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Pierre Bieliavsky

Publications and source records attributed to Pierre Bieliavsky.

At least 19 recordsLinked to original sources
arXiv2026

Kohn--Nirenberg quantization of the affine group and related examples

We show how to construct unitary dual $2$-cocycles for a class of semidirect products that exhibit many similarities with the affine group ${\rm Aff}(V)=\GL(V)\ltimes V$ of a finite dimensional vector space over a local skew field. The primary source of examples comes from Lie groups whose Lie algebras are Frobenius seaweeds. The construction builds on our earlier results and relies heavily on representation theory and an associated quantization procedure of Kohn--Nirenberg type. On the technical side, the key point is the observation that any semidirect product $G=H\ltimes V$ in our class can be presented as a double crossed product $G=P\bowtie N$ with respect to which the unique square-integrable irreducible representation of $G$ takes a particularly nice form. The Kohn--Nirenberg quantization that we construct is intimately related to a scalar Fourier transform $\CF\colon L^2(N)\to L^2(P)$ intertwining the left regular representations of $P$ and $N$ with representations defined by the dressing transformations.

math.OA↗
arXiv2026

Symmetric spaces, non-formal star products and Drinfel'd twists

These notes refer to a minicourse I gave at the occasion of the conference meeting ``Applications of Noncommutative Geometry to Gauge Theories, Field Theories, and Quantum Space-Time'' to be held from 7 April to 11 April 2025 at the Centre International de Rencontres Mathématiques in Luminy. They consist in a review of a long standing work of mine and collaborators (see references therein) in the field of non-formal deformation quantization admitting a large group of symmetries. But they also contain new material and results. More precisely, in a first part, I present a method (called the Retract Method) to define quantizations/symbolic calculi and associated operator symbol composition formulae (non-formal deformations/star products) of symplectic symmetric spaces such as the hyperbolic plane (Kahler) or symmetric co-adjoint orbits of the Poincaré group (non-metric). In a second part, I explain how to derive non-formal Drinfel'd twists for actions of non-Abelian solvable Lie groups (non-Abelian Universal Deformation Formulae) on or Fr échet algebras from the non-formal noncommutative symmetric spaces defined in the first part.

math.QA↗
arXiv2026

Kinematical Lie algebras and symplectic symmetric spaces I: Lie algebraic aspects

We generalize the notion of kinematical Lie algebra introduced in physics for the classification of the various possible relativity algebras an isotropic spacetime can accommodate. We first give an elementary proof of the fact that such a generalized kinematical Lie algebra $\mathfrak{g}$ always carries a canonical structure of symplectic involutive Lie algebra (shortly ``siLa''). In other words, if $G$ is a connected Lie group admitting $\mathfrak{g}$ as Lie algebra, there always exists a Lie subgroup $H$ of $G$ constituted by the elements of $G$ that are fixed under an involutive automorphism of $G$ and such that the homogenenous space $M=G/H$ is a symplectic symmetric space. In particular, the manifold $M$ canonically carries a $G$-invariant linear torsionfree connection $\nabla$ whose geodesic symmetries centered at all points extend as global $\nabla$-affine transformations of $M$. The manifold $M$ is also canonically equipped with a symplectic structure $ω$ which is invariant under every geodesic symmetry, implying in particular that it is parallel w.r.t the linear connection: $\nablaω=0$. In a second part, we give a complete description of the fine structure of our generalized siLa's. Our discussion yields a complete classification of such sila's.

math.DG↗
arXiv2025

Aspects of Warped AdS$_3$ geometries

We discuss the geometry of three-dimensional warped Anti-de Sitter spaces and quotients thereof, paying special attention to their underlying group manifold nature. We perform a systematic analysis of warped Anti-de Sitter geometries, focusing on their global properties and illustrating their occurrence as special solutions of various three-dimensional gravity theories.

gr-qc↗
arXiv2025

Symplectic structures preserved by geodesic symmetries

Answering a conjecture by S. Kobayashi, in 1986, K. Sekigawa and L. Vanhecke proved that an almost hermitian manifold whose local geodesic symmetries preserve the Kähler 2-form is a locally symmetric hermitian space. In the present paper, we relax the hermitean hypothesis by only requiring the manifold to be symplectic. In other words, we study the symplectic manifolds equipped with a symplectic connection whose geodesic symmetries are (local) symplectomorphisms. We call ``S-type'' these affine symplectic manifolds.

