arXiv subjects

Stéphane Baseilhac

Publications and source records attributed to Stéphane Baseilhac.

7 recordsLinked to original sources

arXiv2026

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

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.

math.QA↗

arXiv2025

Noetherian and affine properties of quantum moduli and $\mathfrak{g}$-skein algebras

We prove that the quantum moduli algebra associated to a possibly punctured compact oriented surface and a complex semisimple Lie algebra $\mathfrak{g}$ is a Noetherian and finitely generated ring. If the surface has punctures, we prove also that it has no non-trivial zero divisors (i.e., it is a domain). Moreover, we show that the quantum moduli algebra is isomorphic to the skein algebra of the surface, defined by means of the Reshetikhin-Turaev functor for the quantum group $U_q(\mathfrak{g})$, and which coincides with the Kauffman bracket skein algebra when $\mathfrak{g}=\mathfrak{sl}_2$. We obtain these results by a similar study of quantum graph algebras, which we show to be isomorphic to stated skein algebras.

math.QA↗

arXiv2024

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

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.

math.QA↗

arXiv2022

Unrestricted Quantum Moduli Algebras. I. The Case of Punctured Spheres

Let $Σ$ be a finite type surface, and $G$ a complex algebraic simple Lie group with Lie algebra $\mathfrak{g}$. The quantum moduli algebra of $(Σ,G)$ is a quantization of the ring of functions of $X_G(Σ)$, the variety of $G$-characters of $π_1(Σ)$, introduced by Alekseev-Grosse-Schomerus and Buffenoir-Roche in the mid '90s. It can be realized as the invariant subalgebra of so-called graph algebras, which are $U_q(\mathfrak{g})$-module-algebras associated to graphs on $Σ$, where $U_q(\mathfrak{g})$ is the quantum group corresponding to $G$. We study the structure of the quantum moduli algebra in the case where $Σ$ is a sphere with $n+1$ open disks removed, $n\geq 1$, using the graph algebra of the "daisy" graph on $Σ$ to make computations easier. We provide new results that hold for arbitrary $G$ and generic $q$, and develop the theory in the case where $q=ε$, a primitive root of unity of odd order, and $G={\rm SL}(2,{\mathbb C})$. In such a situation we introduce a Frobenius morphism that provides a natural identification of the center of the daisy graph algebra with a finite extension of the coordinate ring $\mathcal{O}(G^n)$. We extend the quantum coadjoint action of De-Concini-Kac-Procesi to the daisy graph algebra, and show that the associated Poisson structure on the center corresponds by the Frobenius morphism to the Fock-Rosly Poisson structure on $\mathcal{O}(G^n)$. We show that the set of fixed elements of the center under the quantum coadjoint action is a finite extension of ${\mathbb C}[X_G(Σ)]$ endowed with the Atiyah-Bott-Goldman Poisson structure. Finally, by using Wilson loop operators we identify the Kauffman bracket skein algebra $K_ζ(Σ)$ at $ζ:={\rm i}ε^{1/2}$ with this quantum moduli algebra specialized at $q=ε$.

math.QA↗

arXiv2016

Non ambiguous structures on 3-manifolds and quantum symmetry defects

The state sums defining the quantum hyperbolic invariants (QHI) of hyperbolic oriented cusped $3$-manifolds can be split in a "symmetrization" factor and a "reduced" state sum. We show that these factors are invariants on their own, that we call "symmetry defects" and "reduced QHI", provided the manifolds are endowed with an additional "non ambiguous structure", a new type of combinatorial structure that we introduce in this paper. A suitably normalized version of the symmetry defects applies to compact $3$-manifolds endowed with $PSL_2(\mathbb{C})$-characters, beyond the case of cusped manifolds. Given a manifold $M$ with non empty boundary, we provide a partial "holographic" description of the non-ambiguous structures in terms of the intrinsic geometric topology of $\partial M$. Special instances of non ambiguous structures can be defined by means of taut triangulations, and the symmetry defects have a particularly nice behaviour on such "taut structures". Natural examples of taut structures are carried by any mapping torus with punctured fibre of negative Euler characteristic, or by sutured manifold hierarchies. For a cusped hyperbolic $3$-manifold $M$ which fibres over $S^1$, we address the question of determining whether the fibrations over a same fibered face of the Thurston ball define the same taut structure. We describe a few examples in detail. In particular, they show that the symmetry defects or the reduced QHI can distinguish taut structures associated to different fibrations of $M$. To support the guess that all this is an instance of a general behaviour of state sum invariants of 3-manifolds based on some theory of 6j-symbols, finally we describe similar results about reduced Turaev-Viro invariants.

math.GT↗

arXiv2011

Quantum coadjoint action and the $6j$-symbols of $U_qsl_2$

We review the representation theory of the quantum group $U_εsl_2\mathbb{C}$ at a root of unity $ε$ of odd order, focusing on geometric aspects related to the 3-dimensional quantum hyperbolic field theories (QHFT). Our analysis relies on the quantum coadjoint action of De Concini-Kac-Procesi, and the theory of Heisenberg doubles of Poisson-Lie groups and Hopf algebras. We identify the 6j-symbols of generic representations of $U_εsl2\mathbb{C}$, the main ingredients of QHFT, with a bundle morphism defined over a finite cover of the algebraic quotient $PSL_2\mathbb{C}/!/PSL_2\mathbb{C}$, of degree two times the order of $ε$. It is characterized by a non Abelian 3-cocycloid identity deforming the fundamental five term relation satisfied by the classical dilogarithm functions, that relates the volume of hyperbolic 3-polyhedra under retriangulation, and more generally, the simplicial formulas of Chern-Simons invariants of 3-manifolds with flat $sl_2\mathbb{C}$-connections.

math.GT↗

arXiv2004

3D Quantum Hyperbolic Field Theory

We construct a new family of exact quantum field theories modeled on hyperbolic geometry, called {\it quantum hyperbolic field theories} (QHFTs). The QHFTs are defined for a $(2+1)$-bordism category based on the set of compact oriented 3-manifolds $Y$, equipped with properly embedded framed links $L_\Ff$ and with flat connections $ρ$ of principal $PSL(2,\C)$-bundles over $Y \setminus L_\Ff$, with arbitrary holonomy at the link meridians. A main point is the introduction of new parameters for the space of all $PSL(2,\C)$-characters of a punctured surface. Each QHFT associates to a triple $(Y,L_\Ff,ρ)$ as above with parametrized boundary components a tensor, which is generically holomorphic w.r.t. the parameters for the restriction of $ρ$ to $\partial Y \setminus L_\Ff$. This gives new numerical invariants of 3-manifolds, such as Chern-Simons invariants of $PSL(2,\mc)$-characters of arbitrary link complements, or quantum invariants of compact hyperbolic cone manifolds. Also, for any $PSL(2,\mc)$-character of a surface of finite topological type, we obtain new conjugacy classes of linear representations of the mapping class group. Finally, we discuss some evidences showing that the QHFTs are pertinent to 3D gravity.

math.GT↗