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 $X=\{x_1,x_2,\dots \}$. 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 $F⟨X⟩$ generated by all commutators $[a_1,a_2,\dots ,a_n]$ ($a_i\in F⟨X⟩)$. 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 $m=2m^{\prime }$ and $n=2n^{\prime }$ are even then $T^{(m)}{T}^{(n)}⊈T^{(m+n-1)}.$ Our result is valid over any field $F$.

8 pages, remarks and references added, typos fixed

References in corpus (5)

• The additive group of a Lie nilpotent associative ring
• Lower central series of a free associative algebra over the integers and finite fields
• The torsion subgroup of the additive group of a Lie nilpotent associative ring of class 3
• On the Maximal Containments of Lower Central Series Ideals
• Relations in universal Lie nilpotent associative algebras of class 4

Cited by in corpus (6)

• On the Maximal Containments of Lower Central Series Ideals
• On some products of commutators in an associative ring
• Products of several commutators in a Lie nilpotent associative algebra
• On Lie nilpotent associative algebras
• Lie nilpotent Novikov algebras and Lie solvable Leavitt path algebras
• On the $\mathrm{GL}(n)$-module structure of Lie nilpotent associative relatively free algebras