Inner derivations of alternative algebras over commutative rings

Loos, Ottmar; Petersson, Holger; Racine, Michel

doi:10.2140/ant.2008.2.927

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Inner derivations of alternative algebras over commutative rings

Ottmar Loos, Holger P. Petersson and Michel L. Racine

Vol. 2 (2008), No. 8, 927–968

DOI: 10.2140/ant.2008.2.927

Abstract

We define Lie multiplication derivations of an arbitrary non-associative algebra $A$ over any commutative ring and, following an approach due to K. McCrimmon, describe them completely if $A$ is alternative. Using this description, we propose a new definition of inner derivations for alternative algebras, among which Schafer’s standard derivations and McCrimmon’s associator derivations occupy a special place, the latter being particularly useful to resolve difficulties in characteristic $3$ . We also show that octonion algebras over any commutative ring have only associator derivations.

Erhard Neher zum 60. Geburtstag gewidmet

Keywords

inner derivations, alternative algebras, derivation functors, composition algebras, automorphisms

Mathematical Subject Classification 2000

Primary: 17D05

Secondary: 17A36, 17A45, 17B40

Milestones

Received: 6 April 2008

Revised: 26 September 2008

Accepted: 26 October 2008

Published: 23 November 2008

Authors

Ottmar Loos
Fakultät für Mathematik und Informatik FernUniversität in Hagen D-58084 Hagen Germany

Holger P. Petersson
Fakultät für Mathematik und Informatik FernUniversität in Hagen D-58084 Hagen Germany

Michel L. Racine
Department of Mathematics and Statistics University of Ottawa Ottawa, Ontario K1N 6N5 Canada

Vol. 2, No. 8, 2008