Chein Automorphisms of Free Metabelian Anticommutative Algebras

TL;DR

This paper characterizes specific automorphisms, called Chein automorphisms, of free metabelian anticommutative algebras, revealing their structure and complexity for different ranks.

Contribution

It provides a complete description of Chein automorphisms in free metabelian anticommutative algebras and analyzes their properties across different ranks.

Findings

Automorphisms fixing all but one variable are fully described.

All automorphisms of rank 2 are linear.

Chein automorphisms of degree 3 are highly complex for rank ≥ 3.

Abstract

We describe all automorphisms of a free metabelian anticommutative algebra of rank $n\geq 3$ over a field $K$ that move only one variable while fixing the others. Such automorphisms are called Chein automorphisms in the cases of free metabelian groups and free metabelian Lie algebras. We show that all automorphisms of a free metabelian anticommutative algebra of rank $n=2$ are linear, and that the simplest non elementary Chein automorphism of degree $3$ is absolutely wild for all $n\geq 3$.