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

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.