Unrestricted Quantum Moduli Algebras, II: Noetherianity and Simple Fraction Rings at Roots of 1

We prove that the quantum graph algebra and the quantum moduli algebra associated to a punctured sphere and complex semisimple Lie algebra g are Noetherian rings and finitely generated rings over C(q). Moreover, we show that these two properties still hold on C[q,q⁻¹] for the integral version of the quantum graph algebra. We also study the specializations L0,n^ε of the quantum graph algebra at a root of unity ε of odd order, and show that L0,n^ε and its invariant algebra under the quantum group U_ε(g) have classical fraction algebras which are central simple algebras of PI degrees that we compute.