Projective Schur algebras of nilpotent type are Brauer equivalent to radical algebras
type: Item
instance of: scholarly article
title: Projective Schur algebras of nilpotent type are Brauer equivalent to radical algebras
zbMATH Open document ID: 0955.16021
author: Eli Aljadeff
author: Jack Sonn
published in: Journal of Algebra
publication date: 2001-02-27T00:00:00Z
review text: Given a field \(k\), a finite group \(G\) and \(\alpha\in H^2(G,k^\times)\), where \(G\) acts trivially on \(k^\times\), the authors form the twisted group algebra \(k^\alpha G\) and define a projective Schur algebra over \(k\) as a \(k\)-central simple algebra which is a homomorphic image of some \(k^\alpha G\). Such an algebra determines an element in the Brauer group \(\text{Br}(k)\) and the set of these elements forms a subgroup, called the projective Brauer group of \(k\) and denoted by \(\text{PS}(k)\) [see \textit{F. Lorenz} and \textit{H. Opolka}, Math. Z. 162, 175-182 (1978; Zbl 0373.16006)]. The authors study projective Schur algebras defined by a nilpotent group \(G\). The subgroup of \(\text{PS}(k)\) generated by the corresponding elements is written \(\text{PNil}(k)\), with the subgroup \(\text{PAb}(k)\) arising from Abelian groups, i.e. algebras of Abelian type. They show that every projective Schur algebra of Abelian type is isomorphic to a full matrix ring over a tensor product of symbol algebras. -- Next take a finite extension \(K/k\), let \(F\) be the set of elements of \(K^\times\) of finite order \(\text{mod }k^\times\), and form the subfield \(k(F)\) spanned by \(F\). If \(K=k(F)\), \(K/k\) is called a radical extension and a central simple \(k\)-algebra \(A\) is called a radical algebra if it is a crossed product with a radical extension as splitting field. The authors prove that every element of \(\text{PNil}(k)\) can be represented by a radical algebra and they conjecture that this result holds for every member of \(\text{PS}(k)\). The proofs are by a series of reductions, using results from their earlier papers [\textit{E. Aljadeff} and \textit{J. Sonn}, J. Algebra 175, No. 1, 179-187 (1995; Zbl 0828.16017), ibid. 178, No. 2, 530-540 (1995; Zbl 0852.16011)].
Mathematics Subject Classification ID: 16K50
Mathematics Subject Classification ID: 16S35
zbMATH DE Number: 1384465
zbMATH Keywords: central simple algebras
zbMATH Keywords: finite groups
zbMATH Keywords: twisted group algebras
zbMATH Keywords: projective Schur algebras
zbMATH Keywords: projective Brauer groups
zbMATH Keywords: algebras of Abelian type
zbMATH Keywords: full matrix rings
zbMATH Keywords: tensor products of symbol algebras
zbMATH Keywords: crossed products
zbMATH Keywords: radical extensions
zbMATH Keywords: splitting fields
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
OpenAlex ID: W2057822783
cites work: Projective representations and relative semisimplicity
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DOI: 10.1006/JABR.1999.7860
description: scientific article; zbMATH DE number 1384465