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_{m_{i}} denote the degree m_{i} homogeneous component of L. Ralph Stöhr and Micheal Vaughan-Lee derived formulae for the dimension of the subspaces [L_{m₁},L_{m₂}] for all m₁ and m₂. Then, the author and R. Stöhr obtained formulae for the dimension of the products [L_{m₁},L_{m₂},L_{m₃}] under certain conditions on 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 such products.