Projective bases of division algebras and groups of central type.
type: Item
instance of: scholarly article
title: Projective bases of division algebras and groups of central type.
zbMATH Open document ID: 1074.16015
DOI: 10.1007/BF02773539
author: Eli Aljadeff
author: Michael Natapov
author: Darrell Haile
published in: Israel Journal of Mathematics
publication date: 2005-06-09T00:00:00Z
review text: The paper under review continues the study, started by the first two authors, of those twisted group algebras over a field \(k\), which are in fact central division algebras. They have proved [see Isr. J. Math. 121, 173-198 (2001; Zbl 0978.16027)] that the groups corresponding to these algebras are nilpotent, and also, have reduced the presentation of the algebras considered to the description of those finite \(p\)-groups \(P\) for which the twisted group algebra \(k^\alpha P\) (where \(\alpha\) is a \(2\)-cocycle) is of the type dealt with. The present paper provides a short list of such groups, and announces (to appear in a forthcoming paper by the third author) that \(P\) has the required property if and only if it is isomorphic to a direct product of groups from the list. Also, it completes the proof of the assertion that the algebras discussed are isomorphic to tensor products of cyclic division \(k\)-algebras.
reviewed by: Ivan D. Chipchakov
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 20C05
zbMATH DE Number: 2174715
zbMATH Keywords: twisted group algebras
zbMATH Keywords: division algebras
zbMATH Keywords: projective bases
zbMATH Keywords: groups of central type
zbMATH Keywords: projective representations of groups
zbMATH Keywords: central simple algebras
zbMATH Keywords: Schur algebras
zbMATH Keywords: tensor products of cyclic algebras
zbMATH Keywords: symbol algebras
zbMATH Keywords: quaternion algebras
MaRDI profile type: Publication
cites work: Division algebras with a projective basis
cites work: Projective Schur algebras of nilpotent type are Brauer equivalent to radical algebras
cites work: On groups of central type
cites work: Abelian binomials, power residues and exponential congruences
cites work: Q3926778
full work available at URL: https://doi.org/10.1007/bf02773539
OpenAlex ID: W3100933984
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description: scientific article; zbMATH DE number 2174715