On semisimple Hopf algebras of dimension \(2^m\).
type: Item
instance of: scholarly article
title: On semisimple Hopf algebras of dimension \(2^m\).
zbMATH Open document ID: 1040.16025
DOI: 10.1023/B:ALGE.0000003540.74953.97
published in: Algebras and Representation Theory
publication date: 2004-01-05T00:00:00Z
review text: In a previous paper [J. Algebra 232, No. 2, 617--663 (2000; Zbl 0969.16014)] the author obtained the classification of semisimple Hopf algebras of dimension 16 over an algebraically closed field \(k\) of characteristic zero. In this paper she classifies all nontrivial semisimple Hopf algebras \(H\) of dimension \(2^{n+1}\), \(n\geq 4\), such that the group \(G(H)\) of group-like elements in \(H\) is isomorphic to \(\mathbb{Z}_{2^{n-1}}\times\mathbb{Z}_2\). Amongst the class of Hopf algebras classified in this paper, there are two families dual to the examples studied by the author in [Commun. Algebra 31, No. 4, 1643-1668 (2003; Zbl 1040.16025)]. Because \(kG(H)\) is of index 2 and in view of a result of \textit{T. Kobayashi} and \textit{A. Masuoka} [Tsukuba J. Math. 21, No. 1, 55--58 (1997; Zbl 0894.16019)], a (commutative) normal Hopf subalgebra is in \(H\), and thus \(H\) is an (Abelian) extension of \(k(\mathbb{Z}_{2^{n-1}}\times\mathbb{Z}_2)\) by \(k\mathbb{Z}_2\). One of the invariants used for distinguishing isomorphism classes is the Frobenius Schur indicator of irreducible modules, as introduced by \textit{V. Linchenko} and \textit{S. Montgomery} in [Algebr. Represent. Theory 3, No. 4, 347--355 (2000; Zbl 0971.16018)].
reviewed by: Sonia Natale
Mathematics Subject Classification ID: 16W30
zbMATH DE Number: 2021280
zbMATH Keywords: semisimple Hopf algebras
zbMATH Keywords: bicrossed products
zbMATH Keywords: irreducible representations
zbMATH Keywords: power maps
author: Yevgenia Kashina
MaRDI profile type: Publication
full work available at URL: b:alge.0000003540.74953.97
OpenAlex ID: W1981644709
description: scientific article; zbMATH DE number 2021280