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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 $V$?

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  • $\begingroup$ 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. $\endgroup$
    – Theo Johnson-Freyd
    Commented Sep 14, 2011 at 23:32
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    $\begingroup$ I suppose I'm looking for general examples. $\endgroup$
    – Dyke Acland
    Commented Sep 14, 2011 at 23:34
  • $\begingroup$ 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. $\endgroup$
    – Theo Johnson-Freyd
    Commented Sep 14, 2011 at 23:37
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    $\begingroup$ 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. $\endgroup$
    – Dyke Acland
    Commented Sep 15, 2011 at 0:10
  • $\begingroup$ 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) $\endgroup$
    – Manuel Bärenz
    Commented May 6, 2014 at 13:11

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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.

So, your example is so general that there is not more examples..

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).

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