\section*{Introduction}



The {\it quiver Hecke algebras} (or {\it Khovanov-Lauda-Rouquier algebras}), introduced by Khovanov-Lauda (\cite{KL09, KL11}) and Rouquier (\cite{R08}) independently, are $\Z$-graded algebras which provide a categorification for the negative half of a quantum group.
The algebras are a vast generalization of affine Hecke algebras of type $A$ in the direction of categorification (\cite{BK09, R08}), and they have special graded quotients, called {\it cyclotomic quiver Hecke algebras},
 which categorify irreducible integrable highest weight modules (\cite{KK11}).
When the quiver Hecke algebras are {\it symmetric},
we can study them more deeply.

\noi\addtolength{\my}{-2ex}
$\bullet$\
\parbox[t]{\my
{First of all, it is known that
the upper global basis corresponds to the set of the isomorphism classes of simple modules over symmetric quiver Hecke algebras (\cite{R11, VV09}).}


\vs{.5ex}
\noi
$\bullet$\
\parbox[t]{\my}{The KLR-type quantum affine Schur-Weyl duality functor
was constructed in \cite{K^3} using symmetric quiver Hecke algebras
and $R$-matrices of quantum affine algebras.
This functor has been studied in various types (\cite{K^3b,KKKO14c,KKKO15b}).}
\setlength{\my}{\textwidth}


The notion of {\it $R$-matrices} for symmetric quiver Hecke algebras was introduced in \cite{K^3}. The $R$-matrices are special homomorphisms defined by using intertwiners and affinizations.
It turned out that the $R$-matrices have very good properties with {\it real} simple modules (\cite{KKKO14}).
They also had taken an important role as a main tool in studying a monoidal categorification of quantum cluster algebras (\cite{KKKO14b, KKKO15}).

Let us explain the construction of $R$-matrices in \cite{K^3} briefly.
We assume that the quiver Hecke algebra $R$ is symmetric. Let $M$ be an $R$-module and $M_z$ its affinization.
The $R$-module $M_z$ is isomorphic to $\bR[z]\otimes_\bR M$ as a $\bR$-vector space. The actions of $e(\nu)$ and $\tau_i$ on $M_z$ are the same as those on $M$, but
the action of $x_i$ on $M_z$ is equal to
the action $x_i$ on $M$ added by $z$ (see $\eqref{Eq: affinization1}$).
For $R$-modules $M$ and $N$, we next consider the homomorphism $R_{M_z, N_{z'}} \in \tHom_R(M_z \conv N_{z'},  N_{z'} \conv M_z ) $ given by using intertwiners (see $\eqref{Eq: RMN}$).
Here $\tHom$ denotes the non-graded homomorphism space (see \eqref{eq:tHom}).
We set
$$\Rm_{M_z, N_{z'}} := (z'-z)^{-s} R_{M_z, N_{z'}}, \quad \mathbf{r}_{M,N} := \Rm_{M_z, N_{z'}}|_{z=z'=0},$$
where $s$ is the order of zero of $R_{M_z, N_{z'}}$. Then the morphisms $\Rm_{M_z, N_{z'}}$ and $\mathbf{r}_{M,N}$ are non-zero, commute with the spectral parameters $z$, $z'$, and satisfy the braid relations.
Here,  in defining $M_z$ and $\mathbf{r}_{M,N}$,
we crucially use the fact that $R$ is symmetric.



In this paper, we introduce and investigate the notion of
affinizations and $R$-matrices for {\em arbitrary} quiver Hecke algebras, and construct a new duality functor between finitely generated graded module categories of quiver Hecke algebras.
The affinizations defined in this paper generalize the affinizations $M_z$ for symmetric quiver Hecke algebras. The root modules given in \cite{BKM14} are examples of affinizations.

We then define a tensor functor $\F^{\mathcal{D}}: \Modg(R^{\mathcal{D}})\To\Modg(R) $ between graded module categories of quiver Hecke algebras $R$ and $R^{\mathcal{D}}$,
which arises from a {\it duality datum} $\mathcal{D}$ consisting of certain $R$-modules and their homomorphisms. This is inspired by the KLR-type quantum affine Schur-Weyl duality functor in \cite{K^3}.
The functor $\F^{\mathcal{D}}$ sends finite-dimensional modules to finite-dimensional modules. It is exact when $R^{\mathcal{D}}$ is of finite type.
We show that affinizations of real simple modules and their $R$-matrices give a duality datum.
The corresponding duality functor sends a simple module to a simple module or zero when $R^{\mathcal{D}}$ is of finite type.


Here is a brief description of our work. Let $R(\beta)$ be an arbitrary quiver Hecke algebra. We define an affinization $(\Ma, \z)$ of a simple $R(\beta)$-module $\bM$ to be an $R(\beta)$-module $\Ma$ with a homogeneous endomorphism $\z \in \End_R(\Ma)$ and an isomorphism $\Ma/\z\Ma \simeq \bM$ satisfying the conditions in Definition \ref{def:aff}.
However, we do not know if every simple modules admits an affinization.

We then study the endomorphism rings of affinizations and the
homomorphism spaces between convolution products of  simple modules and their affinizations.
For a non-zero $R$-module $N$,
let $s$ be the largest integer such that $R_{\Ma, N} (\Ma \conv N ) \subset \z^s N \conv \Ma$.
We set
$$
\Rm_{\Ma,N} =\z^{-s}R_{\Ma,N}\cl \Ma\conv N\to N\conv \Ma,
$$
and denote by $ \rmat{\bM,N}\cl \bM\conv N\to N\conv \bM $ the homomorphism induced by $\Rm_{\Ma,N}$.
By the definition $\rmat{\bM,N}$ never vanishes. The $R$-matrix
$\rmat{\bM,N}$ has similar properties to $R$-matrices for symmetric quiver Hecke algebras (Proposition \ref{Prop: simple-unique}).
Proposition \ref{prop:MNNM} tells us that, if
$(\Ma,\z)$ and $(\Na,\z[\Na])$ are affinization of simple modules $\bM$ and $\bN$ and one of $ \bM$ and $\bN$ is real (see \eqref{hyp:MN}), then
\bnum
\item $\tHom_{R[\z,\zN]}(\Ma\conv \Na,\Ma\conv \Na) =\cor[\z,\zN]\id_{\Ma\circ \Na},$
\item $\tHom_{R[\z,\zN]}(\Ma\conv \Na,\Na\conv \Ma)$ is a free $\cor[\z,\zN]$-module of rank one.
\ee
Here, $\tHom$ denotes the space of non-graded homomorphisms (see \eqref{eq:tHom}).
We define $\Rm_{\Ma,\Na}$ as a generator of the $\cor[\z,\zN]$-module $\tHom_{R[\z,\zN]}(\Ma\conv \Na,\Na\conv \Ma)$.
Then $\Rm_{\Ma,\Na}$ commutes with $\z$ and $\zN$ by the construction, and
we prove that
$\Rm_{\Ma,\Na}|_{\z=\zN=0} \in\tHom(\bM\conv\bN,\bN\conv\bM)$
does not vanish and coincides with $\rmat{\bM, \bN}$
up to a constant multiple  in Theorem \ref{Thm: specializetion of Rm}.

We next define the duality datum $\mathcal{D}=\{\beta_j, M_j, \zj,\rr,\Rmat_{j,k}\}_{j,k\in J}$ axiomatically. Here,
$J$ is a finite index set, and
\begin{align*}
&M_j \in \Modg(R(\beta_j)), \quad \qquad \mathsf{z}_j \in \tEnd_{R(\beta_j)}(M_j),  \\
&\mathsf{r}_j \in \tEnd_{R(2\beta_j)}(M_j \circ  M_j),\quad \mathsf{R}_{j,k} \in \tHom_{R(\beta_j + \beta_k )}(M_j \circ  M_k, M_k \circ  M_j),
\end{align*}
satisfying certain conditions given in Definition \ref{Def: duality}.
We construct the generalized Cartan matrix $\cmA^{\mathcal{D}}$ and the polynomial parameters $\qQD_{i,j}(u,v)$ out of the duality datum $\mathcal{D}$
and consider the quiver Hecke algebra $R^{\mathcal{D}}$ corresponding to $\cmA^{\mathcal{D}}$ and $\qQD_{i,j}(u,v)$.
For $\gamma \in {\rlQ}_+^{\mathcal{D}}$ with $m = \Ht{\gamma} $, we define
$$ \Delta^{\mathcal{D}}( \gamma )\seteq\soplus_{\mu \in J^{ \gamma}} \Delta^{\mathcal{D}}_{\;\mu}, $$
where
$$ \Delta^{\mathcal{D}}_{\;\mu} \seteq M_{\mu_1} \conv M_{\mu_2} \conv \cdots \conv
M_{\mu_m} \ \text{for $\mu = (\mu_1, \mu_2, \ldots, \mu_m) \in J^{\gamma}$.} $$
It turns out that $\Delta^{\mathcal{D}}( \gamma )$ has an $(R,  R^{\mathcal{D}})$-bimodule structure (Theorem \ref{Thm: bimodule}), and
 we obtain the duality functor $\F^{\mathcal{D}}\cl \Modg(R^{\mathcal{D}})\to\Modg(R) $
by tensoring $\Delta^{\mathcal{D}}( \gamma )$. Theorem \ref{Thm: main thm} tells that $\F^{\mathcal{D}}$ is a tensor functor and sends finite-dimensional modules to finite-dimensional modules.
Moreover it is exact when $\cmA^{\mathcal{D}}$ is of finite type.
Affinizations of real simple modules and their $R$-matrices provide
a duality functor which enjoys extra good properties (Theorem \ref{Thm: duality and affinization}).

Several examples of duality functors $\F^{\mathcal{D}}$ are given in Sections
\ref{Sec: Ex} and \ref{Sec: Ex2}.
In Example \ref{Ex: D}, we construct a duality functor $\F^{\mathcal{D}}$ from the graded module category of the quiver Hecke algebra of type $D_\ell$ to that of type $A_\ell$.
The other examples are in non-symmetric cases.
We discuss a duality functor from type $C_\ell$ to type $A_\ell$
in Example \ref{Ex: C},
 and ones from types $B_{\ell-1}$ and $A_{\ell-1}$ to type $B_\ell$
in Examples \ref{Ex: B1} and \ref{Ex: B2}.


\vs{1em}
\noindent
{\bf Acknowledgments.}
The authors would like to thank Myungho Kim for fruitful discussions.



\vskip 2em



\section{Preliminaries}

\subsection{Quantum groups} Let $I$ be an index set.
\Def\label{def:cartan}
A Cartan datum is a quadruple $\bl \cmA,\wlP,\Pi,\Pi^\vee,(\cdot,\cdot)\br$
consisting of
\bna
\item a free abelian group $\wlP$, called the {\em weight lattice},
\item $\Pi = \set{\alpha_i}{i\in I} \subset \wlP$,
called the set of {\em simple roots},
\item $\Pi^{\vee} = \set{ h_i}{i\in I} \subset \wlP^{\vee}\seteq
\Hom( \wlP, \Z )$, called the set of {\em simple coroots},
\item \label{item:inner}
$(\cdot,\cdot)$ is a $\Q$-valued symmetric bilinear form on $\Po$,
which satisfies
\be[{\rm(1)}]
\item  $(\al_i,\al_i)\in 2\Z_{>0}$ for any $i\in I$, \label{hyp:even}
\item $\lan h_i,\la\ran=\dfrac{2(\al_i,\la)}{(\al_i,\al_i)}$ for any $i\in I$and $\la\in\Po$, \label{item:h}
\item $\cmA \seteq(\lan h_i,\al_j\ran)_{i,j\in I}$ is
a {\em generalized Cartan matrix}, i.e.,
$\lan h_i,\al_i\ran=2$ for any $i\in I$ and
$\lan h_i,\al_j\ran\in\Z_{\le0}$ if $i\not=j$,
\item $\Pi$ is a linearly independent set,
\item for each $i\in I$, there exists $\Lambda_i \in \wlP$
such that $\langle h_j, \Lambda_i \rangle = \delta_{ij}$ for any $j\in I$.
\ee
\ee
\edf
Let us write $\rlQ = \bigoplus_{i \in I} \Z \alpha_i$ and $\rlQ_+ = \sum_{i\in I} \Z_{\ge 0} \alpha_i$.
For $\beta=\sum_{i \in I} k_i \alpha_i \in \rlQ_+$, set $\Ht{\beta}
=\sum_{i \in I} k_i$.
The {\em Weyl group} $\weyl$ associated with the Cartan datum
is the subgroup of $\Aut(\wlP)$ generated by
the reflections $\{r_i\}_{i\in I}$ defined by
$$r_i(\lambda)=\lambda-\langle h_i, \lambda\rangle\alpha_i \qquad
\text{for $\lambda\in \wlP$.}$$

Let $\g$ be the Kac-Moody algebra associated with a Cartan datum
$(\cmA,\wlP, \Pi, \Pi^\vee, (\cdot,\cdot))$
and $\pr$ the set of positive roots of $\g$.
We denote by $U_q(\g)$ the corresponding quantum group, which is an associative algebra over $\Q(q)$ generated by $e_i$, $f_i$ $(i\in I)$ and $q^h$
$(h\in \wlP^\vee)$ with certain defining relations (see \cite[Chap.\ 3]{HK02} for details).
Set $\A=\Z[q,q^{-1}]$. We denote by $U_\A^-(\g)$ the subalgebra of $U_q(\g)$ generated by $f_i^{(n)} := f_i^n / [n]_i!$ for $i\in I$ and $n\in \Z_{\ge0}$,
where  $q_i = q^{ (\alpha_i, \alpha_i)/2 }$ and
\begin{equation*}
 \begin{aligned}
 \ &[n]_i =\frac{ q^n_{i} - q^{-n}_{i} }{ q_{i} - q^{-1}_{i} },
 \ &[n]_i! = \prod^{n}_{k=1} [k]_i.
 \end{aligned}
\end{equation*}


\subsection{Quiver Hecke algebras}\
Let $\bR$ be a field. For $i,j\in I$, we take polynomials
$\qQ_{i,j}(u,v) \in \bR[u,v]$ such that

\noi
(i) $\qQ_{i,j}(u,v) = \qQ_{j,i}(v,u)$, and

\noi
(ii) it is of the form
\begin{align} \label{Eq: Qij}
\qQ_{i,j}(u,v) = \left\{
                 \begin{array}{ll}
                   \sum_{2(\alpha_i , \alpha_j) + p(\alpha_i , \alpha_i) + q(\alpha_j , \alpha_j) = 0} t_{i,j;p,q} u^pv^q & \hbox{if } i \ne j,\\
                   0 & \hbox{if } i=j,
                 \end{array}
               \right.
\end{align}
where $t_{i,j;-a_{ij},0} \in  \bR^{\times}$.
We set
\eq
&&\bQ_{i,j}(u,v,w)=\dfrac{\qQ_{i,j}(u,v)-\qQ_{i,j}(w,v)}{u-w}\in\cor[u,v,w].
\eneq
For $\beta\in\rtl_+$ with $\Ht{\beta}=n$, set
$$I^\beta\seteq\set{\nu=(\nu_1, \ldots, \nu_n ) \in I^n}
{\sum_{k=1}^n\alpha_{\nu_k} = \beta}.$$
The symmetric group $\mathfrak{S}_n = \langle s_k \mid k=1, \ldots, n-1 \rangle$ acts on $I^\beta$ by place permutations.

\Def \ For $\beta\in\rtlp$,
the {\em quiver Hecke algebra} $R(\beta)$ associated with $\cmA$ and $(\qQ_{i,j}(u,v))_{i,j\in I}$
is the $\bR$-algebra generated by
$$\set{e(\nu)}{\nu \in I^\beta}, \;\set{x_k}{1 \le k \le n},
 \;\set{\tau_l}{1 \le l \le n-1} $$
satisfying the following defining relations:

\begin{equation} \label{Eq: defining relations}
\begin{aligned}
& e(\nu) e(\nu') = \delta_{\nu,\nu'} e(\nu),\ \sum_{\nu \in I^{\beta}} e(\nu)=1,\
x_k e(\nu) =  e(\nu) x_k, \  x_k x_l = x_l x_k,\\
& \tau_l e(\nu) = e(s_l(\nu)) \tau_l,\  \tau_k \tau_l = \tau_l \tau_k \text{ if } |k - l| > 1, \\[5pt]
&  \tau_k^2 e(\nu) = \qQ_{\nu_k, \nu_{k+1}}(x_k, x_{k+1}) e(\nu), \\[5pt]
&  (\tau_k x_l - x_{s_k(l)} \tau_k ) e(\nu) = \left\{
                                                           \begin{array}{ll}
                                                             -  e(\nu) & \hbox{if } l=k \text{ and } \nu_k = \nu_{k+1}, \\
                                                               e(\nu) & \hbox{if } l = k+1 \text{ and } \nu_k = \nu_{k+1},  \\
                                                             0 & \hbox{otherwise,}
                                                           \end{array}
                                                         \right. \\[5pt]
&( \tau_{k+1} \tau_{k} \tau_{k+1} - \tau_{k} \tau_{k+1} \tau_{k} )  e(\nu) \\[4pt]
&\qquad \qquad \qquad = \left\{
                                                                                   \begin{array}{ll}
\bQ_{\,\nu_k,\nu_{k+1}}(x_k,x_{k+1},x_{k+2}) e(\nu) & \hbox{if } \nu_k = \nu_{k+2}, \\
0 & \hbox{otherwise}. \end{array}
\right.\\[5pt]
\end{aligned}
\end{equation}
\edf

The algebra $R(\beta)$ has the $\Z$-graded algebra structure given by
\begin{align} \label{Eq: grading}
\deg(e(\nu))=0, \quad \deg(x_k e(\nu))= ( \alpha_{\nu_k} ,\alpha_{\nu_k}), \quad  \deg(\tau_l e(\nu))= -(\alpha_{\nu_{l}} , \alpha_{\nu_{l+1}}).
\end{align}


For $\beta\in \rtl_+$, let us denote by $\Mod(R(\beta))$ the category of $R(\beta)$-modules and by $R(\beta)\smod$ the category of finite-dimensional $R(\beta)$-modules.

