\subsection{Actions of algebraic groups on affine varieties}
We recall the following framework and basic facts about actions of algebraic groups.
Let $\Gamma$ be a reductive algebraic group acting on an affine variety $X$.
The following holds (see Section 8.3 in \cite{Humphreys} and Lemma 3.3 in \cite{Newstead})
\begin{proposition}
	All the orbits of $\Gamma$ in $X$ are locally closed. For a $\Gamma$-orbit $\mathcal{O}$ in $X$, it holds that $\overline{\mathcal{O}}\backslash \mathcal{O}$ is the union of orbits of smaller dimension. 
\end{proposition}
\begin{proposition}\label{prop:sepinvs} If $W_1$ and $W_2$ are two closed disjoint $\Gamma$-stable subsets of $X$, then there is an element $f\in K[X]^{\Gamma}$ such that $f(W_1) = 1$ and $f(W_2)=0$. 
In other words- we can separate the subsets $W_1$ and $W_2$ by an invariant polynomial.
\end{proposition}

\end{section}

\begin{section}{Finite dimensional Hopf algebras in braided fusion categories}\label{sec:hopffusion}
Let $H$ be a finite dimensional Hopf algebra in a braided fusion category $\mathcal{C}$. 
The coradical filtration of $H$ is defined as follows (see Chapter IX of \cite{Sweedler} and \cite{AS98}):
\begin{itemize}
	\item $H_0\subseteq H$ is the coradical. It is the sum of all simple subcoalgebras of $H$. This is dual to the operation of taking the quotient of $H$ by its Jacobson radical. 
	\item For every $n> 0$ we define $H_n = \text{Ker} (H\stackrel{\Delta}{\to} H\otimes H\to (H/H_{n-1})\otimes (H/H_0)$. 
	Equivalently, $H_n = \text{Ker}\big(H\stackrel{\Delta^{n-1}}{\to}H^{\otimes n}\to (H/H_0)^{\otimes n}\big)$.
\end{itemize}
If $H$ satisfy the dual Chevalley property, i.e. the tensor product of semisimple $H$-comodules is again a semisimple $H$-comodule, then $H_0$ is also a subalgebra of $H$, and not only a subcoalgebra (see \cite{AGM}). In this case the filtration is a filtration of algebras as well, that is $H_i\cdot H_j \subseteq H_{i+j}$. 
As a result, the associated graded object \begin{equation}H_{gr} := \oplus H_n/H_{n-1}\end{equation} is a graded Hopf algebra. We will say that $H$ is \textit{coardically graded} if $H\cong H_{gr}$ as Hopf algebras. The grading gives a split surjection $\pi:H_{gr}\to H_0$ of Hopf algebras. 
Using the process of Bosonization (or Radford-Majid biproduct) one can also write this algebra in the form 
$H_{gr} = H_0\# R$ where $R$ is a graded Hopf algebra in the category of Yetter-Drinfeld modules over $H_0$.
As a vector space \begin{equation}R=\{r\in H_{gr}| (1\otimes \pi)\Delta(r) = r\otimes 1\in H_{gr}\otimes H_0\}\end{equation} and $R_0=\textbf{1}$. 
See \cite{AS10} for the description of $R$ as a Hopf algebra in $_{H_0}^{H_0}\mathcal{YD}$. 

The following lemma is the first step in proving Theorem \ref{thm:main2}:
\begin{lemma}\label{lem:genHgenR}
Let $A$ be a finite dimensional semisimple Hopf algebra. 
The following conditions are equivalent:
\begin{enumerate}
\item Every finite dimensional Hopf algebra $H$ in which the coradical $H_0$ is isomorphic to $A$ is generated by the zeroth and first levels of its coradical filtration.
\item Every coardically graded finite dimensional Hopf algebra $H$ in which the coradical $H_0$ is isomorphic to $A$ is generated by the zeroth and first levels of its coradical filtration.
\item Every coradically graded finite dimensional Hopf algebra $R\in ^A_A\mathcal{YD}$ in which $R_0=\textbf{1}$ is generated by its primitive elements.
\end{enumerate}
\end{lemma}
\begin{proof}
The first condition clearly implies the second one. On the other hand, if the second condition holds and $H$ is a Hopf algebra such that $H_0\cong A$, we can pass to the associated graded Hopf algebra $H_{gr}$. Since this Hopf algebra is coardically graded the second condition implies that it is generated by its zeroth and first terms of the coradical filtration, and the same thus holds also for $H$ (see Lemma 2.2. in \cite{AS98}).

We next prove that the second and third conditions are equivalent. 
Indeed, if $H$ is coradically graded then the above discussion implies that $H\cong H_0\# R$ where $R$ is a coradically graded Hopf algebra in $_A^A\mathcal{YD}$. It then holds that $H$ is generated by its first and zeroth terms of its coradical filtration if and only if the same holds for $R$. But this is equivalent to $R$ being generated by its primitive elements.
\end{proof}
\begin{lemma}\label{lem:pgen} (see also Lemma 5.5 in \cite{AS00})
Assume that $R$ is a Hopf algebra inside a braided fusion category $\mathcal{C}$.
If $R$ is generated by $\text{P}(R)$, and $R^*$ is generated by $\text{P}(R^*)$, then $R$ is isomorphic to $\text{B}(\text{P}(R))$ (that is: $R$ is a Nichols algebra).
\end{lemma}
\begin{remark} This proposition also holds if one replaces the braided fusion category with a finite braided tensor category.
\end{remark}
\begin{proof}
Write $\text{P}(R) = V$ and $\text{P}(R^*) = W$.
Write $\pi:T(V)\to R$ for the resulting surjective Hopf-algebra map in $\text{Ind}(\mathcal{C})$.
Let $p:T(V)\to\text{B}(V)$ be the canonical surjection. 
We will show that $\pi$ splits via $p$. 

Assume that $U\subseteq T(V)$ is a primitive subobject of degree $n>1$ (that is: $U\subseteq \text{P}(T(V))$). We will show that $\pi(U)=0$.
The primitivity of $W$ implies that 
$$\langle W,V^{\cdot n}\rangle = \langle \sum_{i=0}^{n-1}\epsilon^i\otimes W\otimes \epsilon^{n-1-i},V^{\otimes n}\rangle = 0.$$
So in particular $\langle W,\pi(U)\rangle = 0$. Using now the primitivity of $U$ (which also implies the primitivity of $\pi(U)$, since $\pi$ is a Hopf-algebra morphism) we get 
$$\langle W^{\cdot m},\pi(U)\rangle = \langle W^{\otimes m},\sum_{i=0}^{m-1}1^i\otimes \pi(U)\otimes 1^{m-i-1}\rangle=0$$
for every $m$. But this implies that $\pi(U)$ is perpendicular to the subalgebra of $R^*$ generated by $W$, which is $R^*$ itself.
This means that $\pi(U)=0$, so $U\subseteq \text{Ker}(\pi)$.  

This implies that the ideal $I_1\subseteq T(V)$ generated by primitive elements of degree $>1$ is contained in $\text{Ker}(\pi)$.
We thus get a surjective Hopf algebra map $\pi_1:T(V)/I_1\to R$. Denote by $I_2$ the ideal of $T(V)/I_1$ generated by primitive elements of degree $>1$ in this algebra. By the same argument, $I_2\subseteq \text{Ker}(\pi_1)$. We define now inductively ideals $I_n$ and Hopf algebra surjections $\pi_n:T(V)/I_n\to R$ such that $I_n\subseteq T(V)/I_{n-1}$ is the ideal generated by all primitive elements of degree >1. This is the same as the chain of ideals which appears after Definition \ref{def:Nichols}.
The union of the inverse images of the ideals $I_n$ inside $T(V)$ is exactly the kernel of the surjection $p:T(V)\to \text{B}(V)$. We thus get a surjective map 
$$\overline{\pi}:\text{B}(V)\to R$$ which is injective on $V$. 
This map must be injective as well, because if $n$ is the minimal number such that $\text{Ker}(\overline{\pi})_n\neq 0$ then $\text{Ker}(\overline{\pi})_n$ must be a primitive object by grading consideration, and by the definition of the Nichols algebra this is impossible. 
\end{proof}

\begin{lemma}\label{lem:pairing}
The pairing $\text{P}(R)\otimes \text{P}(R^*)\to \textbf{1}$ is non-degenerate if and only if $R\cong \text{B}(\text{P}(R))$.
\end{lemma}
\begin{proof}
One direction follows from the fact that the graded dual of $\text{B}(V)$ is $\text{B}(V^*)$, see \ref{lem:nicholsdual}. 
Assume, on the other hand, that the pairing is non-degenerate.
By the previous proposition, it will be enough to prove that $R^*$ is generated by $W=\text{P}(R^*)$.  
By a dual argument, $R$ is then generated by $V=\text{P}(R)$, and we can use the previous proposition to finish the proof.

