Skein and cluster algebras with coefficients for unpunctured surfaces

2023/12/05 by Ishibashi, Tsukasa, Kano, Shunsuke, Yuasa, Wataru
#13F60 #57K20 (Secondary) #57K31 (Primary) #FOS: Mathematics #Geometric Topology (math.GT) #Quantum Algebra (math.QA)

paper · doi:10.48550/arxiv.2312.02861

Abstract

We propose a skein model for the quantum cluster algebras of surface type with coefficients. We introduce a skein algebra \mathscrS_Σ,\mathbbW^A of a walled surface (Σ,\mathbbW), 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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