Tangent Lie Algebras of Automorphism Groups of Free Algebras

TL;DR

This paper introduces tangent Lie algebras for automorphism groups of free algebras, explores their properties, and classifies automorphisms as tame or wild, revealing many automorphisms are absolutely wild with notable exceptions.

Contribution

It defines tangent Lie algebras for automorphism groups of free algebras and applies them to classify automorphisms as tame or wild, including new examples of absolutely wild automorphisms.

Findings

Most known wild automorphisms are absolutely wild.

Bergman automorphism of free matrix algebras of order two is absolutely wild.

Free algebras in non-abelian, non-metabelian varieties have absolutely wild automorphisms.

Abstract

We study an analogue of the Andreadakis-Johnson filtration for automorphism groups of free algebras and introduce the notion of tangent Lie algebras for certain automorphism groups, defined as subalgebras of the Lie algebra of derivations. We show that, for many classical varieties of algebras, the tangent Lie algebra is contained in the Lie algebra of derivations with constant divergence. We also introduce the concepts of approximately tame and absolutely wild automorphisms of free algebras in arbitrary varieties and employ tangent Lie algebras to investigate their properties. It is shown that nearly all known examples of wild automorphisms of free algebras are absolutely wild -- with the notable exceptions of the Nagata and Anick automorphisms. We show that the Bergman automorphism of free matrix algebras of order two is absolutely wild. Furthermore, we prove that free algebras in any variety of polynilpotent Lie algebras -- except for the abelian and metabelian varieties -- also possess absolutely wild automorphisms.