• «  Orthogonal free quantum group factors are strongly 1-bounded »  Orthogonal free quantum group factors are strongly 1-bounded with Michael Brannan
  Files | arxiv : ps, pdf | local : pdf

  In this article we prove that the orthogonal free quantum group factors are stronly 1-bounded in the sense of Jung. This allows to answer in the negative the important question whether these factors are isomorphic to free group factors. The proof relies on a recent rank theorem for free entropy dimension due to Jung and Shlyakhtenko, and on a spectral regularity result for the edge reversing operator in the quantum Cayley graph.

• «  The radial MASA in free orthogonal quantum groups »  The radial MASA in free orthogonal quantum groups with Amaury Freslon
  Journal of Functional Analysis, 271 (2016) 2776-2807
  Files | arxiv : ps, pdf | journal : html | local : pdf

  In this article we prove that the subalgebra generated by the fundamental character in the von Neumann algebra of an orthogonal free quantum group FO(n) is maximal abelian for n≥3, mixing, and we study the corresponding bimodule. In order to do so we establish new properties of the Jones-Wenzl projections in the category of corepresentations of FO(n).

• «  The Connes embedding property for quantum group von Neumann algebras »  The Connes embedding property for quantum group von Neumann algebras with Benoît Collins and Michael Brannan
  Transactions of the AMS, 369 (2017) 3799-3819
  Files | arxiv : ps, pdf | journal : html, pdf | local : pdf

  In this article we show that the von Neumann algebras of orthogonal free quantum groups FO(n), endowed with the Haar state, satisfy Connes' embedding property for n≠3. We work at the level of the dual compact quantum groups and proceed by induction over the dimension n. The main technical result consists in proving that the universal orthogonal quantum group O(n,+) is generated by two copies of O(n-1,+) seen as stabilizer subgroups.

• «  A cocycle in the adjoint representation of the orthogonal free quantum groups »  A cocycle in the adjoint representation of the orthogonal free quantum groups with Pierre Fima
  International Mathematics Research Notices, 2015 (2015) 10069-10094
  Files | arxiv : ps, pdf | journal : html, pdf | local : pdf

  In this article we construct a deformation by automorphisms of the von Neumann algebra M associated to a unimodular orthogonal free quantum group inside M⊗M, and we use it to give a new proof of Haagerup's property for these groups. Moreover we show that the bimodule M⊗M is associated to the adjoint representation, and we prove that the non-trivial part of it is weakly contained in the regular representation. This proves non inner amenability, Ozawa-Popa's property strong (HH), and yields a new proof of strong solidity of M.

• «  The K-theory of free quantum groups »  The K-theory of free quantum groups with Christian Voigt
  Mathematische Annalen, 357 (2013) 355-400
  Files | arxiv : ps, pdf | journal : html, pdf | local : pdf

  In this article we study the K-theory of free products of unitary and orthogonal free quantum groups, using the formulation of the Baum-Connes conjecture in terms of triangulated categories. We show the stability of the Baum-Connes conjecture under free products for torsion-free discrete quantum groups by constructing Dirac and dual-Dirac elements from the action on the quantum Bass-Serre tree. We deduce in particular K-amenability of unitary free quantum groups and compute their K-theory.

• «  Paths in quantum Cayley trees and L²-cohomology »  Paths in quantum Cayley trees and L²-cohomology
  Advances in Mathematics, 229 (2012) 2686-2711
  Files | arxiv : ps, pdf | journal : html, pdf | local : pdf
  There was a first version about the orthogonal case. The second version has been submitted in april 2010.

  In this article I go on with the study of quantum Cayley trees, focusing in particular on the existence and uniqueness of paths to the origin. For the free group F_n, these paths define a proper cocycle, which proves Haagerup's Property. In the case of universal orthogonal quantum groups, I prove on the contrary that the unique path cocycle is trivial. In the unitary case, it is neither bounded nor proper. Finally, I use this geometrical result to establish the vanishing of the first L^2 Betti number for A_o(I_n) — whereas the one of F_n equals n-1.

