Hopf Algebras, Tensor Categories and Related Topics

2021 ◽  
Keyword(s):  
Hopf Algebras ◽  
Tensor Categories

scholarly journals Hochschild homology of Hopf algebras and free Yetter–Drinfeld resolutions of the counit

2012 ◽  
Vol 149 (4) ◽  
pp. 658-678 ◽  
Author(s):  
Julien Bichon
Keyword(s):  
Hopf Algebra ◽  
Quantum Group ◽  
Bilinear Form ◽  
Hopf Algebras ◽  
Betti Numbers ◽  
Free Resolution ◽  
Hochschild Homology ◽  
Invertible Matrix ◽  
Tensor Categories ◽  
Orthogonal Case

AbstractWe show that if$A$and$H$are Hopf algebras that have equivalent tensor categories of comodules, then one can transport what we call a free Yetter–Drinfeld resolution of the counit of$A$to the same kind of resolution for the counit of$H$, exhibiting in this way strong links between the Hochschild homologies of$A$and$H$. This enables us to obtain a finite free resolution of the counit of$\mathcal {B}(E)$, the Hopf algebra of the bilinear form associated with an invertible matrix$E$, generalizing an earlier construction of Collins, Härtel and Thom in the orthogonal case$E=I_n$. It follows that$\mathcal {B}(E)$is smooth of dimension 3 and satisfies Poincaré duality. Combining this with results of Vergnioux, it also follows that when$E$is an antisymmetric matrix, the$L^2$-Betti numbers of the associated discrete quantum group all vanish. We also use our resolution to compute the bialgebra cohomology of$\mathcal {B}(E)$in the cosemisimple case.

scholarly journals Crossed extensions of the corepresentation category of finite supergroup algebras

2015 ◽  
Vol 26 (09) ◽  
pp. 1550067
Author(s):  
Adriana Mejía Castaño ◽  
Martín Mombelli
Keyword(s):  
Hopf Algebras ◽  
Tensor Categories ◽  
Finite Dimensional

We present explicit examples of finite tensor categories that are C2-graded extensions of the corepresentation category of certain finite-dimensional non-semisimple Hopf algebras.

scholarly journals Pointed finite tensor categories over abelian groups

2017 ◽  
Vol 28 (11) ◽  
pp. 1750087
Author(s):  
Iván Angiono ◽  
César Galindo
Keyword(s):  
Abelian Group ◽  
Cohomology Class ◽  
Hopf Algebras ◽  
Abelian Groups ◽  
Tensor Category ◽  
Tensor Categories ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras

We give a characterization of finite pointed tensor categories obtained as de-equivariantizations of the category of corepresentations of finite-dimensional pointed Hopf algebras with abelian group of group-like elements only in terms of the (cohomology class of the) associator of the pointed part. As an application we prove that every coradically graded pointed finite braided tensor category is a de-equivariantization of the category of corepresentations of a finite-dimensional pointed Hopf algebras with abelian group of group-like elements.

scholarly journals Tensor Categories and Hopf Algebras

2019 ◽  
Keyword(s):  
Hopf Algebras ◽  
Tensor Categories

scholarly journals Finite quasi-quantum groups of diagonal type

2020 ◽  
Vol 2020 (759) ◽  
pp. 201-243 ◽  
Author(s):  
Hua-Lin Huang ◽  
Gongxiang Liu ◽  
Yuping Yang ◽  
Yu Ye
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras ◽  
Tensor Categories ◽  
Finite Dimensional

AbstractIn this paper, we give a classification of finite-dimensional radically graded elementary quasi-Hopf algebras of diagonal type, or equivalently, finite-dimensional coradically graded pointed Majid algebras of diagonal type. By a Tannaka–Krein type duality, this determines a big class of pointed finite tensor categories. Some efficient methods of construction are also given.

scholarly journals Tensor Algebras in Finite Tensor Categories

2019 ◽  
Author(s):  
Pavel Etingof ◽  
Ryan Kinser ◽  
Chelsea Walton
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Classification Result ◽  
Tensor Categories ◽  
Tensor Algebras ◽  
Fusion Categories ◽  
Path Algebras ◽  
Finite Dimensional

Abstract This paper introduces methods for classifying actions of finite-dimensional Hopf algebras on path algebras of quivers and more generally on tensor algebras $T_B(V)$ where $B$ is semisimple. We work within the broader framework of finite (multi-)tensor categories $\mathcal{C}$, classifying tensor algebras in $\mathcal{C}$ in terms of $\mathcal{C}$-module categories. We obtain two classification results for actions of semisimple Hopf algebras: the first for actions that preserve the ascending filtration on tensor algebras and the second for actions that preserve the descending filtration on completed tensor algebras. Extending to more general fusion categories, we illustrate our classification result for tensor algebras in the pointed fusion categories $\textsf{Vec}_{G}^{\omega }$ and in group-theoretical fusion categories, especially for the representation category of the Kac–Paljutkin Hopf algebra.

Mini-Workshop: Cohomology of Hopf Algebras and Tensor Categories

2020 ◽  
Vol 16 (1) ◽  
pp. 663-693
Author(s):  
Henning Krause ◽  
Sarah Witherspoon ◽  
James Zhang
Keyword(s):  
Hopf Algebras ◽  
Tensor Categories

scholarly journals From Hopf Algebras to Tensor Categories

2014 ◽  
pp. 1-31 ◽  
Author(s):  
N. Andruskiewitsch ◽  
I. Angiono ◽  
A. García Iglesias ◽  
B. Torrecillas ◽  
C. Vay
Keyword(s):  
Hopf Algebras ◽  
Tensor Categories
Sign in / Sign up

Export Citation Format

Share Document