On the Coradical of a Finite-Dimensional HOPF Algebra

1975 ◽  
Vol 53 (1) ◽  
pp. 9 ◽  
Author(s):  
David E. Radford
Keyword(s):  
Hopf Algebra ◽  
Finite Dimensional ◽  
Finite Dimensional Hopf Algebra

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 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 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.

scholarly journals McKay matrices for finite-dimensional Hopf algebras

2021 ◽  
pp. 1-46
Author(s):  
Georgia Benkart ◽  
Rekha Biswal ◽  
Ellen Kirkman ◽  
Van C. Nguyen ◽  
Jieru Zhu
Keyword(s):  
Hopf Algebra ◽  
Hopf Algebras ◽  
Eigenvalues And Eigenvectors ◽  
Indecomposable Modules ◽  
Finite Dimensional ◽  
Generalized Eigenvectors ◽  
The Matrix ◽  
Dual Module ◽  
The Right ◽  
Finite Dimensional Hopf Algebra

Abstract For a finite-dimensional Hopf algebra $\mathsf {A}$ , the McKay matrix $\mathsf {M}_{\mathsf {V}}$ of an $\mathsf {A}$ -module $\mathsf {V}$ encodes the relations for tensoring the simple $\mathsf {A}$ -modules with $\mathsf {V}$ . We prove results about the eigenvalues and the right and left (generalized) eigenvectors of $\mathsf {M}_{\mathsf {V}}$ by relating them to characters. We show how the projective McKay matrix $\mathsf {Q}_{\mathsf {V}}$ obtained by tensoring the projective indecomposable modules of $\mathsf {A}$ with $\mathsf {V}$ is related to the McKay matrix of the dual module of $\mathsf {V}$ . We illustrate these results for the Drinfeld double $\mathsf {D}_n$ of the Taft algebra by deriving expressions for the eigenvalues and eigenvectors of $\mathsf {M}_{\mathsf {V}}$ and $\mathsf {Q}_{\mathsf {V}}$ in terms of several kinds of Chebyshev polynomials. For the matrix $\mathsf {N}_{\mathsf {V}}$ that encodes the fusion rules for tensoring $\mathsf {V}$ with a basis of projective indecomposable $\mathsf {D}_n$ -modules for the image of the Cartan map, we show that the eigenvalues and eigenvectors also have such Chebyshev expressions.

scholarly journals On the Even Powers of the Antipode of a Finite-Dimensional Hopf Algebra

2002 ◽  
Vol 251 (1) ◽  
pp. 185-212 ◽  
Author(s):  
David E. Radford ◽  
Hans-Jürgen Schneider
Keyword(s):  
Hopf Algebra ◽  
Finite Dimensional ◽  
Finite Dimensional Hopf Algebra

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.

Invariants of FP-Projective Dimensions Under Cleft Extensions

2013 ◽  
Vol 20 (04) ◽  
pp. 689-700
Author(s):  
Xiuli Chen ◽  
Haiyan Zhu ◽  
Fang Li
Keyword(s):  
Hopf Algebra ◽  
Projective Dimension ◽  
Fixed Field ◽  
Finite Dimensional ◽  
Cleft Extensions ◽  
Finite Dimensional Hopf Algebra

Let H be a finite dimensional Hopf algebra and A be an algebra over a fixed field k. Firstly, it is proved that the left FP-projective dimension is invariant under cleft extensions when H is semisimple and A is left coherent. Secondly, using (co)induction functors, we study the relations between FP-projective dimensions in A # H-Mod and the counterparts in AH-Mod. Finally, we characterize the FP-projective preenvelopes (resp., precovers) under H*-extensions and cleft extensions, respectively.

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