Noetherian and affine properties of quantum moduli and g-skein algebras

We prove that the quantum moduli algebra associated to a possibly punctured compact oriented surface and a complex semisimple Lie algebra 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 Uq(g) , and which coincides with the Kauffman bracket skein algebra when g=sl₂ . We obtain these results by a similar study of quantum graph algebras, which we show to be isomorphic to stated skein algebras.