6
votes

Accepted

### Modules over Hopf Algebras and $E_2$-algebras

Let $A$ be a brace algebra and $B$ the Koszul dual bialgebra. There is a natural adjunction
$$
\Omega\colon \mathrm{CoMod}_B\rightleftarrows \mathrm{LMod}_A
$$
where, for instance, the functor $\...

4
votes

Accepted

### Can the ribbon category of f.d. reps of $\mathcal{U}_q(\mathfrak{sl}(2))$ be modified so the twist is trivial on the vector representation?

There is a category built from the HOMFLYPT skein relation in the same way that $\mathcal U_q(\mathfrak sl(2))$ is built from the Kauffman skein relation. It is a version of $\mathcal U_q(\mathfrak ...

3
votes

Accepted

### When are the braid relations in a quasitriangular Hopf algebra equivalent?

The condition $R_{21}\,R=I$ (to be triangular) implies the equivalence of the two equations.
For the second question, let $A$ be a finite abelian group, and $H=k^A$ the Hopf algebra of function on $A$...

2
votes

### When are the braid relations in a quasitriangular Hopf algebra equivalent?

Those conditions alone don't determine a quasitriangular structure.
An old paper of Radford's shows that the quasitriangular structure conditions for a finite dimensional Hopf algebra $H$ define Hopf ...

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