Now for the coradical filtration $(R_i^*)$ of $R^*$ it holds that $R^*_0=K$ and $R^*_1=K\oplus W$. 
Since the coradical filtration is dual to the radical filtration, It holds that $R^*_1\cong (R/J^2)^*$ where $J$ is the Jacobson radical of $R$. The fact that the pairing between $V$ and $W$ is non-degenerate then means that the image of $V$ inside $R/J^2$ must be onto. But this means that $V$ generates $R$ as a unital algebra. 
By a similar argument $W$ generates $R^*$, and by the previous proposition $R$ is a Nichols algebra. 
\end{proof}

We are now ready to prove the equivalence of the third and fourth conditions of \ref{thm:main2}. 
\begin{proof}
Lemma \ref{lem:genHgenR} and Lemma \ref{lem:pgen}shows that the third condition of Theorem \ref{thm:main2} is equivalent to the statement that all finite dimensional coardically graded Hopf algebras $R\in ^A_A\mathcal{YD}$ with $R_0=\textbf{1}$ are Nichols algebras. 
If the fourth condition of Theorem \ref{thm:main2} holds and all the ccc Hopf algebras in $^A_A\mathcal{YD}$ are Nichols algebras, this is in particular true for coardically graded Hopf algebras with $R_0=\textbf{1}$, and the fourth condition thus implies the third condition. 

If on the other hand the third condition holds and $R$ is a ccc Hopf algebra in $^A_A\mathcal{YD}$ then the Hopf algebra $R_{gr}$ arising from the coradical filtration of $R$ is a Nichols algebra. This implies that $R$ is generated by its primitive elements. Similarly, the dual $R^*$ is also generated by its primitive elements, and by Lemma \ref{lem:pgen} $R$ is a Nichols algebra. 
\end{proof}
\end{section}

\begin{section}{Braided fusion categories by linear algebra}\label{sec:catslinear}
Let $\mathcal{C}$ be a braided fusion category.
Write $X_1,\ldots X_s$ for a set of representatives of the isomorphism classes of simple objects of $\mathcal{C}$. We would like to describe all the data encoded in the structure of $\mathcal{C}$ as a braided fusion category using vector spaces and linear maps.
We begin with the hom-spaces, which are very easy to describe. Indeed, it holds that 
\begin{equation}\label{eq:simples}\text{Hom}_{\mathcal{C}}(X_i,X_j) = \begin{cases} 0 & \text{ if } i\neq j \\
									K\cdot Id_{X_i} & \text{ if } i=j \end{cases}
								\end{equation}
since $\mathcal{C}$ is a semisimple category, and the $X_i$ are non-isomorphic simple objects. Every object of $\mathcal{C}$ is isomorphic to a direct sum of simple objects.
Instead of writing an object of $\mathcal{C}$ as $\bigoplus_i X_i^{\oplus a_i}$ we will use the isomorphic object $$\bigoplus_i U_i\otimes X_i$$ where $U_i$ are plain vector spaces. Notice that the fact that $\mathcal{C}$ is a $K$-linear category means that taking tensor products of objects in $\mathcal{C}$ with vector spaces makes sense.

The hom-spaces in $\mathcal{C}$ are then given by 
\begin{equation}\label{eq:homs} \text{Hom}_{\mathcal{C}}(\bigoplus _i V_i\otimes X_i,\bigoplus_j U_j\otimes X_j) = \bigoplus_i \text{Hom}_K(V_i,U_i)\end{equation}
We describe next the tensor product and the associativity constraints.
Assume that $$[X_i]\cdot [X_j] = \sum_k N_{i,j}^k [X_k]$$ in the Grothendieck ring of $\mathcal{C}$. For every three indices $i,j,k\in \{1,\ldots s\}$ fix a vector space $V_{i,j}^k$ of dimension $N_{i,j}^k$. We can then write
$$X_i\otimes X_j \cong  \bigoplus_k V_{i,j}^k\otimes  X_k.$$ Notice that 
$$\big(\bigoplus_i U_i\otimes X_i\big)\otimes \big(\bigoplus_j W_j\otimes X_j\big)\cong 
\bigoplus_{i,j,k} U_i\otimes W_j\otimes V_{i,j}^k\otimes X_k.$$ 
For $i,j,k\in\{1,\ldots s\}$ we then have
$$(X_i\otimes X_j)\otimes X_k\cong \bigoplus_a V_{i,j}^a\otimes X_a\otimes X_k\cong \bigoplus_{a,b} V_{i,j}^a\otimes V_{a,k}^b \otimes X_b$$ while on the other hand
$$ X_i\otimes (X_j\otimes X_k)\cong \bigoplus_c X_i\otimes V_{j,k}^c\otimes X_c\cong \bigoplus_{c,b} V_{i,c}^b\otimes V_{j,k}^c \otimes X_b.$$
The associativity constraints $$\alpha_{i,j,k}:(X_i\otimes X_j)\otimes X_k\to X_i\otimes (X_j\otimes X_k)$$ are then given by a family of linear maps 
\begin{equation}\alpha_{i,j,k}^{a,b,c}: V_{i,j}^a\otimes V_{a,k}^b\to V_{i,c}^b\otimes V_{j,k}^c.\end{equation}
The Pentagon axiom then translates into a list of axioms which say that 
certain sums of compositions of the linear maps $\alpha_{i,j,k}^{a,b,c}$ are equal. More precisely, for every $i,j,k,l\in \{1,\ldots s\}$ writing the pentagon diagram for the tensor product of $X_i\otimes X_j\otimes X_k\otimes X_l$ gives us that for every $a,b,c\in\{1,\ldots s\}$ the composition

$$V_{i,j}^a\otimes V_{a,k}^b\otimes V_{b,l}^c\stackrel{\alpha_{i,j,k}^{a,b,d}}{\longrightarrow} 
\bigoplus_d V_{i,d}^b\otimes V_{j,k}^d\otimes V_{b,l}^c\stackrel{\alpha_{i,d,l}^{b,c,e}}{\longrightarrow} $$
\begin{equation}\bigoplus_{d,e}\otimes V_{i,e}^c\otimes V_{j,k}^d V_{d,l}^e\stackrel{\alpha_{j,k,l}^{d,e,f}}{\longrightarrow} 
\bigoplus_{e,f} V_{i,e}^c \otimes V_{j,f}^c\otimes V_{k,l}^f\end{equation} is equal to the composition
\begin{equation} V_{i,j}^a\otimes V_{a,k}^b\otimes V_{b,l}^c\stackrel{\alpha_{a,k,l}^{b,c,f}}{\longrightarrow}
\bigoplus_f V_{i,j}^a\otimes V_{a,f}^c\otimes V_{k,l}^f\stackrel{\alpha_{i,j,f}^{a,c,e}}{\longrightarrow}q
\bigoplus_{f,e} V_{i,e}^c\otimes V_{j,f}^t\otimes V_{k,l}^f.\end{equation}

Assuming that $X_1=\textbf{1}$ is the tensor unit, the unit axioms for the monoidal category $\mathcal{C}$ can be translated as saying that $V_{1,i}^j$ and $V_{i,1}^j$ are zero if $i\neq j$, are one dimensional in case $i=j$, and that there are distinguished bases $l_i\in V_{1,i}^i$ and $r_i\in V_{i,1}^i$ such that for every $i,j,k\in\{1,\ldots s\}$ it holds that 
\begin{equation}\alpha_{i,1,j}^{i,k,j}:V_{i,1}^i\otimes V_{i,j}^k\to V_{i,j}^k\otimes V_{1,j}^j\end{equation} sends $r_i\otimes v$ to $v\otimes l_i$ for $v\in V_{i,j}^k$. 

The rigidity of the category can be described in this language in the following way:
for every $i\in \{1,\ldots s\}$ there is a unique $\bar{i}\in \{1,\ldots s\}$ such that $V_{i,\bar{i}}^1$ and $V_{\bar{i},i}^1$ are one-dimensional, and $V_{i,j}^1=0$ for $j\neq \bar{i}$.
The evaluation \begin{equation}ev_i:X_{\bar{i}}\otimes X_i\to X_1\end{equation} is then given by a linear isomorphism which we denote by the same symbol $ev_i:V_{\bar{i},i}\to K$ and the coevaluation $X_1\to X_i\otimes X_{\bar{i}}$ is then given by a linear isomorphism $coev_i:K\to V_{i,\bar{i}}$. The rigidity axioms translate again to equality between compositions of linear maps. 
The equality between the composition \begin{equation}X_i\to (X_i\otimes X_{\bar{i}})\otimes X_i\to X_i\otimes(X_{\bar{i}}\otimes X_i)\to X_i\end{equation} and $Id_{X_i}$ translates to the equality \begin{equation}K\to V_{i,\bar{i}}^1\otimes V_{1,i}^i\to V_{i,1}^i\otimes V_{\bar{i},i}^1\to K=K\stackrel{Id_K}{\to} K\end{equation} where the first map sends 1 to $coev_i\otimes l_i$, the second map is $\alpha_{i,\bar{i},i}^{1,1,i}$ and the third map sends $r_i\otimes v$ to $ev_i(v)\in K$. 
 
