Products of commutators in a Lie nilpotent associative algebra

Abstract

Let F be a field and let F〈X〉 be the free unital associative algebra over F freely generated by an infinite countable set X={x1,x2,…}. Define a left-normed commutator [a1,a2,…,an] recursively by [a1,a2]=a1a2−a2a1, [a1,…,an−1,an]=[[a1,…,an−1],an] (n≥3). For n≥2, let T(n) be the two-sided ideal in F〈X〉 generated by all commutators [a1,a2,…,an] (ai∈F〈X〉).Let F be a field of characteristic 0. In 2008 Etingof, Kim and Ma conjectured that T(m)T(n)⊂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)⊂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=2m′ and n=2n′ are even then T(m)T(n)⊈T(m+n−1). Our result is valid over any field F.