On the structure and representations of quantum graph algebras at roots of unity
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866911372485328896 |
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| author | Baseilhac, Stéphane Faitg, Matthieu Roche, Philippe |
| author_facet | Baseilhac, Stéphane Faitg, Matthieu Roche, Philippe |
| contents | We study the specializations $\mathcal{L}_{g,n}^ε$ at roots of unity $ε$ of odd order of the graph algebras, associated to a simply-connected complex semi-simple algebraic group $G$ and a compact oriented surface $Σ_{g,n}^{\circ}$ with genus $g$, $n$ punctures, and one boundary component. We prove that the central localizations of $\mathcal{L}_{g,n}^ε$ and of its subalgebra $\mathcal{L}_{g,n}^{u_ε}$ of invariant elements under the coadjoint action of a small quantum group, are central simple algebras of PI degrees that we compute. Also, we describe their centers, and show they are integrally closed rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_08789 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the structure and representations of quantum graph algebras at roots of unity Baseilhac, Stéphane Faitg, Matthieu Roche, Philippe Quantum Algebra Rings and Algebras Representation Theory 17B37 (Primary) 16T20, 20G42 (Secondary) We study the specializations $\mathcal{L}_{g,n}^ε$ at roots of unity $ε$ of odd order of the graph algebras, associated to a simply-connected complex semi-simple algebraic group $G$ and a compact oriented surface $Σ_{g,n}^{\circ}$ with genus $g$, $n$ punctures, and one boundary component. We prove that the central localizations of $\mathcal{L}_{g,n}^ε$ and of its subalgebra $\mathcal{L}_{g,n}^{u_ε}$ of invariant elements under the coadjoint action of a small quantum group, are central simple algebras of PI degrees that we compute. Also, we describe their centers, and show they are integrally closed rings. |
| title | On the structure and representations of quantum graph algebras at roots of unity |
| topic | Quantum Algebra Rings and Algebras Representation Theory 17B37 (Primary) 16T20, 20G42 (Secondary) |
| url | https://arxiv.org/abs/2601.08789 |