The braided structure is given by maps $\sigma_{X_i,X_j}:X_i\otimes X_j\to X_j\otimes X_i$. This is the same as a collection of linear isomorphisms
\begin{equation}\sigma_{i,j}^k:V_{i,j}^k\to V_{j,i}^k,\end{equation}
which should satisfy the axioms arising from the braid relations.

The introduction of the vector spaces $V_{i,j}^k$ here and the linear maps $\alpha_{i,j,k}^{a,b,c}$ and $\sigma_{i,j}^k$
can be seen as a way to ``introduce coordinates'' on the category $\mathcal{C}$. We use here the fact that as an abelian category, $\mathcal{C}$ is very simple to understand on the level of objects and morphisms. The additional braided monoidal structure is described using linear algebra. This will be used later on in the construction of the variety $X_B$.
This should be seen as more of an auxiliary result, and will not play a dominant role in the sequel.
\end{section}


\begin{section}{The variety $X_B$ and the action of the group}\label{sec:variety}
Usually, when one speaks of ``a Hopf algebra $R$ inside the braided fusion category $\mathcal{C}$'' it is understood that $R$ is an object of $\mathcal{C}$ which is equipped with structure maps which are not written explicitly.
We will take here a different approach. We will fix an object $B$ inside our braided fusion category $\mathcal{C}$, and ask what are \textit{all the possible} Hopf algebra structures one can give on that object. 

A Hopf algebra is given by morphisms $$m: B\otimes B\to B,$$ $$u:K\to B,$$ \begin{equation}\Delta:B\to B\otimes B\end{equation} $$\epsilon:B\to K \text{ and }$$ $$S:B\to B$$ which satisfy some axioms.
We can thus think of a Hopf algebra as a point in the affine space 
$$\mathbb{A}^N = \text{Hom}_{\mathcal{C}}(B\otimes B,B)\oplus\text{Hom}_{\mathcal{C}}(K,B)\oplus\text{Hom}_{\mathcal{C}}(B,B\otimes B)$$ \begin{equation} \oplus\text{Hom}_{\mathcal{C}}(B,K)\oplus\text{Hom}_{\mathcal{C}}(B,B).\end{equation}
We will write a point in this spaces as $(m,u,\Delta,\epsilon,S)$. 
Notice that not all points in this affine space will define Hopf algebra structure, and not all Hopf algebras will be ccc.
We write $$X_B\subseteq \mathbb{A}^N$$ for the subset of all points $(m,u,\Delta,\epsilon,S)$ which define a ccc Hopf algebra structure on $B$.
For $t=(m,u,\Delta,\epsilon,S)\in X_B$ we write $(B,t)$ for the Hopf algebra $B$ with structure given by $t$. When we will say ``$R$ is a Hopf algebra in $X_B$'' we will mean that $R = (B,t)$ and $t\in X_B$. We will also write $R\in X_B$.  

Write $\Gamma_B=\text{Aut}_{\mathcal{C}}(B)$. When no confusion will arise we will write $\Gamma = \Gamma_B$. The group $\Gamma$ acts on the different direct summands in $\mathbb{A}^n$ by conjugation. 
The action of $\Gamma$ on $B\otimes B$ is the diagonal one, and on $K$ is the trivial one. 
This induces a linear action of $\Gamma$ on $\mathbb{A}^N$. We claim the following:
\begin{lemma}
	The action of $\Gamma$ on $\mathbb{A}^N$ stabilizes the subset $X_B$. 
	Two points $(m_1,u_1,\Delta_1,\epsilon_1,S_1)$ and $(m_2,u_2,\Delta_2,\epsilon_2,S_2)$ in $X_B$ define isomorphic Hopf algebras 
	if and only if they lie in the same $\Gamma$-orbit. The stabilizer of $(m,u,\Delta,\epsilon,S)$ in $\Gamma$ can be identified with the group of automorphisms
	of the Hopf algebra that this tuple defines. 
\end{lemma}
\begin{proof}
	Write $t_i = (m_i,u_i,\Delta_i,\epsilon_i,S_i)$ for $i=1,2$. 
	The Hopf algebra axioms can be phrased as equalities between certain linear maps. Associativity of the multiplication, for example, is the equality 
	\begin{equation}m(1_B\otimes m)\alpha_{B,B,B} = m(m\otimes 1_B)\end{equation} as morphisms in $\text{Hom}_{\mathcal{C}}((B\otimes B)\otimes B,B)$. If $\gamma:B\to B$ is an automorphism in $\mathcal{C}$, and if 
	$\gamma(t_1)=(t_2)$ then we have that
	$$m_2(1_B\otimes m_2)\alpha_{B,B,B} = \gamma m_1 (\gamma^{-1}\otimes \gamma^{-1})(1_B\otimes \gamma m_1 (\gamma^{-1}\otimes\gamma^{-1}))\alpha_{B,B,B} = $$
	\begin{equation} \gamma m_1(1_B\otimes m_1)\alpha_{B,B,B} (\gamma^{-1}\otimes \gamma^{-1}\otimes \gamma^{-1})\end{equation} where we used the naturality of $\alpha$ and the definition of the action of $\gamma$. On the other hand
	a similar calculation gives us 
	\begin{equation}m_2(m_2\otimes 1_B) = \gamma m_1(m_1\otimes 1_B)(\gamma^{-1}\otimes\gamma^{-1}\otimes\gamma^{-1}).\end{equation}
	This shows that if $\gamma(t_1) = t_2$ then $m_1$ is associative if and only if $m_2$ is associative. 
	Similarly, all the other Hopf algebra axioms are valid for $t_1$ if and only if they are valid for $t_2$.
	So if $\gamma(t_1) = t_2$ then $t_1$ defines a Hopf algebra if and only if $t_2$ does.
	Moreover, $t_1$ will define a ccc Hopf algebra if and only if $\text{Ker}(\epsilon_1)$ is a nilpotent ideal in $B$ with respect to the multiplication $m_1$ 
	and $\text{Ker}(u_1^*)$ is a nilpotent ideal in $B^*$ with respect to the multiplication $\Delta_1^*$. 
	For the same reason as above, this happens if and only if $\text{Ker}(\epsilon_2)$ is a nilpotent ideal with respect to the multiplication $m_2$
	and $\text{Ker}(u_2^*)$ is a nilpotent ideal in $B^*$ with respect to the multiplication $\Delta_2^*$. 
	In other words, $\Gamma$ stabilizes the subset $X_B$ of $\mathbb{A}^N$. 
	 	
	We will think of the equation $\gamma(t_1) = t_2$ as saying that $\gamma$ defines an isomorphism between $(B,t_1)$ and $(B,t_2)$. 
	Indeed, $\gamma m_1 (\gamma^{-1}\otimes \gamma^{-1})=m_2$ can be rephrased as saying that the diagram 
	\begin{equation}\xymatrix{ B\otimes B\ar[r]^{\gamma\otimes \gamma}\ar[d]^{m_1} & B\otimes B\ar[d]^{m_2} \\ B\ar[r]^{\gamma} & B}\end{equation} is commutative.
	Similar statements hold for $u,\Delta,\epsilon$ and $S$. This implies that if $\gamma(t_1)=t_2$ then $t_1$ and $t_2$ define isomorphic Hopf algebras.
	On the other hand, if $t_1$ and $t_2$ define isomorphic Hopf algebras on $B$, take an isomorphism $\gamma:(B,t_1)\to (B,t_2)$ between these Hopf algebras.
	Then by the same calculations as above we get that $\gamma(t_1)=t_2$. 
	So the orbits of $\Gamma$ in $X_B$ are in one to one correspondence with isomorphism classes of ccc Hopf algebras which are isomorphic to $B$ as an object of $\mathcal{C}$.
	Finally, the equality $\gamma(t) = t$ for $t\in X_B$ just means that $\gamma:(B,t)\to (B,t)$ is an automorphism.
\end{proof}

The rest of this section will be devoted to prove the following claim:
\begin{lemma}
	The subset $X_B$ is an affine sub-variety of $\mathbb{A}^N$. The group $\Gamma$ is isomorphic to a direct product of general linear groups,
	and the action of $\Gamma$ on $\mathbb{A}^N$ is rational.
	\end{lemma}

\begin{proof} The proof of the lemma will be based on analyzing objects and morphisms in the category $\mathcal{C}$.
We begin by writing $B$ as \begin{equation}B=\bigoplus_i B_i\otimes X_i\end{equation} where $X_i$ are representatives of the isomorphism classes of simple objects of $\mathcal{C}$ and $B_i$ are vector spaces.
By Equation \ref{eq:homs} this already gives us an isomorphism 
$\Gamma\cong \prod_i \text{GL}(B_i)$. 
We choose a basis $\{e_{i1},\ldots e_{id_i}\}$ for $B_i$. 

