All Questions
Tagged with subfactors hopf-algebras
10 questions
5
votes
0
answers
606
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Weak Hopf algebra structure on twisted group algebra
A (normalized) $2$-cocycle on a finite group $G$ with values in $S^1$ is a map
$\sigma:G\times G\rightarrow S^1$ such that $$\sigma(g,h)\sigma(gh,k)=\sigma(h,k)\sigma(g,hk)$$ and $$\sigma(g,e)=\sigma(...
- 393
6
votes
2
answers
256
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Are there examples of finite-dimensional weak Hopf C*-algebras with non-involutive antipode?
For finite-dimensional (non-weak) Hopf C*-algebras it is known that the antipode is always involutive, as claimed e.g. in https://arxiv.org/pdf/1007.5283.pdf. I couldn't find the same statement for ...
- 3,001
4
votes
0
answers
160
views
Is the Nichols-Richmond theorem true for integral fusion rings?
The Nichols-Richmond theorem is a result on cosemisimple Hopf algebras, proved in their paper.
It was restated for integral fusion categories by Dong-Natale-Vendramin (Theorem 3.4 here):
Theorem: ...
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 ...
5
votes
0
answers
203
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Is there a perfect integral fusion category with PFdim = 2 mod 4?
A finite group $G$ is perfect if $G = G^{(1)} := \langle [G,G] \rangle$.
Proposition: A finite group $G$ is perfect iff every $1$-dimensional complex representation of $G$ is trivial.
proof: First if $...
8
votes
0
answers
488
views
Is there a non-trivial Hopf algebra without left coideal subalgebra?
Let $H$ be a finite dimensional Hopf ${\rm C}^{\star}$-algebra.
A $\star$-subalgebra $I$ of $H$ is a left coideal if $\Delta(I) \subset H \otimes I$.
$H$ is called maximal if it has no left coideal $\...
5
votes
0
answers
161
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Are the integer index finite depth irreducible subfactors Kac-coideal?
Is every integer index finite depth irreducible subfactors planar algebra, the intermediate of an irreducible finite index depth $2$ subfactors planar algebra?
In other words, of the following form (...
4
votes
0
answers
239
views
Is an integral simple fusion ring, categorifiable?
A fusion ring $\mathcal{F}$ (see here p 28) is integral if the Perron-Frobenius dimension $d(h_i)$ of its basic elements $\{h_1,...,h_r\}$, are integers. Its rank is $r$ and its dimension is $\sum d(...
6
votes
0
answers
239
views
Existence of a Kac algebra for a given fusion ring in a particular class
A $n$-dimensional Kac algebra (i.e., a Hopf C*-algebra), admits finitely many irreducible representations, whose cardinal $r$ is called its rank, the increasing sequence $(d_{1},d_{2},d_{3}, ..., d_{r}...
2
votes
0
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166
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How simplify the pentagonal equation from two fusion rings?
A semi-simple finite dimensional Hopf algebra $\mathbb{A}$, and its dual $\mathbb{A}^{*}$ produce two fusion categories $\mathcal{C}$ and $\mathcal{C}^{*}$ and then two fusion rings $\mathcal{R}_{1}$ ...