Restricted Lie algebras with maximal 0-PIM

TL;DR

This paper investigates the structure of projective covers of trivial modules in finite-dimensional solvable restricted Lie algebras, revealing their induction from maximal tori and bounding the number of irreducible modules with a fixed p-character.

Contribution

It establishes that the projective cover of the trivial module is induced from a maximal torus and characterizes when this occurs in characteristic p>3.

Findings

Projective cover of trivial module is induced from a maximal torus.

Number of irreducible modules with fixed p-character is bounded by p^MT(L).

Induction from a maximal torus characterizes solvability in characteristic p>3.

Abstract

In this paper we show 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, we obtain that 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 largest dimension of a torus in L. Finally, we prove that in characteristic p>3 the projective cover of the trivial irreducible L-module is only induced from the one-dimensional trivial module of a torus of maximal dimension if L is solvable.