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Dixmier conjecture

In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any endomorphism of a Weyl algebra is an automorphism.

Tsuchimoto in 2005, and independently Belov-Kanel and Kontsevich in 2007, showed that the Dixmier conjecture is stably equivalent to the Jacobian conjecture.

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References

Dixmier, Jacques (1968), "Sur les algèbres de Weyl", Bulletin de la Société Mathématique de France, 96: 209–242, doi:10.24033/bsmf.1667, MR 0242897 (problem 1) /wiki/Jacques_Dixmier ↩
Tsuchimoto, Yoshifumi (2005), "Endomorphisms of Weyl algebra and p-curvatures", Osaka J. Math., 42: 435–452 ↩
Belov-Kanel, Alexei; Kontsevich, Maxim (2007), "The Jacobian conjecture is stably equivalent to the Dixmier conjecture", Moscow Mathematical Journal, 7 (2): 209–218, arXiv:math/0512171, Bibcode:2005math.....12171B, doi:10.17323/1609-4514-2007-7-2-209-218, MR 2337879, S2CID 15150838 /wiki/ArXiv_(identifier) ↩