Inner derivations of alternative algebras over commutative rings
type: Item
instance of: scholarly article
title: Inner derivations of alternative algebras over commutative rings
zbMATH Open document ID: 1191.17011
author: Ottmar Loos
author: Holger P. Petersson
author: Michel L. Racine
published in: Algebra \& Number Theory
publication date: 2009-03-16T00:00:00Z
review text: In [Bull. Am. Math. Soc. 55, 769--776 (1949; Zbl 0033.34803)], \textit{R. D. Schafer} introduced a notion of inner derivations for arbitrary non-associative algebras over fields. This new notion reduces to the usual one when dealing with Lie or unital associative (resp. Jordan) algebras. Moreover, inner derivations in this sense always form an ideal in the full derivation algebra. Unfortunately, inner derivations in the sense of Schafer do not satisfy the Mapping Principle even for the class of alternative algebras (the Mapping Principle states that for any homomorphism \(f: A \to B\) of algebras \(A\) and \(B\) and any inner derivation \(D\) of \(A\), there exists an inner derivation \(D'\) of \(B\) with is \(f\)-related to \(D\), i.e., \(f(D(a))=D'(f(a))\) for all \(a\in A\)). In view of this result, E. Neher suggested to come back to the old idea of defining individually inner derivations for each class of relevant nonassociative algebras, and the authors of this paper devote this work to study a suitable notion of inner derivations for alternative algebras. In section 1 the authors introduce for a general nonassociative algebra \(A\) what they call the Lie multiplication derivation algebra of \(A\), \(\text{LMDer}(A)\), which is a particular subset of derivations of \(A\) contained in the Lie multiplication algebra of \(A\). This set is an ideal of the algebra of derivations \(Der(A)\) of \(A\) and it coincides with the usual set of inner derivations for associative and linear Jordan algebras, and with the set of inner derivations in the sense of Schafer if \(A\) is a general non-associative unital algebra. In the rest of this section the authors study the well behavior of \(\text{LMDer}(A)\) under suitable scalar extensions. In section 2, following A5.2 of [\textit{K. McCrimmon}, Alternative algebras, \url{http://www.mathstat.uottawa.ca/\(\sim\)neher/Papers/alternative/} (1980)] the authors describe the Lie multiplication derivation algebra \(\text{LMDer}(A)\) …
Mathematics Subject Classification ID: 17D05
Mathematics Subject Classification ID: 17A36
Mathematics Subject Classification ID: 17A45
Mathematics Subject Classification ID: 17B40
zbMATH DE Number: 5529349
zbMATH Keywords: inner derivations
zbMATH Keywords: alternative algebras
zbMATH Keywords: derivation functors
zbMATH Keywords: composition algebras
zbMATH Keywords: automorphisms.
MaRDI profile type: Publication
full work available at URL: ant.2008.2.927
OpenAlex ID: W2150685975
DOI: 10.2140/ANT.2008.2.927
description: scientific article; zbMATH DE number 5529349