Gradings, Derivations, and Automorphisms of Nearly Associative Algebras

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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 areTheorem2.2.Let A be a G-graded left(α,β,γ)-algebra and V=⊕g∈GVga 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.Theorem2.12.Let A be a G-graded(α,β,γ)-algebra with finite support, where G=T×Zmand 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 isCorollary3.3.Let L=⊕g∈GLgbe 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.