On torsion-free abelian groups and Lie algebras

Abstract

It is known that many of the classes of simple Lie algebras of prime characteristic of nonclassical type have simple infinite-dimensional analogues of characteristic zero (see, for example, [4, p. 518]). We consider here analogues of those algebras which are defined by a modification of the definition of a group algebra. Thus we consider analogues of the Zassenhaus algebras as generalized by Albert and Frank in [1]. The algebras considered are defined as follows. Let G be a nonzero abelian group, F a field, g an additive mapping from G to F, and f an alternate biadditive mapping from G X G to F. We index a basis of an algebra over F by G, denoting by ua the basis element corresponding to a in G, and define multiplication by