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

Abstract

Let L be a free Lie algebra of Önite rank r 2 over a Öeld F and we let Lmidenote the degree mi homogeneous component of L. Ralph Stˆhrand Micheal Vaughan-Lee derived formulae for the dimension of the subspaces[Lm1; Lm2] for all m1 and m2. Then, the author and R. Stˆhr obtained formulae for the dimension of the products [Lm1; Lm2; Lm3] under certain conditionson m1; m2; m3. In this paper, we study on products of n homogeneous components in free Lie algebra and we derive formulae for the dimension of suchproducts.