---
tag: F59TS79V34
kind: conjecture
name: Dixmier conjecture
msc: [16S32, 16W20, 14R15]
uses: [98QE7KY55W, B5JFXEQQD8]
about: [98QE7KY55W]
sources:
  - url: https://en.wikipedia.org/wiki/Dixmier_conjecture
same_as:
  wikidata: Q5284929
  wikipedia: Dixmier conjecture
history:
  posed_by: "Jacques Dixmier"
  first_stated: "1968"
---

## Statement

Let $k$ be a field of characteristic $0$ and let $A_1(k)$ be the first [[Weyl algebra|98QE7KY55W]] over $k$.
Then every endomorphism of $A_1(k)$ as a $k$-algebra is an automorphism.

## Remarks

Dixmier asked the question in a 1968 paper on Weyl algebras.
The same question for the $n$-th Weyl algebra $A_n(k)$ is called the generalized Dixmier conjecture.
An endomorphism of $A_1(k)$ is injective, because $A_1(k)$ is a simple ring, so the question is whether it is also surjective.
Tsuchimoto (2005) and Belov-Kanel and Kontsevich (2007) proved that the generalized Dixmier conjecture for all $n$ is equivalent to the [[Jacobian conjecture|B5JFXEQQD8]] for all $n$.
The case of $A_1(k)$ is open.

## Explanations

### L1 human

The first Weyl algebra is the algebra of polynomial differential operators in one variable.
A map from this algebra to itself that respects sums and products is always one-to-one.
Dixmier asked whether it must also be onto.
The question turned out to be closely tied to the Jacobian conjecture about polynomial maps.