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

Stéphane Baseilhac, Philippe Roche

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 $\mathcal{L}_{0,n}^ε$ of the quantum graph algebra at a root of unity $ε$ of odd order, and show that $\mathcal{L}_{0,n}^ε$ and its invariant algebra under the quantum group $U_ε(\mathfrak{g})$ have classical fraction algebras which are central simple algebras of PI degrees that we compute.

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 are Noetherian rings and finitely generated rings over . Moreover, we show that these two properties still hold on for the integral version of the quantum graph algebra. We also study the specializations of the quantum graph algebra at a root of unity of odd order, and show that and its invariant algebra under the quantum group have classical fraction algebras which are central simple algebras of PI degrees that we compute.

Paper Structure

This paper contains 18 sections, 40 theorems, 181 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Basic notations
  4. Background results
  5. On $\boldsymbol{U_q}$, $\boldsymbol{{\mathcal{O}}_q}$, $\boldsymbol{{\mathcal{L}}_{0,n}}$, $\boldsymbol{{\mathcal{M}}_{0,n}}$, and $\boldsymbol{\Phi_n}$
  6. Integral forms and specializations
  7. Definitions
  8. Canonical bases and modified quantum groups
  9. Some consequences on $\boldsymbol{{\mathcal{L}}_{0,n}^A}$ and $\boldsymbol{{\mathcal{M}}_{0,n}^A}$
  10. Perfect pairings
  11. Integral form and specialization of $\boldsymbol{\Phi_n}$
  12. Structure theorems for $\boldsymbol{U_\epsilon}$ and $\boldsymbol{{\mathcal{O}}_\epsilon}$
  13. Noetherianity and finiteness
  14. Proof of Theorem \ref{['Llibre']}
  15. Proof of Theorem \ref{['degree']}
  16. ...and 3 more sections

Key Result

Theorem 1.1

${\mathcal{L}}_{0,n}$, ${\mathcal{M}}_{0,n}$ and the integral form ${\mathcal{L}}_{0,n}^A$ are Noetherian rings, and finitely generated rings.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Lemma 2.7
  • ...and 43 more