Quantum groups: Galois and integration techniques

Project ANR BLAN07-3_183390


Description of the project. The aim of the project is to contribute to the classification of quantum groups by studying two subproblems arising in representation theory: the classification of Hopf-Galois extensions of a Hopf algebra and the search of explicit formulae for the Haar measure. The project associates mathematicians having different backgroud and expertise: algebra, operator algebras, free probability.

Members of the project.
Conference ''Quantum Groups'' at Clermont-Ferrand : august 30 - september 3, 2010.
Past events.
14-16 may 2009 : ``Quantum Groups Workshop'' at the Mathematics Institute of Toulouse.
14-16 february 2008:
''Quantum groups days'' at Strasbourg, Institut de Recherche Mathématique Avancée.

  • E. Aljadeff and C. Kassel, Polynomial identities and noncommutative versal torsors, Adv. Math. 218 (2008), 1453-1495.
  • N. Andruskiewitsch and J. Bichon and , Examples of inner linear Hopf algebras, Rev. Un. Mat. Argentina 51 (2010), 7-18.
  • T. Banica, A note on free quantum groups, Ann. Math. Blaise Pascal 15 (2008), 135-146.
  • T. Banica, Cyclotomic expansion of exceptional spectral measures, Internat. J. Math. 20 (2009), 275-297.
  • T. Banica, The orthogonal Weingarten formula in compact form, Lett. Math. Phys. 91 (2010), 105-118.
  • T. Banica, S.T. Belinschi, M. Capitaine and B. Collins, Free Bessel laws, Canad. J. Math 63 (2011), 3-37.
  • T. Banica and J. Bichon, Quantum groups acting on 4 points, J. Reine Angew. Math. 626 (2009), 74-114.
  • T. Banica and J. Bichon, Hopf images and inner faithful representations, Glasg. Math. J. 52 (2010), 677-703.
  • T. Banica, J. Bichon and B. Collins, The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22 (2007), 345-384.
  • T. Banica, J. Bichon , B. Collins and S. Curran, A maximality result for orthogonal quantum groups, arxiv:1106.5467.
  • T. Banica, J. Bichon and S. Curran, Quantum automorphisms of twisted group algebras and free hypergeometric laws, Proc. Amer. Math. Soc., to appear.
  • T. Banica, J. Bichon and S. Natale, Finite quantum groups and quantum permutation groups, arxiv:1104.1400.
  • T. Banica, J. Bichon and J.-M. Schlenker, Representations of quantum permutation algebras, J. Funct. Anal. 257 (2009), 2864-2910.
  • T. Banica, B. Collins and J.-M. Schlenker, On orthogonal matrices maximizing the 1-norm, Indiana Univ. Math. J. 59 (2010), 839-856.
  • T. Banica, B. Collins, and J.-M. Schlenker, On polynomial integrals over the orthogonal group, J. Combin. Theory Ser. A 118 (2011), 778-795.
  • T. Banica, B. Collins and P. Zinn-Justin, Spectral analysis of the free orthogonal matrix, Int. Math. Res. Not. 17 (2009), 3286-3309.
  • T. Banica and S. Curran, Decomposition results for Gram matrix determinants, J. Math. Phys. 51 (2010), 1-14
  • T. Banica, S. Curran and R. Speicher, Classification results for easy quantum groups, Pacific J. Math. 247 (2010), 1-26.
  • T. Banica, S. Curran and R. Speicher, De Finetti theorems for easy quantum groups, Ann. Probab. , to appear, arxiv:0907.3314.
  • T. Banica, S. Curran and R. Speicher, Stochastic aspects of easy quantum groups, Probab. Theory Related Fields 149 (2011), 435-462.
  • T. Banica and D. Goswami, Quantum isometries and noncommutative spheres, Comm. Math. Phys 298 (2010), 343-356.
  • T. Banica and I. Nechita, Asymptotic eigenvalue distributions of block-transposed Wishart matrices, arxiv:1105.2556.
  • T. Banica and J.-M. Schlenker, Combinatorial aspects of orthogonal group integrals, Internat. J. Math., to appear.
  • T. Banica and A. Skalski, Two-parameter families of quantum symmetry groups, J. Funct. Anal. 260 (2011), 3252-3282.
  • T. Banica and A. Skalski, Quantum isometry groups of duals of free powers of cyclic groups, Int. Math. Res. Not., to appear.
  • T. Banica and R. Speicher, Liberation of orthogonal Lie groups, Adv. Math 222 (2009), 1461-1501.
  • T. Banica and R. Vergnioux, Growth estimates for discrete quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 12 (2009), 321-340.
  • T. Banica and R. Vergnioux, Fusion rules for quantum reflection groups, J. Noncommut. Geom. 3 (2009), 327-359.
  • T. Banica and R. Vergnioux, Invariants of the half-liberated orthogonal group, Ann. Inst. Fourier 60 (2010), 2137-2164.
  • J. Bichon, Algebraic quantum permutation groups, Asian-Eur. J. Math. 1 (2008), 1-13.
  • J. Bichon, Hopf-Galois objects and cogroupoids, Pub. Mat. Uruguay, to appear.
  • J. Bichon and C. Kassel, The lazy homology of a Hopf algebra, J. Algebra 323(2010), 2556-2590.
  • J. Bichon and S. Natale, Hopf algebra deformations of binary polyhedral groups, Transform. Groups 16 (2011), 339-374.
  • B. Collins, J. Härtel and A. Thom, Homology of free quantum groups, C. R. Acad. Sci. Paris, Ser. I 347 (2009), 271-276.
  • C. Kassel, Hopf algebras and polynomial identities , Proc. Conf. "Quantum Groups and Quantum Topology", RIMS Kokyuroku 1714, 2010, 49-62
  • C. Kassel, Generic Hopf Galois extensions, Proc. of the Workshop on Quantum Groups and Noncommutative Geometry, M. Marcolli and D. Parashar (eds.), Max Planck Institut fur Mathematik, Bonn 2007, Vieweg Verlag (Max-Planck Series), vol. E41, 2011, 104-120.
  • P. Guillot and C. Kassel, Cohomology of invariant Drinfeld twists on group algebras, Int. Math. Res. Not. 2010 (2010), 1894-1939.
  • P. Guillot, C. Kassel and A. Masuoka, Twisting algebras using non-commutative torsors: explicit computations, Math. Z., to appear.
  • C. Kassel and A. Masuoka, Flatness and freeness properties of the generic Hopf Galois extensions , Rev. Un. Mat. Argentina 51(2010), 79-94.
  • S. Vaes and N. Vander Vennet, Poisson boundary of the discrete quantum group Au(F)^, Compositio Math. 146 (2010), 1073-105.