The Conjecture of Dixmier for the first Weyl algebra is true

TL;DR

The paper proves the Dixmier conjecture for the first Weyl algebra $A_1$, showing that every endomorphism is an automorphism, i.e. End$_K(A_1)=$Aut$_K(A_1)$. It develops a comprehensive framework based on generalized Schur theory and normal forms to study endomorphisms via DC-pairs $(Q,P)$ with $[Q,P]=1$, analyzes their Newton polygons, and imposes a head-chopping procedure yielding a maximal distance $p+q-1$. By connecting DC-pairs to string equations and rank-one commuting differential operators through Schur/Sato machinery and spectral data, the authors derive a contradiction from a carefully constructed family of DC-pairs under tame automorphisms, showing that no DC-pairs can exist. The result emphasizes deep links between the Dixmier and Jacobian-type conjectures via the theory of commuting differential operators and their spectral geometry. Consequently, $ ext{End}_K(A_1)$ coincides with $ ext{Aut}_K(A_1)$, resolving the $A_1$ case and contributing to the broader landscape of noncommutative algebraic questions connected to integrable systems and spectral theory.

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.

Theorems & Definitions (112)

• Theorem 1.1
• Theorem 2.1
• Remark 2.1
• Definition 2.1
• Definition 2.2
• Theorem 2.2
• Remark 2.2
• Definition 2.3
• Lemma 2.1
• Proposition 2.1