A local-global principle for algebras with involution and Hermitian forms.
type: Item
instance of: scholarly article
title: A local-global principle for algebras with involution and Hermitian forms.
zbMATH Open document ID: 1092.16011
author: D. W. Lewis
author: Thomas Unger
published in: Mathematische Zeitschrift
publication date: 2003-09-21T00:00:00Z
review text: Let \(A\) be a (finite-dimensional) central simple algebra over a field \(K\) of characteristic different from~\(2\), and let \(D\) be a central division \(K\)-algebra which is Brauer-equivalent to \(A\). Every involution (i.e., involutive anti-automorphism) on \(A\) is adjoint to a Hermitian or skew-Hermitian form on a \(D\)-vector space. An involution \(\sigma\) is called hyperbolic if it is adjoint to a hyperbolic form. It is called weakly hyperbolic if for some \(n\) the involution on the matrix algebra \(M_n(A)\) defined by \((a_{ij})\mapsto(\sigma(a_{ij}))^t\) is hyperbolic. The main result of the paper characterizes weakly hyperbolic involutions in terms of the associated trace form \(T_\sigma\) on \(A\), defined by \(T_\sigma(x)=T(\sigma(x)x)\), where \(T\) is the reduced trace. The authors show that \(\sigma\) is weakly hyperbolic if and only if \(T_\sigma\) has signature \(0\) at each ordering of the subfield \(k\subset K\) of \(\sigma\)-invariant elements. As a consequence, they obtain a similar result for Hermitian forms on \(A\) with respect to any involution \(\sigma\): a Hermitian form defines a torsion element in the Witt group \(W(A,\sigma)\) if and only if its signature is \(0\) at each ordering of \(k\). In the case where \(A=K\) and \(\sigma\) is the identity, this is Pfister's ``local-global principle'' for quadratic forms.
reviewed by: Jean-Pierre Tignol
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 11E39
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16W20
zbMATH DE Number: 1983481
zbMATH Keywords: central simple algebras
zbMATH Keywords: hyperbolic involutions
zbMATH Keywords: Hermitian forms
zbMATH Keywords: orderings
zbMATH Keywords: central division algebras
zbMATH Keywords: anti-automorphisms
zbMATH Keywords: trace forms
zbMATH Keywords: Witt groups
MaRDI profile type: Publication
full work available at URL: s00209-003-0490-6
OpenAlex ID: W2148982725
DOI: 10.1007/S00209-003-0490-6
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description: scientific article; zbMATH DE number 1983481