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Mathematics > Geometric Topology

arXiv:2201.08833 (math)
[Submitted on 21 Jan 2022 (v1), last revised 22 Jan 2024 (this version, v4)]

Title:Consequences of the compatibility of skein algebra and cluster algebra on surfaces

Authors:Han-Bom Moon, Helen Wong
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Abstract:We investigate two algebra of curves on a topological surface with punctures - the cluster algebra of surfaces defined by Fomin, Shapiro, and Thurston, and the generalized skein algebra constructed by Roger and Yang. By establishing their compatibility, we resolve Roger-Yang's conjecture on the deformation quantization of the decorated Teichmuller space. We also obtain several structural results on the cluster algebra of surfaces. The cluster algebra of a positive genus surface is not finitely generated, and it differs from its upper cluster algebra.
Comments: 32 pages, a main result (Theorem C) has been strengthened. Comments are welcome!
Subjects: Geometric Topology (math.GT); Commutative Algebra (math.AC); Algebraic Geometry (math.AG)
MSC classes: 57K31, 13F60, 57K20
Cite as: arXiv:2201.08833 [math.GT]
  (or arXiv:2201.08833v4 [math.GT] for this version)
  https://doi.org/10.48550/arXiv.2201.08833

Submission history

From: Han-Bom Moon [view email]
[v1] Fri, 21 Jan 2022 18:41:48 UTC (3,336 KB)
[v2] Sun, 6 Feb 2022 19:31:47 UTC (3,337 KB)
[v3] Sat, 30 Jul 2022 19:10:22 UTC (3,338 KB)
[v4] Mon, 22 Jan 2024 21:09:54 UTC (3,338 KB)
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