# Construction of “braided” Hopf algebras

Assume $(H,m,\eta, \Delta, \epsilon, S,r)$ to be a coquasitriangular Hopf-Algebra over $\mathbb C$. The category $C(H)$ of $H$-comodules is braided monoidal.

Now consider a coaction $\delta: H \rightarrow H \otimes_{\mathbb C} H$ of $H$ on itself such that $H$ is a comodule-algebra (algebra object in $C(H)$ . Can the antipode $S$ and comultiplication $\Delta$ of $H$ be "braided" in such a way that $\underline H:=(H, m, \eta, \underline \Delta, \epsilon, \underline S)$ in $C(H)$ will be a Hopf-Algebra object in $C(H)$?

To be a bit more precise I am asking for a way to define a new comultiplication $\underline \Delta$ and antipode $\underline S$ using the given braiding and existing comultiplication and antipode such that $\underline H$ is a Hopf algebra in $C(H)$.

This should be "similar" to transmutations of hopf algebras

• In terms of the "transmutation" theory, your problem can be solved in a somewhat different manner: If $\delta$ is the adjoint coaction $Ad$ of $H$ on itself then you can obtain such a braided group; not by redefining the comultiplication but rather by redefining the multiplication to obtain a braided multiplication. However, i am not sure if this is what you are interested in. If yes, i could provide more details on this point. – KonKan Jun 30 '17 at 15:39
• The construction of a braided group through a suitable redefinition of the comultiplication and the antipode is possible in the case which the original hopf algebra is quasitriangular rather than coquasitriangular. (In terms of the usual transmutation theory of hopf algebras). – KonKan Jun 30 '17 at 15:53
• I'm aware of the transmutation of a Hopf algebra (insofar that I know how to define it and how to prove that it's a braided Hopf algebra). It isn't what I'm looking for because it redefines the multiplication on $H$. But if you could give me a more conceptual overview on that topic I'd very much appreciate it. If I assume $H$ to be a quasitriangular Hopf algebra is its comodule category braided (and what is the braiding)? – SeHa Jul 3 '17 at 20:42
• I like the topic and i would like to try providing a more conceptual overview of that. But in order for such a try to be meaningful you should either edit your question or maybe ask a new one, since the question in your last comment seems quite different from your OP. – KonKan Jul 17 '17 at 19:40
• Now, regarding your last question: an "easy" case (in the sense that it admits a quick answer) is the case of self dual Hopf algebras such as for example groupf Hopf algebras $k\mathbb{G}$ of a finite abelian group $\mathbb{G}$. In this case the quasitriangularity (which makes the category of modules braided) is equivalent to coquasitriangularity (which makes the category of comodules braided) and the braiding can be computed via the corresponding R-matrix. – KonKan Jul 17 '17 at 19:43