MONDAY 18.08.2014
09:00
OPENING OF THE CONFERENCE
(P.M. Hajac)
11:00 - 12:00
QUANTUM PERMUTATIONS OF TWO ELEMENTS
(T. Maszczyk)
We introduce a Radon-Nikodym differentiable structure
on a finitely-dimensional algebra.
This allows us to speak about the "quantum fundamental cycle"
for finite-dimensional algebras which exists
for any Frobenius algebra.
For the algebra of functions on a finite set the quantum fundamental cycle
coincides with
its classical counterpart.
For any Hopf algebra H with bijective antipode coacting on a
finite-dimensional algebra A with a fundamental cycle,
one can ask whether the coaction preserves the fundamental cycle
in the same way as classical permutations do. If it is the case,
there is a canonical cohomology class in
H1(H, A×)
which is an obstruction to the existence of
an invariant Frobenius structure on A
supported on the fundamental class. We call it the modular class.
We show that, for every finite-dimensional algebra A
with a fundamental cycle, there exists a universal Hopf algebra
with bijective antipode and coaction preserving the fundamental
class on A. Thus the modular class of the
universal coaction becomes an invariant of a Frobenius algebra.
We show that the universal Hopf algebra
with bijective antipode and coaction on the algebra of functions on
a finite set preserves the fundamental cycle and that,
for a finite set of cardinality bigger than one,
the modular class is non-trivial, although it vanishes
on classical permutations.
14:00 - 15:00
AUTOEQUIVALENCES OF TENSOR CATEGORIES FOR TWISTED q-DEFORMATIONS OF QUANTUM GROUPS
(M. Yamashita)
The isomorphism problem for the compact quantum groups
presented as twisted q-deformations of some semisimple Lie groups
can be reduced to the study of autoequivalences of the associated tensor
categories.
Building on the work of Neshveyev and Tuset on the untwisted case,
we determine such autoequivalences for simple and simply connected Lie
groups.
This talk is based on joint work with S. Neshveyev.
16:00 - 17:00
A QUANTUM VERSION OF DE LEEUW & GLICKSBERG DECOMPOSITION THEOREM
(B. Das)
In 1961, de Leeuw and Glicksberg showed that the C*-algebra of almost
periodic functions on a topological group G
is complemented (as a Banach
space) inside the C*-algebra of weakly almost periodic functions.
Later in 2012, Stokke and Spronk adapted this argument and proved similar
results: almost periodic functions are complemented inside
the Eberlein compactification of G.
Using C*-algebraic arguments, we will generalize this decomposition result
in the context of Kac algebras. In particular, we will recover the classical
decomposition theorem as a corollary. Our methods
give a new proof of this decomposition result for classical group.
(Based on joint work with Matthew Daws.)
TUESDAY 19.08.2014
09:00 - 10:00
SPECTRAL ZETA FUNCTIONS IN NONCOMMUTATIVE GEOMETRY
(M. Khalkhali)
I shall first briefly review recent progress in understanding spectral invariants,
in particular scalar curvature, for noncommutative tori in dimensions up to four
(based on joint work with Fazrad Fathizadeh).
I will then report on ongoing work in further study of conformal anomalies for noncommutative tori.
11:00 - 12:00
EXTREMA OF THE EINSTEIN-HILBERT ACTION FOR NONCOMMUTATIVE 4-TORI
(F. Fathizadeh)
I will present my joint work with M. Khalkhali on conformal geometry of
noncommutative 4-tori. After perturbing the flat geometry of a noncommutative
4-torus by a Weyl conformal factor, we compute the associated Laplacian
and find
an explicit local expression for the scalar curvature appearing
in a small time heat
kernel expansion that depends on the high frequency behavior
of the eigenvalues of the Laplacian. The analog of the Einstein-Hilbert action is then considered,
and flat metrics are shown to be the critical points of this action. We also provide a convenient tool
for computing the Dixmier trace of pseudodifferential operators of order -4 on noncommutative
4-tori by defining a noncommutative residue and by proving an analog of Connes' trace theorem.
14:00 - 15:00
METRIC, TORSION AND SCALAR CURVATURE IN NONCOMMUTATIVE GEOMETRY
(A. Sitarz)
I shall review recent results of scalar curvature computations
and minimality of Laplace-type operators for noncommutative tori in
dimensions 2 and 4.
