Consequences of the compatibility of skein algebra and cluster algebra on surfaces

TL;DR

This paper explores the relationship between skein and cluster algebras on punctured surfaces, proving their compatibility and resolving a conjecture about deformation quantization, while revealing structural properties of the cluster algebra.

Contribution

It establishes the compatibility between skein and cluster algebras on surfaces, resolving a conjecture and uncovering new structural insights about the cluster algebra.

Findings

Confirmed compatibility of skein and cluster algebras on surfaces

Resolved Roger-Yang's conjecture on deformation quantization

Found that the cluster algebra of a positive genus surface is not finitely generated

Abstract

We investigate two algebra of curves on a topological surface with punctures - the cluster algebra of surfaces defined by Fomin, Shapiro, and Thurston, and the generalized skein algebra constructed by Roger and Yang. By establishing their compatibility, we resolve Roger-Yang's conjecture on the deformation quantization of the decorated Teichmuller space. We also obtain several structural results on the cluster algebra of surfaces. The cluster algebra of a positive genus surface is not finitely generated, and it differs from its upper cluster algebra.