Let $V$ be a finite dimensional braided vector space over $\mathbb{C}$. This means that we have a map $$c_{V,V}:V\otimes V\to V\otimes V$$ which gives us an action of the braid group $B_n$ on $V^{\otimes n}$ by letting the generator $\sigma_i$ act on the $i$th and $i+1$st factors in $V^{\otimes n}$: $$\sigma_i\mapsto 1\otimes 1\otimes \cdots\otimes c\otimes\cdots\otimes 1$$ The homomorphism of groups $p_n:B_n\to S_n$ sending $\sigma_i$ to $s_i:=(i,i+1)$ does not split, but there is a canonical set-theoretical lifting $s_n:S_n\to B_n$, sending a permutation $\tau= s_{i_1}\cdots s_{i_k}$ to $\sigma_{i_1}\cdots \sigma_{i_k}$, assuming that $k$ is the minimal number for which such an expression of $\tau$ exists.

Define now the map $$ш_n:V^{\otimes n}\to V^{\otimes n}$$ $$ш_n = \frac{1}{n!}\sum_{\tau\in S_n} s_n(\tau)$$ for every $n$. The map $ш_n$ appears in the construction of the Nichols algebra of $V$. Indeed, it is known that one of the possible ways to define the Nichols algebra $B(V)$ of $V$ is by $$B(V)_n = V^{\otimes n}/ Ker(ш_n)$$

My question is the following:

Question: Is it true that $Ker(ш_n) = Ker(ш_n^2)$?

This must be true in case we know the operator $ш_n$ is semisimple. This happens for example in case the braiding on $V$ comes from a super vector space structure.

For a general braided vector space, the case $n=2$ is equivalent to the fact that the geometric and the algebraic dimension of the eigenvalue -1 of $c$ coincide. Is this still true in more general cases? For example when $V$ arises from some modular fusion category, or when the action of $B_n$ on $V^{\otimes n}$ splits via some finite quotient of $B_n$?

I have asked this more general question: lifting of idempotents in group ring But got no answer, so I am asking this more specific one again.


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