Coxeter quiver representations in fusion categories and Gabriel's theorem
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866929242933035008 |
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| author | Heng, Edmund |
| author_facet | Heng, Edmund |
| contents | We introduce a notion of representation for a class of generalised quivers known as Coxeter quivers. These representations are built using fusion categories associated to $U_q(\mathfrak{s}\mathfrak{l}_2)$ at roots of unity and we show that many of the classical results on representations of quivers can be generalised to this setting. Namely, we prove a generalised Gabriel's theorem for Coxeter quivers that encompasses all Coxeter--Dynkin diagrams -- including the non-crystallographic types $H$ and $I$. Moreover, a similar relation between reflection functors and Coxeter theory is used to show that the indecomposable representations correspond bijectively to the positive roots of Coxeter root systems over fusion rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_01866 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Coxeter quiver representations in fusion categories and Gabriel's theorem Heng, Edmund Representation Theory Category Theory Quantum Algebra 16G20 (Primary), 18M20 (Secondary), 17B22 We introduce a notion of representation for a class of generalised quivers known as Coxeter quivers. These representations are built using fusion categories associated to $U_q(\mathfrak{s}\mathfrak{l}_2)$ at roots of unity and we show that many of the classical results on representations of quivers can be generalised to this setting. Namely, we prove a generalised Gabriel's theorem for Coxeter quivers that encompasses all Coxeter--Dynkin diagrams -- including the non-crystallographic types $H$ and $I$. Moreover, a similar relation between reflection functors and Coxeter theory is used to show that the indecomposable representations correspond bijectively to the positive roots of Coxeter root systems over fusion rings. |
| title | Coxeter quiver representations in fusion categories and Gabriel's theorem |
| topic | Representation Theory Category Theory Quantum Algebra 16G20 (Primary), 18M20 (Secondary), 17B22 |
| url | https://arxiv.org/abs/2302.01866 |