We denote by $\Modg\bl R(\beta)\br$ the category of graded $R(\beta)$-modules
and by $R(\beta)\gmod$ the category of finite-dimensional graded
$R(\beta)$-modules.
We denote by $\Modf\bl R(\beta)\br$ the full subcategory of
$\Modg\bl R(\beta)\br$ consisting of finitely generated graded $R(\beta)$-modules.
Their morphisms are homogeneous of degree zero.
Hence, $\Mod\bl R(\beta)\br$, $R(\beta)\smod$, $\Modg\bl R(\beta)\br$,
$R(\beta)\gmod$ and $\Modf\bl R(\beta)\br$ are abelian categories.
We set $\Modg(R)\seteq\soplus_{\beta\in\rtl^+}\Modg\bl R(\beta)\br$,
$R\smod\seteq\soplus_{\beta\in\rtl^+} R(\beta)\smod$, etc.
The objects of $\Modg(R)$ are sometimes simply called $R$-modules.

We denote by $R(\beta)\text{-proj}$
the full subcategory of $\Modg(R(\beta))$
consisting of finitely generated projective graded $R(\beta)$-modules.


Let us denote by $q$ the {\em grading shift functor},
i.e., $(qM)_k = M_{k-1}$ for a graded module $M = \bigoplus_{k \in \Z} M_k $.


For $\nu\in I^\beta$ and $\nu'\in I^{\beta'}$, let $e(\nu, \nu')$ be the idempotent corresponding to the concatenation
$\nu\ast\nu'$ of
$\nu$ and $\nu'$, and set
$$
e(\beta, \beta') := \sum_{\nu \in I^\beta, \nu' \in I^{\beta'}} e(\nu, \nu').
$$
For an $R(\beta)$-module $M$ and an $R(\beta')$-module $N$, we define an $R(\beta+\beta')$-module $M \conv N$ by
$$ M \conv N := R(\beta+\beta') e(\beta, \beta') \otimes_{R(\beta) \otimes R(\beta')} (M \otimes N). $$

We denote by $M \hconv N$ the head of $M \conv N$.



For a graded $R(\beta)$-module $M$, the {\em $q$-character} of $M$ is defined by
$$ \ch_q(M) \seteq \sum_{\nu \in I^\beta} \dim_q (e(\nu)M) \nu ,$$
Here,  $ \dim_q V\seteq\sum_{k\in \Z} \dim(V_k)q^{k} $ for a graded vector space $V = \bigoplus_{k\in \Z} V_k $. It is well defined
whenever $\dim V_k<\infty$ for all $k\in\Z$.

For $i\in I$, let $L(\alpha_i)$ be the simple graded $R(\alpha_i)$-module
such that $\ch_qL(\al_i)=(i)$. For simplicity, we write $L(i)$ for $L(\alpha_i)$ if there is no confusion.

For graded $R(\beta)$-modules $M$ and $N $, let $\Hom_{R(\beta)}(M,N)$ be the
space of morphisms in $\Modg(R(\beta))$, i.e.\ the $\bR$-vector space of
homogeneous homomorphisms of degree $0$, and set
\addtolength{\my}{-10ex}
\eq&& \label{eq:tHom}
\hs{1ex}\tHom_{R(\beta)}(M,N)=\smash{\mathop\bigoplus\limits_{k \in \Z}} \tHom_{R(\beta)}(M,N)_k\\
&&\hs{35ex}\hfill\text{with $\tHom_{R(\beta)}(M,N)_k\seteq\Hom_{R(\beta)}(q^kM,N)$.}\nn
\eneq
We write $\tEnd_{R(\beta)}(M)$ for $\tHom_{R(\beta)}(M,M)$.
When $f \in \Hom_{R(\beta)}(q^{k}M, N)$, we denote
$$ \deg(f) := k. $$
For simplicity, we write $\tHom_R( M, N )$ for $\tHom_{R(\beta)}( M, N )$ if there is no confusion.

We write $[R\text{-proj}]$ and $[R\gmod]$ for
the (split) Grothendieck group of $R\proj$
and the Grothendieck group of $R\gmod$.
Then, the $\Z$-grading gives a $\Z[q,q^{-1}]$-module structure on
$[R\text{-proj}]$ and $[R\gmod]$,
and the convolution gives an algebra structure.


\Th [{\cite{KL09, KL11, R08}}]
There exist algebra isomorphisms
$$ [R\text{-{\rm proj}}] \simeq  U_\A^-(\g), \quad [R\gmod] \simeq  \Aq.$$
\enth
Here, $\Aq\seteq\set{a\in\Um}{(a,\Uam)\subset\A}$, where $(\cdot,\cdot)$ is the non-degenerate symmetric bilinear form
on $\Um$ defined in \cite{K91}. Note that $\Aq$ is an $\A$-subalgebra of
$\Uam$ (cf.\ \cite{KKKO15} where $\Aq$ is denoted by $\Aq[\mathfrak{n}]_{\Z[q^{\pm1}]}$).

\vskip 0.3em

\Def \label{dfn: grading}
Let $\fc$ be a $\Z$-valued skew-symmetric bilinear form on
$\rtl$.
If we redefine $\deg(\tau_l e(\nu)) $ to be
$-(\al_{\nu_l},\al_{\nu_{l+1}})-\fc(\al_{\nu_l},\al_{\nu_{l+1}})$,
then it gives a well-defined $\Z$-graded algebra structure on $R(\beta)$.
We denote by $R_\fc(\beta)$ the $\Z$-graded algebra thus defined.
\edf
The usual grading $\eqref{Eq: grading}$
is a special case of such  a $\Z$-grading.

We define $R_\fc(\beta)\gmod$, $R_\fc\gmod$, etc., similarly.

Let us denote by $\Modg(R_\fc(\beta))[q^{1/2}]$ the category of
 $(\frac{1}{2}\Z)$-graded modules over $R_\fc(\beta)$.
For $\nu\in I^\beta$ we set
$$H(\nu)=\frac{1}{2}\sum_{1\le a<b\le \height{\beta}}
\fc(\al_{\nu_a},\al_{\nu_b}).$$

\Lemma
For $\beta\in\rtlp$ and $M\in\Modg(R(\beta))[q^{1/2}]$,
set
$$\bl K_\fc(M)\br_n=\soplus_{\nu\in I^\beta}e(\nu)M_{n-H(\nu)}.$$
Then we have
\bnum
\item $K_\fc$ is an equivalence of categories
from $\Modg(R(\beta))[q^{1/2}]$ to $\Modg(R_\fc(\beta))[q^{1/2}]$.
\item For $M\in \Modg(R(\beta))[q^{1/2}]$ and $N\in\Modg(R(\gamma))[q^{1/2}]$,
we have
$$K_\fc(M\conv N)
\simeq q^{\frac{1}{2}\fc(\beta,\gamma)} K_\fc(M)\conv K_\fc(N).$$
\ee
\enlemma
\Proof
(i)\ We have
\begin{align*}
\tau_ke(\nu)\bl K_\fc(M)\br_n
&=\tau_ke(\nu)M_{n-H(\nu)} \\
& \subset e(s_k\nu)M_{n-H(\nu)-(\al_{\nu_k},\al_{\nu_{k+1}})}\\
&= e(s_k\nu)\bl K_\fc(M)\br_{n-H(\nu)-(\al_{\nu_k},\al_{\nu_{k+1}})+H(s_k\nu)}.
\end{align*}
Then (i) follows from
\begin{align*}
H(s_k\nu)-H(\nu)=\frac{1}{2}
\bl\fc(\al_{\nu_{k+1}},\al_{\nu_{k}})-\fc(\al_{\nu_{k}},\al_{\nu_{k+1}})\br
=-\fc(\al_{\nu_{k}},\al_{\nu_{k+1}}).
\end{align*}

\noi
(ii)\ For $\nu\in I^\beta$ and $\mu\in I^\gamma$, we have
\begin{align*}
e(\nu)K_\fc(M)_a\tens e(\mu)K_\fc(N)_b
&=e(\nu)M_{a-H(\nu)}\tens e(\mu)N_{b-H(\mu)}\\
&\subset e(\nu\ast\mu)(M\conv N)_{a+b-H(\nu)-H(\mu)}\\
&=e(\nu\ast\mu)K_\fc(M\conv N)_{a+b-H(\nu)-H(\mu)+H(\nu\ast\mu)}.
\end{align*}
Since
$$H(\nu\ast\mu)-H(\nu)-H(\mu)
=\frac{1}{2}\fc(\beta,\gamma),$$
we have
\eqn
K_\fc(M)_a\tens K_c(N)_b
\subset K_c(M\conv N)_{a+b+\frac{1}{2}\fc(\beta,\gamma)}.
\eneqn
Therefore we have
$$K_\fc(M)_a\tens K_c(N)_b\To
\bl q^{-\frac{1}{2}\fc(\beta,\gamma)}K_c(M\conv N)\br_{a+b},$$
which induces an isomorphism
$$K_\fc(M)\conv K_c(N)\isoto q^{-\frac{1}{2}\fc(\beta,\gamma)}K_c(M\conv N).$$
\QED


We define the algebra
$\Uam_\fc$ as $\Z[q^{\pm1/2}]\otimes_{\Z[q^{\pm1}]}\Uam$ with the new multiplication $\circ_\fc$
given by
$$a\circ_\fc b=q^{-\frac{1}{2}\fc(\al,\beta)}ab\quad
\text{for $a\in \Z[q^{\pm1/2}]\otimes_{\Z[q^{\pm1}]}\Uam_\al$ and
$b\in \Z[q^{\pm1/2}]\otimes_{\Z[q^{\pm1}]}\Uam_\beta$.}$$
We define $\Aq_\fc$ similarly.

\Cor\label{cor:twist}
There is a $\Z[q^{\pm1/2}]$-algebra isomorphism
$$\xi_\fc\col\Aq_\fc\isoto
\Z[q^{\pm1/2}]\tens_{\Z[q^{\pm1}]}[R_\fc\gmod].
$$
\encor


\subsection{Remark on parity}
Under hypothesis \eqref{item:inner}\,\eqref{hyp:even} in Definition \ref{def:cartan},
the category $\Modg(R(\beta))$ is divided
into two parts according to the parity of degrees for any $\beta\in\rtlp$.

\Lemma \label{lem:parity}
Let $\beta\in\rtlp$. Then there exists a map
$$S\cl I^\beta\To \Z/2\Z$$
such that
$$S(s_k\nu)=S(\nu)+(\al_{\nu_k},\al_{\nu_{k+1}})$$
for any
$\nu\in I^\beta$ and any integer $k$
with $1\le k<\Ht{\beta}$.
\enlemma
\Proof Set $n=\Ht{\beta}$.
Let us choose a total order $\prec$ on $I$.
Then we set
$$S(\nu)\seteq\sum_{1\le a<b\le n, \;\nu_{a}\prec\nu_b}(\al_{\nu_a},\al_{\nu_b}).$$
Then we have
\begin{align*}
S(s_k\nu)&=S(\nu)+\bl\delta(\nu_{k+1}\prec\nu_k)-
\delta(\nu_k\prec\nu_{k+1})\br(\al_{\nu_k},\al_{\nu_{k+1}})\\
&\equiv
S(\nu)+\bl1-\delta(\nu_{k}=\nu_{k+1})\br(\al_{\nu_k},\al_{\nu_{k+1}})
\equiv S(\nu)+(\al_{\nu_k},\al_{\nu_{k+1}})  \mod 2.
\end{align*}

Here, for a statement $P$, we set $\delta(P)$ to be $1$ if $P$ is true and $0$ if $P$ is false.
\QED

\Prop\label{prop:paritydec} Let $\beta\in\rtlp$ and $S\cl I^\beta\To \Z/2\Z$ be as in {\rm Lemma \ref{lem:parity}}.
Let $\Modg(R(\beta))^S$ be the full subcategory of
$\Modg(R(\beta))$ consisting of graded $R(\beta)$-modules $M$ such that
$e(\nu)M_k=0$ for any $\nu\in I^\beta$ and $k\equiv S(\nu)+1\;\mathrm{mod}\,2$.
Then we have
$$\Modg(R(\beta))\simeq \Modg(R(\beta))^S\soplus q\Modg(R(\beta))^S.$$
\enprop
\Proof
For any graded $R(\beta)$-module $M$ and $\eps=0,1$ set
$$M^\eps\seteq\soplus_{\substack{
\nu\in I^\beta,\;k\in\Z, \\ k\equiv S(\nu)+\eps\;\mathrm{mod}\,2}}e(\nu)M_k.
$$
Then, we can see easily that  they are $R(\beta)$-submodules of $M$
and $M=M^{0}\soplus M^{1}$.
 Moreover, we have $M^\eps\in q^{\eps}\Modg(R(\beta))^S$.
\QED

Note that $q^2\Modg(R(\beta))^S=\Modg(R(\beta))^S$ and
\eq
&&\text{$\tHom_{R(\beta)}(M,N)_k=0$ if $k$ is odd and  $M$, $N\in\Modg(R(\beta))^S$.}\label{eq:parity}
\eneq

\subsection{$R$-matrices}\label{Sec: R matrix}
Let $\beta\in\rtlp$ and $m=\Ht{\beta}$.
For $k = 1, \ldots, m-1$ and $\nu\in I^\beta$, the {\em intertwiner} $\varphi_k \in R(\beta)$ is defined by
$$ \varphi_k e(\nu) := \left\{
              \begin{array}{ll}
                (\tau_kx_k - x_k\tau_k) e(\nu) & \hbox{ if } \nu_k = \nu_{k+1}, \\
                \tau_k e(\nu) & \hbox{ otherwise.}
              \end{array}
            \right.
$$
\begin{lemma}[{\cite[Lem.\ 1.3.1]{K^3}}] \label{Lem: intertwiners} \
\bnum
\item $\varphi_k^2 e(\nu) = ( \qQ_{\nu_k, \nu_{k+1}} (x_k, x_{k+1}) + \delta_{\nu_k, \nu_{k+1}} )e(\nu). $
\item $\{ \varphi_k \}_{1 \le k \le m-1}$ satisfies the braid relation.
\item For a reduced expression $w = s_{i_1} \cdots s_{i_t}  \in \sg_{m}$, let $\varphi_w = \varphi_{i_1} \cdots \varphi_{i_t}$. Then $\varphi_w$ does not depend on the choice of reduced expressions of $w$.
\item For $w\in \sg_m$ and $1 \le k \le m$, we have $\varphi_w x_k = x_{w(k)} \varphi_w$.
\item For $w \in \sg_m$ and $1 \le k < m$, if $w(k+1) = w(k) + 1$, then $\varphi_w \tau_k = \tau_{w(k)} \varphi_w$.
\item $\varphi_{w^{-1}} \varphi_{w} e(\nu) = \prod_{a < b,\ w(a)>w(b)} ( \qQ_{\nu_a, \nu_{b}} (x_a, x_{b}) + \delta_{\nu_a, \nu_b} )e(\nu)$.
\ee
\end{lemma}

For $m,n \in \Z_{\ge0}$, let $w[m,n]$ be the element of $\sg_{m+n}$ defined by
$$
w[m,n](k) = \left\{
              \begin{array}{ll}
                k+n & \hbox{ if } 1 \le k \le m, \\
                k-m & \hbox{ if } m < k \le m+n.
              \end{array}
            \right.
$$
Let $M$ be an $R(\beta)$-module with $\Ht{\beta}=m$ and $N$ an $R(\beta')$-module with $ \Ht{\beta'}=n$.
The $R(\beta)\otimes R(\beta')$-linear map $M\otimes N \longrightarrow N \conv M$ given by $u\otimes v \mapsto \varphi_{w[n,m]}(v \otimes u)$ can be extended to the $R(\beta+\beta')$-module
homomorphism
\begin{align} \label{Eq: RMN}
R_{M,N}: M\conv N \longrightarrow N \conv M.
\end{align}

For $\beta = \sum_{k=1}^m \alpha_{i_k}$, we set $\supp(\beta)\seteq\{ i_k \mid 1 \le k \le m \}$.
\Def \label{def:symmetric}
The quiver Hecke algebra $R(\beta)$ is said to be {\em symmetric} if $\qQ_{i,j}(u,v)$ is a polynomial in $u-v$ for all $i,j \in \supp(\beta)$.
\edf

Suppose that $R(\beta)$ is symmetric. Let $z$ be an indeterminate. For an $R(\beta)$-module $M$, we define an $R(\beta)$-module structure on $M_z := \bR[z] \otimes_{\bR} M$ by
\begin{equation}\label{Eq: affinization1}
\begin{aligned}
&e(\nu) (a \otimes u) = a \otimes e(\nu)u, \quad
x_j(a \otimes u) = (za)\otimes u + a \otimes x_j u, \\
&\tau_k (a \otimes u) = a \otimes (\tau_k u),
\end{aligned}
\end{equation}
for $\nu \in I^n$, $a\in \bR[z]$ and $u\in M$.
We call $M_z$ the {\em affinization} of $M$.
For a non-zero $R(\beta)$-module $M$ and a non-zero $R(\beta')$-module $N$, let $s$ be the order of zeros of $R_{M_z, N_{z'}}: M_z \conv N_{z'} \rightarrow N_{z'} \conv M_z$, and
$$\Rm_{M_z, N_{z'}} := (z'-z)^{-s} R_{M_z, N_{z'}} .$$
We define $  \mathbf{r}_{M,N} : M \conv N \longrightarrow N\conv M  $ by
$$\mathbf{r}_{M,N} := \Rm_{M_z, N_{z'}}|_{z=z'=0}. $$


We put $R(\beta)[z_1, \ldots, z_k] := \bR[z_1, \ldots, z_k]\otimes_{\bR} R(\beta)$. For simplicity,
we write $R[z_1, \ldots, z_k]$ for $R(\beta)[z_1, \ldots, z_k]$ if there is no afraid of confusion.


