Identities of relatively free algebras of Lie nilpotent associative algebras

TL;DR

The paper analyzes relatively free algebras F_n(N_p) in the Lie-nilpotent PI-varieties N_p over characteristic-zero fields, providing explicit generating sets for their polynomial identities in key cases (notably p=3 and p=4) and establishing parity-based structural results. It combines commutator-ideal theory, multilinearization, and S_m-module decompositions (notably Volichenko and Stoyanova–Venkova) to derive finite bases of identities for F_n(N3) and F_n(N4), plus general bounds for longer commutator products in F_n(N_p). It also studies asymptotic behavior of the varieties, proving asymptotic equivalence of N_2 and N_3 while showing non-equivalence between consecutive odd/even families, and provides conjectures on the precise minimal lengths of identities and standard identities in the general p-case. The results deepen understanding of how finite-rank relatively free algebras reflect the hierarchy among Lie-nilpotent PI-varieties, with explicit, testable polynomial generators and parity-sensitive phenomena.

Abstract

In this paper, we consider the relatively free algebra of rank $n$, $F_n(\mathfrak{N}_p)$, in the variety of Lie nilpotent associative algebras of index $p$, denoted by $\mathfrak{N}_p$, over a field of characteristic zero. We describe an explicit minimal basis for the polynomial identities of $F_n(\mathfrak{N}_p)$ when $p=3$ and $p=4$, for all $n$, except for $F_3(\mathfrak{N}_4)$. In the general case, we exhibit a lower and an upper bound for the minimal $k$ such that $[x_1,x_2]\cdots[x_{2k-1},x_{2k}]$ is an identity for $F_n(\mathfrak{N}_p)$ for all $n$ and for all $p$.

Theorems & Definitions (76)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Lemma 2.2
• proof
• Lemma 2.3
• proof
• Proposition 2.4
• Lemma 2.5