All Questions
Tagged with qa.quantum-algebra braided-tensor-categories
22 questions
2
votes
1
answer
78
views
Does there exist a nontrivial triangular weak Hopf algebra?
Quasitriangular weak Hopf algebras (QWHAs) are defined in Nikshych-Turaev-Vainerman (2000): A QWHA is a pair
($H,\mathcal{R}$) where $H$ is a WHA and
$\mathcal{R} \in \Delta^{op}(1)(H\otimes H)\Delta(...
- 727
4
votes
0
answers
295
views
Is there an integral fusion category of the Ising type?
In [EGNO, Section 8.27.3], we read:
Any braided fusion category ${\mathcal C}$ is obtained from a weakly anisotropic
category (namely, the core of ${\mathcal C}$) using finite groups (via the
...
4
votes
0
answers
82
views
Is the Drinfeld element of a semisimple quasitriangular Hopf algebra invariant under the Drinfeld twist?
Let $A$ be a finite dimensional semisimple quasitriangular Hopf algebra over $\mathbb{C}$ with universal $R$-matrix denoted by $\mathcal{R}\in A\otimes A$. The Drinfeld element of $A$ is defined as
$$...
- 727
4
votes
1
answer
246
views
Is there a non-split super-modular positive integral fusion category?
We reference [EGNO] for the concept of a braided fusion category. Following the conventions in [JFR], let $\mathcal{C}$ denote a braided fusion category equipped with a braiding $\beta$, and let $\...
10
votes
1
answer
380
views
Braidings on Temperley-Lieb Category
Let $k$ be a field, and let $q\in k^{\times}$. We can then consider the Temperley-Lieb category $TL(q)$. The objects of $TL(q)$ are the non-negative integers, and morphisms are roughly isotopy classes ...
- 1,001
4
votes
0
answers
227
views
Is the rank of an integral MTC an upper bound for the prime factors of its FPdim?
In this post, the abbreviation "MTC" and "FPdim" stand for "Modular Tensor Category" and "Frobenius-Perron dimension".
From [DLN, Theorem II (iii)], where the ...
5
votes
1
answer
525
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(\...
- 2,283
12
votes
1
answer
309
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 ...
- 545
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 ...
- 835
5
votes
0
answers
113
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
11
votes
1
answer
356
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 $\...
- 1,981
6
votes
2
answers
286
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}...
- 5,565
10
votes
0
answers
287
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 ...
- 5,565
7
votes
1
answer
273
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. ...
15
votes
1
answer
729
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 $ \...
- 1,033
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 ...
- 18.7k
0
votes
1
answer
257
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 ...
- 1,479
2
votes
1
answer
385
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
24
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" ...
- 18.7k
14
votes
0
answers
792
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 ...
- 8,559
7
votes
1
answer
398
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 ...
- 7,237
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 ...
- 44.7k