## 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 GL _{r}(K), so each homogeneous component M ^{d} is a finite dimensional GL _{r}(K)-module. The homogeneous component M ^{1} of degree 1 is the natural module, and the other M ^{d} are the metabelian Lie powers of the title.This paper investigates the submodule structure of the M ^{d}. In the main result, a composition series is constructed in each M ^{d} 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 M ^{d} is then determined by exploiting a 1975 paper of Yu.A. Bakhturin on identical relations in metabelian Lie algebras.

Original language | English |
---|---|

Pages (from-to) | 232-267 |

Number of pages | 36 |

Journal | Journal of Algebra |

Volume | 352 |

Issue number | 1 |

DOIs | |

Publication status | Published - 15 Feb 2012 |