Hyperbolicity of algebras with involution over a given extension.
type: Item
instance of: scholarly article
title: Hyperbolicity of algebras with involution over a given extension.
zbMATH Open document ID: 1246.16030
author: Jung-Miao Kuo
author: Su Chi Wen
published in: Journal of Pure and Applied Algebra
publication date: 2011-04-29T00:00:00Z
review text: Problem: given a finite extension \(K\supseteq F\), determine which anisotropic central simple algebras with involution over \(F\) become hyperbolic over \(K\). (Recall that \((A,\sigma)\) is isotropic if \(\sigma(a)a=0\) for some \(a\neq 0\), and hyperbolic if for some right ideal \(I\), \(I^\perp=\{x\mid\sigma(x)I=0\}=I\).) This problem was studied by \textit{J.-P. Tignol} [J. Algebra 181, No. 3, 857-875 (1996; Zbl 0847.16019)], who constructed for a field \(F\) (characteristic not \(2\)) and a separable polynomial \(\pi\in F[t]\) of degree \(2n\), an algebra with involution \(H_\pi\), whose simple quotients are the only anisotropic central simple algebras over \(F\) that become hyperbolic over \(F(\pi)=F[t]/F[t]\pi\). In fact, \(H_\pi\) is the free algebra \(F\langle a_1,\dots,a_n,b_1,\dots,b_n\rangle\) with the relations following from asserting the decomposition \(\pi(t)=(t^n+a_1t^{n-1}+\cdots+a_n)(t^n+b_1t^{n-1}+\cdots+b_n)\). \textit{D. E. Haile} and \textit{J.-P. Tignol} [J. Algebra 199, No. 1, 94-115 (1998; Zbl 0897.16018)] showed that when \(n>2\), \(H_\pi\) contains a noncommutative free subalgebra, and thus introduce a more manageable homomorphic image \(C_\pi\), which covers many examples of algebras that become hyperbolic over \(F(\pi)\). This quotient is described in the current paper as follows. Let \(q=\sum a_{ij}x_ix_j\) be the generic quadratic form of dimension \(n\) over \(F\), and \(f=\sum r_it^{n-i}\) the generic polynomial of degree \(n\). Let \(Z_\pi\) be quotient of \(F[a_{ij},r_i]\) defined by the relation \(f(t)-q(t^{n-1},\dots,1)=\pi(t)\). Then \(C_\pi\) is the even Clifford algebra of the form \(x_0^2-q(x_1,\dots,x_n)\) over \(Z_\pi\). It is shown that for every anisotropic quadratic form \(p\) over a finite extension \(K\) of \(F\), if \(p\) represents \(\pi(t)\) over \(K(t)\), then the even Clifford algebra \(C_0(p)\) is a quotient of \(C_\pi\) (as a ring with involution). Partial converse is given, but a complete description of the s…
reviewed by: Uzi Vishne
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 16K20
Mathematics Subject Classification ID: 11E04
Mathematics Subject Classification ID: 11E88
Mathematics Subject Classification ID: 15A66
zbMATH DE Number: 5882427
zbMATH Keywords: algebras with involution
zbMATH Keywords: hyperbolic algebras
zbMATH Keywords: hyperbolic central simple algebras
zbMATH Keywords: anisotropic central simple algebras
zbMATH Keywords: Clifford algebras
MaRDI profile type: Publication
full work available at URL: j.jpaa.2010.08.015
OpenAlex ID: W2064403622
cites work: Separable algebras over commutative rings
cites work: HYPERBOLICITY OF ALGEBRAS WITH INVOLUTION AND CONNECTIONS WITH CLIFFORD ALGEBRAS
cites work: Algebras with involution that become hyperbolic under a given extension
cites work: Q3882586
cites work: Q4210545
cites work: A Cassels-Pfister theorem for involutions on central simple algebras
DOI: 10.1016/J.JPAA.2010.08.015
description: scientific article; zbMATH DE number 5882427