MathDB

The Dixmier conjecture for the first Weyl algebra

About 16 years old · traced to

Let KK be a field of characteristic zero. The Dixmier conjecture. Every endomorphism of the first Weyl algebra

A1(K):=K{t,x}/⟨xt−tx−1⟩A_1(K):=K\{t,x\}/\langle xt-tx-1\rangle

is an automorphism. Here A1(K)A_1(K) is generated by t,xt,x with xt=tx+1xt=tx+1. This is the original Dixmier problem, formulated as a question in 1968 and commonly called a conjecture; its resolution is not established in the supplied text.

References

Primary source

William Fajardo and Oswaldo Lezama, “The Dixmier problem for skew PBW extensions and rings”, arXiv:2506.09285 (2025).

Additional references

4 papers in this index state this conjecture (2010–2025). The statement above is taken from the most recent of them; the others are arXiv:1912.03759, arXiv:1707.06450, arXiv:1010.5189.

endomorphismsnoncommutative-algebraweyl-algebras
Posted

Progress summary

Refreshed
Claimed solved

A 2024 paper and 2026 repository deposits claim a complete proof, but no independent verification has established that the conjecture is solved.

Dixmier asked in 1968 whether every endomorphism of the first Weyl algebra A1(K)A_1(K) over a characteristic-zero field is an automorphism.

Known results

  • Prime or coprime cases of the relevant normal-form obstruction were proved in 2024; these do not settle the conjecture.
  • In 2014, equivalent formulations and several involution-symmetric special cases were established.
  • In 2011, possible counterexamples were shown to require a degree parameter greater than 1515.

October 2024 and August 2026 claimed proofs

The October 2024 paper The Conjecture of Dixmier for the first Weyl algebra is true claims End⁡K(A1)=Aut⁡K(A1) \operatorname{End}_K(A_1)=\operatorname{Aut}_K(A_1), using normal forms and an attempted exclusion of counterexamples. On August 29, 2026, duplicate Zenodo deposits likewise claimed complete proofs. Neither claim has independent verification, peer review, or a documented resolution of possible gaps.

Current status (as of August 2026): The conjecture is not established; complete proofs are claimed, but their correctness remains unverified.

Sources

Solutions 0

No solutions have been posted yet.