Products of commutators in a Lie nilpotent associative algebra

Abstract

Let $F$ be a field and let $F \langle X \rangle$ be the free unital associative algebra over $F$ freely generated by an infinite countable set $X = \{x_1, x_2, \dots \}$. Define a left-normed commutator $[a_1, a_2, \dots, a_n]$ recursively by $[a_1, a_2] = a_1 a_2 - a_2 a_1$, $[a_1, \dots, a_{n-1}, a_n] = [[a_1, \dots, a_{n-1}], a_n]$ $(n \ge 3)$. For $n \ge 2$, let $T^{(n)}$ be the two-sided ideal in $F \langle X \rangle$ generated by all commutators $[a_1, a_2, \dots, a_n]$ ($a_i \in F \langle X \rangle)$. Let $F$ be a field of characteristic $0$. In 2008 Etingof, Kim and Ma conjectured that $T^{(m)} T^{(n)} \subset T^{(m+n -1)}$ if and only if $m$ or $n$ is odd. In 2010 Bapat and Jordan confirmed the "if" direction of the conjecture: if at least one of the numbers $m$, $n$ is odd then $T^{(m)} T^{(n)} \subset T^{(m + n -1)}.$ The aim of the present note is to confirm the "only if" direction of the conjecture. We prove that if $m = 2 m'$ and $n = 2 n'$ are even then $T^{(m)} T^{(n)} \nsubseteq T^{(m +n -1)}.$ Our result is valid over any field $F$.