scholarly journals The antipode of a finite-dimensional Hopf algebra over a field has finite order

1975 ◽  
Vol 81 (6) ◽  
pp. 1103-1106 ◽  
Author(s):  
David E. Radford
Keyword(s):  
Hopf Algebra ◽  
Finite Order ◽  
Finite Dimensional ◽  
Finite Dimensional Hopf Algebra ◽  
Algebra Over A Field

scholarly journals On Hopf DeMeyer-Kanzaki Galois extensions

2003 ◽  
Vol 2003 (26) ◽  
pp. 1627-1632
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Hopf Algebra ◽  
Galois Extension ◽  
Smash Product ◽  
Module Algebra ◽  
Galois Extensions ◽  
Finite Dimensional ◽  
Dual Hopf Algebra ◽  
Finite Dimensional Hopf Algebra ◽  
Algebra Over A Field

LetHbe a finite-dimensional Hopf algebra over a fieldk,Ba leftH-module algebra, andH∗the dual Hopf algebra ofH. For anH∗-Azumaya Galois extensionBwith centerC, it is shown thatBis anH∗-DeMeyer-Kanzaki Galois extension if and only ifCis a maximal commutative separable subalgebra of the smash productB#H. Moreover, the characterization of a commutative Galois algebra as given by S. Ikehata (1981) is generalized.

scholarly journals On Hopf Galois Hirata extensions

2003 ◽  
Vol 2003 (64) ◽  
pp. 4033-4039
Author(s):  
George Szeto ◽  
Lianyong Xue
Keyword(s):  
Hopf Algebra ◽  
Smash Product ◽  
Sufficient Condition ◽  
Separable Extension ◽  
Finite Dimensional ◽  
Dual Hopf Algebra ◽  
Finite Dimensional Hopf Algebra ◽  
Algebra Over A Field

LetHbe a finite-dimensional Hopf algebra over a fieldK,H*the dual Hopf algebra ofH, andBa rightH*-Galois and Hirata separable extension ofBH. ThenBis characterized in terms of the commutator subringVB(BH)ofBHinBand the smash productVB(BH)#H. A sufficient condition is also given forBto be anH*-Galois Azumaya extension ofBH.

Hopf algebra actions and transfer of Frobenius and symmetric properties

2020 ◽  
Vol 126 (1) ◽  
pp. 32-40
Author(s):  
S. Dăscălescu ◽  
C. Năstăsescu ◽  
L. Năstăsescu
Keyword(s):  
Hopf Algebra ◽  
Finite Dimensional Algebra ◽  
Dimensional Algebra ◽  
Finite Dimensional ◽  
Finite Dimensional Hopf Algebra ◽  
Symmetric Properties

If $H$ is a finite-dimensional Hopf algebra acting on a finite-dimensional algebra $A$, we investigate the transfer of the Frobenius and symmetric properties through the algebra extensions $A^H\subset A\subset A\mathbin{\#} H$.

On the Hopf algebra dual of an enveloping algebra

1982 ◽  
Vol 91 (2) ◽  
pp. 215-224 ◽  
Author(s):  
Stephen Donkin
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Algebraic Groups ◽  
Field Extension ◽  
Smash Product ◽  
Enveloping Algebras ◽  
Enveloping Algebra ◽  
Finite Dimensional ◽  
Polycyclic Groups ◽  
Algebra Over A Field

In (1) it is claimed that the main results of that paper have applications to the representation theory of algebraic groups, of polycyclic groups and of Lie algebras. An application to algebraic groups is given in Corollary 6·4 of (1), the applications to polycyclic groups are given in (2), the purpose of this work is to deal with the outstanding case of enveloping algebras. To make use of the results of (1), in this context, we show that the Hopf algebra dual of the enveloping algebra of a finite dimensional Lie algebra over a field of characteristic zero is quasi-affine (see § 1·5). This is done by an easy field extension argument and a generalization, to the Hopf algebra dual of the smash product of Hopf algebras, of Proposition 1·6·3 of (2) on the dual of the group algebra of a semidirect product of groups. Since this paper is aimed at those readers interested in enveloping algebras, the Hopf theoretic aspects are dealt with at a fairly leisurely pace.

On coideal subalgebras of abelian cocentral extensions and a generalization of Wall's conjecture

2014 ◽  
Vol 14 (02) ◽  
pp. 1550021
Author(s):  
Sebastian Burciu
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Abelian Extension ◽  
Cyclic Module ◽  
Finite Dimensional ◽  
Dual Hopf Algebra ◽  
Finite Dimensional Hopf Algebra

It is shown that any coideal subalgebra of a finite-dimensional Hopf algebra is a cyclic module over the dual Hopf algebra. Using this we describe all coideal subalgebras of a cocentral abelian extension of Hopf algebras extending some results from [R. Guralnick and F. Xu, On a subfactor generalization of Wall's conjecture, J. Algebra 332 (2011) 457–468].

scholarly journals Varieties of Null-Filiform Leibniz Algebras Under the Action of Hopf Algebras

2021 ◽  
Author(s):  
Lucio Centrone ◽  
Chia Zargeh
Keyword(s):  
Hopf Algebra ◽  
Free Algebra ◽  
Hopf Algebras ◽  
Leibniz Algebra ◽  
Module Structure ◽  
Leibniz Algebras ◽  
Finite Dimensional ◽  
Filiform Leibniz Algebra ◽  
Algebra Over A Field ◽  
Cocommutative Hopf Algebra

AbstractLet L be an n-dimensional null-filiform Leibniz algebra over a field K. We consider a finite dimensional cocommutative Hopf algebra or a Taft algebra H and we describe the H-actions on L. Moreover we provide the set of H-identities and the description of the Sn-module structure of the relatively free algebra of L.

scholarly journals Critical groups for Hopf algebra modules

2018 ◽  
Vol 168 (3) ◽  
pp. 473-503
Author(s):  
DARIJ GRINBERG ◽  
JIA HUANG ◽  
VICTOR REINER
Keyword(s):  
Finite Group ◽  
Hopf Algebra ◽  
Regular Representation ◽  
Critical Group ◽  
Critical Groups ◽  
Greatest Common Divisor ◽  
Group Representations ◽  
Finite Dimensional ◽  
Projective Representations ◽  
Finite Dimensional Hopf Algebra

AbstractThis paper considers an invariant of modules over a finite-dimensional Hopf algebra, called the critical group. This generalises the critical groups of complex finite group representations studied in [1, 11]. A formula is given for the cardinality of the critical group generally, and the critical group for the regular representation is described completely. A key role in the formulas is played by the greatest common divisor of the dimensions of the indecomposable projective representations.

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