PROGRAM

MONDAY 30
9.15 Coffee and welcome
9.50 - Opening
10 - C. Kassel
11.15 - P. Fima
12.30 Lunch
14. - E. Musicantov
15.10 - G. Carnovale
16.10 - Coffee break
16.40 - M. Caspers
17.50 - C. Voigt

TUESDAY 31
9. - A. Skalski
10. Coffee break
10.25 - M. Müger
11.35 - J. Cuadra
12.45 Lunch
14. - C. Lecouvey
15.10 - B. Collins
16.10 - Coffee Break
16.40 - P. Guillot
17.50 - C. Galindo

WEDNESDAY 1
9. - S. Morrison
10. - Coffee break
10.25 - S. Caenepeel
11.35 - D. Nikshych
12.45 Lunch
Free afternoon

THURSDAY 2
9. - S. Burciu
10. - Coffee break
10.25 - A. Bruguières
11.35 - S. Launois
12.45 Lunch
14. - S. Curran
15.10 - S. Neshveyev
16.10 - Coffee Break
16.40 - E. Meier
20. Conference Dinner

FRIDAY 3
8.30 - K. De Commer
9.30 - Coffee break
9.50 - V. Ostrik
11. - T. Banica
12.15 - Lunch


ABSTRACTS

Teodor Banica. Representations of quantum permutation groups.
Abstract. I will describe some recent advances on the representation theory of the quantum permutation groups, and of their duals. This is based on joint work with Bichon and Curran, Vergnioux, Schlenker.

Alain Bruguières. Tensor functors, Hopf comonads and commutative central algebras.
Abstract. I will explain how a tensor functor $F : C \rightarrow D$ between tensor categories $C$ and $D$ admitting a right adjoint $R$ can be described in terms of a Hopf co-monad on $D$ and also (if $R$ is faithful exact) in terms of a commutative algebra in the categorical center of $C$. This construction generalizes classical facts about Tannaka duality and the theory of Hopf algebra. Indeed, if $D$ is $vect_k$ then $F$ is a fiber functor. In that case, the Hopf comonad on $vect_k$ is the endofunctor $H \otimes ?$, where $H$ is a Hopf algebra, and $C$ is (equivalent to) the category of $H$ comodules. Moreover, $H$ can be seen as a commutative algebra in the center of $C \simeq comod_H$ (it generalizes the 'trivializing algebra' of tannaka theory) and its category of modules is $vect_k$ by Sweedler's theorem of the struture of Hopf modules. The hypothesis on the existence of a right adjoint can be lifted using categories of ind-objects. If time allows, I will introduce the notion of exact sequence of tensor categories - which generalizes a notion of exact sequence of Hopf algebras due to Schneider - and show how these exact sequences are classified by normal Hopf (co)monads. Equivariantizations and in particular, modularizations, provides examples of such exact sequences. The talk is based on the following preprints : Hopf monads on monoidal categories, avec S. Lack et A. Virelizier - preprint arXiv:1003.1920 [math.QA] Exact Sequences of Tensor Categories, avec S. Natale - preprint arXiv:1006.0569 [math.QA]

Sebastian Burciu. Kernels of representations of Hopf algebras
Abstract. The notion of kernel and center of a representation of a Hopf algebra is introduced. Similar properties to the kernel of a group representation are proved in some special cases. In particular, it is shown that every normal Hopf subalgebra of a semisimple Hopf algebra H is the kernel of a representation of H. The normal Hopf subalgebras of semisimple Drinfeld doubles D(H) and of some other Hopf algebra constructions are also discussed.

Stefaan Caenepeel. Harrison cohomology over commutative Hopf algebroids
Abstract. We introduce cohomology with values in a restricted Picard groupoid. Restricted Picard groupoids can be seen as the appropriate categorical generalization of abelian groups. To a complex of restricted Picard groupoids, we associate a sequence of bicategories, and the cohomology groups are the equivalence classes of 0-cells in these bicategories. We can associate a complex of restricted Picard groupoids to a commutative bialgebroid. The associated cohomology groups fit into a long exact sequence. The obtained cohomology generalizes classical cohomology theories, such as group cohomology, Amitsur cohomology and Sweedler cohomology. We can give algebraic interpretations of the cohomology groups in low index: the 0-th cohomology classifies certain invertible modules; the 1st cohomology classifies certain Galois coobjects; the 2nd cohomology group classifies certain monoidal structures.

Giovanna Carnovale. Brauer groups of a Hopf algebra: examples, counterexamples and open problems.
We will recall the different Brauer groups BM, BC and BQ of a finite dimensional Hopf algebra, as defined by Caenepeel, Van Oystaeyen and Zhang, and the relations between them. If the group BM for a triangular Hopf algebra is well understood and can be described in a general framework, this is no longer the case for quasitriangular Hopf algebras, nor for the group BQ. The main difficulties show uo already in low-dimensional examples such as Sweedler's Hopf algebra H4 and Nichols' Hopf algebra E(2). We will show how the group BQ for H_4 is related to the group BM for E(2) and with the groups BM for H_4. The talk will mainly be based on G. Carnovale; J. Cuadra, On the subgroup structure of the full Brauer group of Sweedler Hopf algebra, preprint arXiv:0904.1883, to appear in Israel Journal of Mathematics.

