On the Number of Simple Modules of a Supersolvable Restricted Lie Algebra

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Journal of Algebra


Around 1939 Hans Zassenhaus started to study the representation theory of finite-dimensional Lie algebras over a field of prime characteristic [18, 19]. Since then a lot of progress has been made but nevertheless a classification of the simple modules (up to isomorphism) is not known for many classes of Lie algebras. In the papers cited above Zassenhaus gave such a classification for nilpotent Lie algebras. About 30 years later parts of this were extended to supersolvable Lie algebras by Veisfeiler and Kac (see [17, Sect. 2]). More generally, they introduced a partition of the (infinite) set of isomorphism classes of simple modules into (infinitely many) finite sets, namely, the sets of isomorphism classes of simple modules with a fixed p-character (see [17, Sect. 1]). In particular, for a nilpotent restricted Lie algebra the number of isomorphism classes of simple modules with a fixed p-character is known (see [15, Satz 6]). A fundamental question that still remains open is to determine this number for more general classes of Lie algebras. The aim of this paper is to develop an approach for attacking this problem in the case of supersolvable (restricted) Lie algebras. In the following we will describe the contents of the paper in more detail.

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Mathematics and Statistics