Module decompositions of free lie algebras

In this study, the technique known as elimination to devise some new bases of the free Lie algebra which consist of Lie products of left normed basic Lie monomials is used. In section 2, the direct decomposition as a free K-module of a free Lie algebra is obtained and the direct decompositions of the homogeneous components of the free Lie algebra with direct summands that are tensor products of metabelian Lie powers are obtained. Furthermore, the new filtrations and decompositions of free Lie algebras as modules for groups of graded algebra automorphisms are obtained. Besides, some new decompositions for free Lie algebras as KG- modules over fields of characteristic zero and positive characteristic, respectively, are obtained.