Belgian Bracket and Quantisation Workshop, Brussels April 1-3 2014

Organizing Committee :

Pierre Bieliavsky (UCL), Simone  Gutt (ULB), Pierre Mathonet (ULg)



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Martin Bordemann (Mulhouse) : An unabelian version of T.Voronov's construction of L-infinity structures

Abstract: In 2005 T.Voronov gave a rather useful explicit construction of an L-infinity structure on a graded vector space V which is supposed to be an abelian subalgebra

  complementing a subalgebra H in a graded Lie algebra G which he extended to the  ambient Lie algebra G. His technique gave rise to some L-infinity constructions attached

 to coisotropic submanifolds and the simultaneous deformation of associative or Lie algebras and their morphisms (work of Y.Frégier et al.).

  We generalize his construction to an L-infinity structure on the quotient G/H (and the extension) without assuming that there is an abelian subalgebra complement to H in G. The construction simplifies a bit to some `graded dressing transformation' if there is a (non)abelian subalgebra complement.The main idea is the observation that the quotient U(G)/(U(G)H) of the universal envelopping algebra U(G) of G is a cofree coalgebra on which G acts from the left by coderivations. This quotient had recently been studied in the trivially graded case by Calaque, Caldararu and Tu: using their result we can show that the generalized Voronov L-infinity structure is isomorphic just to a differential (no higher brackets) iff the (graded) Atiyah (or Nguyen-van Hai) class of the Lie algebra pair (G,H) vanishes.

  We shall indicate how the generalization may help to the quantization problem of coisotropic submanifolds as modules.


Marius Crainic (Utrecht) : Jacobi structures and Spencer operators.

Abstract: The aim of this talk is to explain the tight relationship between Jacobi structures and the the Spencer operators coming from the geometric theory of PDEs. In particular, I will explain a new approach to the integrability of Jacobi structures- approach which is considerably shorter and more geometric that previous ones (e.g. it avoids passing to Poisson geometry and it uses directly the Jacobi brackets instead of lengthy algebraic expressions). This is based on joint work with Maria Salzar and Ivan Stuchiner.


Giuseppe Dito (Dijon) : Universal quantization formulas and wheels

Abstract: The celebrated deformation quantization formula of Kontsevich associates a star-product to a Poisson bracket defined on the affine space. This formula is expressed in terms of graphs having oriented cycles (wheels). In this talk, I will provide an elementary argument showing that any universal quantization formula should involve graphs with wheels.



Axel de Goursac (UCL) : Star-exponential of Kählerian Lie Groups

Abstract :  We will see the construction of non-formal deformation quantization on a large class of solvable Lie groups, Kählerian Lie groups with negative curvature. Then, we will present the associated star-exponential and its applications : an adapted Fourier transformation for these Lie groups and the construction of new C noncommutive tori by generators and relations.



Joel Fine (ULB) : Quantisation and canonical metrics in Kähler geometry

Abstract :A large amount of research in Kähler geometry has been done on the subject of "canonical metrics” (Kähler-Einstein metrics, constant scalar curvature metrics, extremal metrics… ) Around 15 years ago Donaldson observed that there was a “quantised” version of this problem, where “Kähler metric” is replaced by a distinguished choice of Hermitian metric on the space of holomorphic sections of powers of an ample line bundle. In the semi-classical limit, as the power tends to infinity, one expects that “the problem of finding a distinguished Hermitian metric” should converge in some sense to “the problem of finding a canonical Kähler metric”. I will explain Donaldson’s original result in this direction and then my own subsequent contributions. 


Kirill Mackenzie (Sheffield) : Double structures and commutativity conditions

Abstract: I will describe the evolution of the notion of double groupoid from early work by Ehresmann, Loday and R. Brown, to the infinitesimal form -- double Lie algebroid -- as defined in  terms of an associated Lie bialgebroid, and as reformulated insuper terms by Th. Voronov.


Jean-Philippe Michel (ULg) : Higher symmetries of Dirac and Laplace operators - towards supersymmetries  

Abstract : The algebra of higher symmetries of Laplace operator appears in higher spin field theory, separation of variables, and minimal representation of the orhtogonal group.

t is an algebra of differential operators, generated by the Lie algebra of conformal Killing vector fields. More precisely, this is isomorphic to a quotient of the universal

envelopping algebra by the Joseph ideal [Eastwood, 2005].

In this talk, I will show how to recover this result thanks to  the conformally equivariant quantization. In particular, the algebra of higher symmetries of the Laplacian appears then as a the unique invariant star-deformation of the algebra  of regular functions on the minimal nilpotent orbit  of the conformal group.

Afterwards, I will discuss higher symmetries of the Dirac operator and of the system Laplace+Dirac operators. In dimension 3 and 4,  the algebra of higher symmetries of this system is a quotient of the universal envelopping algebra of a simple Lie superalgebra.

