The starred Dixmier conjecture for $A_1$

Abstract

Let $A_1(K)=K⟨x,y|yx-xy=1⟩$ 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 $\alpha -J{C}_2$.

arXiv admin note: text overlap with arXiv:1310.7562

References in corpus (2)

• On the equivalence of the Jacobian, Dixmier and Poisson Conjectures in any characteristic
• On the shape of possible counterexamples to the Jacobian Conjecture

Cited by in corpus (1)

• Involutions and the Jacobian conjecture