Braiding on complex oriented Soergel bimodules
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913419131617280 |
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| author | Liu, Yu Leon |
| author_facet | Liu, Yu Leon |
| contents | In this note, we study U(n) Soergel bimodules in the context of stable homotopy theory. We define the $(\infty, 1)$-category $\mathrm{SBim}_E(n)$ of $E$-valued U(n) Soergel bimodules, where $E$ is a connective $\mathbb{E}_\infty$-ring spectrum, and assemble them into a monoidal locally additive $(\infty, 2)$-category $\mathrm{SBim}_E$. When $E$ has a complex orientation, we then construct a braiding, i.e. an $\mathbb{E}_2$-algebra structure, on the universal locally stable $(\infty, 2)$-category $\mathrm{K}^b_{\mathrm{loc}}(\mathrm{SBim}_E)$ associated to $\mathrm{SBim}_E$. Along the way, we also prove spectral analogs of standard splittings of Soergel bimodules. This is a topological generalization of the type $A$ Soergel bimodule theory developed in a previous paper. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_04891 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Braiding on complex oriented Soergel bimodules Liu, Yu Leon Algebraic Topology Quantum Algebra Representation Theory In this note, we study U(n) Soergel bimodules in the context of stable homotopy theory. We define the $(\infty, 1)$-category $\mathrm{SBim}_E(n)$ of $E$-valued U(n) Soergel bimodules, where $E$ is a connective $\mathbb{E}_\infty$-ring spectrum, and assemble them into a monoidal locally additive $(\infty, 2)$-category $\mathrm{SBim}_E$. When $E$ has a complex orientation, we then construct a braiding, i.e. an $\mathbb{E}_2$-algebra structure, on the universal locally stable $(\infty, 2)$-category $\mathrm{K}^b_{\mathrm{loc}}(\mathrm{SBim}_E)$ associated to $\mathrm{SBim}_E$. Along the way, we also prove spectral analogs of standard splittings of Soergel bimodules. This is a topological generalization of the type $A$ Soergel bimodule theory developed in a previous paper. |
| title | Braiding on complex oriented Soergel bimodules |
| topic | Algebraic Topology Quantum Algebra Representation Theory |
| url | https://arxiv.org/abs/2407.04891 |