Identities of relatively free algebras of Lie nilpotent associative algebras

In this paper, we consider the relatively free algebra of rank $n$, $F_n({\mathfrak{N}}_p)$, in the variety of Lie nilpotent associative algebras of index $p$, denoted by ${\mathfrak{N}}_p$, over a field of characteristic zero. We describe an explicit minimal basis for the polynomial identities of $F_n({\mathfrak{N}}_p)$ when $p=3$ and $p=4$, for all $n$, except for $F_3({\mathfrak{N}}_4)$. In the general case, we exhibit a lower and an upper bound for the minimal $k$ such that $[x_1,x_2]\cdots [x_{2k-1},x_{2k}]$ is an identity for $F_n({\mathfrak{N}}_p)$ for all $n$ and for all $p$.