math.SG↗
arXiv2025

Quantization of subgroups of the affine group

Consider a locally compact group $G=Q\ltimes V$ such that $V$ is abelian and the action of $Q$ on the dual abelian group $\hat V$ has a free orbit of full measure. We show that such a group $G$ can be quantized in three equivalent ways: (1) by reflecting across the Galois object defined by the canonical irreducible representation of $G$ on $L^2(V)$; (2) by twisting the coproduct on the group von Neumann algebra of $G$ by a dual $2$-cocycle obtained from the $G$-equivariant Kohn-Nirenberg quantization of $V\times\hat V$; (3) by considering the bicrossed product defined by a matched pair of subgroups of $Q\ltimes\hat V$ both isomorphic to $Q$. In the simplest case of the $ax+b$ group over the reals, the dual cocycle in (2) is an analytic analogue of the Jordanian twist. It was first found by Stachura using different ideas. The equivalence of approaches (2) and (3) in this case implies that the quantum $ax+b$ group of Baaj-Skandalis is isomorphic to the quantum group defined by Stachura. Along the way we prove a number of results for arbitrary locally compact groups $G$. Using recent results of De Commer we show that a class of $G$-Galois objects is parametrized by certain cohomology classes in $H^2(G;\mathbb T)$. This extends results of Wassermann and Davydov in the finite group case. A new phenomenon is that already the unit class in $H^2(G;\mathbb T)$ can correspond to a nontrivial Galois object. Specifically, we show that any nontrivial locally compact group $G$ with group von Neumann algebra a factor of type I admits a canonical cohomology class of dual $2$-cocycles such that the corresponding quantization of $G$ is neither commutative nor cocommutative.

math.OA↗
arXiv2023

Quantization of locally compact groups associated with essentially bijective $1$-cocycles

Given an extension $0\to V\to G\to Q\to1$ of locally compact groups, with $V$ abelian, and a compatible essentially bijective $1$-cocycle $η\colon Q\to\hat V$, we define a dual unitary $2$-cocycle on $G$ and show that the associated deformation of $\hat G$ is a cocycle bicrossed product defined by a matched pair of subgroups of $Q\ltimes\hat V$. We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang--Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of $G$ on $L^2(Q)$ and a unitary quantization map $L^2(G)\to{\rm HS}(L^2(Q))$ of Kohn--Nirenberg type.

math.OA↗
arXiv2021

On deformations of C*-algebras by actions of Kahlerian Lie groups

We show that two approaches to equivariant strict deformation quantization of C*-algebras by actions of negatively curved Kahlerian Lie groups, one based on oscillatory integrals and the other on quantizations maps defined by dual 2-cocycles, are equivalent.

math.OA↗
arXiv2019

Deformation Quantization for Actions of Kählerian Lie Groups

Let $\mathbb B$ be a Lie group admitting a left-invariant negatively curved Kählerian structure. Consider a strongly continuous action $α$ of $\mathbb B$ on a Fréchet algebra $\mathcal A$. Denote by $\mathcal A^\infty$ the associated Fréchet algebra of smooth vectors for the action $α$. In the Abelian case $\mathbb B=\mathbb R^{2n}$ and $α$ isometric, Marc Rieffel proved that Weyl's operator symbol composition formula yields a deformation through Fréchet algebra structures ${\star_θ^α}_{θ\in\mathbb R}$ on $\mathcal A^\infty$. When $\mathcal A$ is a $C^\star$-algebra, every deformed algebra $(\mathcal A^\infty,\star^α_θ)$ admits a compatible pre-$C^\star$-structure. In this paper, we prove both analogous statements in the general negatively curved Kählerian group and (non-isometric) "tempered" action case. The construction relies on the one hand on combining a non-Abelian version of oscillatory integral on tempered Lie groups with geometrical objects coming from invariant WKB-quantization of solvable symplectic symmetric spaces, and, on the second hand, in establishing a non-Abelian version of the Calderòn-Vaillancourt Theorem. In particular, we give an oscillating kernel formula for WKB-star products on symplectic symmetric spaces that fiber over an exponential Lie group.

math.OA↗
arXiv2019

Quantization of Hamiltonian coactions via twist

In this paper we introduce a notion of quantum Hamiltonian (co)action of Hopf algebras endowed with Drinfel'd twist structure (resp., 2-cocycles). First, we define a classical Hamiltonian action in the setting of Poisson Lie groups compatible with the 2-cocycle stucture and we discuss a concrete example. This allows us to construct, out of the classical momentum map, a quantum momentum map in the setting of Hopf coactions and to quantize it by using Drinfel'd approach.

