Quantum Groups and Symplectic Reductions

TL;DR

This paper connects the algebraic structure of a graded Lie algebra associated with a reductive group and its representation to the geometric structure of the symplectic reduction stack, establishing an equivalence of categories.

Contribution

It constructs a sheaf of Lie algebras over the symplectic reduction and proves an equivalence of derived categories, linking algebraic and geometric perspectives.

Findings

Identification of the sheaf of Lie algebras with the tangent Lie algebra of the stack

Establishment of an equivalence between module categories and perfect complexes

Extension of Costello-Gaiotto's algebraic structures to geometric sheaves

Abstract

Let $G$ be a reductive algebraic group with Lie algebra $\mathfrak{g}$ and $V$ a finite-dimensional representation of $G$. Costello-Gaiotto studied a graded Lie algebra $\mathfrak{d}_{\mathfrak{g}, V}$ and the associated affine Kac-Moody algebra. In this paper, we show that this Lie algebra can be made into a sheaf of Lie algebras over $T^*[V/G]=[\mu^{-1}(0)/G]$, where $\mu: T^*V\to \mathfrak{g}^*$ is the moment map. We identify this sheaf of Lie algebras with the tangent Lie algebra of the stack $T^*[V/G]$. Moreover, we show that there is an equivalence of braided tensor categories between the bounded derived category of graded modules of $\mathfrak{d}_{\mathfrak{g}, V}$ and graded perfect complexes of $[\mu^{-1}(0)/G]$.