Skein and cluster algebras with coefficients for unpunctured surfaces

Tsukasa Ishibashi; Shunsuke Kano; Wataru Yuasa

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arXiv · 2312.02861

Skein and cluster algebras with coefficients for unpunctured surfaces

Tsukasa Ishibashi·Shunsuke Kano·Wataru Yuasa

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Abstract

We propose a skein model for the quantum cluster algebras of surface type with coefficients. We introduce a skein algebra $\mathscr{S}_{\Sigma,\mathbb{W}}^{A}$ of a walled surface $(\Sigma,\mathbb{W})$, and prove that it has a quantum cluster structure. The walled surfaces naturally generalize the marked surfaces with multi-laminations, which have been used to describe the quantum cluster algebras of geometric type for marked surfaces by Fomin--Thurston [FT18]. Moreover, we give skein theoretic interpretation for some of quasi-homomorphisms [Fra16] between these quantum cluster algebras.

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Tsukasa Ishibashi, Shunsuke Kano, Wataru Yuasa. 2023-12-05. Skein and cluster algebras with coefficients for unpunctured surfaces. https://arxiv.org/abs/2312.02861

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