Title:

The structure of the second derived ideal of free centrebymetabelian Lie rings

We study the free centrebymetabelian Lie ring, that is, the free Lie ring with the property that the second derived ideal is contained in the centre. We exhibit explicit generating sets for the homogeneous components and the fine homogeneous components of the second derived ideal. Each of these components is a direct sum of a free abelian group and a (possibly trivial) elementary abelian $2$group. Our generating sets are such that some of their elements generate the torsion subgroup while the remaining ones freely generate a free abelian group. A key ingredient of our approach is the determination of the dimensions of the corresponding homogeneous components of the free centrebymetabelian Lie algebra over fields of characteristic other than $2$. For this we exploit a $6$term exact sequence of modules over a polynomial ring that is originally defined over the integers, but turns into a sequence whose terms are projective modules after tensoring with a suitable field. Our results correct a partly erroneous theorem in the literature. Moreover, we study the product of three homogeneous components of a free Lie algebra. Let $L$ be a free Lie algebra of finite rank over a field and let $L_n$ denote the degree $n$ homogeneous component of $L$. Formulae for the dimension of the subspaces $[L_n,L_m]$ for all $n$ and $m$ were obtained by Ralph St\"{o}hr and Michael VaughanLee. Formulae for the dimension of the subspaces of the form $[L_n,L_m,L_k]$ under certain conditions on $n,m$ and $k$ were obtained by Nil Mansuro\u{g}lu and Ralph St\"{o}hr. 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. 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 $2$.
