An Explicit Counterexample to the Dixmier Conjecture in A3

Jeff Pickhardt

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An Explicit Counterexample to the Dixmier Conjecture in A3

Abstract

We exhibit an explicit injective, non-surjective endomorphism of the third Weyl algebra A3(k) over an arbitrary field k of characteristic zero, disproving the Dixmier conjecture in di- mension three. The input is a polynomial map F: A3 k →A3 k , announced by Alpöge and refuting the Jacobian conjecture in dimension three, whose Jacobian determinant is the constant−2 and which sends three distinct rational points to a single image. Since that determinant is a unit, the inverse Jacobian is again a polynomial matrix and lifts F to an endomorphism Φ of A3(k). The extension of a Jacobian map to the Weyl algebra was given earlier by Bavula, and Belov-Kanel and Kontsevich record the order-preserving pullback argument. We supply the explicit instance with a verification that Alpöge’s map satisfies the hypotheses, and a detailed account of the resulting endomorphism, carried out in two ways: by a centralizer computation in PBW normal form, and by the differential-order filtration, the latter also giving exact preservation of differen- tial order. Only the elementary implication from the Dixmier to the Jacobian conjecture in the same dimension is used, through its contrapositive, so no stable equivalence and no doubling of dimension enters.