Proper central and core polynomials of relatively free associative algebras with identity of Lie nilpotency of degrees 5 and 6

We study the centre of a relatively free associative algebra $F^{(n)}$ with the identity $[x_1,\dots,x_n]=0$ of Lie nilpotency of degree $n=5,6$ over a field of characteristic 0. It is proved that the core $Z^*(F^{(5)})$ of the algebra $F^{(5)}$ (the sum of all ideals of $F^{(5)}$ contained in its centre) is generated as a $\mathrm T$-ideal by the weak Hall polynomial $[[x,y]^{2},y]$. It is also proved that every proper central polynomial of $F^{(5)}$ is contained in the sum of $Z^*(F^{(5)})$ and the $\mathrm T$-space generated by $[[x,y]^{2}, z]$ and the commutator $[x_1,\dots, x_4]$ of degree 4. This implies that the centre of $F^{(5)}$ is contained in the $\mathrm T$-ideal generated by the commutator of degree 4.
Similar results are obtained for $F^{(6)}$; in particular, it is proved that the core $Z^{*}(F^{(6)})$ is generated as a $\mathrm T$-ideal by the commutator of degree 5.
Bibliography: 15 titles.