Finitely generated bimodules over Weyl algebras

TL;DR

This paper investigates conditions under which subalgebras of Weyl algebras are equal to the entire algebra, linking module finiteness properties to the Dixmier Conjecture through reduction techniques.

Contribution

It proves that if a subalgebra of a Weyl algebra makes it finitely generated as a module, then the subalgebra must be the whole algebra, connecting to the Dixmier Conjecture.

Findings

If $A$ is finitely generated as a module over $S$, then $S = A$.

Holonomicity implies $A$ is finitely generated as an $S$-bimodule.

Transfer of bimodule property to positive characteristic could imply the Dixmier Conjecture.

Abstract

Let $A$ be the $n$-th Weyl algebra over a field of characteristic zero, and $\varphi:A\rightarrow A$ an endomorphism with $S = \varphi(A)$. We prove that if $A$ is finitely generated as a left or right $S$-module, then $S = A$. The proof involves reduction to large positive characteristics. By holonomicity, $A$ is always finitely generated as an $S$-bimodule. Moreover, if this bimodule property could be transferred into a similar property in large positive characteristics, then we could again conclude that $A=S$. The latter would imply the Dixmier Conjecture.