Irreducible representations of strongly solvable Lie algebras over a field of positive characteristic. (Russian) Zbl 0542.17004

Dzhumadil’daev, A. S.

Mat. Sb., N. Ser. 123(165), No. 2, 212-229 (1984).

Let L be a strongly solvable Lie algebra over an algebraically closed field of characteristic \(p>0\). In this study of irreducible representations of such algebras the author is able to obtain a number of results which are valid even if L is not restricted. This is made possible in part by consideration of the Lie algebra of derivations of the form (ad x)\({}^{p^ k}\), 0\(\leq k\), \(x\in L\). It is also shown that the structure of irreducible representations of L can be deduced from that of irreducible representations of a certain restricted Lie algebra into which L can be mapped by an injective homomorphism. The dimension of an irreducible module of weight \({\mathfrak z}\) depends on the dimension of the stationary subalgebra of a certain linear functional \(\tilde {\mathfrak z}\). A formula is obtained for the number of nonisomorphic representations of a strongly solvable restricted Lie algebra. The results are applied to describe certain irreducible representations of a maximal subalgebra of a Zassenhaus algebra.

Reviewer: G.Brown

MSC:

17B50	Modular Lie (super)algebras
17B30	Solvable, nilpotent (super)algebras

Keywords:

prime characteristic; strongly solvable Lie algebra; irreducible representations; derivations; restricted Lie algebra; number of nonisomorphic representations; maximal subalgebra; Zassenhaus algebra