Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 20 (2024), 047, 70 pages      arXiv:2106.04136      https://doi.org/10.3842/SIGMA.2024.047

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

Stéphane Baseilhac and Philippe Roche
IMAG, Univ Montpellier, CNRS, Montpellier, France

Received May 11, 2023, in final form May 07, 2024; Published online June 06, 2024

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.

Key words: quantum groups; invariant theory; character varieties; skein algebras; TQFT.

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