  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.
