Quantum Groups and Symplectic Reductions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915008102793216 |
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| author | Niu, Wenjun |
| author_facet | Niu, Wenjun |
| contents | 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 $[μ^{-1}(0)/G]$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_04195 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Quantum Groups and Symplectic Reductions Niu, Wenjun Representation Theory Mathematical Physics Quantum Algebra 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 $[μ^{-1}(0)/G]$. |
| title | Quantum Groups and Symplectic Reductions |
| topic | Representation Theory Mathematical Physics Quantum Algebra |
| url | https://arxiv.org/abs/2411.04195 |