 The ideal appearing is a natural generalization of the Joseph ideal.


Eva Miranda (Barcelone) : Toric b-Poisson manifolds and their generalizations.

Abstract :The aim of this talk is to show some examples of simple Poisson manifolds which have some common features with symplectic manifolds (including thestudy of group actions).

I will start presenting a Delzant theorem for toric b-symplectic manifolds (joint work with Victor Guillemin, Ana Rita Pires and Geoffrey Scott) taking the 2-dimensional case as starting point.

Time permitting, I will mention an application to  geometric quantization of b-symplectic manifolds. I also plan to report on an ongoing project with Geoffrey Scott on generalizing the notion of b-symplectic manifold. This notion includes other Poisson manifolds which share good properties with b-symplectic manifoldsand seem to have less topological constraints.


Sylvie Paycha (Potsdam) : Quantisation of toroidal symbols and associated traces in the noncommutative setup.

Abstract :  The  global symbol calculus for pseudodifferential operators on tori can be  generalised to noncommutative tori. In this  global approach, the quantisation map is invertible and   traces are discrete sums. On the noncommutative torus,   Fathizadeh and Wong had characterised the Wodzicki residue   as the unique trace which vanishes on trace-class operators.  In contrast, we build and characterise  the canonical trace on classical pseudodifferential operators on a  noncommutative torus, which extends the ordinary trace on trace-class operators. It can be written as   a canonical discrete sum on the underlying toroidal symbols.  On the grounds of this uniqueness result, we prove that in the commutative setup, this canonical trace on the noncommutative torus reduces to Kontsevich and Vishik's canonical trace which is thereby identified with a discrete sum. By means of the canonical trace, we derive defect formulae for regularised traces on noncommutative toris. The conformal invariance of the zeta-function at zero of the Laplacian on the noncommutative torus   then arises as a straightforward consequence.

This is based on joint work with Cyril Lévy and  Carolina Neira Jiménez.


John Rawnsley (Warwick) : Mp^c quantisation.


Martin Schlichenmaier (Luxembourg) : Some naturally defined star products for Kähler manifolds.

Abstract: We give for the Kähler manifold case an overview of the constructions of some naturally defined star products. In particular, the Berezin-Toeplitz, Berezin, Geometric Quantization, Bordemann-Waldmann, and Karbegov standard star product are introduced. With the exception of the Geometric Quantization case they are of separation of variables type. The classifying Karabegov forms and the Deligne-Fedosov classes are given. Besides the Bordemann-Waldmann star product they are all equivalent.


Ted Voronov ( Manchester) : On volumes of some classical supermanifolds

Abstract: "Volume" in the super case may show unexpected features, due to properties of the Berezin integration. For example, the invariant volume of the unitary supergroup $U(n|m)$ vanishes whenever $nm>0$, i.e., when it does not reduce to the ordinary group. This was discovered by Berezin in 1970s. Recently, a question was put forward by Witten whether the Liouville volume of compact symplectic supermanifolds should always be zero. A counterexample of the complex projective superspace shows that this is not so. An interesting observation is that the explicit formula for the volume in this case can be obtained by analytic continuation of the formula for the ordinary (purely even) complex projective space. This is likely to hold true for other examples, and we shall discuss that in the talk. (Work in progress.)



Stefaan Vaes (KUL) : L^2-Betti numbers for groups and equivalence relations, but what about von Neumann algebras?

 Abstract: I will introduce L^2-Betti numbers for discrete groups and for orbit equivalence relations given by probability measure preserving group actions. I will then discuss an attempt to define L^2-Betti numbers for von Neumann algebras and a negative result in this direction proven recently by Sorin Popa and myself. The talk will be at an introductory level.



Stefan Waldmann (Würzburg) : A nuclear Weyl algebra.

Abstract :  In this talk I will report on a recent construction of a convergent star  product for a symplectic (or Poisson) vector space: in finite dimensions the  relevant locally convex topologies coincide with the ones found earlier by  Omori, Maeda, Miyazaki, and Yoshioka. In infinite dimesions this gives  interesting applications to quantum field theory. Many additional features

like nuclearity and Schauder bases can be discussed in this framework.




Lectures are planned to start at 1pm on Tuesday the 1st in the lecture Hall Forum C.




The Conferences will take place on the Campus Plaine of the Universite Libre de Bruxelles (for a map see 

The metro station on Campus is called delta ; it is on the metro line 5 in the direction of Hermann Debroux.

From Gare centrale, the line 5 is direct and it takes about 17 minutes.

From Brussels National airport, there is a train every 15-20 minutes going to the center of town and stopping at Gare centrale.

From the Airport Brussels South-Charleroi, there is a bus to the Gare du midi.

From the Gare du midi, one can take the metro line 2 or 6 in the direction of Elisabeth and change at the station Arts-Loi for line 5 in the direction of Hermann Debroux, getting off at the station delta.



List of Participants