Supersolvable Malcev algebras

Abstract

Recall that a group G is supersolvable if it possesses a finite increasing maximal chain of normal subgroups of G such that each factor is cyclic. This concept was introduced with the hope that it would be more tractable than solvability in general. Indeed many results on this class have appeared (see [9]). Motivated at least in part by this success, an analogous concept was introduced in Lie algebras where normal subgroups with cyclic factors were replaced by ideals with one dimensional factors. Once again the investigations proved fruitful [l, 2, 31. Adding to the importance of the concept in the Lie setting is the famous theorem of S. Lie which yields that over an algebraically closed field of characteristic 0, all solvable Lie algebras are supersolvable. This fundamental theorem had influence on the development of the structure and representation theory of these algebras. On the other hand, Malcev algebras are a generlization of Lie algebras in which results on the latter often find extensions to the former. Because of all the foregoing, supersolvability shows much promise as an object of study in Malcev algebras. Barnes [ 11 used cohomology theory to find Lie algebra analogues to theorems of Baer, Gaschtitz, and Huppert. These results were used to show various structure theorems and to construct a theory of formations [a]. Eventually, simpler proofs of the Barnes’ theorems were given by Barnes and Newell [3]. Still later the analogues of the theorems of Baer and Gas- chiitz were extended to Malcev algebras [S] and [S]. The final result con- cerns supersolvability and we extend this result to Maicev algebras in this paper where the algebra M is called supersolvable if there exists an increas- ing maximal chain of ideals of M each codimension one in the next. Although this result and related concepts extend to the present case, com- plications arise in the proofs due to the weaker defining identity for Malcev 69