• «  Invariants of the half-liberated orthogonal group »  Invariants of the half-liberated orthogonal group with Teodor Banica
  Annales de l'Institut Fourier, 60 (2010) 2137-2164
  Files | arxiv : ps, pdf | journal : html, pdf | local : pdf

  The half-liberated orthogonal quantum group O_n^* is a intermediate compact quantum group between the compact group O_n and its free version O_n^+ associated with the Woronowicz algebra A_o(n). We discuss in this article the basic algebraic properties of O_n^* and we classify its irreducible representations. This classification relies upon the use of a noncommutative weight lattice and a comparison with the representation theory of U_n. We show in particular that O_n^* has noncommutative fusion rules and has the same exponent of polynomial growth as SU_n.

• «  Fusion rules for quantum reflection groups »  Fusion rules for quantum reflection groups with Teodor Banica
  Journal of Noncommutative Geometry, 3 (2009) 327-359
  Files | arxiv : ps, pdf | journal : html, pdf | local : pdf

  In this article we find the fusion rules of the quantum reflection groups A_h^s, which are free versions of the complex reflection groups G(s,s,n). We describe the category of corepresentation by means of certain subsets of indexed non-crossing partitions, and we show that irreducible corepresentations can be indexed by the words over Z/sZ so as to obtain explicit and "local" fusion rules. We also compute the dimensions of irreducible corepresentations.

• «  Growth estimates for discrete quantum groups »  Growth estimates for discrete quantum groups with Teodor Banica
  Infin. Dimens. Anal. Quantum Probab. Relat. Top., 12 (2009) 321-340
  Files | arxiv : ps, pdf | journal : html, pdf | local : pdf

  We give in this article estimates and exact formulae for the main known classes of discrete quantum groups, including exponential and polynomial examples. In the case of duals of real compact Lie groups we establish a relation between a quantum random walk and a classical random walk on the weight lattice. This allows to check in this dual case the quantum analogue of Gromov's result relating growth to the probability of return to the origin for discrete groups.

• «  The boundary of universal discrete quantum groups, exactness and factoriality »  The boundary of universal discrete quantum groups, exactness and factoriality with Stefaan Vaes
  Duke Mathematical Journal, 140 (2007) 35-84
  Files | arxiv : ps, pdf | journal : html, pdf | local : pdf

  In this article we construct and study the Gromov boundary of Wang-Banica orthogonal free quantum groups. It is a unital C*-algebra, constructed as an inductive limit, which is naturally endowed with an action of the discrete quantum group. We show that this action is amenable, in an appropriate quantum sense, and we deduce the exactness and Property AO for these quantum groups. We also prove simplicity results and hence we see that the von Neumann algebras associated to the quantum groups under consideration are full and prime factors. We finally establish connections with the notions of Poisson and Martin boundaries, and give a direct proof of exactness based on the notion of monoidal equivalence.

• «  The Property of Rapid Decay for Discrete Quantum Groups »  The Property of Rapid Decay for Discrete Quantum Groups
  Journal of Operator Theory, 57 (2007) 303-324
  Files | journal : html, pdf | local : pdf

  In this article I introduce the Property of Rapid Decay for discrete quantum groups by equivalent characterizations that generalize the classical ones, and I show that the application to the K-theory of the reduced C*-algebras of these groups carries over to the quantum case. I also investigate examples, proving in particular the Property of Rapid Decay for unimodular free quantum groups. I turns out that only unimodular quantum groups can have Property RD.

• «  Orientation of quantum Cayley trees and applications »  Orientation of quantum Cayley trees and applications
  Journal für die reine und angewandte Mathematik 580 (2005) 101-138
  Files | journal : html | local : pdf

  In this article I introduce the notion of quantum Cayley graph, and I study it in the case of trees, which corresponds to the case of free quantum groups. I focus in particular on the notion of ascending orientation for these quantum trees, and I describe the associated space of edges at infinity, which is an outcome of the non-involutivity of the edge returning operator and vanishes in the classical case. I give two applications to conclude: a proof of Property AO for free quantum groups, and the construction of their KK-theoretic gamma element. The results concerning the space of edges at infinity have been generalized and ameliorated since my PhD thesis, and the application to Property AO is new.

• «  K-amenability for amalgamated free products of amenable discrete quantum groups »  K-amenability for amalgamated free products of amenable discrete quantum groups
  Journal of Functional Analysis 212 (2004) 206-221
  Files | journal : html, pdf | local : pdf

  In this article I present the results of my thesis about the equivariant KK-theory with respect to locally compact quantum groups, and I demonstrate the K-amenability of amalgamated free products of amenable discrete quantum groups, by constructing a quantum analogue of the Bass-Serre trees. Some proofs have been ameliorated since my PhD thesis.

