Noncommutative Hamiltonian structures and quantizations on preprojective algebras

Given a noncommutative Hamiltonian space $A$, we prove that the conjecture ``{\it quantization commutes with reduction}‘’ holds for $A$. We further construct a semidirect product algebra $A⋊{\text{mG}}^A$, and establish a correspondence between equivariant sheaves on the representation space and left $A⋊{\text{mG}}^A$-modules. In the quiver setting, using the quantum and classical trace maps, we establish the explicit correspondence between quantizations of a preprojective algebra and those of a quiver variety.