Center of stated SL(h)-skein algebras

In the paper, we show some properties of (reduced) stated <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S normal upper L left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {SL}(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S normal upper L left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {SL}(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -skein algebras into quantum tori appearing in higher Teichmüller theory. Thanks to the Unicity theorem by Brown and Goodearl [ <italic>Lectures on algebraic quantum groups</italic> , 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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper S normal upper L left-parenthesis n right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">S</mml:mi> <mml:mi mathvariant="normal">L</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi>n</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">\mathrm {SL}(n)</mml:annotation> </mml:semantics> </mml:math> </inline-formula> -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.