Hereditary orders and embeddings of local fields in simple algebras
type: Item
instance of: scholarly article
title: Hereditary orders and embeddings of local fields in simple algebras
zbMATH Open document ID: 0911.16011
author: Paul Broussous
published in: Journal of Algebra
publication date: 1999-03-04T00:00:00Z
review text: Let \(F\) be a commutative locally compact non-archimedean field, \(D\) a central \(F\)-division algebra of index \(d^2\), and \(V\) a right \(D\)-vector space of finite dimension. Let \(G\) denote the group of units of \(A\), the \(D\)-endomorphism ring of \(V\). The smooth complex representations of \(G\) have been classified if \(A\) is either split or a division algebra. In each case, certain field extensions \(E/F\subset A\), called pure extensions, play a rôle. \textit{A. Fröhlich} [Proc. Lond. Math. Soc., III. Ser. 54, 247-266 (1987; Zbl 0615.12021)] pointed out that certain pure extensions are undesirable when \(A\) is neither split nor a division algebra. Fröhlich described extensions that he called sound extensions, and the current author generalizes this to the notion he calls good extensions. He then gives a complete study of good extensions. Each extension \(E/F\) determines a sequence of lattices in \(V_E\), and hence, a hereditary order. The condition for goodness is explicitly described in terms of an embedding of the residue field \(k_E\) into a \(k_F\) algebra of finite dimension.
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 12F05
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 11S45
zbMATH DE Number: 1218700
zbMATH Keywords: hereditary orders
zbMATH Keywords: locally compact fields
zbMATH Keywords: locally compact groups
zbMATH Keywords: central division algebras
zbMATH Keywords: pure extensions
zbMATH Keywords: good extensions
reviewed by: William H. Gustafson
MaRDI profile type: Publication
full work available at URL: jabr.1997.7345
OpenAlex ID: W2007677080
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DOI: 10.1006/JABR.1997.7345
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description: scientific article; zbMATH DE number 1218700