Inner derivations of alternative algebras over commutative rings

Abstract

We define Lie multiplication derivations of an arbitrary non-associative algebra A over any commutative ring and, following McCrimmon [15], 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 [19, 20] and McCrimmon’s associator derivations [15] occupy a special place, the latter being particularly useful to resolve diculties in characteristic 3. We also show that octonion algebras over any commutative ring have only associator derivations. 2000 Mathematics Subject Classification. Primary 17D05; Secondary 17A36, 17A45, 17B40