The Casimir operation for alternative algebras

Abstract

This note is in the nature of an addendum to our paper, Representations of alternative algebras [6 ].2 The terminology and notations of that paper are used throughout. All algebras considered are finitedimensional over a field F of characteristic 0. We generalize to alternative algebras the Casimir operation for associative algebras [3, p. 682 ]. However, we have been unable to obtain by this means the second Whitehead lemma for alternative algebras. This lemma is actually valid, since it is equivalent to one case of the known Wedderburn principal theorem for alternative algebras [5]. It would be desirable to have a direct proof, since then the proof of the Wedderburn principal theorem would not rely on elementwise construction for each type of split algebra. What we do prove, using the Casimir operation, is a weaker theorem (Theorem 2), which yields certain simplifications in the known proof of the Wedderburn principal theorem.