Minimal k for product-of-commutator identity in Fn(Ap)

Determine the minimal integer k such that the product of simple commutators [x1, x2] ··· [xk−1, xk] is a polynomial identity for the relatively free algebra of rank n in the variety Ap of Lie-nilpotent associative algebras of index p over a field of characteristic zero, denoted Fn(Ap).

Background

Section 5 studies, for general p and n, when a product of simple commutators becomes an identity in Fn(Ap). The authors establish parity-dependent lower bounds showing such products are not identities up to specific lengths (Proposition 5.1), and give a general upper bound showing that for even k > np − n + 1 the product [x1, x2] ··* [xk−1, xk] is an identity (Proposition 5.2).

For p = 3 and p = 4, earlier sections determine exact identities for all n (except for one unresolved finite-rank case in p = 4), motivating a general determination of the precise minimal k. The authors propose a concrete conjectural formula depending on the parity of n and p and certain size conditions (n relative to p).