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Article Dans Une Revue Symmetry, Integrability and Geometry : Methods and Applications Année : 2024

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

Résumé

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 $\mathbb{C}[q, q^{−1}]$ 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.
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Dates et versions

hal-03265204 , version 1 (19-06-2021)
hal-03265204 , version 2 (03-09-2021)
hal-03265204 , version 3 (15-10-2023)
hal-03265204 , version 4 (30-01-2024)
hal-03265204 , version 5 (06-06-2024)

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Stéphane Baseilhac, Philippe Roche. Unrestricted quantum moduli algebras, II: noetherianity and simple fraction rings at roots of 1. Symmetry, Integrability and Geometry : Methods and Applications, 2024, 20 (047), ⟨10.3842/SIGMA.2024.047⟩. ⟨hal-03265204v5⟩
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