Cluster algebras and skein algebras for surfaces
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913738455515136 |
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| author | Karuo, Hiroaki Moon, Han-Bom Wong, Helen |
| author_facet | Karuo, Hiroaki Moon, Han-Bom Wong, Helen |
| contents | We consider two algebras of curves associated to an oriented surface of finite type - the cluster algebra from combinatorial algebra, and the skein algebra from quantum topology. We focus on generalizations of cluster algebras and generalizations of skein algebras that include arcs whose endpoints are marked points on the boundary or in the interior of the surface. We show that the generalizations are closely related by maps that can be explicitly defined, and we explore the structural implications, including (non-)finite generation. We also discuss open questions about the algebraic structure of the algebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_12319 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cluster algebras and skein algebras for surfaces Karuo, Hiroaki Moon, Han-Bom Wong, Helen Geometric Topology Commutative Algebra Algebraic Geometry 57K31, 13F60, 57K20 We consider two algebras of curves associated to an oriented surface of finite type - the cluster algebra from combinatorial algebra, and the skein algebra from quantum topology. We focus on generalizations of cluster algebras and generalizations of skein algebras that include arcs whose endpoints are marked points on the boundary or in the interior of the surface. We show that the generalizations are closely related by maps that can be explicitly defined, and we explore the structural implications, including (non-)finite generation. We also discuss open questions about the algebraic structure of the algebras. |
| title | Cluster algebras and skein algebras for surfaces |
| topic | Geometric Topology Commutative Algebra Algebraic Geometry 57K31, 13F60, 57K20 |
| url | https://arxiv.org/abs/2503.12319 |