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 prove also 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 $\mathfrak{g}={\mathfrak{s}\mathfrak{l}}_2$. We obtain these results by a similar study of quantum graph algebras, which we show to be isomorphic to stated skein algebras.