Skein and cluster algebras of unpunctured surfaces for $\mathfrak{sp}_4$
Abstract
Continuing to our previous work [IY21](arXiv:2101.00643) on the $\mathfrak{sl}_3$case, we introduce a skein algebra $\mathscr{S}_{\mathfrak{sp}_4,\Sigma}^{q}$ consisting of $\mathfrak{sp}_4$webs on a marked surface $\Sigma$ with certain "clasped" skein relations at special points, and investigate its cluster nature. We also introduce a natural $\mathbb{Z}_q$form $\mathscr{S}_{\mathfrak{sp}_4,\Sigma}^{\mathbb{Z}_q} \subset \mathscr{S}_{\mathfrak{sp}_4,\Sigma}^q$, while the natural coefficient ring $\mathcal{R}$ of $\mathscr{S}_{\mathfrak{sp}_4,\Sigma}^q$ includes the inverse of the quantum integer $[2]_q$. We prove that its boundarylocalization $\mathscr{S}_{\mathfrak{sp}_4,\Sigma}^{\mathbb{Z}_q}[\partial^{1}]$ is included into a quantum cluster algebra $\mathscr{A}^q_{\mathfrak{sp}_4,\Sigma}$ that quantizes the function ring of the moduli space $\mathcal{A}_{Sp_4,\Sigma}^\times$. Moreover, we obtain the positivity of Laurent expressions of elevationpreserving webs in a similar way to [IY21](arXiv:2101.00643). We also propose a characterization of cluster variables in the spirit of FominPylyavksyy [FP16](arXiv:1210.1888) in terms of the $\mathfrak{sp}_4$webs, and give infinitely many supporting examples on a quadrilateral.
 Publication:

arXiv eprints
 Pub Date:
 July 2022
 arXiv:
 arXiv:2207.01540
 Bibcode:
 2022arXiv220701540I
 Keywords:

 Mathematics  Geometric Topology;
 Mathematics  Quantum Algebra;
 Mathematics  Representation Theory;
 13F60;
 57K31 (Primary);
 57K20 (Secondary)
 EPrint:
 57 pages, many TikZ figures