GRADINGS, DERIVATIONS, AND AUTOMORPHISMS OF NEARLY ASSOCIATIVE ALGEBRAS

In this paper, we examine a class of algebras which includes Lie algeb ras, Lie color algebras, right alternative algebras, left alternative algebras, antiassociative algebras, and associative algebras. We call this class of algebras (alpha, beta, gamma)-algebras and we examine gr adings of these algebras by groups with finite support. We generalize various results on associative algebras and finite-dimensional Lie alg ebras. Two of our main results are THEOREM 2.2. Let A be a G-graded le ft (alpha, beta, gamma)-algebra and V = +V-g is an element of G(g) a G -graded left A-module with finite support, where G is a torsion free a belian group. If A(0) acts nilpotently on V, then A also acts nilpoten tly on V. THEOREM 2.12. Let A be a G-graded (alpha, beta, gamma)-algeb ra with finite support, where G = T x Z(m) and T is a torsion free abe lian group. If the identity component A(0,0) acts nilpotently on A on both sides, then A is solvable. These results are used to examine the invariants of automorphisms and derivations. One such application is C OROLLARY 3.3. Let L = +(g is an element of G)L(g) be a Lie color algeb ra over a field K of characteristic 0 and let D be a finite-dimensiona l nilpotent Lie algebra of homogeneous derivations of L which are alge braic as K-linear transformations of L. If L(D) = 0 then L is nilpoten t. We conclude this paper with counterexamples to various questions wh ich arise naturally in light of our results. (C) 1996 Academic Press, Inc.