On Lower Central Series Quotients of Finitely Generated Algebras over $\mathbb{Z}$

TL;DR

This paper investigates the structure of lower central series quotients of noncommutative polynomial algebras over integers with a single relation, analyzing torsion and free parts through examples and theorems.

Contribution

It provides new insights into the algebraic and geometric properties of these quotients, including detailed examples, general theorems, and open problems.

Findings

Analysis of torsion and free parts of $B_k$ and $N_k$

Detailed examples illustrating the structure

Several new theorems on algebraic properties

Abstract

Let $A$ be an associative unital algebra, $B_k$ its successive quotients of lower central series and $N_k$ the successive quotients of ideals generated by lower central series. The geometric and algebraic aspects of $B_k$ and $N_k$ have been of great interest since the pioneering work of \cite{feigin2007}. In this paper, we will concentrate on the case where $A$ is a noncommutative polynomial algebra over $\mathbb{Z}$ modulo a single homogeneous relation. Both the torsion part and the free part of $B_k$ and $N_k$ are explored. Many examples are demonstrated in detail, and several general theorems are proved. Finally we end up with an appendix about the torsion subgroups of $N_k(A_n(\mathbb{Z}))$ and some open problems.