Super-easy quantum groups : definition and examples

• Autorzy: Banica T.

Identyfikatory

DOI

10.4064/ba8133-2-2018

• Języki publikacji: EN

Abstrakty

EN

We investigate the “two-parameter” quantum symmetry groups that we previously constructed with Skalski, with the conclusion that some of these quantum groups, namely those without singletons, are “super-easy” in a suitable sense, which we axiomatize here. Our formalism also covers the symplectic group Sp_n and its free version Sp⁺_n, and some other interesting examples. Finally, we address the general problem of classifying the super-easy quantum groups, and we make a few comments on it.

Słowa kluczowe

EN

quantum group; easy quantum group; super-easiness

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2018
• Tom: Vol. 66, no. 1
• Strony: 57--68
• Opis fizyczny: Bibliogr. 22 poz., rys.

Twórcy

autor

Banica T.

• Department of Mathematics, University of Cergy-Pontoise, F-95000 Cergy-Pontoise, France

Bibliografia

• [1] T. Banica, A duality principle for noncommutative cubes and spheres, J. Noncommut. Geom. 10 (2016), 1043-1081.
• [2] T. Banica, J. Bichon and B. Collins, The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22 (2007), 345-384.
• [3] T. Banica, B. Collins and P. Zinn-Justin, Spectral analysis of the free orthogonal matrix, Int. Math. Res. Notices 2009, 3286-3309.
• [4] T. Banica and S. Curran, Decomposition results for Gram matrix determinants, J. Math. Phys. 51 (2010), 1-14.
• [5] T. Banica and A. Skalski, Two-parameter families of quantum symmetry groups, J. Funct. Anal. 260 (2011), 3252-3282.
• [6] T. Banica and A. Skalski, Quantum isometry groups of duals of free powers of cyclic groups, Int. Math. Res. Notices 2012, 2094-2122.
• [7] T. Banica and R. Speicher, Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), 1461-1501.
• [8] J. Bichon, A. De Rijdt and S. Vaes, Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys. 262 (2006), 703-728.
• [9] M. Brannan and B. Collins, Highly entangled, non-random subspaces of tensor products from quantum groups, arXiv:1612.09598 (2016).
• [10] M. Brannan and K. Kirkpatrick, Quantum groups and generalized circular elements, Pacific J. Math. 282 (2016), 35-61.
• [11] R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. 38 (1937), 857-872.
• [12] G. Cébron and M. Weber, Quantum groups based on spatial partitions, arXiv:1609.02321 (2016).
• [13] B. Collins and P. Śniady, Integration with respect to the Haar measure on unitary, orthogonal and symplectic groups, Comm. Math. Phys. 264 (2006), 773-795.
• [14] A. Freslon, On the partition approach to Schur-Weyl duality and free quantum groups, Transform. Groups 22 (2017), 707-751.
• [15] S. Neshveyev and L. Tuset, Compact Quantum Groups and Their Representation Categories, Soc. Math. France, 2013.
• [16] S. Raum and M. Weber, The full classification of orthogonal easy quantum groups, Comm. Math. Phys. 341 (2016), 751-779.
• [17] R. Speicher and M. Weber, Quantum groups with partial commutation relations, arXiv:1603.09192 (2016).
• [18] P. Tarrago and M. Weber, Unitary easy quantum groups: the free case and the group case, Int. Math. Res. Notices 2017, 5710-5750.
• [19] A. Van Daele and S. Wang, Universal quantum groups, Int. J. Math. 7 (1996), 255-263.
• [20] S. Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), 671-692.
• [21] S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665.
• [22] S. L. Woronowicz, Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math. 93 (1988), 35-76.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2018).