Skein and Cluster Algebras with Coefficients for Unpunctured Surfaces

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 cluster algebras of geometric type for marked surfaces by Fomin–Thurston [ 11]. As a result, this paper gives a simultaneous generalization of Fomin–Thurston and Muller [ 20], incorporating both quantization and (not necessarily normalized) coefficients. Moreover, we give skein theoretic interpretation for some of quasi-homomorphisms [ 14] between these quantum cluster algebras.