• Academic Works
• « Groupes quantiques discrets et algèbres d'opérateurs» Groupes quantiques discrets et algèbres d'opérateurs
  Habilitation Thesis
  Files | local : pdf

  This mémoire has been written on the occasion of my Habilitation, that I defended in June 2013. It presents and synthesizes the results I obtained since the end of my PhD Thesis.

• « Haar integrals on finite and compact quantum groups» Haar integrals on finite and compact quantum groups
  Notes of a graduate course given at Caen in 2010
  Files | local : pdf

  These notes were written for a short graduate course given at the University of Caen in September 2010 as part of a “quantum group” semester. It turned out that it was necessary to include some introductory material about tensor products and group algebras. Then we tried to present some nice features of the elementary theory of Hopf algebras, both in the algebraic and the C*-algebraic settings. Our main goal was the existence and unicity of “Haar integrals”, whose proofs are quite different in both settings, and the presentation of some “real-life” examples where computation skills can be trained. There are also some additional exercices (in french).

• « KK-théorie équivariante et opérateur de Julg-Valette pour les groupes quantiques» KK-théorie équivariante et opérateur de Julg-Valette pour les groupes quantiques
  PhD Thesis supervised by G. Skandalis
  Files | theses directory of the IMJ : dvi, pdf (zip), ps (bz2)

  After three chapters of reminders, I study the crossed product construction for locally compact quantum groups, and in particular, its fonctorial properties and its use in equivariant KK-theory (descent, Green-Julg isomorphism, K-amenability). Then I study examples of actions on quantum trees: quantum Bass-Serre and Cayley trees. In the first case, it allows to demonstrate the K-amenability of the amalgamated free products under consideration. The second case is much more complicated, and it is in particular necessary to introduce a space of edges at infinity. One can then define the gamma-elements associated to these quantum Cayley trees, in the case of the orthogonal free quantum groups.

• « Magistère's Thesis» Magistère's Thesis
  Files | local : pdf, dvi

  This thesis contains the texts I wrote at the Ecole Normale Supérieure of Paris: Equivalence between fermions and bosons (Master's Thesis under the supervision of B. Julia), Matrix Compact Quantum Groups (Diplôme d'Etudes Approfondies under the supervision of G. Skandalis), and an introduction to the links between particle physics and non-commutative geometry.

• See also...
• ...my publications on MathSciNet, my preprints on arXiv, my page on HAL.

• « Conferences» Conferences
  
  

