Dixmier Conjecture for the First Weyl Algebra

Let $A_1(\mathbb C)=\mathbb C\langle x,y\rangle/(yx-xy-1)$ be the first Weyl algebra. Dixmier asked in 1968 whether every unital $\mathbb C$-algebra endomorphism of $A_1(\mathbb C)$ is an automorphism [Dixmier1968Weyl]. Since $A_1(\mathbb C)$ is a simple ring, every unital endomorphism is injective, so the content of the problem is surjectivity.

The higher Weyl algebras give analogues $\mathrm{DC}_n$, and the family is tied to the Jacobian conjecture: $\mathrm{DC}_n$ implies $\mathrm{JC}_n$ in the same rank, while $\mathrm{JC}_{2n}$ implies $\mathrm{DC}_n$, making the two families stably equivalent [Tsuchimoto2005] [BelovKontsevich2007]. Structural results on endomorphisms of $A_1$, such as Bavula's work on Dixmier's related problems, have not yielded surjectivity [Bavula2009Dixmier].

The landscape changed in July 2026, when a verified counterexample to $\mathrm{JC}_3$ refuted $\mathrm{DC}_3$ through the same-rank implication, and with it every $\mathrm{DC}_n$ for $n\ge 3$. The original rank-one case, together with $\mathrm{DC}_2$, is untouched by this: no non-automorphic endomorphism of $A_1(\mathbb C)$ is known, and the problem remains open.