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  "body": "\n## Statement\n\nLet $k$ be a field of characteristic $0$ and let $A_1(k)$ be the first [[Weyl algebra|98QE7KY55W]] over $k$.\nThen every endomorphism of $A_1(k)$ as a $k$-algebra is an automorphism.\n\n## Remarks\n\nDixmier asked the question in a 1968 paper on Weyl algebras.\nThe same question for the $n$-th Weyl algebra $A_n(k)$ is called the generalized Dixmier conjecture.\nAn endomorphism of $A_1(k)$ is injective, because $A_1(k)$ is a simple ring, so the question is whether it is also surjective.\nTsuchimoto (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 $n$.\nThe case of $A_1(k)$ is open.\n\n## Explanations\n\n### L1 human\n\nThe first Weyl algebra is the algebra of polynomial differential operators in one variable.\nA map from this algebra to itself that respects sums and products is always one-to-one.\nDixmier asked whether it must also be onto.\nThe question turned out to be closely tied to the Jacobian conjecture about polynomial maps.\n",
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      "first_stated": "1968",
      "posed_by": "Jacques Dixmier"
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      "16W20",
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    "name": "Dixmier conjecture",
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  "msc": [
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  "source": "https://github.com/tamnd/math2/blob/main/nodes/F59/F59TS79V34.md",
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