THE DIMENSION OF PRODUCTS OF n HOMOGENEOUS COMPONENTS IN FREE LIE ALGEBRAS

Abstract

Let L be a free Lie algebra of finite rank r >= 2 over a field F and we let L-mi denote the degree m(i) homogeneous component of L. Ralph Stohr and Micheal Vaughan-Lee derived formulae for the dimension of the subspaces [L-m1, L-m2] for all m(1) and m(2). Then, the author and R. Stohr obtained formulae for the dimension of the products [L-m1, L-m2, L-m3] under certain conditions on m(1), m(2), m(3). In this paper, we study on products of n homogeneous components in free Lie algebra and we derive formulae for the dimension of such products.