Gradings, Derivations, and Automorphisms of Nearly Associative Algebras Original Research Article

In this paper, we examine a class of algebras which includes Lie algebras, Lie color algebras, right alternative algebras, left alternative algebras, antiassociative algebras, and associative algebras. We call this class of algebras (α, β, γ)-algebras and we examine gradings of these algebras by groups with finite support. We generalize various results on associative algebras and finite-dimensional Lie algebras. Two of our main results are Timage2.2. Let A be a G-graded left(α, β, γ)-algebra and V=circled plusgset membership, variantGVga G-graded left A-module with finite support, where G is a torsion free abelian group. If A0acts nilpotently on V, then A also acts nilpotently on V. Timage2.12. Let A be a G-graded(α, β, γ)-algebra with finite support, where G=T×imagemand T is a torsion free abelian 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 Cimage3.3. Let L=circled plusgset membership, variantGLgbe a Lie color algebra over a field K of characteristic0and let D be a finite-dimensional nilpotent Lie algebra of homogeneous derivations of L which are algebraic as K-linear transformations of L. If LD=0then L is nilpotent. We conclude this paper with counterexamples to various questions which arise naturally in light of our results.