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

Abstract

For an unpunctured marked surface $\Sigma$, we consider a skein algebra ${\mathcal{S}}_{{\mathfrak{sl}}_3,\Sigma }^q$ consisting of ${\mathfrak{sl}}_3$-webs on $\Sigma$ with the boundary skein relations at marked points. We construct a quantum cluster algebra ${\mathcal{A}}_{{\mathfrak{sl}}_3,\Sigma }^q$ inside the skew-field $\mathrm{Frac}{\mathcal{S}}_{{\mathfrak{sl}}_3,\Sigma }^q$ of fractions, which quantizes the cluster $K_2$-structure on the moduli space ${\mathcal{A}}_{S{L}_3,\Sigma }$ of decorated $S{L}_3$-local systems on $\Sigma$. We show that the cluster algebra ${\mathcal{A}}_{{\mathfrak{sl}}_3,\Sigma }^q$ contains the boundary-localized skein algebra ${\mathcal{S}}_{{\mathfrak{sl}}_3,\Sigma }^q[{\partial }^{-1}]$ 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.

arXiv comment65 pages, many TikZ figures; v2: minor corrections, references added; v3: Main Theorem on the comparison of skein and cluster algebras is refined, where an inclusion of the boundary-localized skein algebra into the quantum cluster algebra is given. minor corrections, references added; v4:Title changed, Proof of Theorem 3.12 is revised, minor corrections, references added; to appear in Math. Z

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