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Gradings, derivations, and automorphisms of nearly associative algebras. (English) Zbl 0859.17002

Any associative algebra \(A\) graded by a group with finite support, in which the identity component of \(A\) is zero, is nilpotent. However this is not true for Lie algebras. In this paper this fact is extended to a more general class of algebras (which contains associative and Lie algebras) graded by certain types of abelian groups. The motivation of the authors is the generalization of several results on group-gradings and the invariants of derivations and automorphisms from associative algebras to various nonassociative algebras. The group-gradings become an interesting tool for the study of the invariants of derivations and automorphisms of an algebra.
In the first section the concept of \((\alpha, \beta, \gamma)\)-algebra is introduced: an algebra \(A\) over a commutative ring \(K\) is a left (resp. right) \((\alpha, \beta, \gamma)\)-algebra if there exists a multiplicatively closed set \(S=S(A)\) which generates \(A\) as a \(K\)-module and there exist functions \(\alpha, \beta, \gamma:S\times S\to K\) such that \[ x(yz)= \alpha(x,y)y(xz)+ \beta(xy)(xy)z+ \gamma(x,y)(yx)z, \] (resp. \((zx)y= \alpha(x,y)(zy)x+ \beta(xy)z(xy)+ \gamma(x,y)z(yx)\)) for all \(x,y\in S\) and \(z\in A\). Algebras which are both left and right \((\alpha, \beta, \gamma)\)-algebras are simply called \((\alpha, \beta, \gamma)\)-algebras. Lie algebras, Lie color algebras, right alternative algebras, left alternative algebras, antiassociative algebras and associative algebras are \((\alpha, \beta, \gamma)\)-algebras by one or two sides. If \(G\) is an abelian group and \(A=\bigoplus_{g\in G} A_g\) is a \(G\)-graded \(K\)-algebra, \(A\) is a \(G\)-graded (left) \((\alpha, \beta, \gamma)\)-algebra if it is a (left) \((\alpha, \beta, \gamma)\)-algebra with spanning set \(S(A)\) such that \(S(A)= \bigcup_{g\in G} S_g\), where \(S_g= S(A)\cap A_g\).
The second section contains the three main results of this work concerning group-gradings of \((\alpha, \beta, \gamma)\)-algebras and its modules in some particular abelian groups: torsion free groups, finite cyclic groups and the direct product of a torsion-free group with a finite cyclic group.
Finally, some applications of these theorems to the action of automorphisms and derivations of Lie color algebras are obtained in the last section. The paper is completed with several examples and counterexamples which show that the above results are best possible.

MSC:

17A36 Automorphisms, derivations, other operators (nonassociative rings and algebras)
17A30 Nonassociative algebras satisfying other identities
16W20 Automorphisms and endomorphisms
16W25 Derivations, actions of Lie algebras
17B70 Graded Lie (super)algebras
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