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8 votes
2 answers
853 views

Is a Hopf algebra a group object of some category?

The page of ncatlab on group object states that: A group object in $\mathrm{CRing}^{\mathrm{op}}$ is a commutative Hopf algebra. Question: Is a (noncommutative) Hopf algebra a group object of some ...
Sebastien Palcoux's user avatar
18 votes
2 answers
1k views

Why does Drinfeld Unitarization work?

In Drinfeld's paper "Quasi-Hopf Algebras" he illuminates a process by which you can replace the $R \in A \otimes A$ associated to a quasi-Hopf QUE-algebra $(A, \Delta, \varepsilon, \Phi)$ ...
Olivia Borghi's user avatar
3 votes
1 answer
456 views

Rigidity for the category of comodules over a Hopf algebra

On this page https://ncatlab.org/nlab/show/rigid+monoidal+category there is a discussion of rigidity (left-right duality) for the catagory of modules over a Hopf algebra. What happens if we look at ...
Max Schattman's user avatar
4 votes
0 answers
103 views

Scaling Yetter--Drinfeld Modules

A braided vector space is a pair $(V,\sigma)$ consisting of a vector space $V$, and a linear map $\sigma:V \otimes V \to V \otimes V$, satisfying the Yang--Baxter equation. Ee can scale the braiding ...
Nadia SUSY's user avatar
4 votes
0 answers
310 views

Nichols Algebras as Braided Hopf Algebras

Given a Hopf algebra $H$ and a Yetter--Drinfeld module $V$ over $H$, it is well-known that $V$ has an induced braided vector space structure, and so, one can consider it's Nichols algebra which is a ...
Abo Kutis-Felan's user avatar
2 votes
1 answer
197 views

When are Morita classes represented by certain structured algebra objects?

Let $\mathcal{C}$ be a monoidal category. There is a notion of Morita equivalence of algebra objects internal to $\mathcal{C}$. Does each Morita class have a symmetric Frobenius representative? A Hopf ...
Alex Turzillo's user avatar
16 votes
1 answer
431 views

Are there small examples of non-pivotal finite tensor categories?

I'm looking for small concrete examples of non-pivotal finite tensor categories to do some calculations with. Here a finite tensor category is, according to Etingof-Ostrik, a rigid monoidal category ...
Noah Snyder's user avatar
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5 votes
5 answers
767 views

What are the correct axioms for a "weakly associative monoidal functor"?

Definitions and the main question Recall that a category $\mathcal C$ is monoidal if it is equipped with the following data (two functors, three natural transformations, and some properties): a ...
Theo Johnson-Freyd's user avatar