Center of stated $\mathrm{SL}(n)$-skein algebras

Abstract

In the paper, we show some properties of (reduced) stated $\mathrm{SL}(n)$-skein algebras related to their centers for essentially bordered pb surfaces, especially their centers, finitely generation over their centers, and their PI-degrees. The proofs are based on the quantum trace maps, embeddings of (reduced) stated $\mathrm{SL}(n)$-skein algebras into quantum tori appearing in higher Teichm\"uller theory. Thanks to the Unicity theorem in [BG02, FKBL19], we can understand the representation theory of (reduced) stated $\mathrm{SL}(n)$-skein algebras. Moreover, the applications are beyond low-dimensional topology. For example, we can access to the representation theory of unrestricted quantum moduli algebras, and that of quantum higher cluster algebras potentially.