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

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