Involutions on composition algebras
type: Item
instance of: scholarly article
title: Involutions on composition algebras
zbMATH Open document ID: 1053.17006
DOI: 10.1016/S0019-3577(03)90008-5
published in: Indagationes Mathematicae. New Series
publication date: 2004-06-15T00:00:00Z
review text: In this paper involutions on composition algebras are studied. First, let \(R\) be a unital commutative associative ring where 2 is invertible in \(R\), and let \(C\) be a composition algebra over \(R\). Then \(C\) has a standard involution \(\overline{}\) given by \(\overline x= t(x)l_C- x\), where \(t: C\to R\) is the trace map. An involution \(\tau: C\to C\) is an anti-automorphism of period 2, i.e. \(\tau(x+ y)= \tau(x)+ \tau(y)\), \(\tau(xy)=\tau(y)\tau(x)\), \(\tau^2= \text{id}\), for all \(x,y\in C\). If \(\tau|_R= \text{id}\) then the involution \(\tau\) is called of the first kind. The author shows that there is a one-one correspondence between non-standard involutions of the first kind, and composition algebras of half \(\text{rank\,}{r\over 2}\). Moreover, every non-standard involution \(\tau\) of the first kind is isomorphic to the natural generalization of Lewis's hat involution: \(C\cong \text{Cay}(B, P, N)\) and \(\tau((u, v))= (\overline u, v)\) is the hat involution on \(C\). Now let \(S\) be a quadratic étale \(R\)-algebra with standard involution \(s_0\), let \(C\) be a composition algebra over \(S\) of \(\text{rank\,}\geq 4\), and assume \(A= \text{Cay}(c,\mu_1,\dots, \mu_m)\) is a generalized Cayley-Dickson algebra of \(\text{rank\,}2^m\cdot\text{rank\,}C\geq 16\). An involution on C whose restriction to \(S\) is the standard involution is called of the second kind. Now let \(\sigma={}^-\) denote the scalar involution on \(A\), and let \(\tau\) be any involution on \(A\) of the second kind. Then \(\tau\) can be written as the tensor product of the standard involution of a unique \(R\)-composition subalgebra of \(C\) and the standard involution of \(S/R\).
reviewed by: Huberta Lausch
Mathematics Subject Classification ID: 17A75
Mathematics Subject Classification ID: 16W10
zbMATH DE Number: 2075413
zbMATH Keywords: composition algebra
zbMATH Keywords: involution
zbMATH Keywords: Cayley-Dickson algebra
zbMATH Keywords: quadratic étale \(R\)-algebra
zbMATH Keywords: Cayley-Dickson doubling process
author: Susanne Pumplün
MaRDI profile type: Publication
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full work available at URL: s0019-3577(03)90008-5
OpenAlex ID: W1975488670
description: scientific article; zbMATH DE number 2075413