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 ${\mathcal{S}}_{\Sigma ,\mathbb{W}}^A$ 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.

43 pages; v2: Major revision, introduction refreshed, arguments concerning the minimal position moved to Section 5, the Fomin--Thurston's description of the normalized cluster algebras moved to Section 4.2, Examples 2.13 and 2.14 added