On centres of relatively free associative algebras with a Lie nilpotency identity

We study central polynomials of a relatively free Lie nilpotent algebra $F^{(n)}$ of degree $n$. We prove a product theorem, which generalizes the well-known results of Latyshev and Volichenko. We construct generalized Hall polynomials, by using which we prove that the core centre of the algebra $F^{(n)}$ is nontrivial for any $n\geqslant 5$. We obtain a number of special results when $n=5$ and $6$.
Bibliography: 27 titles.