Martijn Caspers. Modular properties of matrix coefficients of corepresentations of a locally compact quantum group.
Abstract. We consider locally compact quantum groups in the von Neumann algebraic setting as introduced by Kustermans and Vaes. We will see how the modular automorphism group of the Haar weights can be expressed in terms of matrix coefficients of corepresentations. As an application this gives a tool to determine the Plancherel transformation for (certain) locally compact quantum groups for which the von Neumann algebra is type I. We also define a distinguished L^p-Fourier transform for p between 1 and 2 and see how this is related to the Plancherel transformation.

Benoît Collins. Spectral properties of generators of A_o(n).
Abstract. We will explain how to compute the spectrum of the generators of A_o(n) with respect to the Haar measure. The techniques rely on Weingarten calculus, orthogonal polynomials and representation theory of SU_q(2). This is joint work with T. Banica and P. Zinn-Justin.

Juan Cuadra. On the finiteness of the coradical filtration of co-Frobenius Hopf algebras.
Abstract. The notion of integral and that of coradical filtration figure among the most basic and important notions in Hopf Algebra Theory. Hopf algebras possessing a non-zero (left) integral are called co-Frobenius and they were widely studied in the literature. Radford showed that a co-Frobenius Hopf algebra whose coradical is a subalgebra has finite coradical filtration. Andruskiewitsch and Dascalescu proved later on that a Hopf algebra with finite coradical filtration is necessarily co-Frobenius and they conjectured that any co-Frobenius Hopf algebra has finite coradical filtration. In this talk we will present several results about this question. We will provide an alternative formulation of this conjecture in terms of an upper bound for the Loewy length of the injective hulls of the simple comodules. Two new sufficient conditions for a co-Frobenius Hopf algebra to have finite coradical filtration will be then derived. We will give a result explaining several facts on the basis of this problem and showing certain heuristic evidences that the conjecture may be not true. An strategy to look for a counterexample based on a generalization of the Andruskiewitsch-Schneider Lifting Method will be discussed. Some of the results presented in this talk are part of a joint work with N. Andruskiewitsch.

Stephen Curran. Probabilistic aspects of easy quantum groups.
Abstract. The class of "easy" quantum groups was introduced by Banica and Speicher to provide a framework for studying certain common probabilistic and representation theoretic aspects of S_n, O_n and their "free versions". In this talk we will survey some recent results obtained in this framework, including de Finetti theorems and an extension of some results of Diaconis-Shahshahani. This is joint work with Teodor Banica and Roland Speicher.

Kenny De Commer. A triple of locally compact quantum groups.
Abstract. I will report on some of my work which links the quantizations of the Lie groups SU(2), SU(1,1) and E(2), the Euclidian transformation group of the plane. The object which establishes this link is a quantum groupoid with three objects, having the duals of the above quantum groups as its `isotropy groups'. This quantum groupoid has a rich analytic structure, of which I will present some facets. I will also show how one can recover from its representation theory some well-known quantum homogeneous spaces. Finally, by looking at an accompanying infinitesimal structure, I can make the link with Hopf-Galois theory for QUE algebras, which was one of the starting points for these investigations.

Pierre Fima. Quantum groups and type III factors.

César Galindo. On the classification of Galois objects over finite groups.
Abstract. The talk will be about a joint work with Manuel Medina. We classify Galois objects for the dual of a group algebra of a finite group over an arbitrary field.

Pierre Guillot. Un invariant des noeuds via l'indice de Maslov.
Abstract. En remarquant que la représentation de Burau du groupe des tresses préserve une forme hermitienne, on peut définir un invariant des noeuds (et entrelacs) à valeur dans le groupe de Witt du corps considéré. La construction se fait à l'aide de l'indice de Maslov. On retrouve de nombreux invariants classiques à partir de celui-ci, qui les subsume : signature(s), invariants métaplectiques de Jones, et polynôme d'Alexander-Conway (ou presque).

Christian Kassel. Any Hopf algebra fibers over an affine variety of the same dimension.
Abstract. In my talk I will attach an affine algebraic variety V to any Hopf algebra H. This variety is the base space of a quantum principal fiber bundle with "structural group" H. If H has finite dimension d, then V is of dimension d as a variety. These results have been obtained in joint work with Eli Aljadeff (Advances in Math. 208 (2008), 1453-1495) and with Akira Masuoka (arXiv:0911.3719).