16:00 - 17:00
HOPF-CYCLIC COHOMOLOGY OF QUANTIZED ENVELOPING ALGEBRAS
(S. Sütlü)
Given a coalgebra coextension, we introduce a filtration whose associated Hochschild-Serre type spectral
sequence computes the coalgebra Hochschild cohomology of the coextension. We then apply this cohomological
machinery to compute the coalgebra Hochschild cohomology, and then the Hopf-cyclic cohomology via the SBI sequence,
of a quantized enveloping algebra of a Lie algebra with coefficients in a modular pair in involution associated to the Lie algebra.
We will also show that our computation coincides with that of Crainic in the case of the Lie algebra sl(2).
Time permitting, we will discuss an application of this method to Connes-Moscovici Hopf algebras.
(Based on an ongoing joint project with A. Kaygun.)
WEDNESDAY 20.08.2014
09:00 - 10:00
MULTIPLIER HOPF ALGEBROIDS AND THEIR DUALITY
(T. Timmermann)
The theory of multiplier Hopf algebras developed by Alfons Van Daele provides
an elegant algebraic framework for the study of quantum groups and their
duality. We extend this theory to quantum groupoids, where the presence of a
non-trivial base leads to several complications, and focus on the appropriate
notion of an integral and the resulting duality. As a next step, we hope to pass
from this algebraic theory to the operator-algebraic theory of measured
quantum groupoids developed by Michel Enock, Franck Lesieur and Jean-Michel
Vallin.
11:00 - 12:00
POISSON-LIE GROUPOIDS AND THE CONTRACTION PROCEDURE
(K. De Commer)
Certain Lie algebras can be joined into a continuous family by seeing
their structure constants as continuous functions on some parameter
domain. In this way, a continuous family of semi-simple Lie algebras
can degenerate ('contract') into a non-semi-simple Lie algebra at a
limit point. In this talk, we will look at how quantum enveloping
algebras of semi-simple Lie algebras admit more contractions than their
classical analogues, and we interpret the classical limit of this
contraction procedure in terms of Poisson-Lie groupoids.
14:00 - 15:00
THE BOHR-SOMMERFELD GROUPOID OF QUANTUM COMPLEX PROJECTIVE SPACES
(N. Ciccoli)
Integrable Poisson manifolds can be quantized by performing geometric quantization
on the symplectic groupoids integrating them.
A possible way to carry through the whole quantization procedure is to try and build,
from a suitable multiplicative Lagrangian distribution, the quantized algebra as a twisted convolution algebra.
However, strong topological obstructions
often appear. For complex projective spaces, seen as Poisson homogeneous spaces of
standard Poisson SU(n)s, we will show how to relax requirements on the Lagrangian distribution, and use a
natural bihamiltonian structure to obtain a noncommutative groupoid C*-algebra. Here the
cocycle integrating the modular class plays a distinct role as a KMS-operator on an etale groupoid.
We will also show how, to a certain extent,
such a construction is functorial allowing for easy quantizations of some distinct Poisson submanifolds.
16:00 - 17:00
GELFAND-CETLIN INTEGRABLE SYSTEM AND QUANTIZATION OF A SYMPLECTIC GROUPOID
(F. Bonechi)
We will discuss the bihamiltonian structure of the Gelfand-Cetlin integrable
model on compact hermitian symmetric spaces, including
Grassmanians. Our motivation is to quantize the symplectic
groupoid of the Poisson-Bruhat structure and of the associated Poisson
pencil.
We will review the definition of the bihamiltonian structure and analyze
its role in the quantization of projective spaces. A basic tool
that we use in order to show the equivalence with the Gelfand-Cetlin
system is the structure of a Poisson vector bundle naturally defined on
tautological line bundles. We will discuss the main difference between the case
of projective spaces and the case of the aforementioned symplectic groupoid
that makes the quantization of the latter still an open problem.
THURSDAY 21.08.2014
09:00 - 10:00
ODD-DIMENSIONAL MULTI-PULLBACK QUANTUM SPHERES
(B. Zieliński)
We construct a noncommutative deformation of odd-dimensional spheres that preserves
the natural partition of the (2n+1)-dimensional sphere into (n+1)-many solid tori.
This generalizes the case n = 1 referred to as the Heegaard quantum sphere. Our
odd-dimensional
quantum sphere C*-algebras are given as multi-pullback C*-algebras. We prove that
they are isomorphic to the universal C*-algebras generated by certain isometries,
and use this result
to compute the K-groups of our odd-dimensional quantum spheres. On the other hand,
we prove that the fixed-point subalgebras under the natural (diagonal) U(1)-action
on our quantum sphere C*-algebras yield the independently defined C*-algebras of
the quantum complex
projective spaces constructed from Toeplitz cubes. This leads to the main result stating that all the
non-trivial winding number noncommutative line bundles over the quantum complex projective spaces
are not stably trivial.
