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 Uₐ (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.
Baseilhac et al. (Fri,) studied this question.