Document Type

Article

Publication Title

Journal of Algebra

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 an application 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.

First Page

303

Last Page

313

DOI

https://doi.org/10.1016/j.jalgebra.2013.01.030

Publication Date

5-2013

Department

Mathematics and Statistics

Comments

This is the pre-print version of this article. The published version can be found at the following links: https://www.sciencedirect.com/science/article/pii/S0021869313000720 or https://doi.org/10.1016/j.jalgebra.2013.01.030

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