The starred Dixmier's conjecture

TL;DR

This paper investigates a specialized version of Dixmier's conjecture, focusing on involution-preserving endomorphisms of the Weyl algebra, and explores whether such endomorphisms are necessarily automorphisms.

Contribution

The paper introduces and analyzes the starred Dixmier's problem, providing new insights into involution-preserving endomorphisms of the Weyl algebra.

Findings

Characterization of $oldsymbol{ ext{$oldsymbol{ ext{starred}}}$}$ Dixmier's problem

Conditions under which involution-preserving endomorphisms are automorphisms

Partial results supporting the conjecture in specific cases

Abstract

Dixmier's famous question says the following: Is every algebra endomorphism of the first Weyl algebra, $A_1(F)$, where $F$ is a zero characteristic field, an automorphism? Let $\alpha$ be the exchange involution on $A_1(F)$: $\alpha(x)= y$, $\alpha(y)= x$. An $\alpha$-endomorphism of $A_1(F)$ is an endomorphism which preserves the involution $\alpha$. Then one may ask the following question, which may be called the "$\alpha$-Dixmier's problem $1$" or the "starred Dixmier's problem $1$": Is every $\alpha$-endomorphism of $A_1(F)$ an automorphism?