Relations in universal Lie nilpotent associative algebras of class 4

TL;DR

This paper investigates the structure of certain ideals in free associative algebras, providing explicit generating sets for the ideals related to Lie nilpotency of class 4 and 5.

Contribution

It explicitly describes generating sets for the ideals $T^{(4)}$ and $T^{(5)}$ in free associative algebras, extending known results for lower classes.

Findings

Generated $T^{(4)}$ with 3 or 5 polynomial types depending on the ring

Presented a generating set for $T^{(5)}$ with 8 polynomial types

Clarified the structure of Lie nilpotent associative algebras of class 4 and 5

Abstract

Let $K$ be a unital associative and commutative ring and let $K \langle X \rangle$ be the free unital associative $K$-algebra on a non-empty set $X$ of free generators. Define a left-normed commutator $[a_1, a_2, \dots , a_n]$ inductively 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 $K \langle X \rangle$ generated by all commutators $[a_1,a_2, \dots , a_n]$ $( a_i \in K \langle X \rangle )$. It can be easily seen that the ideal $T^{(2)}$ is generated (as a two-sided ideal in $K \langle X \rangle$) by the commutators $[x_1, x_2]$ $(x_i \in X)$. It is well-known that $T^{(3)}$ is generated by the polynomials $[x_1,x_2,x_3]$ and $[x_1,x_2][x_3,x_4] + [x_1,x_3][x_2,x_4]$ $(x_i \in X)$. A similar generating set for $T^{(4)}$ contains 3 types of polynomials in $x_i \in X$ if $\frac{1}{3} \in K$ and 5 types if $\frac{1}{3} \notin K$. In the present article we exhibit a generating set for $T^{(5)}$ that contains 8 types of polynomials in $x_i \in X$.