Test elements, retracts and automorphic orbits

TL;DR

This paper investigates properties of test elements, retracts, and automorphic orbits in the rank-two free associative algebra over a field of characteristic zero, establishing criteria for test elements and automorphism preservation.

Contribution

It proves that elements outside proper retracts are test elements and that endomorphisms preserving automorphic orbits are automorphisms in $A_2$.

Findings

Elements not in proper retracts are test elements.

Endomorphisms preserving automorphic orbits are automorphisms.

Degree estimates are crucial in the proofs.

Abstract

Let $A_2$ be a free associative or polynomial algebra of rank two over a field $K$ of characteristic zero. Based on the degree estimate of Makar-Limanov and J.-T.Yu, we prove: 1) An element $p \in A_2$ is a test element if $p$ does not belong to any proper retract of $A_2$; 2) Every endomorphism preserving the automorphic orbit of a nonconstant element of $A_2$ is an automorphism.