  • 1 march 2017, Trieste, Conference "Noncommutative Geometry and Applications" :
    Free entropy dimension and the orthogonal free quantum groups. Presentation :  pdf.
  • 6 february 2017, Chennai, Conference "Non-commutative Analysis" :
    The radial MASA in free orthogonal quantum groups.
  • 11 july 2016, Greifswald, "Compact quantum groups" satellite conference of the 7th ECM :
    The radial MASA in free orthogonal quantum groups. Presentation :  pdf.
  • 19 october 2015, Paris, Meeting of the ANR project OSQPI :
    Stabilizer subgroups of universal compact quantum groups and the Connes embedding property. Presentation :  pdf.
  • 15 july 2015, Herstmonceux, Conference on Quantum Groups and Quantum Information Theory :
    Operator algebraic aspects of quantum groups. Presentation :  pdf.
  • « More» More
  • 23 april 2015, Paris, Conference "Advances in Noncommutative Geometry" :
    Stabilizer subgroups of universal compact quantum groups and the Connes embedding property. Presentation :  pdf.
  • 11 march 2015, Greifswald, Conference on Algebraic and Analytic Aspects of Quantum Lévy Processes :
    Stabilizer subgroups of universal compact quantum groups and the Connes embedding property. Presentation :  pdf.
  • 6 february 2015, Ottawa, Workshop on Recent Developments in Quantum Groups, Operator Algebras and Applications :
    Stabilizer subgroups of universal compact quantum groups and the Connes embedding property.
  • 20 june 2014, Frascati, GDRE meeting "Noncommutative geometry and applications" :
    Cocycles on free quantum groups. Presentation :  pdf.
  • 7 may 2014, Münster, Workshop "Quantum Groups and Operator Algebras" :
    Cocycles on free quantum groups.
  • 3 april 2014, Saarbrücken, Workshop on Quantum symmetries in free probability :
    Cocycles on free quantum groups.
  • 8 october 2013, Caen, Meeting of the ANR project OSQPI :
    Sur la représentation adjointe des groupes quantiques libres orthogonaux. Presentation :  pdf.
  • 28 september 2013, Indianapolis, Wabash extramural modern analysis miniconference :
    On the adjoint representation of orthogonal free quantum groups. Presentation :  pdf.
  • 26-30 august 2013, invitation to Oberwolfach (workshop "C*-algebras")
  • 22 july 2013, Toronto, Workshop on Analytic, Stochastic, and Operator Algebraic Aspects of Noncommutative Distributions and Free Probability :
    On the adjoint representation of orthogonal free quantum groups. Presentation :  pdf.
  • 18 december 2012, Milan, Conference on Noncommutative Geometry and Applications to Physics :
    K-theory of the unitary free quantum groups. Presentation :  pdf.
  • 25 september 2012, Lille, Workshop on Operator Algebras of the inaugural conference of the CEMPI :
    K-theory of the unitary free quantum groups. Presentation :  pdf.
  • 26 july 2012, Shanghai, Franco-chinese summer institute on noncommutative geometry :
    K-theory of the unitary free quantum groups. Presentation :  pdf.
  • 9 june 2011, Le Havre, 3rd "Normandie-Mathématiques" day :
    Sur quelques propriétés analytiques des groupes quantiques libres. Presentation :  pdf.
  • 26 april 2011, Bucarest, 4th meeting of the EU-NCG network :
    An example of quantum group fusion rules.
  • 12 april 2011, Vienne, Programme on Bialgebras in Free Probability :
    Path cocycles in quantum Cayley trees and $L^2$-cohomology. Presentation :  pdf.
  • 27 september 2010, Luminy, Conference "Quantum groups and noncommutative geometry" :
    Path cocycles in quantum Cayley trees and $L^2$-cohomology. Presentation :  pdf.
  • 14 may 2009, Toulouse, Quantum Group Workshop :
    New examples of fusion rules for quantum groups.
  • 5 december 2008, Caen, Journées trans-Couesnon de groupes et algèbres d'opérateurs :
    Path cocycles in quantum Cayley trees and $L^2$-cohomology.
  • 10 november 2008, Leuven, Workshop Operator algebraic aspects of quantum groups :
    Path cocycles in quantum Cayley trees and $L^2$-cohomology.
  • 17-23 august 2008, invitation to Oberwolfach (workshop "non-commutative geometry")
  • 15 february 2008, Strasbourg, journées Groupes Quantiques :
    Cayley graphs and free quantum groups. Slides :  pdf.
  • 18-22 june 2007, invitation to Fleurance (meeting of the GDR "Noncommutative Geometry")
  • 29 august 2005, Oberwolfach, workshop "C*-algebras" :
    The property of rapid decay for free quantum groups. Slides :  dvi, pdf. Notes :  pdf.
  • 15 november 2004, Dublin, meeting of the european network QSNG :
    The Property of Rapid Decay for free quantum groups. Slides :  dvi, pdf.
  • 12-18 september 2004, invitation to Oberwolfach (workshop "non-commutative geometry")
  • 17 june 2004, Marseille, meeting of the GDR "Algèbres d'Opérateurs" :
    The Property of Akemann and Ostrand for free quantum groups. Slides :  pdf.
  • 18 august 2003, Oberwolfach, workshop "C*-algebras" :
    Gamma-Elements For Free Quantum Groups. Slides :  dvi, pdf. Notes :  dvi, pdf.
  • 11 july 2003, Paris, meeting of the GDR "Algèbres d'Opérateurs" :
    Graphes de Cayley des groupes quantiques libres. Slides :  pdf, dvi. Notes :  dvi, pdf.
  • 15 may 2003, Orléans, meeting "Operator Algebras, Groups and K-Theory" :
    KK-Theory for Quantum Groups: functorial and geometrical methods. Slides :  dvi, pdf. Notes :  pdf.
  • 6 march 2002, Marseille, meeting of the GDR "Algèbres d'Opérateurs" :
    KK-théorie et produits croisés par un groupe quantique. Slides :  pdf, dvi.
  • 23-27 april 2001, invitation to Berkeley (29th Canadian Symposium on Operator Algebras)
  • 23-27 april 2001, invitation to Berkeley (Workshop "Quantization and noncommutative geometry")
• « Seminars» Seminars
  
