Algebras with involution that become hyperbolic under a given extension
type: Item
instance of: scholarly article
title: Algebras with involution that become hyperbolic under a given extension
zbMATH Open document ID: 0897.16018
published in: Journal of Algebra
publication date: 1998-03-24T00:00:00Z
review text: Let \(K\) be a field of characteristic not 2. For a central simple algebra with involution, \((A,\sigma)\), the authors allow besides the usual sort also algebras of the form \(B\times B^o\), where \(B\) is a central simple algebra over \(K\) and \(\sigma\) is the switch involution: \(\sigma(a,b^o)=(b,a^o)\). The subfield of the centre of \(A\) fixed by \(\sigma\) is written \(F\). If \(A\) contains \(a\) such that \(\sigma(a)a=0\), \(A\) is called isotropic and \(A\) is hyperbolic if \(A\) contains a right ideal \(I\) such that \(I=I^\perp\), where \(I^\perp=\{y\in A\mid\sigma(y)I=0\}\). E.g. a product algebra with switch involution is hyperbolic. Let \(\pi(t)\) be a monic separable polynomial over \(F\) of even degree \(2n\). Then the ring \(F(\pi)=F[t]/(\pi(t))\) is a direct product of field extensions of \(F\) and \((A\otimes F(\pi),\sigma\otimes 1)\) is a direct product of algebras. If all these algebras are hyperbolic, \((A,\sigma)\) is said to become hyperbolic over \(F(\pi)\). \textit{J.-P. Tignol} [in J. Algebra 181, No. 3, 857-875 (1996; Zbl 0847.16019)]\ gave a hyperbolicity criterion by proving that an anisotropic algebra \((A,\sigma)\) becomes hyperbolic over \(F(\pi)\) if and only if there is a homomorphism \((H_\pi,\sigma_\pi)\to(A,\sigma)\) of \(F\)-algebras with involution, where \(H_\pi\) is the universal algebra over which \(\pi\) splits into two factors of degree \(n\) and \(\sigma_n\) interchanges these factors. The authors' aim is to describe the structure of \(H_\pi\) and reformulate the hyperbolicity criterion. Consider an algebra \((A,\sigma)\) where \(A\) is the endomorphism ring of an anisotropic quadratic space \((V,q)\) over \(F\) and \(\sigma\) is the adjoint map associated to \(q\). The algebra \(A\) becomes hyperbolic over \(F(\pi)\) precisely if \((V,q)\) becomes hyperbolic over \(F(\pi)\); the criterion stated earlier (in terms of \(H_\pi\)) can now be formulated as follows. Theorem. If an anisotropic quadratic space \((V,q)\) becom…
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 12E15
Mathematics Subject Classification ID: 11E04
zbMATH DE Number: 1115984
zbMATH Keywords: central simple algebras with involutions
zbMATH Keywords: direct products of field extensions
zbMATH Keywords: direct products of algebras
zbMATH Keywords: hyperbolicity criterion
zbMATH Keywords: anisotropic quadratic spaces
zbMATH Keywords: quadratic forms
zbMATH Keywords: skew polynomial rings
zbMATH Keywords: Clifford algebras
zbMATH Keywords: splitting criteria for quaternion algebras
author: Darrell Haile
author: Jean-Pierre Tignol
reviewed by: Paul M. Cohn
MaRDI profile type: Publication
full work available at URL: jabr.1997.7193
OpenAlex ID: W2011935043
cites work: Hyperbolic involutions
cites work: Biquaternion algebras and quartic extensions
cites work: Q3708857
cites work: A Cassels-Pfister theorem for involutions on central simple algebras
DOI: 10.1006/JABR.1997.7193
description: scientific article; zbMATH DE number 1115984