A Hopf algebra structure on $B$ will be given by maps $m:B\otimes B\to B$, $u:\textbf{1}\to B$, $\Delta:B\to B\otimes B$, $\epsilon:B\to \textbf{1}$ and $S:B\to B$. By writing tensor products using the spaces $V_{i,j}^k$ from Section \ref{sec:catslinear} we see that the morphism $m$ is given by 
\begin{equation}\bigoplus_{i,j} B_i\otimes B_j\otimes X_i\otimes X_j\to \bigoplus_k B_k\otimes X_k.\end{equation}
Rewriting the first object using the vector spaces $V_{i,j}^k$ gives us \begin{equation}\bigoplus_{i,j,k}B_i\otimes B_j\otimes V_{i,j}^k\otimes X_k\to \bigoplus_k B_k\otimes X_k.\end{equation}
This means that $m$ is equivalent to a collection of linear maps \begin{equation}m_{i,j}^k:B_i\otimes B_j\otimes V_{i,j}^k\to B_k.\end{equation} Similarly, the morphism $u$ is equivalent to a map \begin{equation}u_1:K\to B_1,\end{equation}
the morphism $\Delta$ is equivalent to a collection of maps 
\begin{equation}\Delta_k^{i,j}:B_k\to B_i\otimes B_j\otimes V_{i,j}^k,\end{equation}
the morphism $\epsilon$ is equivalent to a map 
\begin{equation}\epsilon_1:B_1\to K\end{equation} and the antipode $S$ is equivalent to a collection of linear maps 
\begin{equation}S_i:B_i\to B_i.\end{equation}

We rewrite now the affine variety $\mathbb{A}^N$ as 
$$\mathbb{A}^N = \bigoplus_{i,j,k} Hom_K(B_i\otimes B_j\otimes V_{i,j}^k, B_k)\oplus Hom_K(K,B_1) \oplus $$ \begin{equation}\bigoplus_{i,j,k} Hom_K(B_k,B_i\otimes B_j\otimes V_{i,j}^k)\oplus Hom_K(B_1,K)\oplus \bigoplus_i Hom_K(B_i,B_i).\end{equation}

This description of $\mathbb{A}^N$ shows us that the action of  $\text{Aut}_{\mathcal{C}}(B) =\prod_i \text{GL}(B_i)$ on it is rational. Indeed, it is simply given by pre- and post-composing of linear maps.

A choice of bases for $B_i$ and for $V_{i,j}^k$ for all $i,j,k$ will give us a basis for $\mathbb{A}^N$. This enables us to describe the structure we have at hand, the tuple $t=(m,u,\Delta,\epsilon,S)$, as a collection of numbers, the \textit{structure constants} of $t$. Indeed, using the bases for $B_i$ and $V_{i,j}^k$ we can describe the different structure maps as linear maps between vector spaces with given bases, and these are just given by matrices of scalars.

We explain now why the subset $X_B$ is in fact an affine variety.
The idea is to show that all the Hopf algebra axioms can be expressed using polynomial equations. We will also show that the property of being ccc can be described using polynomial equations.

We begin with proving this for the associativity. The proof for the other Hopf algebra axioms is similar. 
The associativity axioms says that $m(m\otimes 1_B) = m(1_B\otimes m)\alpha_{B,B,B}$. Writing this using the maps $\alpha_{i,j,k}^{a,b,c}:V_{i,j}^a\otimes V_{a,k}^b\to V_{i,c}^b\otimes V_{j,k}^c$ and $m_{i,j}^k$ we get that associativity is equivalent to the commutativity of the diagram 
\begin{equation}\xymatrix{ B_i\otimes B_j\otimes B_k\otimes V_{i,j}^a\otimes V_{a,k}^b\ar[rr]^{\sum_c\alpha_{i,j,k}^{a,b,c}}\ar[d]^{m_{i,j}^a} & & \bigoplus_c B_i\otimes B_j\otimes B_k\otimes V_{i,c}^b\otimes V_{j,k}^c\ar[d]^{m_{j,k}^c} \\
B_a\otimes B_k\otimes V_{a,k}^b\ar[d]^{m_{a,k}^b} & & \bigoplus_c B_i\otimes B_c\otimes V_{i,c}^b\ar[d]^{m_{i,c}^b} \\ 
B_b\ar[rr]^{=} & & B_b } \end{equation}
where we simplified the morphisms by writing $\alpha_{i,j,k}^{a,b,c}$ instead of $1_{B_i\otimes B_j\otimes B_k}\otimes \alpha_{i,j,k}^{a,b,c}$ and similarly for the other morphisms. The linear map $m_{i,j}^a$ goes from  $B_i\otimes B_j\otimes V_{i,j}^a$ to $B_a$, but it is clear how to get a linear map from it as shown in the diagram.
If we write now the linear maps $m_{i,j}^k$ and $\alpha_{i,j,k}^{a,b,c}$ in terms of the bases of $B_i$ and $V_{i,j}^k$ we get a set of quadratic polynomials on the structure constants whose vanishing is equivalent to the associativity of $m$. For similar reasons the other Hopf algebra axioms can be written as well as polynomials in the structure constants. 

It is left to show that being ccc is a closed condition for Hopf algebras. For this, write $n=\sum_i dim_K(B_i)$. We claim the following:
\begin{lemma} An ideal $J$ of $(B,t)$ is nilpotent if and only if $J^n=0$.  
\end{lemma}
\begin{proof}
	One direction is clear. For the other direction, if $J$ is nilpotent, the sequence of ideals $J\supsetneq J^2\supsetneq J^3\cdots  $ is 
	strictly monotonic decreasing until it stabilizes at zero. Thus, the sequence $d_a:=\dim \text{Hom}_{\mathcal{C}}(\oplus_i X_i, J^a)$ satisfy 
	\begin{equation}n > d_{1} > d_{2}> \ldots \end{equation} and this sequence of numbers stabilizes at zero. This implies that if $J$ is nilpotent its $n$-th power must already be zero. 
\end{proof}

By definition, a Hopf algebra $(B,t)$ is ccc if it is connected and coconnected. We will show that being connected is a closed condition. The fact that coconnectedness is a closed condition follows from a dual argument. 
By definition of connectedness, $(B,t)$ is connected if the ideal $\text{Ker}(\epsilon)$ is nilpotent. This is equivalent to $\text{Ker}(\epsilon)^n=0$ by the above lemma. It holds that the map \begin{equation}P=1_B-u\epsilon:B\to B \end{equation}is a projection on $\text{Ker}(\epsilon)$ (this holds in any Hopf algebra, and follows from the fact that $\epsilon\circ u=\text{Id}_{\textbf{1}}$).
The nilpotency of $\text{Ker}(\epsilon)$ is thus equivalent to the fact that the map $$B^{\otimes n}\stackrel{P^{\otimes n}}{\longrightarrow} B^{\otimes n}\stackrel{m^{n-1}}{\longrightarrow} B$$ is the zero map.
But again, this can be written as a polynomial equation using the structure constants of $u$, $\epsilon$ and $m$. The subset $X_B$ is thus a subvariety of $\mathbb{A}^N$, and the action of $\Gamma$ on it is rational. The theory of algebraic groups and geometric invariant theory can thus be applied in our setting.
\end{proof}
\begin{definition}\label{def:leadsto}
If $R_1$ and $R_2$ are two Hopf algebras in $X_B$, we say that $R_2$ is a specialization of $R_1$ and write $R_1\leadsto R_2$ if $\mathcal{O}_{R_2}\subseteq \overline{\mathcal{O}_{R_1}}$.
\end{definition}
\begin{remark}
We have used here a slightly heavy categorical language, in order to construct the variety $X_B$ in the most general way possible. 
If, for example, the category $\mathcal{C}$ is $_G^G\YD$ there is a way around this: we can fix only the dimension of $B$ as a vector space, and consider also the action and coaction of $KG$ as part of the structure of $B$, instead of something that is given a-priori, as we have done here. Describing the isomorphism type of $B$ as an object in $\mathcal{C}$ can then be done by declaring what the trace of the operations of the elements of $D(G)$, the Drinfeld double of $G$, on $B$, should be. The construction here relies heavily on the fact that the category we are working in is semisimple. Indeed, the semisimplicity gives us an easy classification of the object of the category and their automorphism groups. See also \cite{AA18} for the study of Nichols algebras in non-semisimple braided monoidal categories.
\end{remark}
\begin{remark}\label{rmk:gradings} The Hopf algebras in $_G^G\YD$ which one encounters in the study of pointed Hopf algebras are usually graded. 
We study here ccc Hopf algebras and not graded algebras for two reasons. Firstly, 
being connected and coconnected is a conditions which can be described by polynomial equations. 
If we consider instead the variety of all graded Hopf algebras we will get something which is too rigid, and we will not be able to see the specializations in the orbits of $X_B$. Secondly, all finite dimensional graded Hopf algebras with $R_0=\textbf{1}$ are automatically ccc. Indeed, this follows from the fact that if $R=\bigoplus_{i\geq 0}R_i$ then the Jacobson radical is $\bigoplus_{i\geq 1}R_i$, and the quotient is isomorphic to $\textbf{1}$. The same holds for the dual. 
\end{remark}

