Noetherian and affine properties of quantum moduli and $\mathfrak{g}$-skein algebras

We prove that the quantum moduli algebra associated to a possibly punctured compact oriented surface and a complex semisimple Lie algebra \mathfrak{g} is a Noetherian and finitely generated ring. If the surface has punctures, we also prove that it has no non-trivial zero divisors (i.e., it is a domain). Moreover, we show that the quantum moduli algebra is isomorphic to the skein algebra of the surface, defined by means of the Reshetikhin–Turaev functor for the quantum group U_{q}(\mathfrak{g}) , and which coincides with the Kauffman bracket skein algebra when \mathfrak{g}=\mathfrak{sl}_{2} . We obtain these results by a similar study of quantum graph algebras, which we show to be isomorphic to stated skein algebras.