Metabelian Lie powers of the natural module for a general linear group

Abstract

Consider a free metabelian Lie algebra M of finite rank r over an infinite field K of prime characteristic p. Given a free generating set, M acquires a grading; its group of graded automorphisms is the general linear group GLr(K), so each homogeneous component Md is a finite dimensional GLr(K)-module. The homogeneous component M1 of degree 1 is the natural module, and the other Md are the metabelian Lie powers of the title.This paper investigates the submodule structure of the Md. In the main result, a composition series is constructed in each Md and the isomorphism types of the composition factors are identified both in terms of highest weights and in terms of Steinberg's twisted tensor product theorem; their dimensions are also given. It turns out that the composition factors are pairwise non-isomorphic, from which it follows that the submodule lattice is finite and distributive. By the Birkhoff representation theorem, any such lattice is explicitly recognizable from the poset of its join-irreducible elements. The poset relevant for Md is then determined by exploiting a 1975 paper of Yu.A. Bakhturin on identical relations in metabelian Lie algebras.