\begin{remark}\label{rmk:duality} The map $R\mapsto R^*$ gives us an isomorphism of varieties $X_B\cong X_{B^*}$ which commutes with the action of $\text{Aut}_{\mathcal{C}}(B)\cong\text{Aut}_{\mathcal{C}}(B^*)$. In particular, if $R_1,R_2\in X_B$ then $R_1\leadsto R_2$ if and only if $R_1^*\leadsto R_2^*$ in $X_{B^*}$.
\end{remark}

The following lemma will be useful for the proof of Theorem \ref{thm:main3}.
\begin{lemma}\label{lem:subobj}
Assume that $R\leadsto R'$. Then $\text{P}(R)$ is isomorphic to a subobject of $\text{P}(R')$.
\end{lemma}
\begin{proof}
The object of primitive elements in $R$ is the same as the kernel of the map $$T_R=\Delta - u\otimes 1_R - 1_R\otimes u : R\to R\otimes R$$
We can write this map as the direct sum of maps $B_i\otimes X_i\to \oplus_{j,k}B_j\otimes B_k\otimes V_{i,j}^k\otimes X_i$. 
Such a map is thus equivalent to a collection of maps $(T_R)_i:B_i\to B_j\otimes B_k\otimes V_{i,j}^k$.
For every linear map $L$ and any natural number $m$ the condition that $rank(L)\leq m$ is a Zariski closed condition. 
This implies that for every $i$ it holds that $$rank((T_{R'})_i)\leq rank((T_R)_i)$$ and therefore
$$\dim(\text{Ker}(T_{R'})_i)\geq \dim(\text{Ker}((T_R)_i).$$
This gives us the desired result.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm:main3}]
If $R\leadsto \text{B}(V)$ then $\text{P}(R)$ is isomorphic to a subobject of $\text{P}(\text{B}(V)) = V$. 
Similarly $R^*\leadsto \text{B}(V)^*\cong \text{B}(V^*)$ by Lemma \ref{lem:nicholsdual} and Remark \ref{rmk:duality} so $\text{P}(R^*)$ is isomorphic to a subobject of $\text{P}(\text{B}(V^*))=V^*$. 
Since $V$ is simple, $V^*$ is simple as well, and it follows that $\text{P}(R)= 0$ or $V$ and $\text{P}(R^*)= 0$ or $V^*$.
The option $\text{P}(R^*)= 0$ is not possible, since this would imply that $J/J^2=0$ where $J$ is the Jacobson radical of $R$, and $R$ must then be the trivial algebra.
In a similar way, $\text{P}(R)=0$ is impossible. We are left with the situation where $\text{P}(R) = V$ and $\text{P}(R^*)=V^*$. 
But this already implies that $\text{P}(R)\otimes \text{P}(R^*)\to \textbf{1}$ is a non-degenerate pairing, which implies by Lemma \ref{lem:pairing} that $R$ is a Nichols algebra, as desired.
\end{proof}

\end{section}


\begin{section}{Filtrations of Hopf algebras}\label{sec:filtrations}
As we have seen in previous sections, understanding specializations is fundamental to understand ccc Hopf algebras. 
Using the Hilbert-Mumford criterion, we will show that if $\mathcal{O}_{R_1}\subseteq \overline{\mathcal{O}_{R_2}}$ and $\mathcal{O}_{R_1}$ is closed, the specialization $R_2\leadsto R_1$ follows from a filtration on $R_2$, in a way which we will describe now. Let $R=(B,t)$ be a ccc Hopf algebra in $\mathcal{C}$. 
\begin{definition}[see also \cite{Makhlouf}] A Hopf algebra filtration on $R$ is a chain of $\mathcal{C}$-subobjects of $R$ 
$$\cdots R_{-2}\subseteq R_{-1}\subseteq R_0\subseteq R_1\subseteq\cdots$$ such that the following properties hold:
$$R_i = R \text{ for } i>>0,$$
$$R_i=0 \text{ for } i<<0,$$
$$\forall i,j: m(R_i\otimes R_j)\subseteq R_{i+j},$$
\begin{equation}\forall k: \Delta(R_k)\subseteq \sum_{i+j= k} R_i\otimes R_j,\end{equation}
$$\epsilon(R_{-1})=0,$$
$$\text{Im}(u)\subseteq  R_0,$$
$$\forall i: S(R_i)\subseteq R_i.$$
We will denote the filtration by $(R_i)$. 
\end{definition}
\begin{lemma} Every ccc Hopf algebra filtration $(R_i)$ of $R$ determines a $\mathbb{Z}$-graded ccc Hopf algebra $R_{gr}$ with the same underlying object $B$, defined as follows:
\begin{enumerate}
\item As an object of $\mathcal{C}$, $$R=\oplus_{i\in \mathbb{Z}} R_i/R_{i-1}.$$ 
\item The unit $\overline{u}\in R_{gr}$ is the image of $u$ in $R_0/R_{-1}$.
\item The counit $\overline{\epsilon}:R_{gr}\to K$ is given by $\oplus R_i/R_{i-1}\to R_0/R_{-1}\stackrel{\epsilon}{\to} K$ where $\epsilon$ here is the map which is induced from $\epsilon$, since $\epsilon(R_{-1})=0$.
\item The condition on the antipode $S$ implies that it defines a collection of induced maps $\overline{S}_i:R_i/R_{i-1}\to R_i/R_{i-1}$. The antipode $\overline{S}$ of $R_{gr}$ is $\sum_i \overline{S}_i$.
\item The condition on $m$ implies that we have an induced map $\overline{m}_{i,j}:R_i/R_{i-1}\otimes R_j/R_{j-1}\to R_{i+j}/R_{i+j-1}$ for every $i,j\in \mathbb{Z}$. We define $\overline{m}=\sum_{i,j} \overline{m}_{i,j}$.
\item The condition on $\Delta$ implies that we have an induced map $\overline{\Delta}_{i,j}:R_{i+j}/R_{i+j-1}\to R_i/R_{i-1}\otimes R_j/R_{j-1}$ for every $i,j\in\mathbb{Z}$. We define $\overline{\Delta}=\sum_{i,j}\overline{\Delta}_{i,j}$. 
\end{enumerate}
\end{lemma}
\begin{proof}
All the maps for $R_{gr}$ are well defined due to the condition the structure maps of $R$ satisfy.
As we will prove in the next lemma, the condition on the filtration implies that $R\cong \oplus_i R_i/R_{i-1}$ as an object of $\mathcal{C}$. This implies that $R_{gr}\cong R\cong B$ as objects in $\mathcal{C}$. 
The fact that $R$ is a ccc Hopf algebra can be verified directly, or using the next lemma.
\end{proof}
\begin{lemma}
Let $(R_i)$ be a Hopf algebra filtration of the ccc Hopf algebra $R$. Then 
$\mathcal{O}_{R_{gr}}\subseteq \overline{\mathcal{O}_R}$. In particular, $\mathcal{O}_{R_{gr}}$ is also contained in the closed subset $X_B$, and as a result $R_{gr}$ is also a ccc Hopf algebra.
\end{lemma}
\begin{proof}
We first claim that we can write $R$ as $R=\oplus_{i\in \mathbb{Z}} T_i$ such that $R_i = \oplus_{j\leq i} T_j$ where $T_i$ are subobjects of $R$ in $\mathcal{C}$. This follows from the semisimplicity of $\mathcal{C}$ together with the conditions on the filtration $R_i$. Indeed, take $i<<0$ for which $R_i=0$. Define $T_j=0$ for all $j\leq i$. Then choose $T_{i+k}$ inductively to be a direct sum complement of $T_{i+k-1}$ in $R_{i+k}$. Notice that this implies that $T_j\cong R_j/R_{j-1}$ for every $j\in \mathbb{Z}$, and that $B\cong R\cong\oplus T_j\cong \oplus R_j/R_{j-1}\cong R_{gr}$ as objects of $\mathcal{C}$.  

