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Quantum Groups and Symplectic Reductions

Wenjun Niu

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 $T^*[V/G]=[μ^{-1}(0)/G]$, where $μ: T^*V\to \mathfrak{g}^*$ 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 $[μ^{-1}(0)/G]$.

Quantum Groups and Symplectic Reductions

Abstract

Let be a reductive algebraic group with Lie algebra and a finite-dimensional representation of . Costello-Gaiotto studied a graded Lie algebra 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 , where is the moment map. We identify this sheaf of Lie algebras with the tangent Lie algebra of the stack . Moreover, we show that there is an equivalence of braided tensor categories between the bounded derived category of graded modules of and graded perfect complexes of .

Paper Structure

This paper contains 18 sections, 16 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Braided tensor categories and TQFT
  3. B twist of 3d ${\mathcal{N}}=4$ theories and quantum groups
  4. Rozansky-Witten TQFT on cotangent stacks
  5. Boundary VOA and Kazhdan-Lusztig equivalences
  6. Organization and sketch of proof strategies
  7. Acknowledgements
  8. Lie algebra $\mathfrak{d}_{\mathfrak g, V}$ and its quantizations
  9. Definition of $\mathfrak{d}_{\mathfrak g, V}$ as a double
  10. Drinfeld's quantization of Lie bi-algebras
  11. Quantization as a quasi-Hopf algebra
  12. Quantization as Hopf algebra
  13. Explicit formula for the double
  14. Equivalence of the two quantizations
  15. Canonical quantization of Lie bi-algebras
  16. ...and 3 more sections

Key Result

Proposition 1.1

The tangent Lie algebra $T_{[{\mathcal{M}}]}[-1]$ is identified with $l_{[{\mathcal{M}}]}$ as a Lie algebra object in $\mathrm{Coh} ([{\mathcal{M}}])$. Under this identification, the quadratic form on $\mathfrak{d}_{\mathfrak g, V}$ is identified with the symplectic form on $T_{[{\mathcal{M}}]}[-1]$

Theorems & Definitions (35)

  • Proposition 1.1: Proposition \ref{['Prop:TM']}
  • Theorem 1.2: Theorem \ref{['Thm:tangentBTC']}
  • Remark 1.3
  • Remark 1.4
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Corollary 2.5
  • Theorem 2.6: drinfeld1991quasi Theorem A
  • ...and 25 more