\Th [{\cite[Section\,1]{K^3}}]  \label{Thm: R matrices}
Suppose that $R(\beta)$ and $R(\beta')$ are symmetric.
Let $M$ be a non-zero $R(\beta)$-module and $N$ a non-zero $R(\beta')$-module.
\bnum
\item $\Rm_{M_z, N_{z'}}$ and $\mathbf{r}_{M,N}$ are non-zero.
\item $\Rm_{M_z, N_{z'}}$ and $\mathbf{r}_{M,N}$ satisfy the braid relations.
\item Set
$$A = \sum_{\mu \in I^\beta, \nu \in I^{\beta'}}  \left(  \prod_{1 \le a \le m, 1 \le b \le n, \mu_a \ne \nu_b }
\qQ_{\mu_a, \nu_b}(x_a \boxtimes e(\beta'),e(\beta)\boxtimes x_b) \right) e(\mu) \boxtimes e(\nu).$$
Then, $A$ is in the center of $R(\beta)\otimes R(\beta')$, and
$$R_{N_{z'}, M_z} R_{M_z, N_{z'}} (u \otimes v)= A (u \otimes v) $$
for $u \in M_z$ and $v\in N_{z'}$.
\item If $M$ and $N$ are simple modules, then
\begin{align*}
\tEnd_{R(\beta + \beta')[z,z']} ( M_{z} \conv N_{z'} ) &\simeq \bR[z,z'],\\
\tHom_{R(\beta+\beta')[z,z']}( M_z \conv N_{z'}, N_{z'} \conv M_z ) &\simeq \bR[z,z'] \Rm_{M_z, N_{z'}}.
\end{align*}
\end{enumerate}
\enth



\section{Affinization}
\subsection{Definition of affinization}
\Def
For any $i\in I$ and $\beta\in \rtl_+$ with $\Ht{\beta}=m$, we set
$$\ap[i,\beta]=\sum_{\nu\in I^\beta}\Bigl(\prod_{a\in [1,m],\,\nu_a=i}x_a\Bigr)e(\nu),$$
where $[1,m] = \{1,2,\ldots,m \}$.
\edf
Note that $\ap[i,\beta]$ belongs to the center of $R(\beta)$.
If there is no afraid of confusion, we simply write
$\ap$ for  $\ap[i,\beta]$.

\Def\label{def:aff} Let $\beta\in\rtl_+$ and $\bM$  a simple $R(\beta)$-module.
An {\em affinization} $\Ma\seteq(\Ma,\z)$ of $\bM$ is an
$R(\beta)$-module $\Ma$ with an injective homogeneous endomorphism
$\z$ of\/ $\Ma$ of degree $\dM\in\Z_{>0}$
and an isomorphism $\Ma/\z\Ma\isoto \bM$
satisfying the following conditions\/{\rm:}

\bna
\item $\Ma$ is a finitely generated free module
over the polynomial ring $\cor[\z]$,\label{hyp:1}
\setcounter{myc}{\value{enumi}}
\item
$\ap\Ma\not=0$ for any $i\in I$.\label{hyp:nd}
\setcounter{mycounter}{\value{enumi}}
\ee
If it satisfies moreover the following additional condition,
we say that $\Ma$ is a {\em strong affinization}{\rm:}
\bna \setcounter{enumi}{\value{mycounter}}
\item the exact sequence
$0\to \z\Ma/\z^2\Ma\to\Ma/\z^2\Ma\to \Ma/\z\Ma\to0$ of $R(\beta)$-modules
does not split.\label{hyp:2}
\ee
We say that an affinization is {\em even} if $\dM$ is even.
\edf
Let us denote by $\pM\cl \Ma\epito\bM$ the composition
$\Ma\epito\Ma/\z\Ma\isoto\bM$.

\Rem\label{rem:aff}\hfill
\bnum
\item\label{it: affeq}
Condition \eqref{hyp:1} is equivalent to the condition

\setcounter{enumi}{\value{myc}}
\be[{(\rm a')}]
\item \label{hyp:1'}The degree of $\Ma$ is bounded from below, that is,
$\Ma_n=0$ for $n\ll0$.
\ee
Moreover, under these equivalent conditions, we have
$$\ch_q(\Ma)=(1-q^{\dM})^{-1}\ch_q(\bM).$$
Note that a finitely generated $R$-module $M$ satisfies the condition
(\ref{hyp:1}').
\item
The non-splitness condition \eqref{hyp:2} is equivalent to saying that
$\z\Ma/\z^2\Ma$ is a unique proper $R(\beta)$-submodule of $\Ma/\z^2\Ma$.
\item If $R(\beta)$ is a symmetric quiver Hecke algebra, then $\bM_z$ is a strong affinization of any simple $R(\beta)$-module $\bM$ for $\beta\not=0$.
\ee
\enrem

\begin{example}
\bnum
\item For $i\in I$,
$\Ma\seteq L(i)_z\conv L(i)$ is not an affinization of $\bM\seteq L(i)\conv L(i)$
In fact, conditions \eqref{hyp:1} and \eqref{hyp:2}
in Definition~\ref{def:aff} hold
but condition \eqref{hyp:nd} does not.

\item
Let $(\Ma,\z)$ be an affinization of $\bM$.
Assume that $\dM=ab$ for $a,b \in \Z_{>0}$
and let $z$ be an indeterminate of homogeneous degree $b$.
Let $\cor[\z]\to\cor[z]$ be the algebra homomorphism given by $\z\mapsto z^a$.
Then $\bl\cor[z]\tens_{\cor[\z]}\Ma,\,z\br$
is an affinization of $\bM$.
If $a>1$ then it is never a strong affinization,
because $$\bl\cor[z]\tens_{\cor[\z]}\Ma\br/\bl z^a\cor[z]\tens_{\cor[\z]}\Ma\br
\simeq (\cor[z]/\cor[z]z^a)\tens_\cor\bM$$
is a semisimple $R(\beta)$-module.
\ee
\end{example}


As seen in the proposition below, every affinization is essentially even.

\Prop
Let $(\Ma,\z)$ be an affinization of a simple module $\bM$.
Assume that the homogeneous degree $\dM$ of $\z$ is odd.
Then there exists an $R(\beta)$-submodule $\Ma'$ of $\Ma$ such that
\bnum
\item $\z^2\Ma'\subset \Ma'$, and
$(\Ma',\,\z^2)$ is an affinization of $\bM$,
\item $\Ma\simeq \cor[\z]\tens_{\cor[\z^2]}\Ma'$ as an $\R(\beta)[\z]$-module.
\ee
\enprop
\Proof
Let $\Modg(R(\beta))\simeq \Modg(R(\beta))^S\soplus q\Modg(R(\beta))^S$
be the decomposition in Proposition~\ref{prop:paritydec}.
We may assume that $\bM$ belongs to $\Modg(R(\beta))^S$.
Let $\Ma=\Ma'\soplus \Ma''$ be the decomposition with
$\Ma'\in \Modg(R(\beta))^S$ and $\Ma''\in q\Modg(R(\beta))^S$.
Then $\z \Ma'\subset \Ma''$ and $\z \Ma''\subset\Ma'$ by \eqref{eq:parity}.
Hence, we have the decomposition
$$  \Ma /\z \Ma=(\Ma'/\z\Ma'')\soplus (\Ma''/\z\Ma'),$$
which implies that
$\Ma'/\z\Ma''\simeq\bM$ and $\Ma''=\z\Ma'$.
Thus we obtain the desired result.
\QED

\subsection{Strong affinization}
Note that Lemmas~\ref{le:affstrong} and \ref{lem:affweek} below
hold without assumption \eqref{hyp:nd} in Definition~\ref{def:aff}.


\Lemma\label{le:affstrong}
Assume that
\begin{equation*}
\addtocounter{equation}{1}
\setcounter{mycounter}{\value{section}}
\setcounter{mycounters}{\value{equation}}
\hs{-3ex}(\themycounter.\themycounters)_{\rm strong}\hs{1ex}
\left\{\parbox{70ex}{$\beta\in\rtl_+$ and $(\Ma,\z)$ is a {\em strong} affinization of a simple
$R(\beta)$-module $\bM$, $\z$ has homogeneous degree $\dM\in\Z_{>0}$,
and $\pM\cl \Ma\to \bM$ is a canonical projection.}\right.
\end{equation*}
\bnum
\item The head of the $R$-module $\Ma$ is isomorphic to $\bM$,
or equivalently $\z\Ma$ is a unique maximal $R(\beta)$-module. \label{le:afhead}
\item Let $s\seteq\min\set{m\in\Z}{\Ma_m\not=0}$, and
$u$ a non-zero element of $\Ma_s$. Then, $\Ma=R(\beta)u$. \label{lem:cyc}
\item  $\tEnd_{R(\beta)}(\Ma)\simeq\cor[\z]\id_{\Ma}$. \label{lem:staffend}
\ee
\enlemma
\Proof
(i) Let $S$ be a simple module and $\vphi\cl \Ma \to S$ be an epimorphism.
By consideration of the homogeneous degree, we may assume that
$\vphi(\z^k\Ma)=0$ for $k\gg0$.
Let us take $k\ge0$ such that $\vphi(\z^k\Ma)=S$ and $\vphi(\z^{k+1}\Ma)=0$.
Since $\z^k\Ma/\z^{k+1}\Ma \simeq \Ma/\z\Ma$ is simple,
$\vphi$ induces an isomorphism $\z^k\Ma/\z^{k+1}\Ma\isoto S$.
It is enough to show that $k=0$.
If $k>0$ then we have a commutative diagram
\eq
&&\xymatrix
{0\ar[r]& \z^{k}\Ma/\z^{k+1}\Ma\ar[r]\ar[dr]^-\sim& \z^{k-1}\Ma/\z^{k+1}\Ma\ar[r]
\ar[d]^\vphi&\z^{k-1}\Ma/\z^{k}\Ma
\ar[r]& 0\\
&&S} .
\eneq
Hence the first row of the above diagram is a split exact sequence,
which contradicts Definition~\ref{def:aff}~\eqref{hyp:2}.

(ii) Since $u\not\in \z\Ma$, (i) implies that $\Ma=R(\beta)u$.

(iii) Let $f\in\tEnd_{R(\beta)}(\Ma)$ be a homogeneous endomorphism of degree $\ell$.
Assume that $f(\Ma)\subset \z^k\Ma$ for $k\in\Z_{\ge0}$.
We shall show $f\in \cor[\z]\id_{\Ma}$ by the descending induction on $k$.
If $\dM k>\ell$, then $f$ should be $0$ since $f(u)\not\in \z^k\Ma$
if $f(u)\not=0$. Here $u$ is as in (ii).
Suppose that $\dM k \le \ell$.
As $\bM$ is the head of $\Ma$,
the composition $\Ma\To[\z^{-k}f]\Ma\To[\pM] \bM$
decomposes as $\Ma\To[\pM]\bM\to\bM$. Hence the composition
must be equal to $c\pM$ for some $c\in \cor$,
which yields
$$ ( \z^{-k}f - c \id_{\Ma} ) (\Ma) \subset \z \Ma. $$
Therefore, we have $(f-c\z^k)(\Ma)\subset \z^{k+1}\Ma$,
and the induction proceeds.
\QED

\subsection{Normalized R-matrices}

\Lemma\label{lem:affweek}
Assume that
\begin{equation*}
\addtocounter{equation}{1}
\setcounter{mycountert}{\value{section}}
\setcounter{mycounterf}{\value{equation}}
\hs{-3.5ex}
(\themycountert.\themycounterf)_{\rm weak}\hs{1ex}\left\{
\parbox{70ex}{$\beta\in\rtl^+$ and $(\Ma,\z)$ is an affinization of a simple
$R(\beta)$-module $\bM$, $\z$ has homogeneous degree $\dM\in\Z_{>0}$,
and $\pM\cl \Ma\to \bM$ is a canonical projection.}\right.
\end{equation*}
\bnum
\item \label{lem:affend} $\tEnd_{R(\beta)[\z]}(\Ma)\simeq\cor[\z]\id_{\Ma}$.
\item \label{cor:MNNM}  For any $i\in I$, there exist $c_i\in\cor^\times$ and $d_i\in\Z_{\ge0}$ such that
$\ap\vert_\Ma=c_i\z^{d_i}$.
\ee
\enlemma
\Proof
(i) The proof is similar to that of Lemma \ref{le:affstrong} $\eqref{lem:staffend}$.
Let $f\in\tEnd_{R(\beta)[\z]}(\Ma)$ be a homogeneous endomorphism of degree $\ell$.
Suppose that $f(\Ma)\subset \z^k\Ma$ for $k\in\Z_{\ge0}$. We shall show $f\in \cor[\z]\id_{\Ma}$ by the descending induction on $k$.

We have $f=0$ if $\dM k>\ell$ by the degree consideration.
If $\dM k \le\ell$, then
the endomorphism $\z^{-k}f$ induces an endomorphism of $\bM$.
Hence it must be equal to $c\id_{\bM}$ for some $c\in \cor$.
Then $(f-c\z^k)(\Ma)\subset \z^{k+1}\Ma$,
and the induction proceeds.

(ii) The assertion follows from $\eqref{lem:affend}$ immediately.
\QED


\Lemma\label{lem:fg}
Let $\beta$, $\Ma$ and $\bM$ be as in
$(\themycountert.\themycounterf)_{\rm weak}$.
Assume further that $\beta\not=0$.
Then $\Ma$ is a finitely generated $\R(\beta)$-module.
\enlemma
\Proof
Since $\beta\not=0$, there exists $i\in I$ such that
$\ap[i,\beta]$ has a positive degree.
Then there exists $m>0$ such that $\z^m\in\cor\, (\ap[i,\beta]\vert_\Ma)\subset
\tEnd_R(\Ma)$.
Since $\Ma$ is finitely generated over $\cor[\z^m]$,
we obtain the desired result.
\QED


\Lemma\label{lem:Riso}
Let $\beta$, $\Ma$ and $\bM$ be as in
$(\themycountert.\themycounterf)_{\rm weak}$.
Let $\gamma\in\rtl_+$ and $N\in R(\gamma)\gmod$.
\bnum
\item
The homomorphisms
$$\text{$R_{\Ma[\z^{-1}],N}\cl\Ma [\z^{-1}] \conv N\to N\conv \Ma[\z^{-1}]$ and
$R_{N,\Ma[\z^{-1}]}\cl N\conv \Ma[\z^{-1}]\to\Ma [\z^{-1}]\conv N$}$$
are isomorphisms.
Here, $\Ma[\z^{-1}] = \bR[\z, \z^{-1}]\otimes_{\bR[\z]}\Ma$.
\item If $N$ is a simple module, there exist $c\in\cor^\times$ and $d\in\Z_{\ge0}$ such that
$R_{N,\Ma}\conv R_{\Ma,N}=c(\z^d\conv N)$ and
$R_{\Ma,N}\conv R_{N,\Ma}=c(N\conv \z^d)$.
\ee
\enlemma
\Proof
(i) is an immediate consequence of (ii).
Let us show (ii).
Set $m=\Ht{\beta}$ and $n=\Ht{\gamma}$.
Then, $(R_{N,\Ma}\conv R_{\Ma,N})\vert_{\Ma\tens N}$ is given by
$$\sum_{\nu\in I^{\beta+\gamma}}\Bigl(\prod_{1\le a\le m<b\le n, \ \nu_a \ne \nu_b } \qQ_{\nu_a,\nu_b}(x_a,x_b)\Bigr)e(\nu).$$
Since any element in the center of $R(\gamma)$ with positive degree acts by zero on $N$,
it is equal to
$$\sum_{\nu\in I^{\beta+\gamma}}\Bigl(\prod_{1\le a\le m<b\le n , \ \nu_a \ne \nu_b }\qQ_{\nu_a,\nu_b}(x_a,0)\Bigr)e(\nu).$$
Hence it is equal to a product of $\ap[i,\beta]\vert_\Ma$'s
up to a constant multiple.
Hence Lemma~\ref{lem:affweek} \eqref{cor:MNNM} implies the desired result.
\QED

Let $\Ma$ and $\bM$ be as in $(\themycountert.\themycounterf)_{\rm weak}$,
and let $N\in R\gmod$ be a non-zero
module.
Let $s$ be the largest integer such that
$R_{\Ma,N}(\Ma\conv N)\subset \z^sN\conv \Ma$.
Then we set
$$
\Rm_{\Ma,N} =\z^{-s}R_{\Ma,N}\cl \Ma\conv N\to N\conv \Ma.
$$
We denote by
$$
\rmat{\bM,N}\cl \bM\conv N\to N\conv \bM
$$
the homomorphism induced by $\Rm_{\Ma,N}$.
By the definition, $\rmat{\bM,N}$ never vanishes.
We set $\Rm_{\Ma,N}=0$, and $\rmat{\bM,N}=0$ when $N=0$.

