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

Stéphane Baseilhac (1) , Philippe Roche (1)

1 IMAG - Institut Montpelliérain Alexander Grothendieck

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 $\mc(q)$. Moreover, we show that these two properties still hold on $\mc[q,q^{-1}]$ for the integral version of the quantum graph algebra. We also study the specializations $\Ll_{0,n}^\e$ 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 under the quantum group $U_\e(\mathfrak{g})$ have classical fraction algebras which are central simple algebras of PI degrees that we compute.

Mots clés: quantum groups; invariant theory; TQFT

Domaines: Physique mathématique [math-ph]; Théorie des groupes [math.GR]

Stéphane Baseilhac, Philippe Roche. Unrestricted quantum moduli algebras, II: Noetherianity and simple fraction rings at roots of $1$. 2023. ⟨hal-03268803⟩