
Belgian
Bracket and Quantisation Workshop, Brussels April 13 2014 


Pierre Bieliavsky (UCL), Simone Gutt (ULB), Pierre Mathonet (ULg) REGISTRATION:
To
register please email to Edwine.Lukamba@ulb.ac.be 

Lecturers:
Martin Bordemann (Mulhouse) : An unabelian version of
T.Voronov's construction of Linfinity structures Abstract: In 2005 T.Voronov gave a rather useful
explicit construction of an Linfinity 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 Linfinity
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 Linfinity
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 Linfinity
structure is isomorphic just to a differential (no higher brackets) iff the
(graded) Atiyah (or Nguyenvan 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 starproduct 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) : Starexponential of Kählerian Lie
Groups Abstract :
We will see the construction of nonformal deformation quantization on
a large class of solvable Lie groups, Kählerian Lie groups with negative curvature.
Then, we will present the associated starexponential 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”
(KählerEinstein 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 semiclassical 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. JeanPhilippe 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 stardeformation 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 bPoisson 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
bsymplectic manifolds (joint work with Victor Guillemin, Ana Rita Pires and
Geoffrey Scott) taking the 2dimensional case as starting point. Time permitting, I will mention an application
to geometric quantization of bsymplectic manifolds. I also plan to
report on an ongoing project with Geoffrey Scott on generalizing the notion
of bsymplectic manifold. This notion includes other Poisson manifolds which
share good properties with bsymplectic 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 traceclass operators. In contrast, we build and
characterise the canonical trace on classical pseudodifferential
operators on a noncommutative torus, which extends the ordinary trace
on traceclass 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 zetafunction 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 BerezinToeplitz, Berezin, Geometric Quantization, BordemannWaldmann,
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 DeligneFedosov classes are given.
Besides the BordemannWaldmann 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 $U(nm)$ 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^2Betti numbers for groups and
equivalence relations, but what about von Neumann algebras? Abstract: I will
introduce L^2Betti 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^2Betti 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. 