Journal of Lie Theory 24 (2014), No. 1, 029--039
Copyright Heldermann Verlag 2014
Split Strongly Abelian p-Chief Factors and First Degree Restricted Cohomology
Jörg Feldvoss
Dept. of Mathematics and Statistics, University of South Alabama, Mobile, AL 36688-0002, U.S.A.
jfeldvoss@southalabama.edu
Salvatore Siciliano
Dip. di Matematica e Fisica "Ennio De Giorgi", Università del Salento, Via Provinciale Lecce-Arnesano, 73100 Lecce, Italy
salvatore.siciliano@unisalento.it
Thomas Weigel
Dip. di Matematica e Applicazioni, Università degli Studi di Milano-Bicocca, Via Roberto Cozzi 53, 20125 Milano, Italy
thomas.weigel@unimib.it
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.
Keywords: Solvable restricted Lie algebra, irreducible module, p-chief factor, strongly abelian p-chief factor, split p-chief factor, multiplicity of a split strongly abelian p-chief factor, restricted cohomology, transgression, principal block, projective indecomp
MSC: 17B05, 17B30, 17B50, 17B55, 17B56
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