We define $\Rm_{N,\Ma}$ and $\rmat{N,\bM}$ similarly.

\medskip
The arguments in \cite{KKKO14b,KKKO15,KKKO14} still work well under these assumptions,
and we obtain similar results.
We list some of them without repeating the proof.
A simple module $S$ is called {\em real} if $S\conv S$ is simple.

\Prop[\protect{\cite[Th.\ 3.2, Prop.\ 3.8]{KKKO14}, \cite[Prop.\ 4.5]{KKKO15}, \cite[Cor.\ 3.2]{KKKO14b}}] \label{Prop: simple-unique}
Assume that
\eq
&&\left\{\parbox{75ex}{
\bna
\item
$M$ and $N$  are simple $R$-modules,
\item one of them is real simple and also admits an affinization.
\ee}\right.\label{hyp:affMN}
\eneq
\bnum
\item $M\conv N$ has a simple head and a simple socle.
Moreover, $\Im(\rmat{M,N})$ is equal to the head of $M\conv N$ and the socle of $N\conv M$. \label{prop:simple}
\item We have
$$\tHom_R(M\conv N,M\conv N)=\cor\id_{M\circ N}$$
and
$$\tHom_R(M\conv N,N\conv M)=\cor\,\rmat{M,N}.$$  \label{prop:unique}
\item $M\hconv N$ appears only once in a
Jordan-H\"older series of $M\conv N$ in $R\smod$.
\ee
\enprop


\Prop \label{prop:uniqueR}
Let $\Ma$ and $\bM$ be as in $(\themycountert.\themycounterf)_{\rm weak}$,
and let $N$ be a simple $R$-module.
Assume that $\bM$ is real. Then we have
\bnum
\item
\eq
&&\tHom_{R[\z]}(\Ma\conv N,\Ma\conv N)=\cor[\z]\id_{\Ma\circ N}
\label{eq:uniend1}
\eneq
and
\eq&&\tHom_{R[\z]}(N\conv\Ma, N\conv \Ma)=\cor[\z]\id_{N\circ\Ma}.
\label{eq:uniend2}
\eneq
\item \hskip 0.05em  $\tHom_{R[\z]}(\Ma\conv N, N\conv\Ma)$ and $\tHom_{R[\z]}(N\conv\Ma,  \Ma\conv N)$ are free $\cor[\z]$-modules of rank one.
\ee
\enprop
\Proof
(i) Let us first show \eqref{eq:uniend1}.
The idea for the proof is similar to that of Lemma \ref{le:affstrong} $\eqref{lem:staffend}$.

Let $f\in\tHom_{R[\z]}(\Ma\conv N,\Ma\conv N)$ of homogeneous degree $\ell$.
We know that $f(\Ma\conv N)\subset \z^s\Ma\conv N$ for some $s\in \Z_{\ge0}$.
We shall show $f\in \cor[\z]\id_{\Ma\circ N}$ by the descending induction on $s$.
If $s\gg0$, then $f$ should be zero by the degree consideration.
Now, we consider $\z^{-s}f$.
As $\z^{-s}f$ induces an endomorphism of $\bM\conv N$,
by Proposition \ref{Prop: simple-unique} $\eqref{prop:unique}$, it is equal to $c\id_{\bM\circ N}$ for $c \in \bR$.
Hence we have
$$
\bl f-c z^s_{\Ma}\br(\Ma\conv N)\subset \z^{s+1}\Ma\conv N.
$$
Thus, the induction hypothesis implies that
$f-c z^s_{\Ma}\in \cor[\z]\id_{\bM\circ N}$.
The proof of \eqref{eq:uniend2} is similar.

(ii) By Lemma~\ref{lem:Riso}, we have an $R[\z]$-linear monomorphism
$N\conv\Ma\monoto \Ma\conv N$.
Hence
we have
$$\tHom_{R[\z]}(\Ma\conv N,N\conv\Ma)\monoto
\tHom_{R[\z]}(\Ma\conv N,\Ma\conv N)\simeq \cor[\z].$$
As $\tHom_{R[\z]}(\Ma\conv N,N\conv\Ma)$ is non-zero,
$\tHom_{R[\z]}(\Ma\conv N,N\conv\Ma)$ is a free $\cor[\z]$-module of rank one.
\QED



\Prop \label{prop:MNNM}
We assume that
\eq
&&\left\{\parbox{70ex}
{\bna
\item $(\Ma,\z)$ and $(\Na,\zN)$ are affinizations
of simple modules $\bM$ and $\bN$, respectively,
\item one of $\bM$ and $\bN$ is real.
\ee}\right.\label{hyp:MN}
\eneq
Then we have
\bnum
\item  $\tHom_{R[\z,\zN]}(\Ma\conv \Na,\Ma\conv \Na) =\cor[\z,\zN]\id_{\Ma\circ \Na}$,
\item $\tHom_{R[\z,\zN]}(\Ma\conv \Na,\Na\conv \Ma)$ is a free $\cor[\z,\zN]$-module of rank one.
\ee
\enprop
\Proof
(i) Assume that $\bM$ is real simple. The other case can be proved similarly.

Let $f$ be a homogeneous element of
$\tHom_{R[\z,\zN]}(\Ma\conv \Na,\Ma\conv \Na)$ of degree $\ell$. Assuming that $\Im(f)\subset \zN^k(\Ma\conv \Na)$, we shall show
$f\in \cor[\z,\zN]\id_{\Ma\circ \Na}$ by the descending induction on $k$.
If $k\gg0$, then $f$ should be zero by the homogeneous-degree consideration.
We now consider $\zN^{-k}f$. The $R[\z,\zN]$-linear homomorphism
$\zN^{-k}f\cl \Ma\conv \Na\to\Ma\conv \Na$ induces an
$R[\z]$-linear homomorphism
$\Ma\conv \bN\to\Ma\conv\bN$.
By Proposition~\ref{prop:uniqueR},
it is equal to $\vphi(\z)\id_{\Ma\circ\bN}$ for some $\vphi(\z)\in\cor[\z]$.
Hence we have
$$\Im(f-\zN^k\vphi(\z)\id_{\Ma\circ\Na})\subset \zN^{k+1}\Ma\conv \Na,$$
which implies
$$f-\zN^k\vphi(\z)\id_{\Ma\circ \Na}\in\cor[\z,\zN]\id_{\Ma\circ \Na}$$
by the induction hypothesis.

(ii) An injective morphism $R_{\Ma,\Na}\cl\Na\conv \Ma\monoto\Ma\conv\Na$
induces an injective homomorphism
$$\tHom_{R[\z,\zN]}(\Ma\conv \Na,\Na\conv \Ma)
\monoto \tHom_{R[\z,\zN]}(\Ma\conv \Na,\Ma\conv \Na)
\simeq\cor[\z,\zN].$$
Since
the non-zero $\cor[\z,\zN]$-module
$L\seteq\tHom_{R[\z,\zN]}(\Ma\conv \Na,\Na\conv \Ma)$ satisfies the condition:
\eqn&
\parbox{75ex}{if non-zero elements $a,b\in \cor[\z,\zN]$
are prime to each other,
then we have
$aL\cap bL=abL$,}
\eneqn
we conclude that $L$ is a free $\cor[\z,\zN]$-module of rank one.
\QED


We define
$\Rm_{\Ma,\Na}$ as a generator of
the $\cor[\z,\zN]$-module $\tHom_{R[\z,\zN]}(\Ma\conv \Na,\Na\conv \Ma)$.
It is uniquely determined up to a constant multiple.
We call it a {\em normalized R-matrix}.

\Th \label{Thm: specializetion of Rm}
Assume \eqref{hyp:MN}.
Then,
$\Rm_{\Ma,\Na}\vert_{\z=\zN=0}\cl\bM\conv\bN\to\bN\conv\bM$
 does not vanish and is equal to $\rmat{\bM,\bN} $
up to a constant multiple.
\enth
\Proof
Since any simple $R$-module is absolutely simple,
we may assume that the base field $\cor$ is algebraically closed
without loss of generality.


By Proposition~\ref{Prop: simple-unique} $\eqref{prop:unique}$, we have
\eq
\tHom_{R}(\bM\conv \bN,\bN\conv \bM)=\cor\,\rmat{\bM,\bN}
\label{eq:rmat}
\eneq
for a non-zero $\rmat{\bM,\bN}\in \tHom_{R}(\bM\conv \bN,\bN\conv \bM)$.
Let $\ell$ be the homogeneous degree of $\rmat{\bM,\bN}$.

For $a\in\Z$, let $\cor[\z,\zN]_a$ be the homogeneous part
of $\cor[\z,\zN]$ of degree $a$ and set
$\cor[\z,\zN]_{\ge a}=\soplus_{k\ge a}\cor[\z,\zN]_k$.
Let $c\in\Z$ be the largest integer such that
$$\Rm_{\Ma,\Na}(\Ma\conv \Na)\subset \cor[\z,\zN]_{\ge c}(\Na\conv \Ma).$$
Then $\Rm_{\Ma,\Na}$ induces a non-zero map
$$\vphi\cl \bM\conv\bN\To \Bigl(\cor[\z,\zN]_{\ge c}(\Na\conv \Ma)\Bigr)/
\Bigl(\cor[\z,\zN]_{\ge c+1}(\Na\conv \Ma)\Bigr).$$
Since $\Na\conv \Ma$ is a free $\cor[\z,\zN]$-module, we have
$$\Bigl(\cor[\z,\zN]_{\ge c}(\Na\conv \Ma)\Bigr)/
\Bigl(\cor[\z,\zN]_{\ge c+1}(\Na\conv \Ma)\Bigr)
\simeq \cor[\z,\zN]_{c}\tens(\bN\conv \bM).$$
By \eqref{eq:rmat}, there exists a non-zero $f(\z,\zN)\in \cor[\z,\zN]_{c}$
such that
$$\text{$\vphi(u)=f(\z,\zN)\rmat{\bM,\bN}(u)$ for any $u\in\bM\conv\bN$.}$$
Hence, the homogeneous degree of $\Rm_{\Ma,\Na}$ is equal to
$c+\ell$, and
$$\Rm_{\Ma,\Na}(\Ma\conv \Na)\subset f(\z,\zN)(\Na\conv \Ma)+\cor[\z,\zN]_{\ge c+1}(\Na\conv \Ma).$$
Let us show that $f(\z,\zN)$ is a constant function (i.e., $c=0$).
Assuming that $c>0$, let us take
a prime divisor $a(\z,\zN)$ of $f(\z,\zN)$.
Let $(x,y)$ be an arbitrary point of $\cor^2$
such that $a(x,y)=0$.
Let $\dM$ and $\dM[\Na]$ be the homogeneous degree of $\z$ and $\zN$,
respectively,
and let $z$ be an indeterminate of homogeneous degree one.
Then $a(xz^{\dM},yz^{\dM[\Na]})=f(xz^{\dM},yz^{\dM[\Na]})=0$.
Let $\cor[\z,\zN]\to\cor[z]$ be the map obtained by the substitution
$\z=xz^{\dM}$ and $\zN=yz^{\dM[\Na]}$.
Set
\begin{align*}
K=\cor[z]\tens_{\cor[\z,\zN]}(\Ma\conv\Na),\quad
K'=\cor[z]\tens_{\cor[\z,\zN]}(\Na\conv\Ma), \quad R'=\cor[z]\tens_{\cor[\z,\zN]}\Rm_{\Ma,\Na}.
\end{align*}
Then we obtain the map
$$R'\cl K\to z^{c+1}K'.$$
Note that $K/zK\simeq\bM\conv \bN$ and $K'/zK'\simeq\bN\conv\bM$.
We shall show $R'(K)\subset z^{k}K'$
for any $k\ge c+1$ by induction on $k$.
Assume that $k\ge c+1$ and
$R'(K)\subset z^{k}K'$.
Then the morphism $ \bM \conv\bN \rightarrow \bN \conv \bM $ induced by $z^{-k}R'$ is equal to $b\,\rmat{\bM,\bN}$ for some $b \in \cor$.
If $b\not=0$, then the homogeneous degree of $R'$
is equal to $k+\ell>c+\ell$, which is a contradiction.
Thus $b=0$ and
$R'(K)\subset z^{k+1}K'$.
Hence the induction proceeds, and we conclude that
$$\Rm_{\Ma,\Na}\vert_{\z=xz^{\dM},\,\zN=yz^{\dM[\Na]}}=0$$
for any $(x,y) \in \bR^2$ such that $a(x,y)=0$,
which implies $\Rm_{\Ma,\Na}$ is divided by $a(\z,\zN)$.
This is a contradiction.

Therefore $f$ is a constant function, and
$\Rm_{\Ma,\Na}$ induces $\rmat{\bM,\bN}$ (up to a constant multiple)
after the specialization $\z=\zN=0$.
\QED

\Cor\label{cor:Rnorm} Assume \eqref{hyp:MN}.
If $\bM$ is real, then $\Rm_{\Ma,\Na}\vert_{\zN=0}=\Rm_{\Ma,\bN}$
\ro up to a constant multiple\rf.
\encor


\Lemma\label{lem:monic}
Assume \eqref{hyp:MN}. Then there exists a homogeneous element $f(\z,\zN)$
such that
\bnum
\item
$\Rm_{\Na,\Ma}\circ\Rm_{\Ma,\Na}=f(\z,\zN)\id_{\Ma\circ\Na}$ and
$\Rm_{\Ma,\Na}\circ\Rm_{\Na,\Ma}=f(\z,\zN)\id_{\Na\circ\Ma}$,
\item $f(\z,0)$ and
$f(0,\zN)$ are non-zero.
\ee
\enlemma
\Proof
It follows from Proposition~\ref{prop:MNNM},
Corollary~\ref{cor:Rnorm} and Lemma~\ref{lem:Riso}.
\QED

\Lemma \label{Lem: affMN}
Let $(\Ma,\z)$ and $(\Na,\zN)$
be affinizations of simple modules $\bM$ and $\bN$, respectively.
Assume that either $\bM$ or $\bN$ is real, and
$\bM\conv\bN\simeq\bN\conv \bM$.
Let $d$ be a common divisor of the homogeneous degree $\dM$ of $\z$
and $\dN$, that of $\zN$.
Let $z$ be an indeterminate of homogeneous degree $d$ and let
$\cor[\z,\zN]\to\cor[z]$ be the algebra homomorphism given by
$\z\mapsto z^{\dM/d}$ and $\zN\mapsto z^{\dN/d}$.
Then, $\cor[z]\tens_{\cor[\z,\zN]}(\Ma\conv \Na)$
is an affinization of $\bM\conv\bN$.
\enlemma
\Proof By the condition, $\bM\conv\bN$ is simple.
Condition \eqref{hyp:1} in Definition~\ref{def:aff}
is obvious. Condition~\eqref{hyp:nd} follows from
$\ap\vert_{\Ma\circ\Na}=(\ap\vert_\Ma)\conv(\ap\vert_\Na)$.
\QED

\Prop\label{prop:MM}
Let $\Ma$ and $\bM$ be as in $(\themycountert.\themycounterf)_{\rm weak}$.
Assume that $\bM$ is real.
We normalize $\Rm_{\Ma,\Ma}$ so that
it induces $\id_{\bM\circ\bM}$ after the specialization
$\z \conv \id_\Ma=\id_\Ma\conv\z=0$.
Then, we have
\bnum
\item
$(\Rm_{\Ma,\Ma}-\id_{\Ma\circ\Ma})(\Ma\conv \Ma)\subset
\bl\z \conv \id_\Ma-\id_\Ma\conv\z\br(\Ma\conv \Ma).$
\item
$\Rm_{\Ma,\Ma}\circ\Rm_{\Ma,\Ma}=\id_{\Ma,\Ma}$.
\ee
\enprop
\Proof
(i) To avoid possible confusions, let $(\Na,\zN)$ be a copy of $(\Ma,\z)$
and we regard $\Rm_{\Ma,\Ma}$ as a homomorphism $\Ma\conv \Na \rightarrow \Na \conv \Ma$.
We denote by $\imath \cl\Ma\conv \Na\isoto\Na\conv \Ma$ the identity.
We regard $\Ma\conv \Na$ and $\Na\conv \Ma$ as $R[\z,\zN]$-modules.
Then, $\Rm_{\Ma,\Na}$ commutes with $\z$ and $\zN$, but $\imath$ does not.
Precisely, we have $\imath\circ \z=\zN\circ \imath$ and $\imath\circ \zN=\z\circ \imath$.

By the assumption we have
$$(\Rm_{\Ma,\Na}- \imath)(\Ma\conv\Na)\subset (\zN\Na)\conv\Ma+\Na\conv(\z\Ma).$$
Let $z$ be another indeterminate with the same homogeneous degree $\dM$,
and let
$$\cor[\z,\zN]\to\cor[z]$$
be the algebra homomorphism
give by $\z\mapsto z$ and $\zN\mapsto z$.
Then, Lemma \ref{Lem: affMN} implies that
$K\seteq \cor[z]\tens_{\cor[\z,\zN]}(\Ma\conv \Na)$ is an affinization of
$\bM\conv\bM$.
The homomorphisms $\Rm_{\Ma,\Na}$ and $\imath$ induce $R[z]$-linear endomorphisms
$R'$ and $\imath'$ of
$K$. By Lemma~\ref{lem:affweek} $\eqref{lem:affend}$, $R'$ and $\imath'$ are powers of $z$
up to a constant multiple.
Since they are $\id_{\bM\circ\bM}$ after the specialization $z=0$, we conclude
that $R'=\imath'$, which completes the proof.

