Math 2.0

Associative rings and algebras Constructions Conjecture

Dixmier conjecture

Builds on Weyl algebraJacobian conjecture

Statement

Let kk be a field of characteristic 00 and let A1(k)A_1(k) be the first Weyl algebra over kk. Then every endomorphism of A1(k)A_1(k) as a kk-algebra is an automorphism.

Remarks

Dixmier asked the question in a 1968 paper on Weyl algebras. The same question for the nn-th Weyl algebra An(k)A_n(k) is called the generalized Dixmier conjecture. An endomorphism of A1(k)A_1(k) is injective, because A1(k)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 nn is equivalent to the Jacobian conjecture for all nn. The case of A1(k)A_1(k) is open.

Reading path

To understand this page from the ground up, read these 12 pages in order. The foundations come first.

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Characteristic of a ring, concept, no evidence levelCharacteristic of a r…conceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptPolynomial ring, concept, no evidence levelPolynomial ringconceptWeyl algebra, concept, no evidence levelWeyl algebraconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptJacobian matrix, concept, no evidence levelJacobian matrixconceptJacobian determinant, concept, no evidence levelJacobian determinantconceptJacobian conjecture, conjecture, no evidence levelJacobian conjectureconjectureDixmier conjecture, conjecture, no evidence levelDixmier conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Characteristic of a ring concept
  2. Euclidean space object
  3. Function concept
  4. Polynomial ring concept
  5. Weyl algebra concept
  6. Total derivative concept
  7. Function of several variables concept
  8. Partial derivative concept
  9. Jacobian matrix concept
  10. Jacobian determinant concept
  11. Jacobian conjecture conjecture
  12. Dixmier conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Dixmier_conjecture
About this page
TagF59TS79V34
URIm2://F59TS79V34
Kindconjecture
WritingW0 stub
MSC16S32, 16W20, 14R15
StatusThis copy of the site has no build data. Run m2 status F59TS79V34.
AboutWeyl algebra
UsesWeyl algebra, Jacobian conjecture
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