All Questions
Tagged with quantum-groups fusion-categories
16 questions
13
votes
1
answer
284
views
Finiteness of the number of Hopf subalgebras
Let $H$ be a finite-dimensional Hopf algebra over the complex field.
Question: Does $ H $ have a finite number of Hopf subalgebras?
In the case where $ H $ is semisimple, the answer is yes. According ...
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
68
views
Smallest finite dimensional $\mathbb{C}^*$-Hopf algebra that is not "strongly group theoretical"
In this question, let us call a finite dimensional $\mathbb{C}^*$ Hopf algebra $H$ strongly group theoretical if there exists a finite group $G$ such that one of the following three equivalent ...
- 727
8
votes
1
answer
331
views
Some fusion rings/categories I don't recognize
Recently (what I believe are) all multiplicity-free fusion categories up to rank 7 have been posted on the AnyonWiki. Most of the fusion rings belonging to these categories belong to one of the ...
- 293
8
votes
1
answer
312
views
How does the Tannaka duality work for weak Hopf algebras and fusion categories?
I'm a physicist and not yet an expert in fusion category. I've heard that it's possible to reconstruct a weak Hopf algebra from its category of representations, and would like to know how this works ...
- 727
2
votes
1
answer
216
views
Relation between the modular categories SU(2)_n and Sp(n)_1
The online database [1] provides a list of some modular tensor categories classified by rank. Let us consider the two modular categories denoted kmA1_$\ell$ and kmC$\ell$_1 (i.e. Kac Moody $A_1$ level ...
3
votes
1
answer
492
views
Calculating fusion rules for $\operatorname{Rep}(G)$ and $G_{k}$ [reference request]
Could somebody please direct me to textbooks / literature that (perhaps lay the foundations for and) detail the method for determining the fusion rules for categories such as
$\operatorname{Rep}(G)$ ...
- 309
4
votes
0
answers
208
views
Are the finite quantum permutation groups, weakly group-theoretical?
Wang defined in Quantum symmetry groups of finite spaces a notion of quantum automorphism group. The application to a finite space of $n$ elements is called the quantum permutation group of $n$ ...
7
votes
0
answers
140
views
Triviality of Semisimple Hopf Algebras of Cyclic Dimension
A cyclic number is a natural number $n$ such that any group of order $n$ is cyclic. A003277
Theorem (T. Szele, 1947): A number $n$ is cyclic if and only if it is coprime to its Euler totient $\varphi(...
3
votes
0
answers
149
views
Symmetries of modular categories coming from quantum groups
This is a request for references about the computation of the braided autoequivalences of fusion categories coming from a quantum group. I could not find even the description of braided ...
- 1,529
7
votes
1
answer
149
views
Fusion Classification of $U_q(sl_N)$ Categories
Frohlich and Kerler classify categories with $SU(2)_k$ fusion rules and Kazdhan-Wenzl expand this to $SU(N)_k$ categories. In both cases, unless I am missing something, the classification are ...
- 1,089
8
votes
1
answer
432
views
Brauer-Picard for a fusion category coming from a quantum group
In Fusion Categories and Homotopy Theory, ENO attatch a 3-groupoid to a fusion category. In the case of A graded vector spaces they further compute it's truncation as an orthogonal group $O(A \...
- 993
7
votes
0
answers
331
views
An alternative Cauchy theorem on Hopf algebras
Let $\mathbb{A}$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
There already exists a generalization of Cauchy theorem using exponent, see [KSZ06].
We are interesting in an alternative ...
10
votes
2
answers
363
views
Symmetries of module categories over the category of representations of quantum $sl(2)$
The category $\mathcal{C}_l$ of tilting modules of the quantum group $U_q(sl_2)$ quotiented out by the modules of zero quantum dimension has a natural structure as a semisimple monoidal category when ...
- 103
3
votes
0
answers
229
views
The convolution on a semisimple finite quantum groupoid
Let $\mathbb{A}$ be a finite dim. weak Hopf $C^*$-algebra (or semisimple finite quantum groupoid) and $\hat{\mathbb{A}}$ its dual.
Let the Fourier transform $\mathcal{F}: \mathbb{A} \to \hat{\mathbb{...
26
votes
1
answer
2k
views
Does the quantum subgroup of quantum su_2 called E_8 have anything at all to do with the Lie algebra E_8?
The ordinary McKay correspondence relates the subgroups of SU(2) to the affine ADE Dynkin diagrams. The correspondence is that the vertices correspond to irreducible representations of the subgroup, ...
- 28.1k