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Cluster algebras and skein algebras for surfaces

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arxiv 2503.12319 v1 pith:WUGJZ5PQ submitted 2025-03-16 math.GT math.ACmath.AG

Cluster algebras and skein algebras for surfaces

classification math.GT math.ACmath.AG
keywords algebrasalgebraclustergeneralizationsskeinfinitesurfacealgebraic
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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.

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  1. Quantum cluster algebra realization for stated ${\rm SL}_n$-skein algebras and rotation-invariant bases for polygons

    math.QA 2026-05 unverdicted novelty 6.0

    For polygonal surfaces, the localized stated SL_n-skein algebra equals the associated quantum cluster algebra, producing a rotation-invariant basis.