math.QA↗
arXiv2018

Quantum differential surfaces of higher genera

We first construct a real family of $SL(2,\mathbb{R})$-invariant symbol composition product $\{\sharp_θ\}_{θ\in,\mathbb{R}}$ on the analogue of the Schwartz space $S(\mathbb{D})$ on the hyperbolic plane $\mathbb{D}\;:=\;SL(2,\mathbb{R})/SO(2)$. The value $θ=0$ consists in the pointwise commutative product of functions on $\mathbb{D}$. And admits an asymptotic expansion that deforms the pointwise product in the direction of the canonical $SL(2,\mathbb{R}) $-invariant Kahler two form on $\mathbb{D}$. We then extend this construction to any (non-homogeneous) compact surface by considering the left action of an arithmetic Fuschian group $Γ\subset SL(2,\mathbb{R})$ on $\mathbb{D}$ with associated Riemann surface $Σ_Γ\;:=\;Γ\backslash\mathbb{D}$. More precisely, the product $\sharp_θ$ extends from $S(\mathbb{D})$ to a smooth $SL(2,\mathbb{R})$- sub-module of $C^\infty(\mathbb{D})$ that contains the $Γ$-invariants $C^\infty(\mathbb{D})^Γ\simeq C^\infty(Σ_Γ)$ in $C^\infty(\mathbb{D})$. In particular, $\sharp_θ$ defines a Fréchet algebra structure on $C^\infty(Σ_Γ)$. The resulting algebra is pre - $C^\ast$ and admits a continuous trace.

math.OA↗
arXiv2016

Noncommutative principal bundles through twist deformation

We construct noncommutative principal bundles deforming principal bundles with a Drinfeld twist (2-cocycle). If the twist is associated with the structure group then we have a deformation of the fibers. If the twist is associated with the automorphism group of the principal bundle, then we obtain noncommutative deformations of the base space as well. Combining the two twist deformations we obtain noncommutative principal bundles with both noncommutative fibers and base space. More in general, the natural isomorphisms proving the equivalence of a closed monoidal category of modules and its twist related one are used to obtain new Hopf-Galois extensions as twists of Hopf-Galois extensions. A sheaf approach is also considered, and examples presented.

math.QA↗
arXiv2016

Obstructions for Twist Star Products

In this short note we point out that not every star product is induced by a Drinfel'd twist by showing that not every Poisson structure is induced by a classical $r$-matrix. Examples include the higher genus symplectic Pretzel surfaces and the symplectic sphere $\mathbb{S}^2$.

math.QA↗
arXiv2015

Affine connections and symmetry jets

We establish a bijective correspondence between affine connections and a class of semi-holonomic jets of local diffeomorphisms of the underlying manifold called symmetry jets in the text. The symmetry jet corresponding to a torsion free connection consists in the family of $2$-jets of the geodesic symmetries. Conversely, any connection is described in terms of the geodesic symmetries by a simple formula involving only the Lie bracket of vector fields. We then formulate, in terms of the symmetry jet, several aspects of the theory of affine connections and obtain geometric and intrinsic descriptions of various related objects involving the gauge groupoid of the frame bundle. In particular, the property of uniqueness of affine extension admits an equivalent formulation as the property of existence and uniqueness of a certain groupoid morphism. Moreover, affine extension may be carried out at all orders and this allows for a description of the tensors associated to an affine connections, namely the torsion, the curvature and their covariant derivatives of all orders, as obstructions for the affine extension to be holonomic. In addition this framework provides a nice interpretation for the absence of other tensors.

math.DG↗
arXiv2015

Non-formal star-exponential on contracted one-sheeted hyperboloids

In this paper, we exhibit the non-formal star-exponential of the Lie group SL(2,R) realized geometrically on the curvature contraction of its one-sheeted hyperboloid orbits endowed with its natural non-formal star-product. It is done by a direct resolution of the defining equation of the star-exponential and produces an expression with Bessel functions. This yields a continuous group homomorphism from SL(2,R) into the von Neumann algebra of multipliers of the Hilbert algebra underlied by this natural star-product. As an application, we prove a new identity on Bessel functions.

math.OA↗
arXiv2013

Bargmann-Fock realization of the noncommutative torus

We give an interpretation of the Bargman transform as a correspondence between state spaces that is analogous to commonly considered intertwiners in representation theory of finite groups. We observe that the non-commutative torus is nothing else that the range of the star-exponential for the Heisenberg group within the Kirillov's orbit method context. We deduce from this a realization of the non-commutative torus as acting on a Fock space of entire functions.

math.QA↗