Classification of finite-dimensional basic Hopf algebras according to their representation type

2009 ◽  
pp. 103-124 ◽  
Author(s):  
Gongxiang Liu
Keyword(s):  
Hopf Algebras ◽  
Representation Type ◽  
Finite Dimensional

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 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 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 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.

scholarly journals Finite Cartan Graphs Attached to Nichols Algebras of Diagonal Type

2021 ◽  
Vol 2021 ◽  
pp. 1-12
Author(s):  
Chen Qian ◽  
Jing Wang
Keyword(s):  
Root System ◽  
Hopf Algebras ◽  
Enveloping Algebras ◽  
Finite Dimensional ◽  
Quantized Enveloping Algebras ◽  
Pointed Hopf Algebras ◽  
Nichols Algebras ◽  
Simply Connected

Nichols algebras are fundamental objects in the construction of quantized enveloping algebras and in the classification of pointed Hopf algebras by the lifting method of Andruskiewitsch and Schneider. The structure of Cartan graphs can be attached to any Nichols algebras of diagonal type and plays an important role in the classification of Nichols algebras of diagonal type with a finite root system. In this paper, the main properties of all simply connected Cartan graphs attached to rank 6 Nichols algebras of diagonal type are determined. As an application, we obtain a subclass of rank 6 finite dimensional Nichols algebras of diagonal type.

scholarly journals ON NICHOLS ALGEBRAS OVER PGL(2, q) AND PSL(2, q)

2010 ◽  
Vol 09 (02) ◽  
pp. 195-208 ◽  
Author(s):  
SEBASTIÁN FREYRE ◽  
MATÍAS GRAÑA ◽  
LEANDRO VENDRAMIN
Keyword(s):  
Hopf Algebra ◽  
Group Algebra ◽  
Hopf Algebras ◽  
Necessary Conditions ◽  
Drinfeld Modules ◽  
Finite Dimensional ◽  
Pointed Hopf Algebras ◽  
Mathieu Groups ◽  
Nichols Algebras

We compute necessary conditions on Yetter–Drinfeld modules over the groups PGL(2, q) = PGL(2, 𝔽q) and PSL(2, q) = PSL(2, 𝔽q) to generate finite-dimensional Nichols algebras. This is a first step towards a classification of pointed Hopf algebras with group of group-likes isomorphic to one of these groups. As a by-product of the techniques developed in this work, we prove that any finite-dimensional pointed Hopf algebra over the Mathieu groups M20 or M21 = PSL(3, 4) is the group algebra.

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