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 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.

References in corpus (20)

• Combinatorial quantization of the Hamiltonian Chern-Simons theory I
• Combinatorial Quantization of the Hamiltonian Chern-Simons Theory II
• Two dimensional lattice gauge theory based on a quantum group
• Braided Matrix Structure of the Sklyanin Algebra and of the Quantum Lorentz Group
• 3-Dimensional TQFTs From Non-Semisimple Modular Categories
• Classical and quantum dilogarithmic invariants of flat PSL(2,C)-bundles over 3-manifolds
• Kitaev lattice models as a Hopf algebra gauge theory
• Link Invariants and Combinatorial Quantization of Hamiltonian Chern-Simons Theory
• Integrability in Hamiltonian Chern-Simons theory
• Quantum hyperbolic geometry
• Finite presentations for stated skein algebras and lattice gauge field theory
• Analytic families of quantum hyperbolic invariants
• Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres
• Stated SL(n)-Skein Modules and Algebras
• Holonomy and (stated) skein algebras in combinatorial quantization
• O_e(G) is a free module over O(G)
• On stated $SL(n)$-skein modules
• Mapping class groups, skein algebras and combinatorial quantization
• Classification of semi-weight representations of reduced stated skein algebras
• On the Adjoint Representation of a Hopf Algebra