On the automorphisms of a free Lie algebra of rank 3 over an integral domain

TL;DR

This paper investigates the structure of automorphism groups of free Lie algebras of rank 3 over integral domains, revealing their decomposition and providing examples of complex automorphisms.

Contribution

It establishes that tame automorphisms form an amalgamated free product and constructs a wild automorphism example in this setting.

Findings

Tame automorphisms form an amalgamated free product.

Constructed a wild automorphism analogous to the Anick automorphism.

Extended results to free anticommutative algebras.

Abstract

We prove that the group of tame automorphisms of a free Lie algebra (as well as of a free anticommutative algebra) rank 3 over an arbitrary integral domain has the structure of an amalgamated free product. We construct an example of a wild automorphism of a free Lie algebra (as well as of a free anticommutative algebra) of rank 3 over an arbitrary Euclidean ring analogous to the Anick automorphism for free associative algebras.