Laboratoire de Mathématiques de Besançon - UMR 6623 CNRS
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Professional webpage of François Lemeux

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François Lemeux

I completed my PhD in mathematics at the Université de Franche comté, in the Laboratoire de mathématiques de Besançon. My advisors are Uwe Franz and Roland Vergnioux. Here is a CV.

You can contact me at :

  • Laboratoire de mathématiques, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France.
  • Email : francois.lemeux"AT"univ-fcomte.fr
  • Phone : + 33 3 81 66 63 17

Defense

I publicly defended my PhD thesis on the 28th of May 2014 at the Laboratoire de Mathématiques de Besançon. Up to date version of my PhD thesis entitled "On the theory of representations and the operator algebras of free wreath products". Slides of the defense.

Interests

I am interested in analytical, algebraic and combinatorial aspects of quantum groups. I study for instance algebraic operator properties of the C*/von Neumann algebras associated to (locally) compact quantum groups (factoriality, approximation properties...). I also investigate the description of the representation categories of certain compact quantum groups and try to provide a combinatorial approach of them.

Articles

In this paper, we find the fusion rules for the free wreath product quantum groups of CMQG of Kac type by the quantum permutation group S_N^+, N>3. This is based on a combinatorial description of the intertwiner spaces between certain generating representations in this free wreath product. The combinatorial properties of the associated intertwiner spaces then allows us to obtain several probabilistic applications. We then prove the monoidal equivalence between this free wreath product and a compact quantum group whose dual is a discrete quantum subgroup of the free product of the CMQG by SU_q(2), for some q in ]0,1[. We obtain as a corollary certain stability results for the operator algebras associated with the free wreath products of quantum groups such as Haagerup property, weak amenability and exactness.

In this paper, we describe the fusion rules of the free wreath product of any classical discrete group by the quantum permutation group. This generalizes a result by Banica and Vergnioux for the quantum reflection groups which is the free wreath product of the cyclic group by the quantum permutation group. We give several applications of this result. We first prove, in most cases, the simplicity of the associated reduced C*-algebra and the uniqueness of the trace. We also prove that the associated von Neumann algebra is, in most cases, a type a full II_1 factor. We conclude by proving that the dual of the free wreath product of any finite group by the quantum permutation group has the Haagerup property.

In this paper we prove that the duals of the quantum reflection groups have the Haagerup property. We use the canonical arrow onto the quantum permutation groups, and we describe how the characters of the quantum reflection groups behave with respect to this canonical morphism thanks to the description of the fusion rules binding their irreducible corepresentations. This allows us to construct states on the central algebra of the quantum reflection group and to use a fundamental theorem proved by M.Brannan giving a method to construct nets of trace-preserving, normal, unital and completely positive maps on the von Neumann algebra of a compact quantum group of Kac type.

Talks

Some academic works

Ce travail de mémoire revient sur la définition des fonction définies positives et conditionnellement de type négatif sur les groupes discrets. On donne une preuve, en particulier, du théorème de Schoenberg. On voit alors une construction qui permet d’obtenir des groupes vérifiant la propriété de Haagerup : le produit en couronnes. Cette construction permet, en particulier, de construire des exemples afin de démontrer que la propriété de Haagerup n’est pas équivalente à la propriété de moyennabilité faible.

On veut déterminer les représentations unitaires irréductibles du groupe "ax + b". Pour cela, on étudie plus généralement ces représentations sur des produits semi-directs. On étudie en outre la notion de représentation induite, et le théorème d’imprimitivité.

Attended conferences, workshops and summer schools