  

  • 4 may 2017, Cambridge, Semester "Operator algebras: subfactors and their applications" :
    Free entropy dimension and the orthogonal free quantum groups.
  • 25 november 2016, Clermont-Ferrand, Seminar GAAO :
    La sous-algèbre radiale dans les groupes quantiques libres orthogonaux. Presentation :  pdf.
  • 10 march 2016, Paris, Operator Algebras Seminar :
    La sous-algèbre radiale dans les groupes quantiques libres orthogonaux.
  • 26 february 2016, Lille, Functional Analysis Seminar :
    Algèbres d'opérateurs associées aux groupes quantiques discrets.
  • 12 february 2016, College Station, Linear Analysis Seminar :
    The radial MASA in free orthogonal quantum groups.
  • « More» More
  • 12 november 2013, Glasgow, Analysis Seminar :
    On the adjoint representation of orthogonal free quantum groups.
  • 26 september 2013, Urbana-Champaign, Analysis Seminar :
    The Baum-Connes conjecture for free quantum groups.
  • 22 november 2012, Metz, Séminaire "Analyse, Géométrie et Algèbre" :
    Sur certaines propriétés analytiques des $C^*$-algèbres associées aux groupes quantiques libres.
  • 21 november 2012, Metz, Groupe de travail "Géométrie non commutative" :
    K-théorie des groupes quantiques libres.
  • 12 january 2012, Paris, Séminaire "Algèbres d'Opérateurs" :
    K-Théorie des groupes quantiques libres unitaires. Presentation :  pdf.
  • 15 november 2011, Besançon, Séminaire d’Analyse Fonctionnelle :
    K-Théorie des groupes quantiques libres unitaires.
  • 6 july 2009, Clermont-Ferrand, Workgroup GAAO :
    Cocycle chemin dans les arbres de Cayley quantiques et cohomologie $L^2$.
  • 15 january 2009, Paris, seminar "Operatoralgebras" :
    Cocycle chemin dans les arbres de Cayley quantiques et cohomologie $L^2$.
  • 15 december 2008, Rennes, seminar of Ergodic Theory :
    Croissance des groupes quantiques discrets. Notes :  pdf, dvi.
  • 5 february 2008, Besançon, seminar of Functional Analysis :
    Géométrie des groupes quantiques libres. Slides :  pdf.
  • 7 june 2007, Leuven, seminar "Operator Algebras" :
    Paths in quantum Cayley trees.
  • 16 november 2006, Toulouse, seminar "Geometry and Physics" :
    Géométrie des groupes quantiques discrets.
  • 21 february 2005, Toulouse, seminar "Non-commutative Geometry" :
    Propriété de décroissance rapide et groupes quantiques discrets.
  • 15 february 2005, Toulouse, seminar "Groups and Geometry" :
    Longueurs dans les groupes quantiques discrets et applications. Notes :  pdf.
  • 11 february 2005, Orléans, seminar "Operator Algebras" :
    Propriété de décroissance rapide et groupes quantiques discrets. Notes :  pdf.
  • 9 february 2005, Caen, seminar on non-commutative harmonic analysis :
    Propriété de décroissance rapide et groupes quantiques discrets. Slides :  pdf, dvi. Notes :  pdf.
  • 19 april 2004, Toulouse, workgroup "Non-commutative Geometry" :
    Produits libres amalgamés de groupes quantiques discrets. Slides :  pdf. Notes :  pdf.
  • 27 january 2004, Clermont-Ferrand, seminar of Pure Mathematics :
    Graphes de Cayley quantiques et espace des arêtes à l'infini. Slides :  pdf. Notes :  pdf.
  • 20 january 2004, Lyon, seminar of Analysis :
    Espace des arêtes à l'infini pour les groupes quantiques libres. Slides :  dvi, pdf.
  • 9 april 2003, Caen, workgroup on quantum groups :
    Graphes de Cayley pour les groupes quantiques discrets. Slides :  dvi, pdf. Notes :  dvi, pdf.
  • 31 march 2003, Toulouse, workgroup "Non-commutative Geometry" :
    Graphes de Cayley des groupes quantiques libres et KK-théorie. Slides :  pdf, dvi. Notes :  dvi, pdf.
  • 7 march 2003, Orléans, seminar "Operator Algebras" :
    K-moyennabilité pour des produits libres de groupes quantiques discrets. Notes :  pdf.
• « Local talks» Local talks
  
