{SL}_2}$ the associated character. Let $\beta$ be a pure braid fixing $\chi_0$ and $\beta_{\mathbf{p}} \in \operatorname{End}_{\mathcal{B}_n^{\operatorname{SL}_2}}(\chi_{\mathbf{p}})$ the corresponding endomorphism. The reduced quantum hyperbolic operator $\mathcal{H}_q(\beta)$ is the direct sum $\mathcal{H}^{red}_q(\beta)= \oplus_{\mathbf{p}} L_{\omega}^{\operatorname{SL}_2}(\beta_{\mathbf{p}}) \in \operatorname{End}{W^{loc}}$. If $L\subset S^3$ is the closure of $\beta$ and $\rho$ the associated $\operatorname{PSL}_2$ representation, the Baseilhac-Benedetti QHI write $\mathcal{H}_q(L,\rho) := \alpha_q(L, \rho) \operatorname{Tr} (\mathcal{H}^{red}_q(\beta))$, where $\alpha_q(L,\rho)$ is a symmetrization factor. We refer to \cite{BaseilhacBenedetti05} for a detailed discussion on the geometric necessity of this factor, in particular taking only the trace of $\mathcal{H}^{red}_q(\beta)$ would not lead to well defined invariant of the link but would depend on the choice of the link (see \cite{BaseilhacBenedetti_NonAmbiguousStructures} for a discussion in terms of layered triangulations and taut structure). On the other hand, we have defined our invariants as $\operatorname{Tr}^R(L_{\omega}^{\operatorname{SL}_2}(\beta_{\mathbf{p}}))$ and again, taking only the usual trace instead of the renormalized trace would not lead to links invariants (it would not be invariant under Markov stabilisations). This observation leads to the 

\begin{problem}\label{problem_trace}
Relate the Baseilhac-Benedetti symmetrization factor to the renormalized trace.
\end{problem}

This might lead to a more geometric interpretation of the (purely algebraic) renormalized trace.



 \vspace{2mm}
\par $(4)$ Let $\rho: \pi_1(M) \to \operatorname{PSL}_2(\mathbb{C})$ be representation. It is well-known that the (so-called trace-)field $k_{\rho}$ generated over $\mathbb{Q}$ by the traces $\operatorname{Tr}(\rho(\gamma))$ is a finite extension of $\mathbb{Q}$ (\textit{i.e.} is a number field). For instance if $\mathcal{T}$ is an ideal triangulation of $\overline{M}$ and  $(\rho, l)=\mathcal{D}(\{\mathbf{z}_i\})$, then $k_{\rho}$ is generated over $\mathbb{Q}$ by the elements $z_i, z_i', z_i''$ (see \cite[Theorem $5.1.1$]{MaclachlanReid_ArithmeticDim3}). For $N\geq 1$ odd, we define $k_{\rho, N}$ to be the field generated over $\mathbb{Q}$ by the $N$-th root of $1$ and of the $z_i, z_i', z_i''$ (\textit{i.e.} the q-flattenings $w_i, w_i', w_i''$ and $q$). Assuming that the quantum Thurston equations defining $\mathcal{G}_q(\mathcal{T}^{4T}(D))$ are indeed satisfy by the quantum shape parameters as in Problem \ref{problem1}, the explicit formula in Theorem \ref{theorem_ClosedFormula} shows that $\left<L,[\rho,h]\right>_{\omega} \in k_{\rho, N}$. 

\begin{question}\label{question}
Can we renormalized the links invariants in such a way that $\left<L,[\rho,h]\right>_{\omega} $ belongs to the ring of integers of $k_{\rho, N}$ ?
\end{question}

For instance, the Kashaev invariant is a particular case where $\rho=\rho^{trivial}$ is the trivial representation sending any curve to the identity so $k_{\rho^{trivial}}= \mathbb{Q}$ and $k_{\rho^{trivial}, N}$ is the cyclotomic field $\mathbb{Q}[q]$ whose ring of integers is the cyclotomic ring $\mathbb{Z}[q]$. In this case, the answer to Question \ref{question} is positive and we can say even better: the family of Kashaev invariants for all odd roots of unity are evaluations of a universal invariant, defined by Habiro in \cite{Habiro_UniversalInvariants}, living in the so-called Habiro ring $\varprojlim_n \quotient{\mathbb{Z}[q]}{(q-1)(q^2-1)\ldots (q^n-1)}$ of \cite{Habiro_HabiroRing}. 

\begin{problem}\label{problem3}
Assuming that the answer to Question \ref{question} is positive, find a universal invariant for the $\left<L,[\rho,h]\right>_{\omega} $ living is some generalization of the Habiro ring.
\end{problem}
 An important consequence (and original motivation) for such universal Habiro invariants is the possibility to develop an element of the Habiro ring into a power series around a root of unity. The relations between such Ohtsuki series and the combinatorics of Feynman-Jacobi diagrams given by evaluations of the Kontsevich invariant is the object of a vast literature in finite type invariants theory (see \textit{e.g.} \cite{OhtsukiBook} for an introduction) which deserves to be generalized to our context in order to get a deeper understanding of our invariants.
 
 
 
 
 
 

 
 