(Based on joint work with P.M.
Hajac, D. Pask and A. Sims.)
11:00 - 12:00
BRAIDED JOIN COMODULE ALGEBRAS OF GALOIS OBJECTS
(L. Dąbrowski)
We construct the join of noncommutative Galois objects (quantum torsors)
over a Hopf algebra H. To ensure that the join algebra enjoys the natural
(diagonal) coaction of H, we braid the tensor product of the Galois objects.
Then we show that this coaction is principal. Our examples are built from
the noncommutative torus with the natural free action of the classical torus,
and arbitrary anti-Drinfeld doubles of finite-dimensional Hopf algebras.
The former yields a noncommutative deformation of a non-trivial torus bundle,
and the latter a finite quantum covering.
(Based on joint work with T. Hadfield, P.M. Hajac and E. Wagner.)
14:00 - 15:00
RIEFFEL DEFORMATION OF THE TENSOR FUNCTOR AND BRAIDED QUANTUM GROUPS
(P. Kasprzak)
Rieffel deformation of tensor functor and braided quantum groups
In this talk we apply Rieffel deformation to the tensor product considered as a
functor in the category of C*-algebras with an abelian group action.
In the case of the Rieffel deformation applied to a quantum group with the abelian group
acting by automorphisms the deformed tensor product enables us to view the deformed object
as a braided quantum group. We shall link our construction with quantum
groups with projections and category of Yetter-Drinfeld modules of a given quantum group.
16:00 - 17:00
DEFORMATIONS OF BRAIDINGS BY MONOIDAL CATEGORIES
(E. Meir)
In this talk, I will present a new equivalence relation on solutions to
the Yang-Baxter equation arising from a construction of certain braided (and
symmetric) monoidal categories.
I will also explain how can one use Deligne's theory of symmetric
monoidal categories to study a special class of symmetric solutions to
the Yang-Baxter equation.
FRIDAY 22.08.2014
09:00 - 10:00
FORMALITY AND DEFORMATION QUANTIZATION OF GERBES
(R. Nest)
We will formulate the formality theorem for gerbes and sketch its proof. Then
we shall explain an application of the theorem for the
deformation quantization of twisted
Poisson structures.
11:00 - 12:00
QUANTIZATIONS OF POISSON-LIE HAMILTONIAN SYSTEMS
(C. Esposito)
In this talk, we will introduce the concept of a Hamiltonian system in the
canonical and Poisson settings. We will discuss the formal deformation
quantization of
Hamiltonian systems in the Poisson context, and propose a non-formal
approach of the Poisson-Hamiltonian
spaces for triangular Poisson-Lie groups.
14:00 - 15:00
CLASSIFICATION OF GROUP ACTIONS ON DEFORMATION QUANTIZATION
(N. De Kleijn)
Given a symplectic manifold one can consider corresponding
formal deformation quantizations in terms of the Fedosov construction.
This construction allows one to formulate a definition of extension of a
symplectic automorphism of the manifold to an automorphism of the
deformation quantization, as Fedosov already points out in his paper: a
simple geometrical construction of deformation quantization. A natural
question is: what happens when we have a group of symplectic
automorphisms? More specifically can we choose extensions such that
these form a group of automorphisms of the deformation and if so in how
many ways? The answers of existence and classification of such actions
seems to allow an expression in terms of group cohomology. In this talk
we would like to show in what way to extract this expression and comment
on computability of the corresponding cohomology groups/pointed sets.
16:00 - 17:00
QUANTIZATION BY CATEGORIFICATION
(T. Maszczyk)
A unified approach to noncommutative geometry is presented in terms of
monoidal categories.
The classical commutative case is categorified to symmetric monoidal
structure, and next symmetry is dropped.
Then it is shown how to redefine classical structures to make them
resistible to the loss of symmetry while
preserving their classical meaning in the commutative case. This is applied
to affine morphisms and coordinate rings,
flat covers, symmetries, quotients, principal fibrations, classifying maps,
global homological invariants,
infinitesimals and differential operators. In particular,
a categorification of Hopf-cyclic (co)homology is constructed.
The categorification
has the known algebraic theory as a particular component corresponding
to the monoidal unit. Finally,
the loss of commutativity under deformation quantization is categorified
and a corresponding Gerstenhaber algebra
for such categorified deformations is constructed.