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On the structure and representations of quantum graph algebras at roots of unity

Stéphane Baseilhac, Matthieu Faitg, Philippe Roche

TL;DR

This work develops a uniform, Hopf-algebraic framework for quantum graph algebras $\\mathcal{L}_{g,n}$ associated to a simply-connected complex semisimple group and a punctured surface, and analyzes their specializations at odd-order roots of unity. By introducing a co-R-matrix datum, the modified Alekseev morphism, and a quantum moment map, the authors prove that both $\\mathcal{L}_{g,n}^{\\varepsilon}$ and its invariant subalgebra $\\mathcal{L}_{g,n}^{u_{\\varepsilon}}$ are domains with central localizations that are central simple algebras; they compute the PI-degrees as $l^{g\\dim\\mathfrak{g}+N(n-1)-m}$ (and related formulas), and provide explicit descriptions of centers as integrally closed rings. A crucial intermediate step is the surjective map to the small quantum-group graph algebra, tying these algebras to Kernler–Lyubashenko TQFTs and invariant theory. The centers are shown to be trace rings generated by Frobenius-type and trace elements, enabling a detailed account of irreducible representations and the Azumaya locus. Overall, the paper uniformly handles all complex simple Lie algebras, connecting quantum-group methods with representation theory and non-semisimple topology, and sets the stage for further explorations of specialization phenomena and indecomposable representations in low-rank cases (e.g., $\mathfrak{sl}_2$).

Abstract

We study the specializations $\mathcal{L}_{g,n}^ε$ at roots of unity $ε$ of odd order of the graph algebras, associated to a simply-connected complex semi-simple algebraic group $G$ and a compact oriented surface $Σ_{g,n}^{\circ}$ with genus $g$, $n$ punctures, and one boundary component. We prove that the central localizations of $\mathcal{L}_{g,n}^ε$ and of its subalgebra $\mathcal{L}_{g,n}^{u_ε}$ of invariant elements under the coadjoint action of a small quantum group, are central simple algebras of PI degrees that we compute. Also, we describe their centers, and show they are integrally closed rings.

On the structure and representations of quantum graph algebras at roots of unity

TL;DR

This work develops a uniform, Hopf-algebraic framework for quantum graph algebras associated to a simply-connected complex semisimple group and a punctured surface, and analyzes their specializations at odd-order roots of unity. By introducing a co-R-matrix datum, the modified Alekseev morphism, and a quantum moment map, the authors prove that both and its invariant subalgebra are domains with central localizations that are central simple algebras; they compute the PI-degrees as (and related formulas), and provide explicit descriptions of centers as integrally closed rings. A crucial intermediate step is the surjective map to the small quantum-group graph algebra, tying these algebras to Kernler–Lyubashenko TQFTs and invariant theory. The centers are shown to be trace rings generated by Frobenius-type and trace elements, enabling a detailed account of irreducible representations and the Azumaya locus. Overall, the paper uniformly handles all complex simple Lie algebras, connecting quantum-group methods with representation theory and non-semisimple topology, and sets the stage for further explorations of specialization phenomena and indecomposable representations in low-rank cases (e.g., ).

Abstract

We study the specializations at roots of unity of odd order of the graph algebras, associated to a simply-connected complex semi-simple algebraic group and a compact oriented surface with genus , punctures, and one boundary component. We prove that the central localizations of and of its subalgebra of invariant elements under the coadjoint action of a small quantum group, are central simple algebras of PI degrees that we compute. Also, we describe their centers, and show they are integrally closed rings.
Paper Structure (36 sections, 64 theorems, 279 equations)

This paper contains 36 sections, 64 theorems, 279 equations.

Table of Contents

  1. Introduction
  2. General preliminaries
  3. Simple Lie algebras
  4. Parameters q and qD
  5. Specialization of A-modules
  6. Central localization of a domain
  7. General facts on graph algebras
  8. A substitute for quasitriangularity
  9. Definition of the graph algebra
  10. Quantum moment map
  11. The handle algebra and the Heisenberg double
  12. Alekseev morphism and representations
  13. Preliminaries on quantum groups
  14. The quantum enveloping algebra
  15. The quantum function algebra
  16. ...and 21 more sections

Key Result

Theorem 1.2

(see Th. centralextLgn) The algebra $\mathcal{L}_{g,n}^{\epsilon}$ is a central extension of the $l^{(2g+n)\dim(\mathfrak{g})}$-dimensional algebra $\mathcal{L}_{g,n}(u_\epsilon^{\hbox{${\rm Q}$}})$ by $\mathcal{O}^+(G)^{\otimes (2g+n)}$, where $l$ is the order of the root of unity $\epsilon$.

Theorems & Definitions (147)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Corollary 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 137 more