Next, we write all the structure maps of $R$ in terms of the direct sum decomposition $R=\oplus_i T_i$. The conditions on the filtration gives us 
$$m=\sum_{i+j\geq k}m_{i,j}^k, 
\Delta= \sum_{i+j\leq k} \Delta_k^{i,j},$$
\begin{equation}u = \sum_{i\leq 0} u^i, \quad
\epsilon = \sum_{i\geq 0}\epsilon_i\end{equation}
$$S = \sum_{i\geq j}S_i^j$$
 where 
$$m_{i,j}^k:T_i\otimes T_j\to T_k,\quad \Delta_k^{i,j}: T_k\to T_i\otimes T_j,$$
\begin{equation}u^i:K\to T_i, \quad \epsilon_i:T_i\to K,\text{ and }\end{equation} $$S_i^j:T_i\to T_j.$$ 
Using the identification $R\cong \oplus_i T_i\cong \oplus_i R_i/R_{i-1}\cong R_{gr}$ we see that the multiplication in $R_{gr}$ is given by $\sum_{i,j}m_{i,j}^{i+j}$, the comultiplication is given by $\sum_{i,j}\Delta_{i+j}^{i,j}$, the unit by $u=u_0$, the counit is $\epsilon^0$, and the antipode is $\sum_i S_i^i$. We thus see that in passing from $R$ to $R_{gr}$ we ``deleted'' all the parts of the structure maps which are of positive degree and stayed with maps of degree zero (maps of negative degree do not appear here at all), where the degree of a map $T_{i_1}\otimes \cdots\otimes T_{i_r}\to T_{j_1}\otimes\cdots\otimes T_{j_m}$ is $i_1+i_2+\cdots + i_r-j_1-j_2-\cdots-j_m$. 

We use this idea to prove that $R_{gr}\in \mathbb{A}^N$ is in the closure of the orbit of $R$. This will already imply that $R_{gr}$ is a ccc Hopf algebra, because $\mathcal{O}_R\subseteq X_B$, and $X_B$ is closed. To prove this, we will use a so called \textit{one-parameter subgroup} of $\Gamma$. That is: a group homomorphism $\phi:\mathbb{G}_m\to \Gamma= \text{Aut}_{\mathcal{C}}(B)$. We define $\phi$ as follows
\begin{equation}\phi(\lambda) = \sum_{i\in \mathbb{Z}}\lambda^{-i}Id_{T_i}.\end{equation}
We claim that $\phi(\mathbb{G}_m)(R)$ contains $R_{gr}$ in its closure.
Since $\phi(\mathbb{G}_m)$ is a subgroup of $\Gamma$ this will be enough.
To prove this write \begin{equation}\phi(\lambda)(R) = (B,t_{\lambda}) =  (B,m_{\lambda},u_{\lambda},\Delta_{\lambda},\epsilon_{\lambda},S_{\lambda}).\end{equation} 
We get $$m_{\lambda} = \sum_{i+j\geq k } \lambda^{i+j-k}m_{i,j}^k,\quad
\Delta_{\lambda} = \sum_{i+j\leq k} \lambda^{k-i-j}\Delta_k^{i,j}$$
\begin{equation}u_{\lambda} = \sum_{i\leq 0} \lambda^{-i}u^i,\quad 
\epsilon_{\lambda} = \sum_{i\geq 0} \lambda^i\epsilon_i\end{equation}
$$S_{\lambda} = \sum_{i\geq j}\lambda^{i-j}S_i^j.$$
This description shows us that $\lim_{\lambda\to 0}\phi(\lambda)(R)$ exists, since in all the above expression $\lambda$ appears only with non-negative powers. Taking the limit $\lambda\to 0$ gives us $\lim_{\lambda \to 0}\phi(\lambda)(B,t)= (B,m',u',\Delta',\epsilon',S')$
where 
$$m' = \sum_{i,j}m_{i,j}^{i+j} = \overline{m}, \Delta' = \sum_{i,j}\Delta_{i+j}^{i,j}=\overline{\Delta}$$
\begin{equation}u' = u_0 = \overline{u}, \epsilon' = \epsilon_0 = \overline{\epsilon}\end{equation}
$$S' = \sum_i S_i^i = \overline{S}$$
This shows us that the limit point is exactly the structure constants of $R_{gr}$. We are done.
\end{proof}
\begin{remark}
If $R\in X_B$ and $\phi:\mathbb{G}_m\to \Gamma$ is any one-parameter subgroup for which $\lim_{\lambda\to 0}\phi(\lambda)(R)$ exists, 
then we get a filtration on $R$ by setting $$T_i= \text{Ker}(\phi(\lambda)-\lambda^{-i})\subseteq R\text{ for a generic } \lambda$$ $$R_i = \oplus_{j\leq i} T_j.$$
The fact that $\lim_{\lambda\to 0} \phi(\lambda)(R)$ exists implies, by the same argument as above, that $(R_i)$ is a Hopf algebra filtration. We thus see that the isomorphism classes of ccc Hopf algebras which appear on the boundary of $R$ by the action of a 1-parameter subgroup are exactly the ccc Hopf algebra arising from $R$ by a Hopf algebra filtration.
Moreover, by the Hilbert-Mumford criterion if $\mathcal{O}_{R'}\subseteq \overline{\mathcal{O}_R}$ and $\mathcal{O}_{R'}$ is closed, then there exists a one-parameter subgroup $\phi:\mathbb{G}_m\to \Gamma$ such that $R'=\lim_{\lambda\to 0}\phi(\lambda)(R)$ (see Theorem 1.4. in \cite{Kempf}). This implies that in order to understand specializations, and especially specializations to ccc Hopf algebras with closed orbits, we need to study Hopf algebra filtrations.
\end{remark}

We finish this section with two filtrations which are canonically associated to any ccc Hopf algebra: the radical and the coradical filtration. They are dual to one another, in a way which we shall explain below. As was explained in the introduction, the coradical filtration of a Hopf algebra (not necessarily a ccc one) is used in a fundamental way in the classification of non-semisimple Hopf algebras. The use of both filtrations together will play an important role in studying closure of orbits in this paper.

For the radical filtration, let $J=\text{Ker}(\epsilon)$ be the Jacobson radical of $R$. We claim the following:
\begin{lemma}
The filtration $R_i = J^{-i}$ for $i<0$ and $R_i=R$ for $i\geq 0$ is a Hopf algebra filtration.
\end{lemma}
\begin{proof}
Since $J$ is a nilpotent ideal, the condition $R_i=0$ for $i<<0$ holds. It is clear that the condition $R_i=R$ for $i>>0$ holds as well (it holds, in fact, for $i=0$).
The fact that $J^i\subseteq J^j$ when $j\leq i$ implies that $R_i\cdot R_j\subseteq R_{i+j}$. 
The condition $u\in R_0$ is immediate, and the condition $\epsilon(R_{-1})=0$ follows from the fact that $J=\text{Ker}(\epsilon)$. For the condition on $\Delta$, notice that $\Delta(J) \subseteq J\otimes R + R\otimes J$. For any $i,j\geq 0$ the braiding $\sigma:R\otimes R\to R\otimes R$ satisfies $\sigma(J^i\otimes J^j) = J^j\otimes J^i$. A direct calculation implies that $\Delta(J^k) \subseteq \sum_{i+j=k} J^i\otimes J^j$, which is what we wanted to prove. 
\end{proof}
We write $R_{gra}$ for the graded Hopf algebra arising from $R$ via the radical filtration.