(ii) The assertion follows from Lemma~\ref{lem:monic} immediately.
\QED

\Ex\label{Ex:Affn}
Let $i\in I$.
Let $P(i^n)$ be a projective cover of the simple module $L(i)^{\circ\, n}$.
Then $P(i^n)$ is an $R(n\al_i)$-module generated by an element $u$ of degree
$0$ with the defining relation
$\tau_ku=0$ ($1\le k<n$).
Let $e_k(x_1,\ldots, x_n)$ be the elementary symmetric function of degree $k$.
The center of $R(n\al_i)$ is equal to
$\cor[e_k(x_1,\ldots, x_n)\mid k=1,\ldots,n]=\cor[x_1,\ldots,x_n]^{\sym_n}$.
Then we have
$$L(i)^{\circ\,n}\simeq P(i^n)/\Bigl(\sum_{k=1}^{n}R(n\al_i)e_k(x_1,\ldots, x_n)u\Bigr).$$
Set
$$\KM(i^n)\seteq P(i^n)/\Bigl(\sum_{k=1}^{n-1}R(n\al_i)e_k(x_1,\ldots, x_n)u\Bigr),$$
and define $\zj[{\KM(i^n)}]\in\tEnd_{R(n\al_i)}(\KM(i^n))$ by
$$\zj[{\KM(i^n)}]u=e_n(x_1,\ldots, x_n)u.$$
Then $(\KM(i^n), \zj[{\KM(i^n)}])$ is a strong affinization of $L(i)^{\circ\, n}$.
Note that $\ap\vert_{\KM(i^n)}=\zj[{\KM(i^n)}]$.
The homogeneous degree of $\zj[{\KM(i^n)}]$ is $n(\al_i,\al_i)$.
\enEx
\section{Root modules}\label{Sec: root modules}\
In this section, we shall review results of
McNamara (\cite{Mc12}) and Brundan-Kleshchev-McNamara (\cite{BKM14}).
Throughout this section, we assume that the Cartan matrix $\cmA$ is of finite type.
Fix a reduced expression $w_0 = r_{i_1}r_{i_2} \ldots r_{i_N}$ of the longest element $w_0 \in \weyl$.
This expression gives a convex total order $\prec$ on the set
$\pr$ of positive roots: $\al_{i_1}\prec r_{i_1}\al_{i_2}\prec\cdots\prec
r_{i_1}\cdots r_{i_{N-1}}\al_{i_N}$.
 To each positive root $\beta \in \pr$, McNamara defined
a simple $R(\beta)$-module $L(\beta)$,
which he called the {\em cuspidal module} (\cite{KR11,Mc12}).

\Lemma[{\cite[Lem.\ 3.4]{Mc12}}] For any $\beta\in\rtlp$,
$L(\beta)$ is a real simple module.
\enlemma
\begin{lemma} [{\cite[Lem.\ 3.2]{BKM14}}] For $n\ge 0$, there exist unique
\ro up to isomorphism\rf\  $R(\beta)$-modules $\Delta_n(\beta)$ with $\Delta_0(\beta) = 0$ such that
there are short exact sequences
\begin{align*}
&0 \To q_\beta^{2(n-1)} L(\beta) \To[\ i_n\ ] \Delta_n(\beta) \To[\ p_n\ ] \Delta_{n-1}(\beta) \To 0,\\
&0 \To q_\beta^{2} \Delta_{n-1}(\beta) \To[\ j_n\ ]\Delta_n(\beta) \To[\ q_n\ ]L(\beta)
\To 0 \hs{6ex}\text{for $n\ge1$,}
\end{align*}
where $q_\beta = q^{(\beta,\beta)/2}$. Moreover,
\bnum
\item $[\Delta_n(\beta)] = \frac{1-q_\beta^{2n}}{1-q_\beta^{2}} [L(\beta)]$,
\item $\Delta_n(\beta)$ is a cyclic module with simple head isomorphic to $L(\beta)$ and socle isomorphic to $q_\beta^{2(n-1)}L(\beta)$,
\item for $n\ge1$, we have
\eqn
&&\Ext_{R(\beta)}^k( \Delta_n(\beta), L(\beta) )
\simeq \left\{
 \begin{array}{ll}
    q_\beta^{-2n} \bR & \hbox{ if } k =1, \\
  0 & \hbox{ if } k \ge 2.
  \end{array}
 \right.
\eneqn
\end{enumerate}
\end{lemma}

Define the {\em root module}
\begin{align*}
\Delta(\beta) := \prolim[n] \Delta_n(\beta).
\end{align*}

\Th[{\cite[Th.\ 3.3]{BKM14}}] \label{Thm: Root module}
There is a short exact sequence
$$
0 \To q_\beta^{2} \Delta(\beta)\To[z_\beta]\Delta(\beta)\To L(\beta) \To 0.
$$
Moreover,
\bnum
\item $\Delta(\beta)$ is a cyclic module with $[\Delta(\beta)] = [L(\beta)]/(1-q_\beta^2)$,
\item $L(\beta)$ is the head of $\Delta(\beta)$,

\item $\tEnd_{R(\beta)}(\Delta(\beta)) \simeq \bR[{z_\beta}]$.
\end{enumerate}
\enth

\Cor [{\cite[Cor.\ 3.5]{BKM14}}] \label{Cor: Root module}
Any finitely generated graded $R(\beta)$-module with all simple subquotients isomorphic to $L(\beta)$ \ro up to a grading shift\/\rf\ is a finite direct sum of grade-shifted copies
of the indecomposable modules $\Delta_n(\beta)$ $(n \ge 1)$ and $\Delta(\beta)$.
\encor

\Prop\label{prop: Root module}
For any $\beta\in\pr$,
$(\Delta(\beta),z_\beta)$ is a strong affinization of $L(\beta)$.
\enprop
\Proof
We can easily check that
conditions~\eqref{hyp:1} and \eqref{hyp:2} in Definition~\ref{def:aff}
are satisfied.

We shall show \eqref{hyp:nd}
by induction on $\Ht{\beta}$.
If $\beta$ is a simple root, then \eqref{hyp:nd} is obvious.
Assume that $\height{\beta}>1$. Then, by \cite[Lemma 4.9, Theorem 4.10]{BKM14},
there exist $\al$, $\gamma\in\pr$ such that
$\al+\gamma=\beta$ and there exists an exact sequence
$$0\to q^{-(\al,\gamma)}\Delta(\gamma)\conv\Delta(\al)
\To[\vphi] \Delta(\al)\conv\Delta(\gamma)\To{[1+p]}\Delta(\beta)\To0.$$
Here $p$ is some non-negative integer and $[1+p]$ is the $q$-integer
with respect to the short root.
Moreover $\vphi$ is given by
\eq
&&\vphi(u\tens v)=\tau_{w[m,n]}(v\tens u)\label{eq:vphiw}
\eneq
for any $u\in\Delta(\gamma)$ and $v\in\Delta(\al)$.
Here $m=\Ht{\al}$ and $n=\Ht{\gamma}$.

By the induction hypothesis, $(\Delta(\al),z_\al)$ and $(\Delta(\gamma),z_\gamma)$
are affinizations.
By \eqref{eq:vphiw},
we see that $\vphi$ commutes with $z_\al$ and $z_\gamma$.
Then $\vphi=a(z_\al,z_\gamma)\Rm_{\Delta(\gamma),\Delta(\al)}$ for some
$a(z_\al,z_\gamma)\in\cor[z_\al,z_\gamma]$ by Proposition~\ref{prop:MNNM}.


Note that $\ap\vert_{\Delta(\al)\circ \Delta(\gamma)}=
\bl \ap\vert_{\Delta(\al)}\br\conv\bl\ap\vert_{\Delta(\gamma)}\br$,
and $\ap\vert_{\Delta(\al)}=c_1z_{\al}^{s_1}$ and $\ap\vert_{\Delta(\gamma)}=c_2z_{\gamma}^{s_2}$ for $c_1,c_2\in\cor^\times$ and $s_1,s_2\in\Z_{\ge0}$.
Hence, if \eqref{hyp:nd} failed, then
$(z_\al z_\gamma)^s\vert_{\Delta(\beta)}=0$ for some $s>0$.
Then we have
$$(z_\al z_\gamma)^s \Delta(\al)\conv\Delta(\gamma)
\subset \Im(\vphi)\subset \Im\bl\Rm_{\Delta(\gamma),\Delta(\al)}\br.$$
Let us take $f(z_\al,z_\gamma)\in\cor[z_\al,z_\gamma]$
such that
$\Rm_{\Delta(\gamma),\Delta(\al)}\Rm_{\Delta(\al),\Delta(\gamma)}
=f(z_\al,z_\gamma)\id_{\Delta(\al)\conv\Delta(\gamma)}$.
Then we have
$$(z_\al z_\beta)^s \Im\bl\Rm_{\Delta(\al),\Delta(\gamma)}\br
\subset f(z_\al,z_\gamma)\Delta(\gamma)\conv\Delta(\al).$$
By Lemma~\ref{lem:monic}, we have
$ f(z_\al,0)\not=0$ and $ f(0,z_\gamma)\not=0$,
which implies
$$ \Im\bl\Rm_{\Delta(\al),\Delta(\gamma)}\br
\subset f(z_\al,z_\gamma)\Delta(\gamma)\conv\Delta(\al).$$
Therefore $f(z_\al,z_\gamma)^{-1}\Rm_{\Delta(\al),\Delta(\gamma)}$ is well defined, which implies that $f$ is an invertible element of $\cor$.
Hence $\Rm_{\Delta(\al),\Delta(\gamma)}$ is an isomorphism.
Then $L(\al)\conv L(\gamma)$ is simple, which is a contradiction.
\QED

Note that
$[L(\beta)]\in[R\gmod]\simeq\Aq$
coincides with the dual PBW vector $E^*(\beta)\in\Aq$.
It is known that $\{E^*(m_1,\ldots, m_N)\}_{(m_1,\ldots, m_N)\in\Z_{\ge0}^N}$ is a basis of $\Aq$, which is called the
{\em dual PBW basis}.
Here, we set
$$E^*(m_1,\ldots, m_N)\seteq
\Bigl( q_{\beta_1} ^{m_1(m_1-1)/2}E^*(\beta_1)^{m_1}\Bigr)\cdots
\Bigl( q_{\beta_N} ^{m_N(m_N-1)/2}E^*(\beta_N)^{m_N}\Bigr)$$
with $\beta_k\seteq r_{i_1}\cdots r_{i_{k-1}}\al_{i_k}$ and
$q_{\beta} = q^{(\beta, \beta)/2}$ ($k=1,\ldots,N$).
On the other hand, $E(\beta)=\dfrac{E^*(\beta)}{(E^*(\beta),E^*(\beta))}$
is called the PBW vector and
$$\{E(\beta_1)^{(m_1)}\cdots E(\beta_N)^{(m_N)}\}_{(m_1,\ldots, m_N)\in\Z_{\ge0}^N}$$
is the basis of $\Uam$ called the {\em PBW basis}.
Here $E(\beta)^{(m)}=\dfrac{E(\beta)^m}{[m]_i!}$ with
$i\in I$ such that $(\beta,\beta)=(\al_i,\al_i)$.
Note that the PBW basis and the dual PBW basis are dual to each other.

\section{The duality functor}

\subsection{Duality data}
Let $R$ be the quiver Hecke algebra associated with a generalized Cartan matrix $\cmA$ and polynomials $\qQ_{i,j}(u,v)$.



\Def \label{Def: duality}
Let $J$ be a finite index set.
We say that $\mathcal{D}=\{\beta_j, M_j, \zj,\rr,\Rmat_{j,k}\}_{j,k\in J}$
is a {\em duality datum} if $\beta_j\in\rtlp\setminus\{0\}$,
$M_j \in \Modg(R(\beta_j))$ and
homogeneous homomorphisms
\begin{equation}\label{Eq: endmorphims}
\begin{aligned}
&\mathsf{z}_j \in \tEnd_{R(\beta_j)}(M_j), \quad \mathsf{r}_j \in \tEnd_{R(2\beta_j)}(M_j \circ  M_j), \\
&\mathsf{R}_{j,k} \in \tHom_{R(\beta_j + \beta_k )}(M_j \circ  M_k, M_k \circ  M_j)\quad
\text{for $j,k\in J$}
\end{aligned}
\end{equation}
satisfy the following conditions:
\begin{enumerate}
\item[$\Fa$] For $j\in J$, $\deg \mathsf{z}_j \in 2\Z_{>0}$.
In addition, $M_j$ is a finitely generated free module
over the polynomial ring $\bR[\mathsf{z}_j]$.
\item[$\Fb$] For $j\in J$, we have $\mathsf{r}_j \in \tEnd_{R(2\beta_j)}(M_j \circ  M_j)_{- \deg \mathsf{z}_j}$ and
$$ \mathsf{R}_{j,j} = \bl\mathsf{z}_j \conv {M_j}-{M_j}  \conv \mathsf{z}_j\br
\, \mathsf{r}_j
+ \id_{M_j\circ M_j}. $$
\item[$\Fc$] For $k,l \in J$,
\bna
\item $ (\mathsf{z}_l \conv M_k)\, \mathsf{R}_{k,l}  =
\mathsf{R}_{k,l}\,( M_k \circ \mathsf{z}_l) $ in
$ \tHom_{R(\beta_k + \beta_l)} (M_k \conv M_l, M_l \conv M_k  )$,
\item $(M_l \conv \mathsf{z}_k)\, \mathsf{R}_{k,l}  = \mathsf{R}_{k,l}\,
 (\mathsf{z}_k \conv M_l) $ in
$\tHom_{R(\beta_k + \beta_l)} (M_k \conv M_l, M_l \conv M_k  )$.
\end{enumerate}
\item[$\Fd$] There exist polynomials $\qQ^{\mathcal{D}}_{k,l}(u,v) \in \bR[u,v]$ $(k,l\in J )$
such that
\bna
\item $\qQ^{\mathcal{D}}_{k,k}(u,v)=0$, and
$\qQ^{\mathcal{D}}_{k,l}(u,v)$ $(k\ne l)$ is of the form
$$ \sum_{ \deg \mathsf{R}_{k,l} + \deg \mathsf{R}_{l,k} - p \deg \mathsf{z}_k  - q \deg \mathsf{z}_l=0  } t_{k,l;p,q} u^p v^q, $$
where $t_{k,l;\; (\deg \mathsf{R}_{k,l} + \deg \mathsf{R}_{l,k})/\deg \mathsf{z}_k ,\, 0} \in \bR^\times$,
\item $\qQ^{\mathcal{D}}_{k,l}(u,v) = \qQ^{\mathcal{D}}_{l,k}(v,u) $,
\item $ \mathsf{R}_{l,k} \mathsf{R}_{k,l} = \left\{
                                                     \begin{array}{ll}
                                                       1 & \hbox{ if } k=l, \\
                                                       \qQ^{\mathcal{D}}_{k,l}(\mathsf{z}_k \conv M_l , M_k\conv \mathsf{z}_l ) & \hbox{ if } k \ne l.
                                                     \end{array}
                                                   \right.
$
\end{enumerate}
\item[$\Fe$]
For any $j,k,l \in J$,
$$
(\mathsf{R}_{k,l} \conv M_j) ( M_k\conv\mathsf{R}_{j,l})
(\mathsf{R}_{j,k} \conv M_l) =
( M_l\conv \mathsf{R}_{j,k})(\mathsf{R}_{j,l} \conv M_k) ( M_j\conv \mathsf{R}_{k,l})
$$
holds in $\tHom_{R(\beta_j + \beta_k + \beta_l)}
( M_j \conv M_k \conv M_l, M_l \conv M_k \conv M_j )$.
\end{enumerate}
\edf

For simplicity, we write shortly $\{ M_j, \zj,\Rmat_{j,k}\}_{j,k\in J} $ for $\{\beta_j, M_j, \zj,\rr,\Rmat_{j,k}\}_{j,k\in J}$ if there is no afraid of confusion.


\medskip
We now construct a Cartan datum corresponding to the duality datum
$\mathcal{D}$ as follows.
Let $\{\alD_j\}_{j\in J}$ be the simple roots.
Then we define the weight lattice $\PoD$ by $\PoD=\rtlD\seteq\soplus_{j\in J}\Z\alD_j$,
and define a symmetric bilinear form on $\PoD$ by
\eq
&&
(\alD_j,\alD_k) =
\bc
\deg \zj&\text{if $j=k$,}\\[1ex]
-\bl\deg\qQD_{j,k}(\zj,\zj[k])\br/2=-\bl\deg \Rmat_{j,k}+\deg \Rmat_{k,j}\br/2
&\text{otherwise.}
\ec
\eneq
Define $\hD_j$ by \eqref{item:inner}\;\eqref{item:h} in Definition~\ref{def:cartan}.
Then the corresponding generalized Cartan matrix
$ \cmA^{\mathcal{D}}\seteq( a^{\mathcal{D}}_{jk})_{j,k\in J}$ is given by
$a^{\mathcal{D}}_{jk}=\dfrac{2(\alD_j,\alD_k)}{(\alD_j,\alD_j)}$.
Since $\qQD_{j,k}(\zj,0)\in \cor^\times
\zj^{-a^{\mathcal{D}}_{jk}}$ for $j\not=k$,  $-a^{\mathcal{D}}_{jk}$
 is a non-negative integer. Therefore, $\cmA^{\mathcal{D}}$ is a
generalized Cartan matrix.
We now define $R^{\mathcal{D}}$
as the quiver Hecke algebra   corresponding to the datum
$\{\qQD_{j,k}\}_{j,k\in J}$.

