
Journal of Lie Theory 24 (2014), No. 1, 029039 Copyright Heldermann Verlag 2014 Split Strongly Abelian pChief Factors and First Degree Restricted Cohomology Jörg Feldvoss Dept. of Mathematics and Statistics, University of South Alabama, Mobile, AL 366880002, U.S.A. jfeldvoss@southalabama.edu Salvatore Siciliano Dip. di Matematica e Fisica "Ennio De Giorgi", Università del Salento, Via Provinciale LecceArnesano, 73100 Lecce, Italy salvatore.siciliano@unisalento.it Thomas Weigel Dip. di Matematica e Applicazioni, Università degli Studi di MilanoBicocca, Via Roberto Cozzi 53, 20125 Milano, Italy thomas.weigel@unimib.it We investigate the relation between the multiplicities of split strongly abelian pchief factors of finitedimensional 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 pchief 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 finitedimensional restricted Lie algebra. In particular, we obtain a characterization of finitedimensional 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, pchief factor, strongly abelian pchief factor, split pchief factor, multiplicity of a split strongly abelian pchief factor, restricted cohomology, transgression, principal block, projective indecomp MSC: 17B05, 17B30, 17B50, 17B55, 17B56 [ Fulltextpdf (268 KB)] for subscribers only. 