Fusion-stable structures on triangulated categories
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866915121166548992 |
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| author | Qiu, Yu Zhang, Xiaoting |
| author_facet | Qiu, Yu Zhang, Xiaoting |
| contents | Let $\mathcal{G}$ be a fusion category acting on a triangulated category $\mathcal{D}$, in the sense that $\mathcal{D}$ is a $\mathcal{G}$-module category. Our motivation example is fusion-weighted species, which is essentially Heng's construction. We study $\mathcal{G}$-stable tilting, cluster and stability structures on $\mathcal{D}$. In particular, we prove the deformation theorem for $\mathcal{G}$-stable stability conditions.
A first application is that Duffield-Tumarkin's categorification of cluster exchange graphs of finite Coxeter-Dynkin type can be naturally realized as fusion-stable cluster exchange graphs. Another application is that the universal cover of the hyperplane arrangements of any finite Coxeter-Dynkin type can be realized as the space of fusion-stable stability conditions for certain ADE Dynkin quiver. This provides an alternative uniform proof of $K(π,1)$-conjecture in the finite Coxeter-Dynkin case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_02917 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Fusion-stable structures on triangulated categories Qiu, Yu Zhang, Xiaoting Representation Theory Quantum Algebra Let $\mathcal{G}$ be a fusion category acting on a triangulated category $\mathcal{D}$, in the sense that $\mathcal{D}$ is a $\mathcal{G}$-module category. Our motivation example is fusion-weighted species, which is essentially Heng's construction. We study $\mathcal{G}$-stable tilting, cluster and stability structures on $\mathcal{D}$. In particular, we prove the deformation theorem for $\mathcal{G}$-stable stability conditions. A first application is that Duffield-Tumarkin's categorification of cluster exchange graphs of finite Coxeter-Dynkin type can be naturally realized as fusion-stable cluster exchange graphs. Another application is that the universal cover of the hyperplane arrangements of any finite Coxeter-Dynkin type can be realized as the space of fusion-stable stability conditions for certain ADE Dynkin quiver. This provides an alternative uniform proof of $K(π,1)$-conjecture in the finite Coxeter-Dynkin case. |
| title | Fusion-stable structures on triangulated categories |
| topic | Representation Theory Quantum Algebra |
| url | https://arxiv.org/abs/2310.02917 |