Algebra & Number Theory

Inner derivations of alternative algebras over commutative rings

Ottmar Loos, Holger Petersson, and Michel Racine

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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.

Article information

Algebra Number Theory, Volume 2, Number 8 (2008), 927-968.

Received: 6 April 2008
Revised: 26 September 2008
Accepted: 26 October 2008
First available in Project Euclid: 20 December 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 17D05: Alternative rings
Secondary: 17A36: Automorphisms, derivations, other operators 17A45: Quadratic algebras (but not quadratic Jordan algebras) 17B40: Automorphisms, derivations, other operators

inner derivations alternative algebras derivation functors composition algebras automorphisms


Loos, Ottmar; Petersson, Holger; Racine, Michel. Inner derivations of alternative algebras over commutative rings. Algebra Number Theory 2 (2008), no. 8, 927--968. doi:10.2140/ant.2008.2.927.

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