Discriminant and Clifford algebras
type: Item
instance of: scholarly article
title: Discriminant and Clifford algebras
zbMATH Open document ID: 1009.16021
author: Anne Quéguiner-Mathieu
author: Jean-Pierre Tignol
published in: Mathematische Zeitschrift
publication date: 2002-09-17T00:00:00Z
review text: Let \(F\) be a field of characteristic not equal to \(2\), and let \(A\) be a central simple \(F\)-algebra of degree \(n=4m\), for some positive integer \(m\), endowed with an orthogonal involution \(\sigma\). An element \(\theta\in A\) is said to be a square-central skew-symmetric unit, if \(\sigma(\theta)=-\theta\) and \(\theta^2\in F^*\). It is known that if \(\theta\) is such a unit, then \(\sigma\) induces on the centralizer \(\widetilde A=C_A(\theta)\) a unitary involution \(\widetilde\sigma\), and \(\widetilde A\) is a central simple algebra of degree \(2m\) over the etale quadratic extension \(F(\theta)\) of \(F\); also, one can associate with these data two different central simple algebras with involutions, namely, the Clifford algebra \(C(A,\sigma)\) of \((A,\sigma)\), and the discriminant algebra \(D(\widetilde A,\widetilde\sigma)\) of \((\widetilde A,\widetilde\sigma)\), both endowed with their canonical involutions [see \textit{M.-A. Knus, A. S. Merkurjev, M. Rost} and \textit{J.-P. Tignol}, The book of involutions, Colloq. Publ. Am. Math. Soc. 44, Providence, RI (1998; Zbl 0955.16001)]. The paper under review concentrates on the study of the special case \(m=2\). It shows that then \(A\) contains a square-central skew-symmetric unit if and only if the discriminant of \(\sigma\) is trivial, so \(C(A,\sigma)=C_1\times C_2\), for some central simple \(F\)-algebras \(C_1\) and \(C_2\) of degree \(8\), such that at least one of them is of index \(1\) or \(2\). As pointed out by the authors, this result enables one to find without difficulty examples of degree \(8\) central simple algebras with orthogonal involutions that do not contain such a unit. The main theorem of the paper states (for \(m=2\)) that \((A,\sigma)\) is decomposable, i.e. \((A,\sigma)\cong(H,\sigma_H)\otimes_F(A_0,\sigma_0)\) where \(H\) is a quaternion algebra, \(A_0\) is of degree \(4\) and \(\sigma=\sigma_H\otimes\sigma_0\), if and only if the discriminant of \(\sigma\) is trivial, so …
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 15A66
zbMATH DE Number: 1801562
zbMATH Keywords: square-central skew-symmetric units
zbMATH Keywords: Clifford algebras
zbMATH Keywords: central simple algebras
zbMATH Keywords: orthogonal involutions
zbMATH Keywords: discriminant algebras
zbMATH Keywords: quaternion algebras
reviewed by: Ivan D. Chipchakov
MaRDI profile type: Publication
full work available at URL: s002090100385
OpenAlex ID: W2079116352
DOI: 10.1007/S002090100385
description: scientific article; zbMATH DE number 1801562