Split strongly abelian p-chief factors and first degree restricted cohomology

TL;DR

This paper explores the relationship between split strongly abelian p-chief factors and first degree restricted cohomology in finite-dimensional restricted Lie algebras, providing new characterizations of solvability and insights into their representation theory.

Contribution

It introduces new characterizations of solvable restricted Lie algebras based on multiplicities of split strongly abelian p-chief factors and cohomological properties.

Findings

Characterization of solvable restricted Lie algebras via p-chief factors

Relation between cohomology and multiplicities of p-chief factors

Results on the principal block and projective covers in representation theory

Abstract

In this paper we investigate the relation between the multiplicities of split strongly abelian p-chief factors of finite-dimensional restricted Lie algebras and first degree restricted cohomology. As an application we obtain a characterization of solvable restricted Lie algebras in terms of the multiplicities of split strongly abelian p-chief factors. Moreover, we derive some 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 finite-dimensional solvable restricted Lie algebras in terms of the second Loewy layer of the projective cover of the trivial irreducible module. The analogues of these results are well known in the modular representation theory of finite groups.