Nonassociative algebras with scalar involution

Abstract

The classical theory of nondegenerate quadratic forms permitting composition has recently been generalized in several directions: Kunze and Scheinberg considered degenerate forms on alternative algebras over fields of characteristic 2; Petersson and Racine briefly considered nondegenerate forms over general rings of scalars; the generalized Cayley-Dickson algebras of dimension 2" carry a scalar involution, but are not alternative and do not admit composition for n > 3. In this paper we study general algebras with scalar involution (where all norms xx* and traces x + x* are scalars) over arbitrary rings of scalars. We locate these among all degree 2 algebras, and derive conditions for them to be flexible, alternative, or composition algebras. We consider Cayley elements and Cayley ^'representations, recovering the results of Kunze and Scheinberg on radicals of norm forms. We also investigate the Cayley-Dickson doubling process for constructing new scalar involutions out of old ones.