In this paper it is shown that the projective cover of the trivial irreducible module of a finite-dimensional solvable restricted Lie algebra is induced from the one dimensional trivial module of a maximal torus. As a consequence, the number of the isomorphism classes of irreducible modules with a fixed p-character for a finite-dimensional solvable restricted Lie algebra L is bounded above by p MT(L), where MT(L) denotes the maximal dimension of a torus in L. Finally, it is proved that in characteristic p > 3 the projective cover of the trivial irreducible L-module is induced from the one-dimensional trivial module of a torus of maximal dimension, only if L is solvable.
Mathematics and Statistics
FELDVOSS, J., SICILIANO, S. & WEIGEL, T. RESTRICTED LIE ALGEBRAS WITH MAXIMAL 0-PIM. Transformation Groups 21, 377–398 (2016). https://doi.org/10.1007/s00031-015-9362-5