Dixmier conjecture

In algebra the Dixmier conjecture, asked by Dixmier (1968, problem 1), is the conjecture that any endomorphism of a Weyl algebra is an automorphism.

Belov-Kanel & Kontsevich (2007) showed that the Dixmier conjecture (generalized to Weyl algebras with more generators) is equivalent to the Jacobian conjecture.

References

• Dixmier, Jacques (1968), "Sur les algèbres de Weyl", Bulletin de la Société Mathématique de France 96: 209–242, MR0242897, http://www.numdam.org/item?id=BSMF_1968__96__209_0 
• Tsuchimoto, Yoshifumi. Endomorphisms of Weyl algebra and p-curvatures. Osaka J. Math. 42 (2005), 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, MR2337879