Hopf algebra extensions arising from semi-direct products of groups
type: Item
instance of: scholarly article
title: Hopf algebra extensions arising from semi-direct products of groups
zbMATH Open document ID: 1010.16040
author: Mitja Mastnak
published in: Journal of Algebra
publication date: 2002-09-08T00:00:00Z
review text: Let \(G=N\rtimes H\) be a semidirect product of finite groups and let \(k\) be a field. The object of study in this paper is the cohomology classifying Hopf algebra extensions of the form \((kN)^*\to C\to kH\). Such extensions, arising from a ``matched pair'' of groups, were studied by \textit{G. I. Kac} [Mat. Sb., N. Ser. 76(118), 473-496 (1968; Zbl 0205.03301)], and more recently by, for example, \textit{A. Masuoka} [Trans. Am. Math. Soc. 352, No. 8, 3837-3879 (2000; Zbl 0960.16045)]. An exact sequence (first studied by Kac) connecting group cohomology with the above Hopf algebra extension is studied here in an explicit way, using e.g. work of \textit{K.-i. Tahara} [Math. Z. 129, 365-379 (1972; Zbl 0238.20068)]. Several classes of examples are calculated in detail.
reviewed by: Kenneth A. Brown
Mathematics Subject Classification ID: 16W30
Mathematics Subject Classification ID: 20J06
zbMATH DE Number: 1797480
zbMATH Keywords: Hopf algebras
zbMATH Keywords: Hopf algebra extensions
zbMATH Keywords: cohomology of groups
zbMATH Keywords: matched pairs of groups
zbMATH Keywords: exact sequences
MaRDI profile type: Publication
OpenAlex ID: W4253334608
cites work: Extensions of Hopf algebras and their cohomological description
cites work: EXTENSIONS OF GROUPS TO RING GROUPS
cites work: Q4693723
cites work: Q4422540
cites work: Calculations of some groups of Hopf algebra extensions
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cites work: On the second cohomology groups of semidirect products
DOI: 10.1006/JABR.2002.9145
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description: scientific article; zbMATH DE number 1797480