Ideal Classes of the Weyl Algebra and Noncommutative Projective Geometry (with an Appendix by M. Van den Bergh)

TL;DR

This paper establishes a bijection between ideal classes of the Weyl algebra and certain triples involving endomorphisms, using noncommutative geometric methods and homological algebra.

Contribution

It constructs an explicit inverse map between ideal classes and triples, linking algebraic and geometric perspectives in noncommutative projective geometry.

Findings

Proves the bijection between ideal classes and triples.

Provides an explicit description of the inverse map.

Shows equivariance of the map under automorphism group actions.

Abstract

Let R be the set of isomorphism classes of ideals in the Weyl algebra $A=A_{1}$, and let C be the set of isomorphism classes of triples (V; X, Y), where V is a finite-dimensional (complex) vector space, and X, Y are endomorphisms of V such that [X,Y]+I has rank 1. Following a suggestion of L. Le Bruyn, we define a map $\theta: R \to C$ by appropriately extending an ideal of A to a sheaf over a quantum projective plane, and then using standard methods of homological algebra. We prove that $\theta$ is inverse to a bijection $\omega: C \to R$ constructed in \cite{BW} by a completely different method. The main step in the proof is to show that $\theta$ is equivariant with respect to natural actions of the group G=Aut(A) on R and C: for that we have to study also the extensions of an ideal to certain weighted quantum projective planes. Along the way, we find an elementary description of \theta.