Dixmier conjecture

The Dixmier conjecture asserts that every endomorphism of a Weyl algebra over a field of characteristic zero is an automorphism. It remains open in general.

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Weyl algebra: An algebra generated by position and differentiation operators subject to canonical commutation relations. The conjecture asks whether every endomorphism of this algebra is invertible.
Jacques Dixmier: A French mathematician known for work in operator algebras and representation theory. He formulated the conjecture in his study of Weyl algebras.
Differential operator: An operator built from differentiation and multiplication by functions. Weyl algebras model polynomial differential operators and their commutation rules.
Jacobian conjecture in dimension two: The Jacobian conjecture restricted to polynomial maps of the affine plane. By equivalence, resolving this case would resolve the two-generator-pair Dixmier conjecture.
Algebra endomorphism: A structure-preserving map from an algebra to itself or another algebra. The conjecture concerns endomorphisms, asking whether each must also have an inverse.
Enveloping algebra: An associative algebra that encodes representations of a Lie algebra. Dixmier’s work on endomorphisms of enveloping algebras provided the conjecture’s setting.
Characteristic of a field: The least positive integer annihilating the field’s multiplicative identity, or zero if none exists. The conjecture’s stated setting excludes positive characteristic, where behavior can differ sharply.
Jacobian conjecture in positive characteristic: Versions of the Jacobian conjecture formulated over fields of positive characteristic. The characteristic-zero equivalence does not remove the distinct technical issues of positive characteristic.
Jacobian conjecture: The conjecture that a polynomial map with constant nonzero Jacobian determinant has a polynomial inverse. For Weyl algebras over characteristic-zero fields, it is equivalent to the Dixmier conjecture.
Heisenberg Lie algebra: A nilpotent Lie algebra whose central commutators encode the canonical commutation relations. Its enveloping algebra yields the Weyl algebra underlying the conjecture.