Questions tagged [ribbon-hopf-algebras]

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Superfluous axioms for ribbon Hopf algebra

In his book Foundations of quantum group theory, Majid defines (2.1.10) a ribbon Hopf algebra as a quasi-triangular Hopf algebra $(H, R)$ with a special central element $v \in H$ satisfying (1) $v^2 ...
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7 votes
1 answer

Modules over Hopf Algebras and $E_2$-algebras

Justin Young has a paper on the brace bar-cobar duality between hopf algebras and $E_2$-algebras: I was wondering if anybody knows of a nice relationship between ...
6 votes
2 answers

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

Quasitriangular Hopf algebras have to satisfy, amongst other conditions, the following equations: $$(\Delta \otimes \mathrm{id}) (R) = R_{13} R_{23}$$ $$(\mathrm{id} \otimes \Delta) (R) = R_{13} R_{12}...
4 votes
1 answer

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?

Consider the ribbon category of finite-dimensional representations of $\mathcal{U}_q(\mathfrak{sl}(2))$, with twist $\theta$. If $V$ is the vector representation, then $\theta_V$ is multiplication by $...
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10 votes
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What's the relation between half-twists, star structures and bar involutions on Hopf algebras?

A star structure on a Hopf algebra is an antilinear antiautomorphism squaring to 1 and satisfying some further axioms. A Hopf algebra with a star structure is then a star algebra and a Hopf algebra in ...
3 votes
3 answers

When does a monoidal functor between ribbon categories preserve cups and caps, but not necessarily braidings?

Imagine a monoidal functor between ribbon categories (i.e. monoidal, with a braiding, a twist and compatible left and right duals). An important example would be the restriction functor from the ...
4 votes
0 answers

Ribbon Algebras and Co-(dual)-quasi-triangular Hopf Algebras

As is well-known, one can use the coquasi-triangular structure $\cal R$ of $U_q(\frak{g})$ to give it's category of (right) modules $\cal{M}_{U_q(\frak{g})}$ the structure of a braided monoidal ...
6 votes
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Do purification and equivariantization commute?

Suppose that we have an action of a group $G$ on a (quasi-)Hopf algebra $H$, so that we can construct $H\rtimes G$ as in Majid's Cross Products by Braided Groups and Bosonization. It is known that $...
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4 votes
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Reshetikhin-Turaev and links with a distinguished component

Hi, This question came up to me when reading the paper of Cartier on Vassiliev invariants, but it can probably be turned into a more general question. Let $T$ be the category whose objects are ...
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4 votes
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Is there a good reference for how ribbon structures change when one switches coproducts?

I'm just going assume readers are familiar with the notions of R-matrix and ribbon categories. Given a quasi-triangular Hopf algebra $A$ with $R$-matrix $R$, one can construct the co-opposite Hopf ...
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6 votes
2 answers

How do quantum knot invariants change when I pick a funny ribbon element?

So, there's a construction of Reshetikhin and Turaev which extracts knot invariants from ribbon monoidal categories, which are (usually) the representation category a Hopf algebra with a choice of ...
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