# When is the action of the braid group on tensor powers of Yetter-Drinfeld modules faithful?

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.

• 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. – Adrien Sep 4 '20 at 12:03
• Thank you for your thoughtful comment and the literature @Adrien. I edited my question to make it fit better. – user66288 Sep 5 '20 at 7:18
• 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. – user66288 Sep 5 '20 at 7:22
• 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. – Adrien Sep 5 '20 at 9:55