Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sl}_3$

TL;DR

This paper constructs a quantum cluster algebra related to $ ext{SL}_3$-webs on unpunctured surfaces, connecting skein algebras with moduli spaces and providing algorithms for explicit computations and positivity results.

Contribution

It introduces a new quantum cluster algebra inside the skein algebra of $ ext{SL}_3$-webs, linking it to moduli spaces and establishing positivity of certain Laurent polynomials.

Findings

Constructed a quantum cluster algebra inside the skein algebra of $ ext{SL}_3$-webs.

Proved the inclusion of boundary-localized skein algebra as a subalgebra.

Provided an algorithm for computing Laurent expressions and demonstrated positivity of bracelets and bangles.

Abstract

For an unpunctured marked surface $\Sigma$, we consider a skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}$ consisting of $\mathfrak{sl}_3$-webs on $\Sigma$ with the boundary skein relations at marked points. We construct a quantum cluster algebra $\mathscr{A}^q_{\mathfrak{sl}_3,\Sigma}$ inside the skew-field $\mathrm{Frac}\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}$ of fractions, which quantizes the cluster $K_2$-structure on the moduli space $\mathcal{A}_{SL_3,\Sigma}$ of decorated $SL_3$-local systems on $\Sigma$. We show that the cluster algebra $\mathscr{A}^q_{\mathfrak{sl}_3,\Sigma}$ contains the boundary-localized skein algebra $\mathscr{S}_{\mathfrak{sl}_{3},\Sigma}^{q}[\partial^{-1}]$ as a subalgebra, and their natural structures, such as gradings and certain group actions, agree with each other. We also give an algorithm to compute the Laurent expressions of a given $\mathfrak{sl}_3$-web in certain clusters and discuss the positivity of coefficients. In particular, we show that the bracelets and the bangles along an oriented simple loop in $\Sigma$ have Laurent expressions with positive coefficients, hence give rise to quantum GS-universally positive Laurent polynomials.