dixmier_conjecture

⚠️ Retired — specification defect

The Lean statement below does not encode the problem shown on this page, so its Disproved status carries no information about that problem. Do not import this node or use it as a dependency.

Dixmier conjecture (1968): Every endomorphism of the nth Weyl algebra Aₙ(ℂ) is an automorphism. Equivalent to the Jacobian conjecture. Proved for n=1; open for n≥2 in characteristic 0.

Why this node was retired

The posted statement is

import Mathlib

theorem dixmier_conjecture (n : ℕ) (hn : 1 ≤ n)
    (A : Type*) [Ring A] [Algebra ℂ A]
    (comm_rels : ∀ i : Fin n,
      ∃ (xi yi : A), xi * yi - yi * xi = 1)
    (phi : A →ₐ[ℂ] A) :
    Function.Bijective phi := by
  sorry

The algebra is arbitrary and only contains a commutator-one pair; it is not required to be the Weyl algebra.

The recorded counterexample refutes the statement as encoded. It says nothing about the problem shown above, which is a different proposition.

Proposed corrected statement

For n≥1, define the nth Weyl algebra A_n(ℂ) by generators x_i,∂_i and the full Weyl commutation relations, with no additional unrelated elements; conjecture every ℂ-algebra endomorphism of A_n(ℂ) is an automorphism.

Diagnosis and correction from the public Prove2Me statement audit (wamlat/prove2me-errors). The correction is natural-language mathematics and is not Lean-verified — it is a specification for a corrected node, not a drop-in replacement. No corrected replacement node exists yet.