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Gradings, Derivations, and Automorphisms of Nearly Associative Algebras

✍ Scribed by Jeffrey Bergen; Piotr Grzeszczuk

Publisher: Elsevier Science
Year: 1996
Tongue: English
Weight: 260 KB
Volume: 179
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1996.0034

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✦ Synopsis

In this paper, we examine a class of algebras which includes Lie algebras, Lie color algebras, right alternative algebras, left alternative algebras, antiassociative Ž . algebras, and associative algebras. We call this class of algebras ␣, ␤, ␥ -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

G-graded left A-module with finite support, where G is a torsion free abelian group. If A acts nilpotently on V, then A also acts nilpotently on V. 0 Ž . T HEOREM 2.12. Let A be a G-graded ␣ , ␤, ␥ -algebra with finite support, where G s T = ‫ޚ‬ and T is a torsion free abelian group. If the identity component A m Ž0, 0.

acts nilpotently on A on both sides, then A is sol¨able.

These results are used to examine the invariants of automorphisms and derivations. One such application is COROLLARY 3.3. Let L s [ L be a Lie color algebra o¨er a field K of g g G g characteristic 0 and let D be a finite-dimensional nilpotent Lie algebra of homogeneous deri¨ations of L which are algebraic as K-linear transformations of L. If L D s 0 then L is nilpotent.

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