In the paper, we show some properties of (reduced) stated S L (n) 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) 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) 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.
Karuo et al. (Tue,) studied this question.