The Conjecture of Dixmier for the first Weyl algebra is true

TL;DR

This paper proves the Dixmier conjecture for the first Weyl algebra over a characteristic zero field, establishing that every algebra endomorphism is an automorphism, thus confirming a long-standing mathematical conjecture.

Contribution

It provides a proof that all endomorphisms of the first Weyl algebra are automorphisms, confirming the conjecture in this specific algebraic setting.

Findings

All algebra endomorphisms of the first Weyl algebra are automorphisms.

The Dixmier conjecture holds for the first Weyl algebra.

This result advances understanding of automorphism groups in algebraic structures.

Abstract

Let $K$ be a field of characteristic zero, let $A_1=K[x][\partial ]$ be the first Weyl algebra. In this paper we prove that the Dixmier conjecture for the first Weyl algebra is true, i.e. each algebra endomorphism of the algebra $A_1$ is an automorphism.