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 ON SECOND COHOMOLOGY OF DUALS OF COMPACT GROUPS

2011 ◽  
Vol 22 (09) ◽  
pp. 1231-1260 ◽  
Author(s):  
SERGEY NESHVEYEV ◽  
LARS TUSET
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras ◽  
Cohomology Group ◽  
Tensor Category ◽  
Compact Groups ◽  
Tensor Categories ◽  
Connected Group ◽  
Compact Quantum Groups ◽  
Algebraic Counterpart

We show that for any compact connected group G the second cohomology group defined by unitary invariant two-cocycles on Ĝ is canonically isomorphic to [Formula: see text]. This implies that the group of autoequivalences of the C*-tensor category Rep G is isomorphic to [Formula: see text]. We also show that a compact connected group G is completely determined by Rep G. More generally, extending a result of Etingof–Gelaki and Izumi–Kosaki we describe all pairs of compact separable monoidally equivalent groups. The proofs rely on the theory of ergodic actions of compact groups developed by Landstad and Wassermann and on its algebraic counterpart developed by Etingof and Gelaki for the classification of triangular semisimple Hopf algebras. We give a self-contained account of amenability of tensor categories, fusion rings and discrete quantum groups, and prove an analog of Radford's theorem on minimal Hopf subalgebras of quasitriangular Hopf algebras for compact quantum groups.

scholarly journals INTRODUCTION TO QUANTIZED LIE GROUPS AND ALGEBRAS

1992 ◽  
Vol 07 (25) ◽  
pp. 6175-6213 ◽  
Author(s):  
T. TJIN
Keyword(s):  
Lie Groups ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Baxter Equation ◽  
Product Decomposition ◽  
R Matrix ◽  
Conformal Field ◽  
Finite Dimensional ◽  
Lie Groups And Algebras ◽  
Poisson Lie Groups

We give a self-contained introduction to the theory of quantum groups according to Drinfeld, highlighting the formal aspects as well as the applications to the Yang-Baxter equation and representation theory. Introductions to Hopf algebras, Poisson structures and deformation quantization are also provided. After defining Poisson Lie groups we study their relation to Lie bialgebras and the classical Yang-Baxter equation. Then we explain in detail the concept of quantization for them. As an example the quantization of sl2 is explicitly carried out. Next we show how quantum groups are related to the Yang-Baxter equation and how they can be used to solve it. Using the quantum double construction we explicitly construct the universal R matrix for the quantum sl2 algebra. In the last section we deduce all finite-dimensional irreducible representations for q a root of unity. We also give their tensor product decomposition (fusion rules), which is relevant to conformal field theory.

HOPF ALGEBRAS OF DIMENSION 30

2010 ◽  
Vol 09 (01) ◽  
pp. 11-15 ◽  
Author(s):  
DAIJIRO FUKUDA
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Finite Dimensional

This paper contributes to the classification of finite dimensional Hopf algebras. It is shown that every Hopf algebra of dimension 30 over an algebraically closed field of characteristic zero is semisimple and thus isomorphic to a group algebra or the dual of a group algebra.

scholarly journals ON REPRESENTATIONS OF QUANTUM GROUPS Uq(fm(K,H))

2008 ◽  
Vol 78 (2) ◽  
pp. 261-284 ◽  
Author(s):  
XIN TANG ◽  
YUNGE XU
Keyword(s):  
Quantum Groups ◽  
Irreducible Representations ◽  
Whittaker Model ◽  
Finite Dimensional ◽  
Completely Reducible ◽  
The Relationship ◽  
Natural Families

AbstractWe construct families of irreducible representations for a class of quantum groups Uq(fm(K,H). First, we realize these quantum groups as hyperbolic algebras. Such a realization yields natural families of irreducible weight representations for Uq(fm(K,H)). Second, we study the relationship between Uq(fm(K,H)) and Uq(fm(K)). As a result, any finite-dimensional weight representation of Uq(fm(K,H)) is proved to be completely reducible. Finally, we study the Whittaker model for the center of Uq(fm(K,H)), and a classification of all irreducible Whittaker representations of Uq(fm(K,H)) is obtained.

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.

On the classification of finite quasi-quantum groups

2017 ◽  
Vol 29 (10) ◽  
pp. 1730003 ◽  
Author(s):  
Mamta Balodi ◽  
Hua-Lin Huang ◽  
Shiv Datt Kumar
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Tools And Methods

We give an overview of the classification results obtained so far for finite quasi-quantum groups over an algebraically closed field of characteristic zero. The main classification results on basic quasi-Hopf algebras are obtained by Etingof, Gelaki, Nikshych, and Ostrik, and on dual quasi-Hopf algebras by Huang, Liu and Ye. The objective of this survey is to help in understanding the tools and methods used for the classification.

scholarly journals On the classification of finite-dimensional pointed Hopf algebras

2010 ◽  
Vol 171 (1) ◽  
pp. 375-417 ◽  
Author(s):  
Nicolás Andruskiewitsch ◽  
Hans-Jürgen Schneider
Keyword(s):  
Hopf Algebras ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras

scholarly journals Pointed Hopf algebras: a guided tour to the liftings

2019 ◽  
Vol 53 (supl) ◽  
pp. 1-44 ◽  
Author(s):  
Iván Angiono ◽  
Agustín García Iglesias
Keyword(s):  
Hopf Algebras ◽  
The Other ◽  
Cartan Type ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
The One ◽  
Guided Tour

This article serves a two-fold purpose. On the one hand, it is asurvey about the classification of finite-dimensional pointed Hopf algebras with abelian coradical, whose final step is the computation of the liftings or deformations of graded Hopf algebras. On the other, we present a step-by-step guide to carry out the strategy developed to construct the liftings. As an example, we conclude the work with the classification of pointed Hopf algebras of Cartan type B2.

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