ON THE DIMENSION OF THE PRODUCT [L-2, L-2, L-1] IN FREE LIE ALGEBRAS

Abstract

Let L be a free Lie algebra of rank r >= 2 over a field F and let L-n denote the degree n homogeneous component of L. By using the dimensions of the corresponding homogeneous and fine homogeneous components of the second derived ideal of free centre-by-metabelian Lie algebra over a field F, we determine the dimension of [L-2, L-2, L-1]. Moreover, by this method, we show that the dimension of [L-2, L-2, L-1] over a field of characteristic 2 is different from the dimension over a field of characteristic other than 2.