The starred Dixmier conjecture for $A_1$

TL;DR

This paper proves that every involution-preserving endomorphism of the first Weyl algebra over a characteristic zero field is an automorphism, providing a significant step towards the Dixmier conjecture and related Jacobian conjecture in dimension 2.

Contribution

It establishes that all involution-preserving endomorphisms of the Weyl algebra are automorphisms, advancing the understanding of the Dixmier conjecture.

Findings

All $ ext{α}$-endomorphisms of $A_1(K)$ are automorphisms.

Proves an analogue of the Jacobian conjecture in dimension 2 for involution-preserving maps.

Supports the conjecture that involution-preserving endomorphisms are automorphisms in Weyl algebras.

Abstract

Let $A_1(K)=K \langle x,y | yx-xy= 1 \rangle$ be the first Weyl algebra over a characteristic zero field $K$ and let $\alpha$ be the exchange involution on $A_1(K)$ given by $\alpha(x)= y$ and $\alpha(y)= x$. The Dixmier conjecture of Dixmier (1968) asks: Is every algebra endomorphism of the Weyl algebra $A_1(K)$ an automorphism? The aim of this paper is to prove that each $\alpha$-endomorphism of $A_1(K)$ is an automorphism. Here an $\alpha$-endomorphism of $A_1(K)$ is an endomorphism which preserves the involution $\alpha$. We also prove an analogue result for the Jacobian conjecture in dimension 2, called $\alpha-JC_2$.