Dixmier's Conjecture

Let \(A_1\) be the first Weyl algebra over a field of characteristic zero (generated by \(x, y\) with \([y, x] = 1\)). Then every algebra endomorphism of \(A_1\) is an automorphism.

Posed by Jacques Dixmier in 1968 as the noncommutative analogue of the Jacobian conjecture: an endomorphism of the Weyl algebra that is onto on generators ought to be an automorphism. Dixmier himself settled low degrees; the general case resisted all direct attack until Tsuchimoto (2005) and Belov-Kanel–Kontsevich (2007) showed it equivalent to the Jacobian conjecture in a precise stable sense — so the two great conjectures stand or fall together, and both remain open.