We recall here also the definition of the dual filtriation, the coradical filtration, from Section \ref{sec:hopffusion}:
We define $R_i=0$ for $i<0$, $R_0 = \text{Im}(u)$ and $$R_n = \text{Ker}\Big(R\stackrel{\Delta^{n-1}}{\to}R^{\otimes n}\to (R/R_0)^{\otimes n}\Big)\text{ for } n>0.$$
Again, a direct verification, using the fact that the dual algebra is connected, reveals the fact that this is a filtration of Hopf algebras as well. We write $R_{grc}$ for the graded Hopf algebra associated to this filtration. We thus see that $R\leadsto R_{gra}$ and $R\leadsto R_{grc}$ for every ccc Hopf algebra $R$ in $X_B$. 
The two filtrations are dual to one another in the following sense: For $i\in \mathbb{Z}$ let $R_i$ be the $i$-th level of the coradical filtration, and let $X_i\subseteq R^*$ be defined as \begin{equation}X_i = \text{Ker}(R^*\to R_{-i}^*).\end{equation}
Then it holds that $(X_i)$ is the radical filtration on $R^*$. 
This duality induces isomorphisms $(R_{gra})^*\cong (R^*)_{grc}$ and $(R_{grc})^*\cong (R^*)_{gra}$.  
 \end{section}


\begin{section}{A proof of Theorem \ref{thm:main1} and \ref{thm:main4}}\label{sec:proofthm1}
In this section we prove that a ccc Hopf algebra in $X_B$ has a closed orbit if and only if it is isomorphic to a Nichols algebra.
Let $R$ be a Hopf algebra in $X_B$. Recall the associated graded Hopf algebras $R_{gra}$ and $R_{grc}$ from Section \ref{sec:filtrations}.
We know that $R\leadsto R_{gra}$ and $R\leadsto R_{grc}$. In particular, if the orbit of $R$ is closed we get that $R\cong R_{grc}$ and $R\cong R_{gra}$. The following proposition shows that if $\mathcal{O}_R$ is closed then $R$ is a Nichols algebra. 
\begin{proposition}\label{prop:gragrc}
Assume that the Hopf algebras $R_{gra}$ and $R_{grc}$ are isomorphic as Hopf algebras. 
Then $R$ is a Nichols algebra.
\end{proposition}
\begin{proof}
Assume that the two Hopf algebras are isomorphic.
Write $R_{gra} = \bigoplus_{i=0}^n A_i$ where $A_i = J^i/J^{i+1}$ (for the sake of simplicity, we use here positive grdaing instead of the negative grading from Section \ref{sec:filtrations} for the radical filtrations). Then the algebra $R_{gra}$ is generated by the elements in $A_1$. These elements are also primitive, since $\Delta(A_1)\subseteq A_1\otimes A_0 + A_0\otimes A_1$ and $A_0\cong R/J\cong \textbf{1}$. 
This implies, in particular, that $R_{gra}$ is generated by primitive elements.
The algebra $R_{grc} = \oplus_{i=0}^m C_i$ has all primitives in degree 1. Since it is isomorphic to the algebra $R_{gra}$, it is also generated by these elements. This implies that $R_{grc}$ is a graded Hopf algebra which is generated in degree 1 and has all its primitive elements in degree 1. By Definition \ref{def:Nichols} this implies that $R_{grc}$ is a Nichols algebra. It follows that $R$ is generated by primitive elements.
By a dual argument, and by using the fact that $(R_{gra})^*\cong (R^*)_{grc}$ and $(R_{grc})^*\cong (R^*)_{gra}$ we get that $R^*$ is also generated by its primitive elements. Lemma \ref{lem:pgen} gives us the desired result.
\end{proof}

This finishes the proof that if $\mathcal{O}_R$ is closed then $R$ is a Nichols algebra, because the fact that $R\leadsto R_{grc}$ and $R\leadsto R_{gra}$ together with the closure of $\mathcal{O}_R$ implies that $R_{grc}\cong R\cong R_{gra}$. 
Next, we will show that if $R$ is a Nichols algebra then $\mathcal{O}_R$ is closed.
Assume that this is not the case and let $R=\text{B}(V)$ be a Nichols algebra with a non-closed orbit. The closure $\overline{\mathcal{O}_R}$ is the union of the orbit of $R$ with orbits of smaller dimension.
An orbit of minimal dimension in $\overline{\mathcal{O}_R}$ is closed. It follows that $R\leadsto R'$ for some $R'\in X_B$ with $\mathcal{O}_{R'}$ closed. 
But we already know that this implies that $R'$ is a Nichols algebra. 

Write $R'=\text{B}(V')$. Then we have $\text{B}(V)\leadsto \text{B}(V)'$. 
By \ref{lem:subobj} we know that this implies that $\text{P}(\text{B}(V))=V$ is isomorphic to a subobject of $\text{P}(\text{B}(V'))=V'$. 
Since $R\ncong R'$ the object $V$ must be isomorphic to a proper subobject of $V'$. Write $V'=V\oplus V''$. 
The split inclusion of objects in $\mathcal{C}$, $V\to V'\to V$, induces a split inclusion of Nichols algebras $\text{B}(V)\to \text{B}(V')\to \text{B}(V)$. 
This implies that $\text{B}(V)\ncong \text{B}(V')$ because then $\text{B}(V')$ properly contains $\text{B}(V)$, and this contradicts the fact that $R$ and $R'$ are isomorphic to the same object of $\mathcal{C}$.
This finishes the proof of Theorem \ref{thm:main1}

The proof of Theorem \ref{thm:main1} gives us a description of the irreducible components of $X_B$:
\begin{theorem}
Let $B\in \mathcal{C}$. For every $V\in \mathcal{C}$ such that $\text{B}(V)\in X_B$ write $X_V= \{R| R\in X_B\text{ and } R \leadsto \text{B}(V)\}$. 
Then the subsets $X_V$ are stable under the action of $\Gamma$ and are exactly the connected components of $X_B$. 
\end{theorem}
\begin{proof}
The fact that $X_V$ is stable under $\Gamma$ is immediate.
Notice that for dimension considerations the number of objects $V$ such that $\text{B}(V)\in X_B$ is finite.
We denote these objects by $V_1,V_2,\ldots V_m$.
We claim now that the dimension of the invariant subalgebra $K[X_B]^{\Gamma}$ is finite.
Indeed, if $R\leadsto \text{B}(V)$ then continuity considerations imply that $f(R)=f(\text{B}(V))$ for every $f\in K[X_B]^{\Gamma}$.
This implies that $$\phi: K[X_B]^{\Gamma}\to K^m$$ $$f\mapsto (f(\text{B}(V_1)),\ldots, f(\text{B}(V_m)))$$ is an injective algebra homomorphism.
Since $\text{B}(V_i)\ncong \text{B}(V_j)$ when $i\neq j$ and $\mathcal{O}_{\text{B}(V_i)}$ and $\mathcal{O}_{\text{B}(V_j)}$ are closed and disjoint Proposition \ref{prop:sepinvs} implies that for every $i\neq j$ there is a function $f_{ij}\in K[X_B]^{\Gamma}$ such that $f_{ij}(\text{B}(V_i))\neq f_{ij}(\text{B}(V_j))$. Since $\phi$ is an algebra map this implies that $\phi$ is also surjective and thus an isomorphism.
We thus see that all the $X_{V_i}$ are closed, since $X_{V_i}$ is the zero set of the polynomial $1-\phi^{-1}(e_i)$ (where $\{e_i\}$ is the standard basis of $K^m$).

We next claim that $X_{V_i}$ is connected for every $i$. Indeed, assume that $X_{V_i}= Y_1\sqcup Y_2$ with $Y_1$ and $Y_2$ closed and nonempty. 
Take $y_1\in Y_1$. Then $\mathcal{O}_{y_1}=\Gamma\cdot y_1$ is connected, contained in $X_{V_i}$ and intersects $Y_1$, so $\mathcal{O}_{y_1}\subseteq Y_1$. But then $\mathcal{O}_{\text{B}(V_i)}\subseteq \overline{\mathcal{O}_{y_1}}\subseteq Y_1$. By a similar argument $\mathcal{O}_{\text{B}(V_i)}\subseteq Y_2$ and this is a contradiction.
\end{proof}

Due to the last theorem, we can focus our attention on the different subvarieties $X_V$. These subvarieties are stable under the action of $\Gamma$. 
The conditions in Theorem \ref{thm:main5} then boil down to the statement that if $\text{B}(V)$ is finite dimensional, then the variety $X_V$ has a single orbit under the action of $\Gamma$. 
We finish with a proof of Theorem \ref{thm:main4}:
\begin{proof}[Proof of Theorem \ref{thm:main4}]
Assume that $\text{B}(V)$ is not rigid. In other words, assume that there are non-closed orbits in $X_V$. 
Take a non-closed orbit $\mathcal{O}_R$ of minimal dimension in $X_V$. We will prove that such an orbit is the orbit of a pre-Nichols algebra or a post-Nichols algebra.
For this consider the Hopf algebras $R_{gra}$ and $R_{grc}$. If both these Hopf algebras are isomorphic to $R$, then $R\cong R_{gra}\cong R_{grc}$, which implies that $R$ itself is a Nichols algebra by Proposition \ref{prop:gragrc}. This is a contradiction.
If both these algebras are non-isomorphic to $R$, then from the fact that $R_{grc},R_{gra}\in \overline{\mathcal{O}_R}$ it follows that the dimensions of the orbits $\mathcal{O}_{R_{gra}}$ and $\mathcal{O}_{R_{grc}}$ are smaller than the dimension of $\mathcal{O}_R$. By the minimality condition on $R$, this implies that $R_{grc}\cong R_{gra}\cong \text{B}(V)$. But By Proposition \ref{prop:gragrc} again, this implies that $R$ itself is a Nichols algebra, which leads again to a contradiction.

