Center of stated SL(h)-skein algebras

In the paper, we show some properties of (reduced) stated S L ( n ) \mathrm {SL}(n) -skein algebras related to their centers for essentially bordered punctured bordered 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 S L ( n ) \mathrm {SL}(n) -skein algebras into quantum tori appearing in higher Teichmüller theory. Thanks to the Unicity theorem by Brown and Goodearl [ Lectures on algebraic quantum groups , Birkhäuser Verlag, Basel, 2002] and Frohman, Kania-Bartoszynska, and Lê [Invent. Math. 215 (2019), pp. 609–650], we can understand the representation theory of (reduced) stated S L ( n ) \mathrm {SL}(n) -skein algebras. Moreover, the applications are beyond low-dimensional topology. For example, we can access the representation theory of unrestricted quantum moduli algebras, and that of quantum higher cluster algebras potentially.