  

  • 22 october and 26 november and 10 december 2014, Caen, seminar on non-commutative harmonic analysis :
    Frontière de Poisson des catégories monoïdales.
  • 23 october 2013, Caen, seminar on non-commutative harmonic analysis :
    Moyennabilité des groupes quantiques localement compacts.
  • 30 april 2013, Caen, seminar on non-commutative harmonic analysis :
    Algèbres propres et K-moyennabilité.
  • 12 december 2012 and 30 january 2013, Caen, seminar on non-commutative harmonic analysis :
    Bi-exactitude des groupes quantiques libres.
  • 20 june and 4 july 2012, Caen, seminar on non-commutative harmonic analysis :
    Propriété de Baum-Connes forte et K-théorie de FU(n).
  • « More» More
  • 30 may and 6 june 2012, Caen, seminar on non-commutative harmonic analysis :
    Arbres de Bass-Serre quantiques et propriété de Baum-Connes forte.
  • 28 june 2011, Caen, day of the APS team :
    Échangeabilité libre et permutations quantiques.
  • 10 november 2010, Caen, seminar on non-commutative harmonic analysis :
    Propriété (T) et automorphismes extérieurs, d'après Connes.
  • 7 october 2009, Caen, seminar on non-commutative harmonic analysis :
    Autour de la conjecture de Baum-Connes forte pour les groupes quantiques discrets.
  • 23 september 2009, Caen, seminar on non-commutative harmonic analysis :
    KK-théorie : définition et structure triangulée.
  • 6 january 2009, Caen, seminar "Algebra and geometry" :
    Groupes de réflexions complexes quantiques. Notes :  pdf, dvi.
  • 15 october 2008, Caen, seminar on non-commutative harmonic analysis :
    Cocycle chemin et cohomologie $L^2$ des groupes quantiques libres.
  • 11, 18 and 25 june 2008, Caen, Workgroup on tensor categories :
    Algèbres enveloppantes quantifiées. Notes :  dvi, pdf.
  • 10 and 17 october 2007, Caen, seminar on non-commutative harmonic analysis :
    Longueur quantique dans les arbres de Cayley quantiques.
  • 31 january 2007, Caen, seminar on non-commutative harmonic analysis :
    Introduction à la K-théorie des C*-algèbres.
  • 22 and 29 november 2006, Caen, seminar on non-commutative harmonic analysis :
    Graphes de Cayley et croissance des groupes quantiques discrets.
  • 8 february 2006, Caen, seminar on non-commutative harmonic analysis :
    Groupes quantiques libres.
  • 26 january 2006, Paris, seminar "Operatoralgebras" :
    Cohomologie bornée et rigidité (d'après Monod et Shalom). Notes :  pdf, dvi.
  • 15 november 2005, Caen, seminar "discrete structures" :
    Arbres de Bass-Serre quantiques. Slides :  dvi, pdf.
  • 12, 19 and 26 october, 2 and 9 november 2005, Caen, seminar on non-commutative harmonic analysis : Groupes libres et moyennabilité.
  • 7 and 14 december 2004, Münster, seminar "C*-algebras" :
    Higson's proof of the Atiyah-Singer index theorem. Notes :  pdf.
  • 13 july 2004, Münster, seminar "C*-algebras" :
    The Property of Rapid Decay for free quantum groups. Slides :  pdf, dvi. Notes :  pdf.
  • 3 february 2004, Münster, seminar "C*-algebras" :
    The Property of Akemann and Ostrand for free quantum groups.
  • 25 october 2003, Münster, post-docs workshop :
    KK-Theory for Quantum Groups: functorial and geometrical methods. Slides :  pdf, dvi. Notes :  pdf.

See the french version of this page.

A generator of matrices diagonalizable in M_n(Z).