We thus see that either $R\cong R_{gra}$ and $R\ncong R_{grc}$, or $R\ncong R_{gra}$ and $R\cong R_{grc}$. Assume first that $R\cong R_{gra}$ and $R\ncong R_{grc}$. 
Since $R\cong R_{gra}$, $R$ has a grading $R= \oplus R_i$ such that the Jacobson radical $J$ satisfy $J^i = \oplus_{j\geq i} R_j$ for every $i\geq 1$. The grading implies that $R_1$, which generate $R$ as an algebra, is a primitive object. But this already implies that $R$ is a pre-Nichols algebra, as it lies between $T(R_1)$ and $\text{B}(R_1)$. Notice that it is impossible that $R_1\cong V$. Indeed, if this was the case then from dimension considerations the fact that $R$ projects onto $\text{B}(R_1)$ would imply that $R\cong \text{B}(R_1) = \text{B}(V)$, contradicting the fact that the orbit of $R$ is not closed. 
By Lemma \ref{lem:subobj} it follows that $R_1$ is isomorphic to a subobject of $V$. We thus see that it must be a proper subobject. 

This shows that if $R\cong R_{gra}$ then $R$ is a pre-Nichols algebra.
If $R\cong R_{grc}$ then by duality of the radical and coradical filtrations we get that $R^*$ is a pre-Nichols algebra. This finishes the proof of Theorem \ref{thm:main4} and also of \ref{thm:main5}
\end{proof}
\end{section}

\begin{section}{Different notions of rigidity}\label{sec:rigidity}
In this paper we call a ccc Hopf algebra $R$ rigid if any Hopf algebra which speciailzes to it is isomorphic to it. 
There are other notions of rigidity, using deformations by a one-parameter family. We will explain here the relations between them. 

In \cite{AKM} and \cite{DCY} a deformation of a graded bialgebra (not necessarily a finite dimensional one) $B=\oplus_{i\geq 0} B_{(i)}$ by a parameter $\lambda$ is defined as a pair $(m_{\lambda},\Delta_{\lambda})$ such that \begin{equation}m_{\lambda} = \sum_{i=0}^{\infty} m_{(i)}\lambda^i\text{ and }\Delta_{\lambda} = \sum_{i=0}^{\infty} \Delta_{(i)}\lambda^i\end{equation} where $m_{(i)},\Delta_{(i)}$ are maps of degree $-i$, $m_{(0)}=m$ and $\Delta_{(0)}=\Delta$, and $(m_{\lambda},\Delta_{\lambda},u,\epsilon)$ defines a bialgebra structure on $B$ for every $\lambda$. (Bialgebra deformations of Hopf algebras are automatically Hopf algebras as well. Due to the uniqueness of the antipode, we do not need to consider it as part of the deformation data). 
It is shown that this is the same as a filtered Hopf algebra $U$ which satisfies $grU\cong B$. In \cite{AKM} and \cite{DCY} a graded Hopf algebra $B$ is called rigid if it has no non-trivial deformations. We will call it here deformation rigid. 
We claim the following:
\begin{lemma}
A finite dimensional Nichols algebra $\text{B}(V)$ is rigid with respect to Definition \ref{def:rigidity} if and only if both $\text{B}(V)$ and $\text{B}(V^*)$ are deformation rigid.
\end{lemma}
\begin{proof}
Remark \ref{rmk:duality} implies that $\text{B}(V)$ is rigid if and only if $\text{B}(V^*)$ is rigid (with respect to Definition \ref{def:rigidity}).
To prove the first direction it will thus be enough to show that if $\text{B}(V)$ is rigid with respect to Definition \ref{def:rigidity} then it is deformation rigid.

Assume that $(m_{\lambda},\Delta_{\lambda})$ is a one-parameter deformation of $\text{B}(V)$. 
Using the grading $\text{B}(V)=\oplus \text{B}(V)_i$ we have a one-parameter family $\phi:\mathbb{G}_m\to \text{Aut}_{\mathcal{C}}(B)$ which sends $\lambda$ to the automorphism which acts on $\text{B}(V)_i$ by the scalar $\lambda^{-i}$. We then have that 
$$(m_{\lambda},\Delta_{\lambda}) = \phi(\lambda)\cdot (m_1,\Delta_1)\text{ and }(m_0,\Delta_0)=\lim_{\lambda\to 0}\lambda\cdot (m_1,\Delta_1).$$ 
Since $\text{B}(V)$ is rigid, any algebra specializing to it is isomorphic to it. So we get a Hopf algebra isomorphism $\Psi:\text{B}(V)\cong (B,m_1,\Delta_1,u,\epsilon,S_1)$ where $S_1$ is the uniquely defined antipode. 

Next, we claim that $\text{B}(V)_1$ consists of primitive elements with respect to the comultiplication $\Delta_1$ (and therefore, with respect to $\Delta_{\lambda}$ for every $\lambda\in K^{\times}$ as well). Indeed, by grading consideration we have that $\Delta_1|_{\text{B}(V)_1} = \Delta{(0)} + \Delta_{(1)}$, where $\Delta_{(0)}=\Delta$. The map $\Delta_{(1)}:\text{B}(V)_1\to \text{B}(V)_0\otimes \text{B}(V)_0=\textbf{1}\otimes\textbf{1}= \textbf{1}$ must be zero since otherwise this will contradict the fact that $\epsilon(\text{B}(V)_1)=0$. 
This implies that the isomorphism $\Psi$ will map $\text{B}(V)_0$ to $\text{B}(V)_0$ and $\text{B}(V)_1$ to 
$\text{B}(V)_1$. Without loss of generality we can assume that $\Psi|_{\text{B}(V)_1} = Id_{\text{B}(V)_1}$
(recall that both $\text{B}(V)$ and the deformed algebra have the same underlying object $B$).
Since $\text{B}(V)_n$ is the image of $\text{B}(V)_1^{\otimes n}\to \text{B}(V)$ with respect to the iterated multiplication $m$, and since $\bigoplus_{i\leq n}\text{B}(V)_i$ contains the image of $\text{B}(V)_1^{\otimes n}\to \text{B}(V)$ under the iterated multiplication $m_1$, we get that $\Psi$ preserves the filtration $B_{(0)}\subseteq \text{B}_{(1)}\subseteq\cdots $ of $B$ given by $\text{B}_{(n)} = \bigoplus_{i\leq n}\text{B}(V)_i$. But this already implies that the deformation $(m_{\lambda},\Delta_{\lambda})$ is trivial with respect to the definition in Section 2.3 of \cite{AKM}.

In the other direction, assume that $\text{B}(V)$ and $\text{B}(V^*)$ are deformation rigid. 
By Theorem \ref{thm:main5} we know that if $\text{B}(V)$ is not rigid then there is a pre-Nichols Hopf algebra $R$ such that $R\leadsto \text{B}(V)$ or $R\leadsto \text{B}(V^*)$. Since both $\text{B}(V)$ and $\text{B}(V^*)$ are deformation rigid, we can assume without loss of generality that the first case holds.
The algebra $R$ is generated by its primitive elements, and therefore admits a filtration $$R_{(0)}\subseteq R_{(1)}\subseteq\cdots $$ where $R_{(n)} = \sum_{i=0}^n \text{P}(R)^i$. This can easily be seen to be a Hopf algebra filtration of $R$. 
The associated graded object is then $\text{B}(V)$ with its usual grading. Since we assumed that $\text{B}(V)$ is deformation rigid we get that $R\cong \text{B}(V)$, which is a contradiction.
\end{proof}

We recall here Theorem 6.2 from \cite{AKM}. This result relies on the classification result from \cite{A1}.
\begin{theorem} Assume that $\text{B}(V)$ is finite dimensional and that the braiding on $V$ is of diagonal type. Then $\text{B}(V)$ is deformation rigid.
\end{theorem}
This theorem, combined with Theorem \ref{thm:main5} gives a new proof for the generation of pointed Hopf algebras with abelian groups of group-like elements by group-like elements and skew-primitives.
Indeed, the above theorem implies that all the finite dimensional Nichols algebra in $_G^G\YD$ for $G$ abelian are rigid, and therefore by Theorem \ref{thm:main5} all finite dimensional ccc Hopf algebras in $_G^G\YD$ are Nichols algebras, and every Hopf algebras $H$ such that $H_0=KG$ with $G$ abelian is generated by group-like elements and skew-primitives.

\end{section}
\begin{section}*{Acknowledgments}
I would like to thank Nicol{\'a}s Andruskiewitsch, Iv{\'a}n Angiono and Istvan Heckenberger for their useful comments.
\end{section}

