Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sp}_4$

Abstract

Continuing to our previous work [IY21] on the sl 3 -case, we introduce a skein algebra S q sp 4 ,Σ consisting of sp 4 -webs on a marked surface Σ with certain "clasped" skein relations at special points, and investigate its cluster nature. We also introduce a natural Z q -form S Zq sp 4 ,Σ ⊂ S q sp 4 ,Σ , while the natural coefficient ring R of S q sp 4 ,Σ includes the inverse of the quantum integer [2] q . We prove that its boundary-localizationis included into a quantum cluster algebra A q sp 4 ,Σ that quantizes the function ring of the moduli space A × Sp4,Σ . Moreover, we obtain the positivity of Laurent expressions of elevation-preserving webs in a similar way to [IY21]. We also propose a characterization of cluster variables in the spirit of Fomin-Pylyavksyy [FP16] in terms of the sp 4 -webs, and give infinitely many supporting examples on a quadrilateral.