Skein algebras and cluster algebras of marked surfaces

Abstract

This paper defines several algebras associated to an oriented surface $S$ with a finite set of marked points on the boundary. The first is the skein algebra $S{k}_q(S)$, 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 $S$ is triangulable, the quantum cluster algebra $A_q(S)$ and quantum upper cluster algebra U_q(S) can be defined. These are algebras coming from the triangulations of S and the elementary moves between them. Natural inclusions $A_q(S)$ into $S{k}_q^o(S)$ into $U_q(S)$ are shown, where $S{k}_q^o(S)$ is a certain Ore localization of $S{k}_q(S)$. When $S$ has at least two marked points in each component, these inclusions are strengthened to equality, exhibiting a quantum cluster structure on $S{k}_q^o(S)$. The method for proving these equalities has potential to show $A_q=U_q$ for other classes of cluster algebras. As a demonstration of this fact, a new proof is given that $A_q=U_q$ for acyclic cluster algebras

60 pages, 13 figures. Proof of Lemma 4.6 fixed and moved to an appendix

References in corpus (2)

• Cluster algebras and triangulated surfaces. Part II: Lambda lengths
• Locally acyclic cluster algebras

Cited by in corpus (14)

• A positive basis for surface skein algebras
• Quantum Teichmüller spaces and quantum trace map
• A=U for Locally Acyclic Cluster Algebras
• Classical shadows of stated skein representations at roots of unity
• Categorified canonical bases and framed BPS states
• Snake graph calculus and cluster algebras from surfaces III: Band graphs and snake rings
• Skein algebras of surfaces
• Snake graph calculus and cluster algebras from surfaces II: Self-crossing snake graphs
• Poisson algebras of curves on bordered surfaces and skein quantization
• The geometry of cluster varieties from surfaces
• A comparison between $S{L}_n$ spider categories
• Acyclic cluster algebras from a ring theoretic point of view
• Stated skein algebras and their representations
• Cluster algebras and their bases