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

Abstract

We prove that the quantum graph algebra and the quantum moduli algebra associated to a punctured sphere and a 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 𝓛ϵ₀,ₙ of the quantum graph algebra at a root of unity ϵ of odd order, and show that 𝓛ϵ₀,ₙ and its invariant algebra under the quantum group 𝑈ϵ(𝖌) have classical fraction algebras which are central simple algebras of PI degrees that we compute.