The Conjecture of Dixmier for the first Weyl algebra is true

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.

25 p; v2: minor changes; V3: improved exposition, more details to the proof added, 34 p; V4: significantly improved version: investigation of solutions to the string equation is added, 67 p; V5: extended introduction, more comments and references added, calculations at the last stage of the proof are simplified, some typos fixed, 78 p