Questions tagged [ribbon-hopf-algebras]
The ribbon-hopf-algebras tag has no usage guidance.
11
questions
10
votes
0
answers
278
views
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 ...
- 5,485
7
votes
1
answer
697
views
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: https://arxiv.org/pdf/1309.2820.pdf
I was wondering if anybody knows of a nice relationship between ...
- 507
6
votes
2
answers
273
views
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}...
- 5,485
6
votes
2
answers
511
views
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 ...
- 43.4k
6
votes
0
answers
129
views
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 $...
- 1,629
4
votes
1
answer
253
views
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 $...
- 1,444
4
votes
0
answers
133
views
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 ...
- 1,383
4
votes
0
answers
295
views
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 ...
- 7,962
4
votes
0
answers
105
views
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 ...
- 43.4k
3
votes
3
answers
907
views
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 ...
- 5,485
1
vote
0
answers
69
views
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 ...
- 569