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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 $\...
Sebastien Palcoux's user avatar
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}...
Sebastien Palcoux's user avatar
5 votes
0 answers
203 views

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 $...
Sebastien Palcoux's user avatar
3 votes
0 answers
276 views

Is there a non-pointed simple integral modular fusion category?

The complex field $\mathbb{C}$ is assumed to be the base field. Let WGT stand for weakly group-theoretical; then [ENO11, Question 2] asks whether the following holds: Statement 1 (open): There is a ...
Sebastien Palcoux's user avatar
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{...
Sebastien Palcoux's user avatar
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(...
Zhiyuan Wang's user avatar
1 vote
0 answers
107 views

Weakly group theoretical fusion category and subsystems

Let $\mathcal{C}$ be a fusion category and $H$ a semi-simple finite weak Hopf algebra such that $\mathcal{C}(H) = \mathcal{C}$. Suppose that for every nontrivial left coideal subalgebras $S$ of $H$ or ...
Sebastien Palcoux's user avatar