References on $\text{Rep}(H)$ is a braided tensor category

Question

There is the statement:

Let $H$ be a Hopf algebra, then $\text{Rep}(H)$ is a braided tensor category.

Does anybody know some references on this? Is it covered in Kassel's 'Quantum Groups'?

Answer 1

The correct statement is

Let $H$ be a quasi-triangular Hopf-algebra, then $\text{Rep}(H)$ is a braided monoidal category.

This statement is covered in Kassel, see proposition XIII 1.4 in his chapter on braided categories. The literal statement is stronger:

Let $H$ be a bialgebra. Then $\text{Rep}(H)$ is braided if an only if $H$ is a braided algebra (also called a quasi-triangular structure).

The easiest braiding obviously is a flip map. Notice that if $H$ is a cocommutative Hopf-algebra (i.e. the quasi-triangular structure is trivial), then $\text{Rep}(H)$ is trivially braided.

Answer 2

The following is supplementary to the answer provided by user Mathematician 42 (in the sense that it provides an alternative citation apart from Kassel's book already mentioned above and some further remarks):

Theorem: Let $B$ a bialgebra. Then

1. $B$ is quasitriangular $\Leftrightarrow$ the category ${}_{B}\mathcal{M}$ of left $B$-modules, is a braided monoidal category.
2. $B$ is coquasitriangular $\Leftrightarrow$ the category $\mathcal{M}^{B}$ of right $B$-comodules, is a braided monoidal category.

Moreover, $B$ is triangular (resp. cotriangular) $\Leftrightarrow$ ${}_{B}\mathcal{M}$ (resp. $\mathcal{M}^{B}$) is symmetric monoidal.

A proof of the above theorem can be found at Montgomery's book "Hopf algebras and their actions on rings", ch. 10, p. 199-201, Theorem 10.4.2. (a detailed discussion on the conceptual and formal origins of this idea and some of its descendants is also included, see p.201-203).

What essentially happens here is that given the $R$-matrix $R=\sum R^{(1)}\otimes R^{(2)}$ then the the family of $H$-module isomorphisms $\psi_{V,W}:V\otimes W\rightarrow W\otimes V$ given explicitly by $\psi_{V,W}(x\otimes y)=\sum R^{(2)}\cdot y\otimes R^{(1)}\cdot x$ constitutes a braiding. (If you are interested in further details for the special case where the original hopf algebra is a group hopf algebra, see also p.80-81 of this article and the references therein).
(In the special case where the original hopf algebra is trivially quasitriangular (i.e. cocommutative) then the braided monoidal category is actually a symmetric monoidal category and the braiding degenerates into the usual flip map, as has already been mentioned both in the comments and in Mathematician 42's answer as well).