ON THE DIMENSION OF PRODUCTS OF HOMOGENEOUS SUBSPACES IN FREE LIE ALGEBRAS

Let $L$ be a free Lie algebra of finite rank over a field $K$ and let $L_{n}$ denote the degree $n$ homogeneous component of $L$. Formulae for the dimension of the subspaces $[L_m,L_n]$ for all $m$ and $n$ were obtained by the second author and Michael Vaughan-Lee. In this note we consider subspaces of the form $[L_m,L_n,L_k]=[[L_m,L_n],L_k]$. Surprisingly, in contrast to the case of a product of two homogeneous components, the dimension of such products may depend on the characteristic of the field $K$. For example, the dimension of $[L_2,L_2,L_1]$ over fields of characteristic $2$ is different from the dimension over fields of characteristic other than $2$. Our main result are formulae for the dimension of $[L_m,L_n,L_k]$. Under certain conditions on $m$, $n$ and $k$ they lead to explicit formulae that do not depend on the characteristic of $K$, and express the dimension of $[L_m,L_n,L_k]$ in terms of Witt's dimension function