Vol. 2, No. 8, 2008

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ISSN: 1944-7833 (e-only)
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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
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