Decomposition of Mal'cev-Neumann division algebras with involution
type: Item
instance of: scholarly article
title: Decomposition of Mal'cev-Neumann division algebras with involution
zbMATH Open document ID: 0812.16023
DOI: 10.1007/BF02572343
published in: Mathematische Zeitschrift
publication date: 1995-05-09T00:00:00Z
full work available at URL: https://eudml.org/doc/174672
review text: We consider a crossed product \(A\) with center \(F\) and which is elementary abelian of exponent 2. Then \(A\) is endowed with an involution \(\sigma\) leaving the maximal subfield \(K\) elementwise invariant. On the other hand, we consider the Mal'cev-Neumann series division algebra associated to \(A\) and endow it with an involution \(\sigma^*\) associated to \(\sigma\). The crossed product generators of \(A\) and of the series division algebra can be chosen fixed by \(\sigma\) and \(\sigma^*\) respectively. This allows us to compute the discriminants of \(\sigma\) and \(\sigma^*\). Then we give necessary and sufficient conditions for the series fields to have a \(\sigma^*\)-invariant quaternion subalgebra or to be decomposable as a tensor product of \(\sigma^*\)-invariant quaternion algebras. Finally we show that if \(A = \text{End}_ FK\) then \(\sigma\) is the adjoint involution with respect to the quadratic form \(q(x) = \text{Tr}_{K/F}(\alpha x^ 2)\) for some \(\alpha\) in \(K\). In this case, if \(\alpha\) is such that \(N_{K/F}(\alpha)\) is in \(F^{*2}\) but \(N_{K/L}(\alpha)\) is not in \(L^{*2}\) for some quadratic subfield \(L\) of \(K\), then the series field is not a tensor product of \(\sigma^*\)-invariant quaternion algebras but the discriminant of \(\sigma^*\) is trivial. Moreover, if the Clifford algebra of \(q\) has index at least 4, the series field has no \(\sigma^*\)-invariant quaternion algebra. We give such an example where the discriminant of \(\sigma^*\) remains trivial.
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16W60
Mathematics Subject Classification ID: 16S35
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 703849
zbMATH Keywords: crossed product
zbMATH Keywords: involution
zbMATH Keywords: Mal'cev-Neumann series division algebra
zbMATH Keywords: discriminants
zbMATH Keywords: quaternion subalgebra
zbMATH Keywords: tensor product
zbMATH Keywords: Clifford algebra
author: Hélène Dherte
MaRDI profile type: Publication
cites work: Cohomologische Invarianten quadratischer Formen
cites work: Division algebras of degree 4 and 8 with involution
cites work: Discriminants of involutions on Henselian division algebras
cites work: Q4861157
cites work: Clifford algebras for algebras with involution of Type D
cites work: Q4006651
cites work: Q4313530
cites work: A question on the discriminants of involutions of central division algebras
cites work: Q3910639
cites work: Central simple algebras
cites work: Q3708857
cites work: Formes quadratiques, groupes orthogonaux et algèbres de Clifford
cites work: Cyclic and elementary abelian subfields of Malcev-Neumann division algebras
description: scientific article; zbMATH DE number 703849