Products of several commutators in a Lie nilpotent associative algebra

Abstract

Let [Formula: see text] be a field of characteristic [Formula: see text] and let [Formula: see text] be a unital associative [Formula: see text]-algebra. Define a left-normed commutator [Formula: see text] [Formula: see text] recursively by [Formula: see text], [Formula: see text] [Formula: see text]. For [Formula: see text], let [Formula: see text] be the two-sided ideal in [Formula: see text] generated by all commutators [Formula: see text] ([Formula: see text]. Define [Formula: see text]. Let [Formula: see text] be integers such that [Formula: see text], [Formula: see text]. Let [Formula: see text] be positive integers such that [Formula: see text] of them are odd and [Formula: see text] of them are even. Let [Formula: see text]. The aim of the present note is to show that, for any positive integers [Formula: see text], in general, [Formula: see text]. It is known that if [Formula: see text] (that is, if at least one of [Formula: see text] is even), then [Formula: see text] for each [Formula: see text] so our result cannot be improved if [Formula: see text]. Let [Formula: see text]. Recently, Dangovski has proved that if [Formula: see text] are any positive integers then, in general, [Formula: see text]. Since [Formula: see text], Dangovski’s result is stronger than ours if [Formula: see text] and is weaker than ours if [Formula: see text]; if [Formula: see text], then [Formula: see text] so both results coincide. It is known that if [Formula: see text] (that is, if all [Formula: see text] are odd) then, for each [Formula: see text], [Formula: see text] so in this case Dangovski’s result cannot be improved.