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### 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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### What's the relation between half-twists and star structures 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 ...

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### 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 ...

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### 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 ...

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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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### 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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### 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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### 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 ...