Upper central series for the group of unitriangular automorphisms of a free associative algebra

TL;DR

This paper investigates the structure of the group of unitriangular automorphisms of a free associative algebra, identifying centers and hypercenters, and demonstrating the non-linearity of these groups for all n ≥ 2.

Contribution

It determines the centers and hypercenters of the automorphism groups and proves the infinite length of the upper central series for U_2, showing non-linearity for all n ≥ 2.

Findings

Identified the center of U_n.

Described hypercenters of U_2 and U_3.

Proved U_2 has an infinite upper central series.

Abstract

We study some subgroups of the group of unitriangular automorphisms $U_n$ of a free associative algebra over a field of characteristic zero. We find the center of $U_n$ and describe the hypercenters of $U_2$ and $U_3$. In particular, we prove that the upper central series for $U_2$ has infinite length. As consequence, we prove that the groups $U_n$ are non-linear for all $n \geq 2$.