Commutativity of Quantization and Reduction for Quiver Representations

TL;DR

This paper proves that for quiver representations, the processes of non-commutative quantization and reduction commute, linking quantum and classical perspectives through trace maps.

Contribution

It establishes the commutativity of quantization and reduction in the context of quiver representation spaces, connecting non-commutative and classical geometric methods.

Findings

Quantization and reduction commute for quiver representation spaces.

Trace maps induce classical-quantum correspondence.

Results unify non-commutative symplectic geometry with representation theory.

Abstract

Given a finite quiver, its double may be viewed as its non-commutative "cotangent" space, and hence is a non-commutative symplectic space. Crawley-Boevey, Etingof and Ginzburg constructed the non-commutative reduction of this space while Schedler constructed its quantization. We show that the non-commutative quantization and reduction commute with each other. Via the quantum and classical trace maps, such a commutativity induces the commutativity of the quantization and reduction on the space of quiver representations.