Skein and cluster algebras of marked surfaces

Abstract

This paper considers several algebras associated to an oriented surface $\Sigma$ with a finite set of marked points on its boundary. The first is the skein algebra $\mathsf {Sk}\_q(\Sigma)$, which is spanned by links in the surface which are allowed to have endpoints at the marked points, modulo several locally defined relations. The product is given by superposition of links. A basis of this algebra is given, as well as several algebraic results. When $\Sigma$ is triangulable, a quantum cluster algebra $\mathcal A\_q(\Sigma)$ and quantum upper cluster algebra $\mathcal U\_q(\Sigma)$ can be defined. These are algebras coming from the triangulations of $\Sigma$ and the elementary moves between them. Cluster algebras have been a subject of significant recent interest, due in part to their extraordinary positivity and Laurent properties. Natural inclusions $\mathcal A\_q(\Sigma) \subseteq \mathsf {Sk}\_q^o (\Sigma) \subseteq \mathcal U\_q(\Sigma)$ are shown, where $\mathsf {Sk}\_q^o(\Sigma)$ is a certain Ore localization of $\mathsf {Sk}\_q(\Sigma)$. When $\Sigma$ has at least two marked points in each component, these inclusions are strengthened to equality, exhibiting a quantum cluster structure on $\mathsf {Sk}\_q^o(\Sigma)$. The method for proving these equalities has the potential to show $\mathcal A\_q = \mathcal U\_q$ for other classes of cluster algebras. As a demonstration of this fact, a new proof is given that $\mathcal A\_q = \mathcal U\_q$ for acyclic cluster algebras.