We now have two different quiver Hecke algebras $R$ and $R^{\mathcal{D}}$. To distinguish them,
we write
$$\text{$\xd_k$ ($1 \le k \le \Ht{\gamma})$ and  $\td_l $
($1 \le l \le \Ht{\gamma}-1$)} $$
for the generators $x_k$ ($1 \le k \le \Ht{\gamma}$)
and $\tau_l$ ($1 \le j \le \Ht{\gamma}-1$) of $R^{\mathcal{D}}(\gamma)$
($\gamma\in \rtlpD$).

The $\Z$-grading on $R^{\mathcal{D}}(\gamma) $ is give as follows:
\begin{align*}
\deg(e(\mu))=0, \quad \deg(e(\mu) \xd_k )= \deg \zj[{\mu_k}] , \quad
\deg(e(\mu) \td_l )=
\bc
 -\deg \zj[\mu_l]& \text{if $\mu_{l} = \mu_{l+1}$,} \\
\deg \Rmat_{\mu_{l}, \mu_{l+1}} & \text{if $\mu_{l} \ne \mu_{l+1}$,}
\ec
\end{align*}
which is well-defined (see Definition \ref{dfn: grading}).


Let $\gamma \in {\rlQ}_+^{\mathcal{D}}$ with $m = \Ht{\gamma} $, and define
$$ \Delta^{\mathcal{D}}( \gamma )\seteq\soplus_{\mu \in J^{ \gamma}}
\Delta^{\mathcal{D}}_{\;\mu}, $$
where
$$ \Delta^{\mathcal{D}}_{\;\mu} \seteq M_{\mu_1} \conv M_{\mu_2} \conv \cdots \conv
M_{\mu_m} \quad \text{for $\mu = (\mu_1, \mu_2, \ldots, \mu_m) \in J^{\gamma}$.} $$
Let
$$\phi\cl \rtlD\to\rtl$$
be the linear map defined by $\phi(\alD_j)=\beta_j$ for $j\in J$.
Then, it is clear that $\Delta^{\mathcal{D}}( \gamma)$ is a left
$R(\phi(\gamma))$-module.




We define a right $R^{\mathcal{D}}(\gamma)$-module structure on $\Delta^{\mathcal{D}}(\gamma)$ as follows:
\bna
\item
$e(\mu)$ is the projection to the component $\Delta^{\mathcal{D}}_{\;\mu}$,
\item
the action of $\xd_k$ on $\Delta^{\mathcal{D}}_{\;\mu}$
is given by $M_{\mu_1}\conv \cdots \conv M_{\mu_{k-1}}\conv
\zj[{\mu_k}]\conv M_{\mu_{k+1}}\conv\cdots M_{\mu_{m}}$,
\item if $\mu_{k}\not=\mu_{k+1}$,
the action of $\td_k$ on $\Delta^{\mathcal{D}}_{\;\mu}$ is given by
$$M_{\mu_1}\conv \cdots \conv M_{\mu_{k-1}}\conv
\Rmat_{\mu_k,\,\mu_{k+1}}\conv M_{\mu_{k+2}}\conv\cdots M_{\mu_{m}},$$
\item
if $\mu_{k}=\mu_{k+1}$, the action of $\td_k$ on $\Delta^{\mathcal{D}}_{\;\mu}$
is given by
$$M_{\mu_1}\conv \cdots \conv M_{\mu_{k-1}}\conv
\rr[{\mu_k}]
\conv M_{\mu_{k+2}}\conv\cdots M_{\mu_{m}}.$$
\ee

\Th \label{Thm: bimodule}
The right $R^{\mathcal{D}}(\gamma)$-module structure on $\Delta^{\mathcal{D}}( \gamma )$
is well-defined.
\enth
\Proof
Since the proof is easy and similar to the arguments in \cite{K^3}, we omit it.
\QED



By the construction, the right $R^{\mathcal{D}}(\gamma)$-module action commutes with the left $R(\phi(\gamma))$-module action, which means that
\begin{align*}
\text{ $\Delta^{\mathcal{D}}(\gamma)$ has an $( R(\phi(\gamma)), R^{\mathcal{D}}(\gamma) )$-bimodule structure.}
\end{align*}
We now define the functor
\begin{align*}
\F^{\mathcal{D}}_{\gamma} \cl \Modg(R^{\mathcal{D}}(\gamma))\To
\Modg\bl R(\phi(\gamma))\br
\end{align*}
by
$$ \F^{\mathcal{D}}_{\gamma}(M):= \Delta^{\mathcal{D}}(\gamma) \otimes_{R^{\mathcal{D}}(\gamma)} M. $$
Set
\begin{align*}
\F^{\mathcal{D}}= \bigoplus_{\gamma \in \rtlpD} \FD_{\gamma}.
\end{align*}

For $j\in J$, we shall write
$\LD(j)$ for the simple $R^\D(\alD_j)$-module $R^\D(\alD_j)/ R^\D(\alD_j)\xd_1$.


\Th\label{Thm: main thm} Let  $\mathcal{D}=\{\beta_j, M_j, \zj,\rr,\Rmat_{j,k}\}_{j,k\in J}$ be a duality datum.
\bnum
\item The functor
$$ \F^{\mathcal{D}} : \Modg(R^{\mathcal{D}})\To\Modg(R) $$
is a tensor functor.
\item
For $j\in J$,
$$\text{
$\F^{\mathcal{D}} \bl R^{\mathcal{D}}(\alD_j)\br\simeq \Ma_j$ and
$\F^{\mathcal{D}} (\LD(j)) \simeq \Ma_j / \mathsf{z}_j \Ma_j$}. $$


\item If $\cmA^\mathcal{D}$ is of finite type, then the functor $\F^{\mathcal{D}}$ is exact.
\item If a graded $R^{\mathcal{D}}(\gamma)$-module $L$ is finite-dimensional, then so is $\F^{\mathcal{D}}(L)$. Thus, we have the induced functor
$$ \F^{\mathcal{D}} \cl R^\mathcal{D}\gmod \longrightarrow
R\gmod.$$
\end{enumerate}
\enth
\begin{proof}
Since the proof is easy and similar to \cite{K^3}, we omit it.
\end{proof}


\subsection{Construction of duality data from affinizations}


Let $J$ be a finite index set.
Let $ \{\beta_j, \Ma_j, \zj\}_{j\in J}$ be a datum such that
\bna
\item $\beta_j\in\rtlp\setminus\{0\}$,
\item $(\Ma_j,\zj)$ is an even affinization of a real simple $R(\beta_j)$-module
$\bM_j\seteq \Ma_j/\zj \Ma_j$.
\setcounter{mycounter}{\value{enumi}}
\ee
Then we take $\Rmat_{j,k}$ as follows:
\bna\setcounter{enumi}{\value{mycounter}}
\item $\Rmat_{j,k}=\Rm_{\Ma_j,\Ma_k}$.
Furthermore, we normalize $\Rmat_{j,j}$
such that $\Rmat_{j,k}\vert_{\zj=\zj[k]=0}=\id_{\bM_j\circ\bM_j}$ when $j=k$.
\ee

Then, Proposition~\ref{prop:MM}
implies that
\eq
\rr\seteq(\zj\conv \Ma_j-\Ma_j\conv \zj)^{-1}\bl \Rmat_{j,j}-\id_{\Ma_j\circ \Ma_j}\br\label{def:rr}
\eneq
is a well-defined endomorphism of $\Ma_j\conv\Ma_j$.

Note that for any $ \{\beta_j,  \Ma_j, \zj\}_{j\in J}$ satisfying
(a) and (b), we can always choose  $\Rmat_{j,k}$'s.
Moreover, $\Rmat_{j,j}$ is unique and
$\Rmat_{j,k}$ ($j\not=k$) is unique up to constant multiple.

\Th \label{Thm: duality and affinization}
Under the above assumptions {\rm (a), (b), (c)}, we have the following.
\bnum
\item The datum $\mathcal{D}=\{\beta_j, \Ma_j, \zj,\rr,\Rmat_{j,k}\}_{j,k\in J}$
is a duality datum.
\item \label{item:finite} Assume that $\cmA^\mathcal{D}$ is of finite type.
Then, we have the following.
\bna
\item
$\F^\mathcal{D}(M)$ is either a simple module or vanishes
for any simple  $R^\mathcal{D}$-module $M$.

Moreover, if $M$ is a real simple module and $\F^{\mathcal{D}}(M)$ is non-zero, then $\F^{\mathcal{D}}(M)$ is real.

\item \label{item:aff}
Let $(\Na, \zN)$ is an affinization of a simple $R^{\mathcal{D}}$-module $\bN$.
If $\F^{\mathcal{D}}(\bN)$ is simple, then $(\F^{\mathcal{D}} (\Na), \F^{\mathcal{D}}(\zN))$ is an affinization of $\F^{\mathcal{D}}(\bN)$.

\item  Let $M$ and $N$ be simple $R^\mathcal{D}$-modules, and assume that
one of them is real and also admits an affinization.
Then $\F^\mathcal{D}(M\hconv N)$ is zero or isomorphic to
$\F^\mathcal{D}(M)\hconv \F^\mathcal{D}(N)$.
\ee
\ee
\enth

\Proof (i)\
Let us prove that $\mathcal{D}$ is a duality datum.
Since axioms $\Fa$--$\Fd$ are obvious,
we only gives the proof of the braid relation $\Fe$:
\eq
&&\Rmat_{jk}\circ\Rmat_{ik}\circ\Rmat_{ij}
=\Rmat_{ij}\circ\Rmat_{ik}\circ\Rmat_{jk}\label{eq:braidR}
\eneq
as a morphism from
$\Ma_i\conv \Ma_j\conv \Ma_k\to \Ma_k\conv \Ma_j\conv \Ma_i$ for $i,j,k\in J$.
By the definition, we have
$R_{\Ma_i,\Ma_j}=a(\zj[i],\zj)\Rmat_{i,j}$ for a non-zero polynomial $a(\zj[i],\zj)$.
The R-matrices $R_{\Ma_i,\Ma_j}$ satisfy the braid relation:
$$R_{\Ma_j,\Ma_k} \circ R_{\Ma_i,\Ma_k}\circ R_{\Ma_i,\Ma_j}=
R_{\Ma_i,\Ma_j}\circ R_{\Ma_i,\Ma_k}\circ R_{\Ma_j,\Ma_k}.$$
The calculation
\begin{align*}
R_{\Ma_j,\Ma_k} \circ R_{\Ma_i,\Ma_k}\circ R_{\Ma_i,\Ma_j}
&=a(\zj,\zj[k])\Rmat_{j,k}\circ a(\zj[i],\zj[k])\Rmat_{i,k}
\circ a(\zj[i],\zj)\Rmat_{i,j}\\
&=a(\zj,\zj[k])a(\zj[i],\zj[k])a(\zj[i],\zj)\Rmat_{j,k}\circ\Rmat_{i,k}
\circ\Rmat_{i,j},
\end{align*}
and a similar calculation for $R_{\Ma_i,\Ma_j}\circ R_{\Ma_i,\Ma_k}\circ R_{\Ma_j,\Ma_k}$
show that
$$a(\zj,\zj[k])a(\zj[i],\zj[k])a(\zj[i],\zj)
\Bigl(\Rmat_{jk}\circ\Rmat_{ik}\circ\Rmat_{ij}
-\Rmat_{ij}\circ\Rmat_{ik}\circ\Rmat_{jk}\Bigr)=0.$$
Hence we obtain \eqref{eq:braidR}.

\medskip\noi
(ii) (a)\hs{1.5ex}Let us prove that
$\F^\mathcal{D}(M)$ is a simple module or zero
for a simple $R^\mathcal{D}(\gamma)$-module $M$ by induction on
$\height{\gamma}$.
Let us assume $M\simeq N\hconv \LD(j)$ for some $j\in J$ and
a simple $R^\mathcal{D}(\gamma-\alD_j)$-module $N$.
By the induction hypothesis, $\F^\mathcal{D}(N)$ is a simple module or zero.
Let $r\col N\conv \LD(j)\to \LD(j)\conv N$ be a non-zero homomorphism
of $R^\mathcal{D}(\gamma)$-modules.
Then $\Im(r)$ is isomorphic to $N\hconv \LD(j)$.
Since $\F^\mathcal{D}$ is exact, $\F^\mathcal{D}(\Im(r))\simeq
\Im\bl \F^\mathcal{D}(r)\br\simeq \F^\mathcal{D}(M)$.
If $\F^\mathcal{D}(N)\simeq 0$, then
$\F^\mathcal{D}(M)\simeq 0$.
Assume that $\F^\mathcal{D}(N)$ is a simple module.
Then $\Im\bl \F^\mathcal{D}(r)\br$ is isomorphic to
$\F^\mathcal{D}(N)\hconv
\F^\mathcal{D} (\LD(j))$ or $0$ according that
$\F^\mathcal{D}(r)$ is non-zero or zero by Proposition~\ref{Prop: simple-unique}.

If $M$ is real simple and $\F^\mathcal{D}{M}$ is simple, then
$(\F^\mathcal{D}{M})\conv (\F^\mathcal{D}M)\simeq \F^\mathcal{D}(M\conv M)$
is simple and hence $\F^\mathcal{D}{M}$ is real.

Thus we obtain  (ii) (a).

\medskip\noi
(ii) (b)\
Let $\bN$ be a simple $R^{\mathcal{D}} (\gamma)$-module and set $m = \Ht{\gamma}$.
We put $ \Na_\F = \F^{\mathcal{D}}(\Na)$ and $z_\F = \F^{\mathcal{D}}(\zN)$.
Applying the functor $\F^{\mathcal{D}}$ to the exact sequence
$$ 0 \longrightarrow \Na \To[\;\zN\; ] \Na \longrightarrow \bN \longrightarrow 0,  $$
we obtain the exact sequence
$$ 0 \longrightarrow \Na_\F \To[\;z_\F\;] \Na_\F \longrightarrow \F^{\mathcal{D}}(\bN) \longrightarrow 0.  $$
Thus, we have an injective homogeneous endomorphism $z_\F$ of $\Na_\F$ and $ \Na_\F / z_\F \Na_\F \simeq  \F^{\mathcal{D}}(\bN) $.
Since $\Ma_j$ is a finitely generated $R(\beta_j)$-module for any $j$ by
Lemma~\ref{lem:fg}, $\Na_\F$ is a finitely generated graded $R$-module
and $\F^{\mathcal{D}}(\bN)$ is a finite-dimensional $R$-module.
Hence, condition \eqref{hyp:1} of Definition \ref{def:aff} holds
(see Remark~\ref{rem:aff} \eqref{it: affeq}).


Let us show \eqref{hyp:nd} of Definition \ref{def:aff}.
Let $i\in I$.
By  Lemma \ref{lem:affweek} \eqref{cor:MNNM},
for any $j\in J$, there exist
$d_j\in\Z_{\ge0}$ and  $c_j\in\cor^\times$
such that
$ \ap |_{\Ma_{j}}=c_j\z[{{j}}]^{d_j}$.
Since
$ \ap |_{\Ma_{\mu_1} \circ \cdots \circ \Ma_{\mu_m}  } =  (\ap|_{\Ma_{\mu_1}}) \conv \cdots \conv
(\ap|_{\Ma_{\mu_m}}  )=\prod_{k=1}^mc_{\mu_k}(\xd_k)^{d_{\mu_k}} $, we obtain
\begin{align*}
\ap\vert_{ \Na_\F} &= \sum_{\mu \in J^\gamma} \Bigl(\ap \vert_{
\Ma_{\mu_1} \circ \cdots \circ \Ma_{\mu_m} }\Bigr) \tens_{R^{\mathcal{D}}(\gamma)} \Na   \\
&= \sum_{\mu \in J^\gamma}  ( \Ma_{\mu_1} \conv \cdots \conv \Ma_{\mu_m} )
\tens_{R^{\mathcal{D}}(\gamma)} \Bigl( e(\mu) c (\xd_1)^{d_{\mu_1}}
\cdots (\xd_m)^{d_{\mu_m}} \Bigr)\big\vert _\Na   \\
&= \sum_{\mu \in J^\gamma}  ( \Ma_{\mu_1} \conv \cdots \conv \Ma_{\mu_m} )
\tens_{R^{\mathcal{D}}(\gamma)} \Bigl(c e(\mu)
\prod_{j\in J}\bl\smash{\prod_{\substack{k\in[1,m],\;\mu_k=j}}}(\xd_k)^{d_j} \br
\Bigr)\big\vert _\Na   \\
&=   \Delta^{\mathcal{D}}(\gamma) \tens_{R^{\mathcal{D}}(\gamma)} (  c \prod_{j\in J} \ap[j]^{d_{j}} ) \big\vert_ \Na
\end{align*}
with $c=\prod_{k=1}^mc_{\mu_k}$ which does not depend on $\mu\in J^\gamma$.
 Therefore, condition \eqref{hyp:nd}
of Definition \ref{def:aff} holds.



