# Braidings for Comodules of Co-quasi-triangular Hopf algebra

Let $V$ be a (right-)$H$ comodule wrt a coaction $\Delta_R$, where $H$ is a co-quasi-triangular Hopf algebra with co-quasi-triangular Hopf algebra structure $R$. It is well-known that $V$ has a braiding $$\sigma : V \otimes V \to V \otimes V, ~~~~ v \otimes w \mapsto w^{(0)} \otimes v^{(0)} R(v^{(1)}\otimes w^{(1)}),$$ that commutes (of course) with the tensor product coaction; that is, $$\Delta_R \otimes \Delta_R \circ \sigma = (\sigma \otimes \text{id} ) \circ (\Delta_R \otimes \Delta_R).$$

Do there exist any other such braidings for $V$?

• I'm not entirely sure what you are asking. The vector space $V\otimes V$ is of course an $H$-comodule, and you can ask for all of its endomorphisms as an $H$-comodule, but at the level of generality of your question there's not much to say. Probably you mean to ask that the endomorphism extend to a representation of the Braid group on $V^{\otimes n}$. In any case, still in this generality I believe the answer is that there can be plenty, but there also can be some rigidity, so you may find more interesting answers by narrowing the Hopf algebras considered. Sep 14, 2011 at 23:32
• I suppose I'm looking for general examples. Sep 14, 2011 at 23:34
• As a somewhat trivial example, consider the Hopf algebra which is the algebra of functions on $\mathbb Z/2$. This Hopf algebra has two distinct co-triangular structures, one of which makes its corepresentation theory into the usual symmetric $\otimes$ category of $\mathbb Z/2$ representations, and the other makes it into the category of supervector spaces. These two coquasitriangular structures can be distinguished by their action on, say, the sign representation of $\mathbb Z/2$. For similar examples, braidings are classified by some homology group. Sep 14, 2011 at 23:37
• Sorry, I meant an example of another braiding for all comodules of a co-quasi Hopf alg. Or are any such braidings known? This is really my question. Sep 15, 2011 at 0:10
• Would you be happy with just three more, quite canonical examples? In that case you can invert $R$ or swap its components in some cases. Or do you want a complete classification of all possible braidings? (The latter is quite hard, I think) May 6, 2014 at 13:11

If $$c:V\otimes V\to V\otimes V$$ is a braiding on a finite dimensional vector space $$V$$ then $$A=A(V,c)$$, the FRT construction (equal to the free algebra on symbols $$t_i^j$$, $$i,j=1,\dots\dim V$$, modulo some quadratic relations using the $$t_i^j$$'s and the matrix coefficients of the map $$c$$) is a co-quasi-triangular bialgebra, and the map $$c$$ is realized as the example you mention. You can also add extra generators in order to get a Hopf algebra as well.
What you may have is maybe several co-quasi-triangular structures on the same Hopf algebra (think of $$k[\mathbb Z^n]$$ and braidings of diagonal type).