On Anti-Commutative Algebras with an Invariant Form

Abstract

In this paper we consider anti-commutative algebras with an invariant form, that is, an algebra A over a field F such thatand A possesses a symmetric bilinear form f(x, y) such thatLie and Malcev algebras (2, 3) are examples of such algebras and we shall consider generalizations of these algebras obtained by introducing commutation, x o y = xy — yx, as a new multiplicative operation in the non-commutative Jordan algebras of (1).