Groups of triangular automorphisms of a free associative algebra and a polynomial algebra

TL;DR

This paper analyzes the structure of the group of unitriangular automorphisms in free associative and polynomial algebras, revealing its semi-direct product form, subgroup properties, and non-linearity for higher ranks.

Contribution

It provides a new decomposition of the automorphism group, describes its central series, and simplifies the generating system for tame automorphisms.

Findings

The automorphism group is a semi-direct product of abelian groups.

Every element of the derived subgroup is a commutator.

The group of automorphisms of rank > 2 is not linear.

Abstract

We study a structure of the group of unitriangular automorphisms of a free associative algebra and a polynomial algebra and prove that this group is a semi direct product of abelian groups. Using this decomposition we describe a structure of the lower central series and the series of derived subgroups for the group of unitriangular automorphisms and prove that every element from the derived subgroup is a commutator. In addition we prove that the group of unitriangular automorphisms of a free associative algebra of rang more than 2 is not linear and describe some two-generated subgroups from these group. Also we give a more simple system of generators for the group of tame automorphisms than the system from Umirbaev's paper.