Bi-genic Hopf algebras in characteristic \(p\)
type: Item
instance of: scholarly article
title: Bi-genic Hopf algebras in characteristic \(p\)
zbMATH Open document ID: 1365.16017
DOI: 10.1216/RMJ-2012-42-6-1919
author: Alan Koch
published in: Rocky Mountain Journal of Mathematics
publication date: 2013-04-09T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.rmjm/1361800611
review text: In this paper, the author computes Ext\(^1(M,M^\prime)\) in the category of modules over the Dieudonné ring \(E\), where \(M,M^\prime\) are Dieudonné modules which correspond to monogenic Hopf algebras \(H,H^\prime\), respectively. Then, the author explicitly describes the Dieudonn\(\acute{e}\) modules that arise as extensions of \(M\) by \(M^\prime\), giving a 2-element generating set. In the case \(M\) and \(M^\prime\) are Dieudonné modules, one can quickly compute Ext\(^1(M,M^\prime)\) using the long exact Hom-Ext sequence, describe the extension group using this technique and show how it agrees with author's classification. Finally, the author determines the algebra structure of a bi-genic Hopf algebra from its Dieudonné module and gives an explicit example of the coalgebra structure for a simple but nontrivial class of Hopf algebras. We hope that future work will examine the lifting questions as in the monogenic case.
Mathematics Subject Classification ID: 16T05
Mathematics Subject Classification ID: 16E40
Mathematics Subject Classification ID: 16E30
zbMATH DE Number: 6152191
zbMATH Keywords: Dieudonné module
zbMATH Keywords: monogenic Hopf algebra
zbMATH Keywords: bi-genic Hopf algebra
reviewed by: Liangyun Zhang
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6152191