\medskip\noi
(ii) (c) immediately follows from (a) and
the epimorphism
$$ \F^\mathcal{D}(M)\conv \F^\mathcal{D}(N)
\epito \F^\mathcal{D}(M\hconv N)$$
because $M\hconv N$ is simple.
\QED






\section{Examples} \label{Sec: Ex}

Let $\g$ be a Kac-Moody Lie algebra associated with a Cartan matrix $\cmA$
of finite type. Suppose that
\begin{equation} \label{Eq: condiotions on beta}
\left\{\parbox{72ex}{
\bna
\item
$\{ \beta_j\}_{ j \in J }$  is a family of elements of $\rtlp$,
which is linearly independent in $\rlQ$,
\item
$\beta_j - \beta_k \notin \Phi$ for any $j,k \in J$, where $\Phi$ is the set of roots of $\g$.
\ee
}\right.
\end{equation}
Let $\overline{\g}$ be the Lie subalgebra of $\g$ generated by
the root vectors of weight $\beta_j$ and $-\beta_j$ (cf.\ \cite[Th.\ 1.1]{Naito91}).
Then $\overline{\g}$ is a Kac-Moody Lie algebra associated to
\begin{align} \label{Eq: bar A}
\overline{\cmA}:= (\overline{a}_{j,k})_{j,k\in J}\quad
\text{with $\overline{a}_{j,k}\seteq 2(\beta_j, \beta_k)/(\beta_j,\beta_j)$.}
\end{align}
We have an injective algebra homomorphism
\begin{align} \label{Eq: embedding}
\xymatrix@C=2.5ex{U^-(\overline{\g})\;\; \ar@{>->}[r]& U^-(\g).}
\end{align}


Choosing a convex order of the set $\pr$
of positive roots,
let $\bl\Delta(\beta_j),\zj\br$ be the affinization of $L(\beta_j)$
given in Proposition~\ref{prop: Root module}.
Then, we have the duality datum
$$\mathcal{D} := \{ \Delta(\beta_j), \mathsf{z}_j, \mathsf{R}_{k,l}  \}_{j,k,l\in J  } .$$
Let $\g^{\mathcal{D}}$ be the Kac-Moody Lie algebra associated with
$\cmA^{\mathcal{D}}$.
Suppose that $\cmA^{\mathcal{D}}$ is of finite type.
Then, the functor $\F^{\mathcal{D}}$ is exact, and
gives a $\Z[q^{\pm1}]$-algebra homomorphism:
$$[R^{\mathcal{D}}\gmod]\To {[R\gmod]}$$
which gives the $\Z[q^{\pm1/2}]$-algebra homomorphism (see Corollary~\ref{cor:twist}):
\eq
&&\Aq[{(\g^{\mathcal{D}})^+}]_\fc\to \Z[q^{\pm1/2}]\otimes_{\Z[q^{\pm1}]}\Aq.
\label{eq:Aqhom}
\eneq
sending $f_j$ to the dual PBW generator $E^*(\beta_j)$
corresponding to $ [\Delta(\beta_j)]$.
Here $\fc$ is the bilinear form on $\rtl^{\D}$
given by $\fc(\al^{\D}_j,\al^{\D}_k)=\frac{1}{2}(\deg \mathsf{R}_{k,j}-\deg \mathsf{R}_{j,k})$.

By applying the exact functor $ \Q(q^{1/2})\tens_{\Q[q^{\pm1/2}]}\scbul$ to \eqref{eq:Aqhom},
we obtain a $\Q(q^{1/2})$-algebra homomorphism
\eq\Uqm[{\g^\D}]_\fc\To \Q(q^{1/2})\tens_{\Q(q)}\Uqm.
\label{mor:mD}
\eneq
Set $c_\beta\seteq(E^*(\beta),E^*(\beta))^{-1}$.
Then $E(\beta)=c_{\beta}E^*(\beta)$ is the PBW vector corresponding to $\beta\in\pr$.
Let $\psi$ be the algebra automorphism of
$\Uqm[{\g^\D}]_\fc$ sending $f_j$ to $c_{\beta_j}f_j$.
Then the composition
$$\xymatrix{
\Uqm[{\g^\D}]_\fc\ar[r]_-\psi^\sim&\Uqm[{\g^\D}]_\fc\ar[r]&
\Q(q^{1/2})\tens_{\Q(q)}\Uqm}$$
sends $f_j$ to $E(\beta_j)$.
Since $\deg \z=(\beta_j,\beta_j)$ by Theorem~\ref{Thm: Root module},
the above homomorphism sends  the divided power $f_j^{(m)}$ to the divided power
$E(\beta_j)^{(m)}$.
Moreover, the $f_j^{(m)}$'s generate the $\A$-algebra
$\Uam[{\g^\D}]_\fc$,
and the $E(\beta_j)^{(m)}$'s are contained in $\Uam$.
Hence we obtain the algebra homomorphism
\eq\Uam[{\g^\D}]_\fc\To \Q[q^{\pm1/2}]\tens_{\Q[q^{\pm1}]}\Uam.
\label{mor:UqmD}
\eneq
Taking the classical limit $q^{1/2}=1$, we obtain the induced algebra homomorphism
\eq U^- ( \g^{\mathcal{D}}  ) \longrightarrow U^- ( \g  )
\label{mor:Fdlass}
\eneq
sending $f_j$ to the root vector corresponding to $-\beta_j$ for $j\in J$.

\Prop\label{prop:injcr} If
$\cmA^{\mathcal{D}} = \overline{\cmA}$,
then the morphism induced by $\FD$
$$[R^{\mathcal{D}}\gmod]\To {[R\gmod]}$$
is injective.
In particular $\FD$ sends the simple $\RD$-modules to simple $R$-modules.
\enprop
In such a case,
the functor $\F^{\mathcal{D}}$ categorifies the homomorphism $\eqref{Eq: embedding}$.
\Proof
By the condition, we have
$U^- ( \g^{\mathcal{D}}  ) \simeq U^- (\ol{\g})$.
Hence the map \eqref{mor:Fdlass} is injective, which implies that
\eqref{mor:UqmD} is injective.
Hence
\eqref{mor:mD}  as well as  \eqref{eq:Aqhom} is injective.
\QED


Let us give several examples of such duality data.
\Ex \label{Ex: D}
Let $I = \{1,2,\ldots ,\ell \}$ and $\cmA$ the Cartan matrix of type $A_\ell$.
Hence $(\al_i,\al_j)=2\delta(i=j)-\delta(|i-j|=1)$ for $i,j\in I$.
Let $R$ be the quiver Hecke algebra associated with $\cmA$ and the parameter $\qQ_{i,j}(u,v)$ defined as follows: for $i,j \in I$ with $i < j$,
$$ \qQ_{i,j}(u,v) = \left\{
                           \begin{array}{ll}
                             u-v & \text{ if } j = i+1, \\
                             1 & \text{ otherwise. }
                           \end{array}
                         \right.
$$

Let $J = \{1,2,\ldots ,\ell \}$ and $\beta_1\seteq\alpha_1 + \alpha_2$,
$\beta_j \seteq\alpha_j$ for $j\in J\setminus\{1\}$.
Note that the $\beta_j$'s do not satisfy condition \eqref{Eq: condiotions on beta} (b). We put
\begin{align*}
\Delta(\beta_1) := L(1,2)_{\mathsf{z}_1}, \quad \Delta(\beta_j) := L(j)_{\mathsf{z}_j} \quad (j \in J \setminus \{ 1 \}),
\end{align*}
where $L(1,2) := \bR v$ is
the 1-dimensional $R(\beta_1)$-module with the actions
$$ e(\nu) v = \delta_{\nu, (1,2)}v, \quad x_1 v = x_2 v = \tau_1 v = 0 \qquad \text{ for } \nu \in I^{\alpha_1 + \alpha_2} .  $$
Note that $\deg(\mathsf{z}_j)=2$ for $j\in J$ and the $\Delta(\beta_j)$'s are root modules.
We set $\mathsf{R}_{j,k}= \Rm_{\Delta(\beta_j), \Delta(\beta_k)}$.
By direct computations, the $R$-matrix $R_{\Delta(\beta_j), \Delta(\beta_k)}$ $(j\ne k)$ is given as follows: for $u \otimes v \in \Delta(\beta_j) \otimes \Delta(\beta_k)$,
\begin{align*}
R_{\Delta(\beta_j), \Delta(\beta_k)} (u \otimes v) =
\left\{
     \begin{array}{ll}
        (\tau_2\tau_1(\mathsf{z}_2 - \mathsf{z}_1) + \tau_1)(v \otimes u)  & \hbox{ if $j=1$ and $k = 2$},  \\
        \tau_2\tau_1 (v \otimes u) & \hbox{ if $j=1$ and $k > 2$}, \\
        \tau_1 \tau_2 (\mathsf{z}_1 - \mathsf{z}_2) (v \otimes u) & \hbox{ if $j=2$ and $k = 1$}, \\
        \tau_1 \tau_2 (v \otimes u) & \hbox{ if $j>2$ and $k =1$}, \\
        \tau_1 (v \otimes u) & \hbox{ otherwise, }
     \end{array}
   \right.
\end{align*}
which yields
$$
\mathsf{R}_{j,k} =
\left\{
  \begin{array}{ll}
    (\mathsf{z}_1 - \mathsf{z}_2)^{-1} R_{\Delta(\beta_j), \Delta(\beta_k)}
& \hbox{ if $ j=2$ and $k=1$},  \\[1ex]
    R_{\Delta(\beta_j), \Delta(\beta_k)} & \hbox{ otherwise,}
  \end{array}
\right.
$$
and
$$ \deg (\mathsf{R}_{j,k} ) = \left\{
                                \begin{array}{ll}
          1 & \hbox{if $ |j-k|=1$ and $(j,k)\not= (2,1)$,}\\
1&\text{$(j,k)=(1,3), (3,1)$,}\\
                                  -1 & \hbox{ if $(j,k) = (2,1)$}, \\
                                  0 & \hbox{ otherwise.}
                                \end{array}
                              \right.
 $$
Thus, we have
$$\cmA^{\mathcal{D}} = \left(
                         \begin{array}{ccccccc}
                           2 & 0 & -1 & 0 &\cdots & 0 &0\\
                           0 & 2 & -1 & 0 & \cdots  &0&0  \\
                           -1 & -1 & 2 & -1 & \cdots &  0&0 \\
                           0 & 0 & -1 & 2 &\cdots & 0 &0 \\
                           \vdots & \vdots & \vdots & \vdots & \ddots & \vdots& \vdots\\
                           0 & 0 & 0 & 0 & \cdots & 2 & -1 \\
                           0 & 0 & 0 & 0 & \cdots & -1 &2
                         \end{array}
                       \right)
$$
which is of type $D_\ell$, i.e., the quiver Hecke algebra $R^{\mathcal{D}}$ is of type $D_\ell$.
Note that $\deg(e(1,2) \td_1 ) = 1$ and $\deg(e(2,1) \td_1 ) = -1$
(see Definition \ref{dfn: grading}).
By Theorem \ref{Thm: duality and affinization}, we have a functor $ \F^{\mathcal{D}} $ between quiver Hecke algebras of type $D_\ell$ and $A_\ell$
such that
$$ \F^{\mathcal{D}}( L^{\mathcal{D}}(j)) \simeq L(\beta_j) \quad  \text{ for $j\in J.$} $$

Let us consider the $R^{\mathcal{D}}$-module
$\LD(1,3) :=\LD(1) \hconv \LD(3)$ and the one-dimensional  $R$-module
$L(1,2,3) := L(1,2) \hconv L(3)$. Applying the functor $\F^{\mathcal{D}}$ to the exact sequence
$$ 0 \rightarrow \LD(1,3) \rightarrow \LD(3)\conv \LD(1) \rightarrow
\LD(1)\conv \LD(3) \rightarrow \LD(1,3) \rightarrow 0, $$
we have
$$ 0 \rightarrow \F^{\mathcal{D}} (\LD(1,3)) \rightarrow L(3)\conv L(1,2) \rightarrow L(1,2)\conv L(3) \rightarrow \F^{\mathcal{D}}(\LD(1,3)) \rightarrow 0. $$
Since $\F^{\mathcal{D}}$ sends a simple module to a simple module or zero by Theorem \ref{Thm: duality and affinization} and $L(3)\conv L(1,2)$
is not isomorphic to $L(1,2)\conv L(3)$, we have
$$ \F^{\mathcal{D}}(\LD(1,3)) \simeq L(1,2,3). $$
Set $\LD(1,3,2)\simeq\LD(1,3)\hconv \LD(2)$, which
is one-dimensional.
It is isomorphic to the image of the composition of
\eqn
&&\LD(1)\conv \LD(3)\conv \LD(2)
\To \LD(1)\conv \LD(2)\conv \LD(3)
\To\LD(2)\conv \LD(1)\conv \LD(3)\\
&&\hs{57ex}\hfill\To\LD(2)\conv\LD(1,3).\eneqn
By applying $\FD$,
we obtain the diagram
\eq
&&\xymatrix{
L(1,2)\conv L(3)\conv L(2)\ar[r]^{f_1}&L(1,2)\conv L(2)\conv L(3)
\ar[r]^{f_2}& L(2)\conv L(1,2)\conv L(3)\ar@{->>}[d]^{f_3}\\
&&L(2)\conv L(1,2,3).}
\label{dia:Rcomp}
\eneq
Hence $\FD(\LD(1,3,2))$ is isomorphic to the image of $f_3f_2f_1$.
Let $u_{1,2}$, $u_2$, and $u_3$ be the generator of
$L(1,2)$, $L(2)$ and $L(3)$, respectively.
Then we have
\eqn
&&f_1(u_{1,2}\tens u_3\tens u_2)=\tau_3(u_{1,2}\tens u_2\tens u_3),\\
&&f_2(u_{1,2}\tens u_2\tens u_3)=\tau_1(u_{2}\tens u_{1,2}\tens u_3).
\eneqn
Therefore, we obtain
\eqn
&&f_2f_1(u_{1,2}\tens u_3\tens u_2)=
\tau_3\tau_1(u_{2}\tens u_{1,2}\tens u_3)
=\tau_1\tau_3(u_{2}\tens u_{1,2}\tens u_3),
\eneqn
which is killed by $f_3$.
Thus $f_3f_2f_1=0$, and hence we conclude
$$\FD(\LD(1,3,2))\simeq 0.$$
Therefore, $\FD$ can send simple modules to zero in this example.
\enEx

\vskip 1em

\section{Further examples for non-symmetric types} \label{Sec: Ex2}


Let $\beta\in\rtlp$ and let
$(\Ma,\z)$ be an affinization of a real simple $R(\beta)$-module $\bM$.
We set $J=\{\zr\}$, $\beta_\zr=\beta$, $\Ma_\zr=\Ma$.
Then
$$\mathcal{D}= \{ \Ma_\zr, \z, \Rm_{\Ma,\Ma}\} $$
is a
duality datum. Then the corresponding simple root $\alD_\zr$ satisfies
$(\alD_\zr,\alD_\zr)=\deg \z$.
Let $\bl \KM(\zr^n),\zj[{\KM(\zr^n)}]\br$ be the affinization of the simple
$\RD(n\alD_\zr)$-module $L(\alD_\zr)^{\circ \,n}$ given in Example~\ref{Ex:Affn}.

Now $\Ma^{\circ\mspace{1mu}n}\seteq\underbrace{\Ma\conv\cdots\conv\Ma}_{\text{$n$ times}}$
has a structure of $(R(n\beta),\RD(n\alD_\zr))$-bimodule.
We set
 $$\Cv_n(\Ma)=\Ma^{\circ\,n}\tens_{\RD(n\alD_\zr)}\KM(\zr^n)
\simeq\FD(\KM(\zr^n)).$$
Then $\zj[{\KM(\zr^n)}]\in\tEnd(\KM(\zr^n))$ induces an endomorphism
$\zj[{\Cv_n(\Ma)}]\in\tEnd(\Cv_n(\Ma))_{n\deg \z}$.
By Theorem~\ref{Thm: duality and affinization}~
\eqref{item:finite}\,\eqref{item:aff},
we obtain the following lemma.
\Lemma $(\Cv_n(\Ma),\zj[{\Cv_n(\Ma)}]) $ is an affinization of
the real simple module $\bM^{\circ\,n}$.
\enlemma

For example,
\begin{align} \label{Eq: def of tilde M}
\Cv_2(\Ma) =
\frac{ \Ma \conv \Ma }{
(\z\conv \Ma + \Ma \conv \z ) (\Ma\conv \Ma) + \rr[](\Ma \conv\Ma)  }  ,
\end{align}
where $\mathsf{r}$ is the endomorphism given in \eqref{def:rr}.
$\zj[{\Cv_2(\Ma)}]$ is the endomorphism induced by $\z\conv \z$.

\medskip
Let $M \in R(\beta)\gmod$ and $N \in R(\beta')\gmod$
be real simple modules.
Suppose that $R(\beta)$ and $R(\beta')$ are symmetric, and
$$
\Rm_{N_{\mathsf{z}'}, M_{\mathsf{t}}} \Rm_{M_{\mathsf{t}}, N_{\mathsf{z}'}} = c (\mathsf{t} - \mathsf{z}')^p
\in \End_{R(\beta+\beta')}( M_{\mathsf{t}} \circ N_{\mathsf{z}'} )
$$
for some $c\in \bR^\times$ and $p \in \Z_{\ge0}$. Set
\begin{align*}
R_1 &:= (\Rm_{M_{\mathsf{t}_1}, N_{\mathsf{z}'}} \conv 1)
(1 \conv\Rm_{M_{\mathsf{t}_2}, N_{\mathsf{z}'}}) \in
\tHom_{R(2\beta+\beta')} (M_{\mathsf{t}_1}\conv M_{\mathsf{t}_2} \conv N_{\mathsf{z}'} ,\,
 N_{\mathsf{z}'} \conv M_{\mathsf{t}_1}\conv M_{\mathsf{t}_2}), \\
R_2 &:= (1 \conv\Rm_{N_{\mathsf{z}'}, M_{\mathsf{t}_2}} ) (\Rm_{N_{\mathsf{z}'}, M_{\mathsf{t}_1}} \conv 1) \in \tHom_{R(2\beta+\beta')} ( N_{\mathsf{z}'} \conv M_{\mathsf{t}_1}\conv
M_{\mathsf{t}_2}, M_{\mathsf{t}_1}\conv M_{\mathsf{t}_2} \conv N_{\mathsf{z}'} ).
\end{align*}
Setting $\mathsf{t}_1+\mathsf{t}_2=0$ and
$\mathsf{t}_1\mathsf{t}_2=\widehat{\mathsf{z}}\seteq\zj[\Cv_2(M)]$,
we regard $R_1$ and $R_2$ as homomorphisms in
$\tHom_{R(2\beta+\beta')} \bl \Cv_2(M_\mathsf{z}) \conv N_{\mathsf{z}'}, N_{\mathsf{z}'} \conv
\Cv_2(M_\mathsf{z}) \br$ and
$\tHom_{R(2\beta+\beta')} \bl N_{\mathsf{z}'} \conv \Cv_2(M_\mathsf{z}),\, \Cv_2(M_{\mathsf{z}}) \conv N_{\mathsf{z}'}\br$, respectively.
Then, we have
\begin{equation}  \label{Eq: R1 R2}
\begin{aligned}
R_2 R_1 &= c^2 ( \mathsf{t}_1 - \mathsf{z}')^p ( \mathsf{t}_2 - \mathsf{z}')^p \\
&= c^2 ( \mathsf{t}_1\mathsf{t}_2 - ( \mathsf{t}_1+\mathsf{t}_2 )\mathsf{z}' + \mathsf{z'}^{\,2} )^p \\
&= c^2 ( \widehat{\mathsf{z}} + \mathsf{z'}^{\,2} )^p
\end{aligned}
\end{equation}
in $ \End_{R(2\beta+\beta')}( \Cv_2(M_\mathsf{z})\conv N_{\mathsf{z}'} ) $.

