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We prove that the quantum graph algebra and the quantum moduli algebraassociated to a punctured sphere and complex semisimple Lie algebra$\mathfrak{g}$ are Noetherian rings and finitely generated rings over $\mc(q)$.Moreover, we show that these two properties still hold on $\mc[q,q^{-1}]$ forthe integral version of the quantum graph algebra. We also study thespecializations $\Ll_{0,n}^\e$ of the quantum graph algebra at a root of unity$\e$ of odd order, and show that $\Ll_{0,n}^\e$ and its invariant algebra underthe quantum group $U_\e(\mathfrak{g})$ have classical fraction algebras whichare central simple algebras of PI degrees that we compute.

Unrestricted quantum moduli algebras, II: Noetherianity and simple fraction rings at roots of $1$