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Let $V$ be a Yetter-Drinfeld module over a Hopf algebra $H$ with invertible antipode. Recall that $V$ is a braided vector space with braiding $\Psi\colon V\otimes V\to V\otimes V, v\otimes w\mapsto v_{(-1)}w\otimes v_{(0)}$. The braid group $\mathbb{B}_r$ with $r-1$ generators $\sigma_1,\ldots,\sigma_{r-1}$ acts on $V^{\otimes r}$ by $\sigma_ix=\Psi_{i,i+1}x$ where $x\in V^{\otimes r}$ and where $\Psi_{i,i+1}$ is the braiding of the $i$th and $(i+1)$th tensor factor.

Question. Does there exist a Yetter-Drinfeld module $V$ over some Hopf algebra $H$ with invertible antipode such that the action of $\mathbb{B}_r$ on $V^{\otimes r}$ is faithful for some $r>1$?

A proof in special cases, e.g. $H=\mathbb{C}\Gamma$ for a group $\Gamma$, is welcome. I am interested in the ground field $\mathbb{C}$ and don't want positive characteristic examples.

If you know related literature, this is accepted as an answer as well.

EDIT. What I proved so far and which leads me to this question is as follows:

For each $r>1$, there exists a Yetter-Drinfeld module $V$ over $\mathbb{S}_{r+1}$ such that $t(\sigma)x=x$ for all $x\in V^{\otimes r}$ and some $\sigma\in\mathbb{S}_r$ implies $\sigma=1$. Here, $t\colon\mathbb{S}_r\to\mathbb{B}_r$ is the Matsumoto section.

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    $\begingroup$ The answer at this level of generality is clearly no (e.g. because the trivial module is tautologically a Yetter-Drinfeld module). Even excluding this trivial case I don't see why you'd expect these representations to be faithful. Representations coming from f.d. modules over quantum groups aren't faithful either. Constructing faithful representations of $B_n$ is a hard problem, see e.g. numdam.org/item/SB_1999-2000__42__389_0. In particular I doubt you'll find an example of a group $\Gamma$ which works. $\endgroup$ – Adrien Sep 4 '20 at 12:03
  • $\begingroup$ Thank you for your thoughtful comment and the literature @Adrien. I edited my question to make it fit better. $\endgroup$ – user66288 Sep 5 '20 at 7:18
  • $\begingroup$ Note that the Yetter-Drinfeld module $V$ can well be infinite dimensional. I have no problem with that. If the answer is affirmative for all $r>1$ simultaneously, it would be even better, but I doubt that this is possible. $\endgroup$ – user66288 Sep 5 '20 at 7:22
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    $\begingroup$ The Lawrence--Krammer representations which is faithful can apparently be obtained from the braid action on some infinite dimensional $U_q(\mathfrak{sl}_2)$-modules arxiv.org/abs/0912.2114. Modulo subtleties I guess those can be thought of as Yetter-Drinfeld modules over $U_q(\mathfrak{b}^+)$ where $\mathfrak b^+$ is the positive Borel sub-Lie algebra. Note that more generally representations coming from quasi-triangular Hopf algebras can all be interpreted as YD modules over an appropriate sub-Hopf algebra, so you could as well ask the question for those. $\endgroup$ – Adrien Sep 5 '20 at 9:55

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