Stéphane Launois. From total positivity to quantum algebras.
Abstract. In recent publications, the same combinatorial description has arisen for three separate objects of interest: non-negative cells in the real Grassmannian (Postnikov, Williams); torus orbits of symplectic leaves in the classical Grassmannian (Brown, Goodearl and Yakimov); and torus invariant prime ideals in the quantum Grassmannian (Launois, Lenagan and Rigal). The aim of this talk is to present these results and explore the reasons for this coincidence.

Cédric Lecouvey. Cristaux et marches aléatoires dans les chambres de Weyl
Résumé : On se donne une algèbre de Lie simple g et une représentation irréductible V du groupe quantique associé. Il s'agira dans un premier temps de montrer comment la théorie des bases cristallines permet d'associer à V une marche aléatoire M dans le réseau des poids de g. Cette marche peut être vue comme une généralisation du problème des urnes qui correspond à la représentation de dimension n de sl(n). On verra ensuite que, lorsque V est une représentation minuscule, la loi de la chaîne de Markov correspondant la restriction de M à la chambre de Weyl peut être totalement explicitée. Il s'agit d'un travail en collaboration avec E. Lesigne et M. Peigné.

Ehud Meir. On the Hopf Schur group of a field.
Abstract. The talk is partially based on a joint work with Eli Aljadeff, Juan Cuadra and Shlomo Gelaki. Let k be a field. We ask what k-central simple algebras can we get as quotients of Hopf algebras. We call the corresponding subgroup of Br(k) the Hopf Schur group of k. This generalizes the question of what central simple k-algebras are quotients of group algebras (the so called Schur algebras) and the Schur subgroup of Br(k). Since group algebras are Hopf algebras, every Schur algebra is a Hopf Schur algebra. The Schur subgroup might be a very small subgroup of Br(k), and we ask what other Hopf Schur algebras can we find. In this talk I will try to explain why actually every k-central simple algebra is a Hopf Schur algebra up to Brauer equivalence. We shall do so by considering forms of Hopf algebras.

Scott Morrison. Classifying small index subfactors.
Abstract. I'll report on progress towards the classification of subfactors with index at at most 5. There are 10 known examples, and we can now prove that any further examples must lie in two unlikely-looking families. This prompts the conjecture that the 10 examples exhaust subfactors with index at most 5. To date, most of these examples come from ad-hoc and unsatisfying constructions. We need to look for new constructions and better explanations.

Michael Müger. On gradings and factorizations.

Evgeny Musicantov. Reducing extensions to equivariantizations.

Sergey Neshveyev. Cohomology of quantum groups.
Abstrac. By a result of Drinfeld and Kazhdan-Lusztig, the category of representations of the q-deformation G_q of a compact simple Lie group G is equivalent to that of G equipped with nontrivial associativity morphisms defined via monodromy of KZ-equations. It turns out that such an equivalence of categories is essentially unique. This is equivalent to a vanishing result for the second cohomology of the dual of G_q. The uniqueness implies in particular that $G_q$ has a canonical spin structure. (Joint work with Lars Tuset)

Dmitri Nikshych. Fusion categories graded by finite groups.
Abstract. This is a report on a joint work with Pavel Etingof and Victor Ostrik. Given a fusion category C we classify fusion categories graded by a finite group G and having C as the trivial component (i.e., extensions of C by G). This classification is given in terms of the group of invertible C-bimodule categories, called the Brauer?Picard group of C. I will describe obstructions to existence of such extensions and the data parameterizing them. I will also construct an isomorphism between the Brauer-Picard group of C and the group of braided autoequivalences of the Drinfeld center of C.

Victor Ostrik. Deligne's category Rep(S_t).
Abstract. I will report on my joint work with J.Comes. Deligne defined tensor category Rep(S_t) which in a sense interpolates between representation categories of the symmetric groups. Our main result is a description of blocks in this category. I will also explain the similarity of Deligne's category and representation category of quantum SL(2).

Adam Skalski. On some convolution equations for states on (locally) compact quantum groups.
Abstract. The set of states on a (locally) compact quantum group with natural convolution multiplication is a semigroup generalising the classical semigroup of probability measures on a group. Certain convolution equations have natural interpretations - for example the Haar state can be defined as a (unique) state h such that h \star \phi = \phi \star h = h for all states \phi. In this talk we will present certain results on idempotent states (i.e. solutions of the equation \phi \star \phi = \phi) and square roots of Haar states (solutions of the equation \phi \star \phi = h). It is based on joint work with Uwe Franz, Reiji Tomatsu and Pekka Salmi.

Christian Voigt. Quantum groups and the Baum-Connes conjecture.
Abstract. The Baum-Connes conjecture plays an important role in the study of group C*-algebras. In this talk I describe first how the conjecture can be extended to the setting of quantum groups, based on work of Meyer and Nest. Then I will discuss the conjecture in some concrete cases, mostly in connection with free quantum groups.