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

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 \(\text {Frac} {\mathscr {S}}_{{\mathfrak {sl}}_{3},\Sigma }^{q}\) 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.