Split abelian chief factors and first degree cohomology for Lie algebras

TL;DR

This paper explores the relationship between split abelian chief factors and first degree cohomology in finite-dimensional Lie algebras, providing characterizations of modular solvable Lie algebras and applications to restricted Lie algebra representation theory.

Contribution

It offers new characterizations of modular solvable Lie algebras using cohomology and chief factors, extending known results and refining non-vanishing theorems.

Findings

Characterization of modular solvable Lie algebras via cohomology and chief factors

Refinement of Seligman's non-vanishing theorem for cohomology

Results on the structure of projective covers in restricted Lie algebra representations

Abstract

In this paper we investigate the relation between the multiplicities of split abelian chief factors of finite-dimensional Lie algebras and first degree cohomology. In particular, we obtain a characterization of modular solvable Lie algebras in terms of the vanishing of first degree cohomology or in terms of the multiplicities of split abelian chief factors. The analogues of these results are well known in the modular representation theory of finite groups. An important tool in the proof of these results is a refinement of a non-vanishing theorem of Seligman for the first degree cohomology of non-solvable finite-dimensional Lie algebras in prime characteristic. As applications we derive several results in the representation theory of restricted Lie algebras related to the principal block and the projective cover of the trivial irreducible module of a finite-dimensional restricted Lie algebra. In particular, we obtain a characterization of solvable restricted Lie algebras in terms of the second Loewy layer of the projective cover of the trivial irreducible module.