All Questions
Tagged with braided-tensor-categories qa.quantum-algebra
16
questions
22
votes
3
answers
3k
views
How many definitions are there of the Jones polynomial?
Even with the connection to quantum groups being made clearer (I believe it was not known when the Jones polynomial was first introduced), it seems to me that still we don't have the "right" ...
- 18.1k
14
votes
1
answer
685
views
Associators, Grothendieck-Teichmüller group and monoidal categories
The standard definition of an associator seems to be that it a a grouplike power series in two variables $x$ and $ y $ satisfying some pentagon and hexagon relations.
In other words, denoting by $ \...
- 908
14
votes
0
answers
767
views
Splitting of homomorphism from cactus group to permutation group
We all learned in kindergarten that the category of finite-dimensional (type I, say) $U_q(\mathfrak{g})$-modules is braided monoidal for $\mathfrak{g}$ a complex semisimple Lie algebra. This gives an ...
- 8,369
12
votes
1
answer
283
views
Can one show corbordism invariance of the Crane-Yetter state-sum using simplicial methods / are there 'Pachner-like' moves for cobordisms?
Let $\mathcal{C}$ denote some Unitary Braided Modular Fusion Category. It is well known that the Crane-Yetter state-sum, $Z_{CY}(\bullet|\mathcal{C})$ is an oriented-cobordism invariant. In other ...
- 525
11
votes
1
answer
336
views
What is the relation between 2-Gerstenhaber, CohFT, and Gerstenhaber geometrically?
Background. As we know from Fred Cohen's Thesis, taking homology of the little 2-discs operad $\mathcal{D}_2$ with coefficients in a field of characteristic zero produces the Gerstenhaber operad $\...
- 1,971
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
267
views
Geometric Intuition of $P^+$ in Modular Tensor Categories
I'm currently reading through Bakalov and Kirillov's "Lectures on Tensor Categories and Modular Functors," and I am having some difficulty understanding the definition of $p^\pm$ given on page 49. ...
7
votes
1
answer
385
views
Do dualizable Hopf algebras in braided categories have invertible antipodes?
A classical result of Larson and Sweedler says that a finite dimensional Hopf algebra over a field has invertible antipode. Does this result extend to the setting of Hopf algebras in braided ...
- 7,097
6
votes
1
answer
1k
views
Kontsevich Integral without associators?
Recall the fact that the representations of a quantum group form a braided tensor category, and this corresponds to the fact that $U_q(\mathfrak g)$ is a quasi-triangular Hopf algebra. The braiding ...
- 18.1k
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
5
votes
4
answers
1k
views
An inner product that makes the R-matrix unitary
So, if you talk to the right people, they will tell you that the braiding of the category of representations of a quantum group are not unitary and that one can fix this by taking a different commutor ...
- 43.4k
5
votes
0
answers
104
views
Non-semisimple representations of the braid group in a semisimple braided category
Suppose $\mathcal{C}$ is a semisimple braided tensor category (over $\mathbb{C}$, with finite dimensional hom spaces) and $X$ an object in $\mathcal{C}$.
Then for each n > 0 the braid group $B_n$ ...
- 303
4
votes
1
answer
377
views
Motivating quantum groups from knot invariants
Quantum groups are useful for making knot/link invariants: for example, $U_q(\mathfrak{sl}_2$) you get the Jones polynomial. This boils down to the fact that $\mathcal C = \operatorname{rep }U_q(\...
- 2,213
4
votes
0
answers
101
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 ...
- 815
2
votes
1
answer
376
views
Is there a notion of partial trace in a ribbon category?
I've seen some definitions of "right partial trace" and "left partial trace" in http://arxiv.org/abs/1103.1660, but these don't seem canonical in any way.
The motivation for this questions is that I'...
- 21
0
votes
1
answer
230
views
Braidings for Comodules of Co-quasi-triangular Hopf algebra
Let $V$ be a (right-)$H$ comodule wrt a coaction $\Delta_R$, where $H$ is a co-quasi-triangular Hopf algebra with co-quasi-triangular Hopf algebra structure $R$. It is well-known that $V$ has a ...
- 1,429