\smallskip
Using $\eqref{Eq: R1 R2}$, one can construct functors $\F^{\mathcal{D}}$ between symmetric and non-symmetric quiver Hecke algebras.
In particular, the functor from type $C_\ell$ (resp.\ $C_\ell^{(1)}$, $A_{2\ell-1}^{(2)}$) to type $A_\ell$ (resp.\ $A_{\ell+1}$, $D_{\ell+1}$) can be constructed.
We give such constructions in the following examples.

\Ex \label{Ex: C}
We take $I$, $\cmA$, and $\qQ_{i,j}(u,v)$ given in Example \ref{Ex: D}.
In particular, $\g$ is of type $A_\ell$.

Let $J = \{1,2,\ldots, \ell \}$ and
$$
\beta_1 = 2\alpha_1,\ \beta_j = \alpha_j\ \text{for $j\in J\setminus\{1\}$}.
$$
Let us denote
\begin{align*}
\Ma_1 = \KM(1^2),\quad \Ma_j = L(j)_{\mathsf{z}_j} \ (j \in J \setminus \{1 \}),
\end{align*}
and $\zj[1] := \zj[K(1^2)]$. Then $\deg \zj[1]=4$ and $\deg \zj=2$ for $j\not=1$.
Note that
$$
\Rm_{L(j)_{\mathsf{z}}, L(k)_{\mathsf{w}} } =
R_{L(j)_{\mathsf{z}}, L(k)_{\mathsf{w}} }.
$$
We put $ \mathsf{R}_{j,k} := R_{\Ma_j, \Ma_k} $.
It follows from $\eqref{Eq: R1 R2}$ that, for $j, k\in J$ with $j < k$,
\begin{align}
\mathsf{R}_{k,j} \mathsf{R}_{j,k} =
\bc
\mathsf{z}_1 \circ 1 + 1 \circ \mathsf{z}_2^2 & \hbox{ if } (j,k)=(1,2), \\
\mathsf{z}_j \circ 1 - 1 \circ \mathsf{z}_k
& \hbox{ if $k=j+1$ and $(j,k)\not=(1,2)$,}  \\
                                        1 & \hbox{ otherwise.}
   \ec
\label{eq:AD}
\end{align}

We now set
$$ \mathcal{D} = \{ \Ma_j, \zj, \mathsf{R}_{j,k} \}_{j,k\in J}. $$
Then $\mathcal{D}$ is a dual datum, and \eqref{eq:AD} implies that
$$\cmA^{\mathcal{D}} = \left(
                         \begin{array}{cccccc}
                           2 & -1 & 0  &\cdots & 0 &0\\
                           -2 & 2 & -1  & \cdots  &0&0  \\
                           0 & -1 & 2  & \cdots &  0&0 \\
                           \vdots &  \vdots & \vdots & \ddots & \vdots& \vdots\\
                           0 & 0 &  0 & \cdots & 2 & -1 \\
                           0 & 0 &  0 & \cdots & -1 &2\\
                         \end{array}
                       \right)
$$
which is of type $C_\ell$. Therefore, we have the quiver Hecke algebra $R^{\mathcal{D}}$ of type $C_\ell$ and
the functor $ \F^{\mathcal{D}}  $ from
the category of modules over  quiver Hecke algebras of type $C_\ell$
to that of type $A_\ell$
sending
$$
\F^{\mathcal{D}}( \LD(j)) \simeq \left\{
                                                        \begin{array}{ll}
                                       L(1) \circ L(1)  & \hbox{ if } j=1, \\
                                                          L(j) & \hbox{ otherwise.}
                                                        \end{array}
                                                      \right.
$$
We conjecture that $\FD$ sends simple to simple.
\enEx



We make the following examples for type $B_\ell$ by constructing affinizations directly.

\Ex \label{Ex: B1}
Let $I = \{1,2, \ldots ,\ell \}$ and $\cmA$ the Cartan matrix of type $B_\ell$:
$$ \cmA =
                        \left(
                         \begin{array}{cccccc}
                           2 & -2 & 0  &\cdots & 0 &0\\
                           -1 & 2 & -1  & \cdots  &0&0  \\
                           0 & -1 & 2  & \cdots &  0&0 \\
                           \vdots &  \vdots & \vdots & \ddots & \vdots& \vdots\\
                           0 & 0 &  0 & \cdots & 2 & -1 \\
                           0 & 0 &  0 & \cdots & -1 &2\\
                         \end{array}
                       \right),$$
and $(\alpha_i, \alpha_j)=2\delta(i=j=1)+4\delta(i=j\not=1)-2\delta(|i-j|=1)$ for $i,j\in I$.

Let $R$ be the quiver Hecke algebra associated with $\cmA$ and the parameter $\qQ_{i,j}(u,v)$ defined as follows: for $i,j \in I$ such that $i < j$,
$$ \qQ_{i,j}(u,v) = \left\{
                           \begin{array}{ll}
                             u^2-v & \text{ if } (i,j) = (1,2), \\
                             u-v & \text{ if $j=i+1$ and $(i,j) \ne (1,2)$}  , \\
                             1 & \text{ otherwise. }
                           \end{array}
                         \right.
$$

Let $J = \{1,2,\ldots, \ell-1 \}$ and
$$
\beta_1 = \alpha_1 + \alpha_2,\qquad \beta_j = \alpha_{j+1} \quad (j \in J\setminus \{ 1 \}).
$$
Note that $(\beta_1, \beta_1) = 2$ and $(\beta_j, \beta_j)  = 4$ for $j \ne 1$.
We put
\begin{align*}
\Delta(\beta_1) = L(1,2)_{\mathsf{z}_1}, \qquad \Delta(\beta_j) = L(j+1)_{\mathsf{z}_j} \quad (j \ne 1),
\end{align*}
where the $R(\beta_1)$-module
$L(1,2)_{\mathsf{z}_1} \seteq\bR[\mathsf{z}_1]v$ is defined by
\begin{align*}
e(\nu)v = \delta_{\nu, (1,2)}v,\quad x_j v = \mathsf{z}_1^{(\al_j,\al_j)/2}v, \quad
\tau_1 v = 0.
\end{align*}
Note that $\Delta(\beta_j)$'s are root modules and
$\deg(\mathsf{z}_j)$ is $2$ or $4$ according that
$j=1$ or not.
For $j,k\in J$ with $j\ne k$, we define
$$
\mathsf{R}_{j,k} := R_{ \Delta(\beta_j),\Delta(\beta_k)}
\in \Hom_{R(\beta_j + \beta_k)}( \Delta(\beta_j)\circ \Delta(\beta_k), \Delta(\beta_k)\circ \Delta(\beta_j)  ),
$$
that is,
$$
\mathsf{R}_{j,k} (p \otimes q) = \left\{
                                                      \begin{array}{ll}
                                                        \tau_2 \tau_1 (q \otimes p) & \hbox{ if } j=1, \\
                                                        \tau_1 \tau_2 (q \otimes p) & \hbox{ if } k=1, \\
                                                        \tau_1 (q \otimes p) & \hbox{ otherwise}
                                                      \end{array}
                                                    \right.
$$
for $p\otimes q \in \Delta(\beta_j) \otimes_\bR \Delta(\beta_k) $.
For $j,k \in J$ with $j < k$, we have
\begin{align*}
\mathsf{R}_{k,j}\mathsf{R}_{j,k} =
\left\{
  \begin{array}{ll}
    \mathsf{z}_j^2 - \mathsf{z}_k & \hbox{ if } (j,k) = (1,2), \\
    \mathsf{z}_j- \mathsf{z}_k & \hbox{ if $k=j+1$ and $(j,k) \ne (1,2)$},  \\
    1 & \hbox{ otherwise.}
  \end{array}
\right.
\end{align*}
Then we have the duality datum $ \mathcal{D} = \{ \Delta(\beta_j), \mathsf{z}_j, \mathsf{R}_{j,k}  \}_{j,k\in J } $
and
$$\cmA^{\mathcal{D}} =
                        \left(
                         \begin{array}{cccccc}
                           2 & -2 & 0  &\cdots & 0 &0\\
                           -1 & 2 & -1  & \cdots  &0&0  \\
                           0 & -1 & 2  & \cdots &  0&0 \\
                           \vdots &  \vdots & \vdots & \ddots & \vdots& \vdots\\
                           0 & 0 &  0 & \cdots & 2 & -1 \\
                           0 & 0 &  0 & \cdots & -1 &2\\
                         \end{array}
                       \right)
$$
which is of type $B_{\ell-1}$. Therefore, we have the functor $ \F^{\mathcal{D}}  $ from the category of modules over quiver Hecke algebra of type $B_{\ell-1}$ to that of type $B_{\ell}$
sending
$$
\F^{\mathcal{D}}( L^{\mathcal{D}}(j)) \simeq \left\{
                                                      \begin{array}{ll}
                                                        L(1,2)  & \hbox{ if } j =1, \\
                                                        L(j+1) & \hbox{ otherwise,}
                                                      \end{array}
                                                    \right.
$$
where $L(1,2) = L(1,2)_{\mathsf{z}_1}/ \mathsf{z}_1 L(1,2)_{\mathsf{z}_1}$.


It is easy to check that $\{ \beta_1, \ldots, \beta_{\ell-1} \}$ satisfies $\eqref{Eq: condiotions on beta}$ and
$\cmA^{\mathcal{D}}$ is equal to the matrix $\overline{\cmA}$ defined by $\eqref{Eq: bar A}$. Thus, Proposition~\ref{prop:injcr} implies that
the functor $\F^{\mathcal{D}}$ categorifies
the injective homomorphism $U^-(\g^{\mathcal{D}}) \simeq U^-(\overline{\g}) \longrightarrow U^-(\g) $ and
 $\F^{\mathcal{D}}$ sends the simple modules to simple modules.
By Theorem \ref{Thm: duality and affinization}, for a simple $R^{\mathcal{D}}$-module $N$,
$$ \F^{\mathcal{D}} (  N \hconv \LD(j) ) \simeq \left\{
                                                         \begin{array}{ll}
                                                           \F^{\mathcal{D}}(N) \hconv L(1,2)  & \hbox{ if } j=1, \\
                                                            \F^{\mathcal{D}}(N) \hconv L( j+1)  & \hbox{ otherwise.}
                                                         \end{array}
                                                       \right.
 $$


\enEx

\vskip 0.5em

\Ex \label{Ex: B2}
We use the same notations $I$, $\cmA$, and $\qQ_{i,j}(u,v)$ as in Example \ref{Ex: B1}.

Let $J = \{1,2,\ldots, \ell-1 \}$ and
$$
\beta_1 = 2\alpha_1 + \alpha_2,\qquad \beta_j = \alpha_{j+1} \quad (j \in J\setminus \{1 \}).
$$
Note that $(\beta_j, \beta_j) = 4$ for all $j\in J$.
We define an $R(\beta_1)$-module structure on $L(1,1,2)_{\mathsf{z}_1} := \bR[\mathsf{z}_1] \otimes_{\bR} ( \bR u \oplus \bR v)$ by
\begin{align*}
&e(\nu)(a \otimes u) = \delta_{\nu, (1,1,2)} a \otimes u,
\qquad \qquad \quad \  e(\nu)(a \otimes v) = \delta_{\nu, (1,1,2)} a \otimes v,
 \\
&   x_j(a \otimes u) = \left\{
                       \begin{array}{ll}
                        -\mathsf{z}_1 a \otimes v & \text{if $j=1$,} \\
                         \mathsf{z}_1 a \otimes v  & \text{if $j=2$,} \\
                          \mathsf{z}_1 a \otimes u &\text{otherwise,}
                       \end{array}
                     \right.
 \quad
x_j(a \otimes v) = \left\{
                   \begin{array}{ll}
                     -  a \otimes u  &\text{if $j=1$,} \\
                     a \otimes u & \text{if $j=2$,}\\
                      \mathsf{z}_1 a \otimes v & \text{otherwise,}
                   \end{array}
                 \right.
\\
&  \tau_k (a \otimes u) = \left\{
                            \begin{array}{ll}
                              a \otimes v &\text{if $k=1$,} \\
                              0 & \text{if $k\not=1$,}
                            \end{array}
                          \right.
\qquad \quad  \tau_k (a \otimes v) = 0 \quad\text{for any $k$.}
\end{align*}
We put
\begin{align*}
\Delta(\beta_1) = L(1,1,2)_{\mathsf{z}_1}, \qquad \Delta(\beta_j) = L(j+1)_{\mathsf{z}_{j}} \quad (j\ne 1).
\end{align*}
Note that $\Delta(\beta_j)$'s are root modules and $\deg(\mathsf{z}_j) = 4$ for $j\in J$.
For $j,k\in J$ with $j\ne k$ and $p\otimes q \in \Delta(\beta_j) \otimes_\bR \Delta(\beta_k) $, we define
$$\mathsf{R}_{j,k} := R_{\Delta(\beta_j),\,\Delta(\beta_k)}
 \in \Hom_{R(\beta_j + \beta_k)}( \Delta(\beta_j)\circ \Delta(\beta_k), \Delta(\beta_k)\circ \Delta(\beta_j)  ).$$
Then,
\begin{align*}
\mathsf{R}_{k,j}\mathsf{R}_{j,k} =
\left\{
  \begin{array}{ll}
    \mathsf{z}_j - \mathsf{z}_k & \hbox{ if $k=j+1$,} \\
    1 & \hbox{ otherwise,}
  \end{array}
\right.
\end{align*}
for $j,k \in J$ with $j < k$.


Then, we have a duality datum $ \mathcal{D} = \{ \Delta(\beta_j), \mathsf{z}_j,  \mathsf{R}_{j,k}  \}_{j,k\in J} $
and
$$\cmA^{\mathcal{D}} =
                        \left(
                         \begin{array}{cccccc}
                           2 & -1 & 0  &\cdots & 0 &0\\
                           -1 & 2 & -1  & \cdots  &0&0  \\
                           0 & -1 & 2  & \cdots &  0&0 \\
                           \vdots &  \vdots & \vdots & \ddots & \vdots& \vdots\\
                           0 & 0 &  0 & \cdots & 2 & -1 \\
                           0 & 0 &  0 & \cdots & -1 &2\\
                         \end{array}
                       \right)
$$
is of type $A_{\ell-1}$. Therefore, we have the quiver Hecke algebra
$R^{\mathcal{D}}$ of type $A_{\ell-1}$ and
the functor $ \F^{\mathcal{D}}  $ between quiver Hecke algebras of type $A_{\ell-1}$ and $B_\ell$.
Moreover,
$$
\F^{\mathcal{D}}(\LD(j)) \simeq \left\{
                                                      \begin{array}{ll}
                                                        L(1,1,2)  & \hbox{ if } j =1, \\
                                                        L(j+1) & \hbox{ otherwise,}
                                                      \end{array}
                                                    \right.
$$
where $L(1,1,2) = L(1,1,2)_{\mathsf{z}_1}/ \mathsf{z}_1 L(1,1,2)_{\mathsf{z}_1}$.


One can easily show that $\{ \beta_1, \ldots, \beta_{\ell-1} \}$ satisfies $\eqref{Eq: condiotions on beta}$ and
$\cmA^{\mathcal{D}}$ is equal to the matrix $\overline{\cmA}$ defined by $\eqref{Eq: bar A}$. Thus,  Proposition~\ref{prop:injcr} implies
that the functor $\F^{\mathcal{D}}$ categorifies
the injective homomorphism $U^-(\overline{\g}) \longrightarrow U^-(\g) $
and $\F^{\mathcal{D}}$ preserves simple modules.
We have
$$ \F^{\mathcal{D}} (  N \hconv \LD(j) ) \simeq \left\{
                                                         \begin{array}{ll}
                                                           \F^{\mathcal{D}}(N) \hconv L(1,1,2)  & \hbox{ if } j=1, \\
                                                            \F^{\mathcal{D}}(N) \hconv L(j+1)  & \hbox{ otherwise,}
                                                         \end{array}
                                                       \right.
 $$
for a simple $R^{\mathcal{D}}$-module $N$ by Theorem \ref{Thm: duality and affinization}.



\enEx



\vskip 2em


