Dixmier conjecture

{{distinguish|text=the Dixmier Problem in representation theory}}

In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968,{{Citation | last1=Dixmier | first1=Jacques | authorlink=Jacques Dixmier |title=Sur les algèbres de Weyl | url=http://www.numdam.org/item?id=BSMF_1968__96__209_0 | mr=0242897 | year=1968 | journal=Bulletin de la Société Mathématique de France | volume=96 | pages=209–242 | doi=10.24033/bsmf.1667 | doi-access=free }} (problem 1) is the conjecture that any endomorphism of a Weyl algebra is an automorphism.

Tsuchimoto in 2005,{{Citation | last1=Tsuchimoto | first1=Yoshifumi | title=Endomorphisms of Weyl algebra and p-curvatures | year=2005 | journal=Osaka J. Math.| volume=42 | pages=435–452}} and independently Belov-Kanel and Kontsevich in 2007,{{Citation | last1=Belov-Kanel | first1=Alexei | last2=Kontsevich | first2=Maxim | title=The Jacobian conjecture is stably equivalent to the Dixmier conjecture | arxiv=math/0512171 | mr=2337879 | year=2007 | journal=Moscow Mathematical Journal | volume=7 | issue=2 | pages=209–218| doi=10.17323/1609-4514-2007-7-2-209-218 | bibcode=2005math.....12171B | s2cid=15150838 }} showed that the Dixmier conjecture is stably equivalent to the Jacobian conjecture.