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

TL;DR

This paper proves that certain quantum algebras related to punctured spheres and semisimple Lie algebras are Noetherian and finitely generated, and analyzes their structure at roots of unity, including their classical fraction algebras.

Contribution

It establishes Noetherianity and finite generation of quantum graph and moduli algebras, and characterizes their structure at roots of unity, extending previous understanding of these quantum algebras.

Findings

Quantum graph and moduli algebras are Noetherian and finitely generated over c(q)

These properties hold for integral versions over c[q,q^{-1}]

Specializations at roots of unity yield classical fraction algebras that are central simple

Abstract

We prove that the quantum graph algebra and the quantum moduli algebra associated to a punctured sphere and complex semisimple Lie algebra $\mathfrak{g}$ are Noetherian rings and finitely generated rings over $\mathbb{C}(q)$. Moreover, we show that these two properties still hold on $\mathbb{C}\big[q,q^{-1}\big]$ for the integral version of the quantum graph algebra. We also study the specializations $\mathcal{L}_{0,n}^\epsilon$ of the quantum graph algebra at a root of unity $\epsilon$ of odd order, and show that $\mathcal{L}_{0,n}^\epsilon$ and its invariant algebra under the quantum group $U_\epsilon(\mathfrak{g})$ have classical fraction algebras which are central simple algebras of PI degrees that we compute.