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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.14012 |
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| _version_ | 1866916696608997376 |
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| author | Francone, Luca Leclerc, Bernard |
| author_facet | Francone, Luca Leclerc, Bernard |
| contents | We introduce new objects, called $(G,c)$-bands, associated with a simple simply-connected algebraic group $G$, and a Coxeter element $c$ in its Weyl group. We show that bands of a given type are the $K$-points of an infinite dimensional affine scheme, whose ring of regular functions has a cluster algebra structure. We also show that two important invariant sub-algebras of this ring are cluster sub-algebras. These three cluster structures have already appeared in different contexts related to the representation theories of quantum affine algebras, their Borel sub-algebras, and shifted quantum affine algebras. In this paper we show that they all belong to a common geometric setting. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_14012 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cluster structures on schemes of bands Francone, Luca Leclerc, Bernard Representation Theory Quantum Algebra Rings and Algebras 13F60, 20G05, 17B37 We introduce new objects, called $(G,c)$-bands, associated with a simple simply-connected algebraic group $G$, and a Coxeter element $c$ in its Weyl group. We show that bands of a given type are the $K$-points of an infinite dimensional affine scheme, whose ring of regular functions has a cluster algebra structure. We also show that two important invariant sub-algebras of this ring are cluster sub-algebras. These three cluster structures have already appeared in different contexts related to the representation theories of quantum affine algebras, their Borel sub-algebras, and shifted quantum affine algebras. In this paper we show that they all belong to a common geometric setting. |
| title | Cluster structures on schemes of bands |
| topic | Representation Theory Quantum Algebra Rings and Algebras 13F60, 20G05, 17B37 |
| url | https://arxiv.org/abs/2504.14012 |