Quantum groups under very strong axioms

• Autorzy: Banica Teo

Identyfikatory

DOI

10.4064/ba190128-8-3

• Języki publikacji: EN

Abstrakty

EN

We study the intermediate quantum groups H_N ⸦ G ⸦ U⁺_N. The basic examples are H_N, K_N, O_N, U_N, H⁺_N, K⁺_N, O⁺_N, U⁺_N, which form a cube. Any other example G sits inside the cube, and by using standard operations, namely intersection ∩ and generation <, >, can be projected on the faces and edges. We prove that under the strongest possible axioms, namely (1) easiness, (2) uniformity, and (3) geometric coherence of the various projection operations, the eight basic solutions are the only ones.

Słowa kluczowe

EN

quantum isometry; quantum reflection

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2019
• Tom: Vol. 67, no. 1
• Strony: 83--99
• Opis fizyczny: Bibliogr. 30 poz., rys.

Twórcy

autor

Banica Teo

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

Bibliografia

• [1] T. Banica, Liberation theory for noncommutative homogeneous spaces, Ann. Fac. Sci. Toulouse Math. 26 (2017), 127-156.
• [2] T. Banica, Unitary easy quantum groups: geometric aspects, J. Geom. Phys. 126 (2018), 127-147.
• [3] T. Banica, Quantum groups, from a functional analysis perspective, Adv. Oper. Theory 4 (2019), 164-196.
• [4] T. Banica, Homogeneous quantum groups and their easiness level, Kyoto J. Math., to appear.
• [5] T. Banica, S. T. Belinschi, M. Capitaine and B. Collins, Free Bessel laws, Canad. J. Math. 63 (2011), 3-37.
• [6] T. Banica and J. Bichon, Complex analogues of the half-classical geometry, Münster J. Math. 10 (2017), 457-483.
• [7] T. Banica, J. Bichon and B. Collins, The hyperoctahedral quantum group, J. Ramanujan Math. Soc. 22 (2007), 345-384.
• [8] T. Banica, J. Bichon, B. Collins and S. Curran, A maximality result for orthogonal quantum groups, Comm. Algebra 41 (2013), 656-665.
• [9] T. Banica and A. Skalski, Two-parameter families of quantum symmetry groups, J. Funct. Anal. 260 (2011), 3252-3282.
• [10] T. Banica, A. Skalski and P. M. Sołtan, Noncommutative homogeneous spaces: the matrix case, J. Geom. Phys. 62 (2012), 1451-1466.
• [11] T. Banica and R. Speicher, Liberation of orthogonal Lie groups, Adv. Math. 222 (2009), 1461-1501.
• [12] T. Banica and R. Vergnioux, Invariants of the half-liberated orthogonal group, Ann. Inst. Fourier (Grenoble) 60 (2010), 2137-2164.
• [13] H. Bercovici and V. Pata, Stable laws and domains of attraction in free probabilisty theory, Ann. of Math. 149 (1999), 1023-1060.
• [14] J. Bichon, Free wreath product by the quantum permutation group, Algebras Represent. Theory 7 (2004), 343-362.
• [15] M. Brannan, A. Chirvasitu and A. Freslon, Topological generation and matrix models for quantum reflection groups, preprint, 2018.
• [16] R. Brauer, On algebras which are connected with the semisimple continuous groups, Ann. of Math. 38 (1937), 857-872.
• [17] 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.
• [18] V. G. Drinfeld, Quantum groups, in: Proc. ICM (Berkeley, 1986), 798-820.
• [19] A. Freslon, On the partition approach to Schur-Weyl duality and free quantum groups, Transform. Groups 22 (2017), 707-751.
• [20] A. Freslon, On two-coloured noncrossing quantum groups, preprint, 2017.
• [21] M. Jimbo, A q-difference analog of U(g) and the Yang-Baxter equation, Lett. Math. Phys. 10 (1985), 63-69.
• [22] S. Malacarne, Woronowicz’s Tannaka-Krein duality and free orthogonal quantum groups, Math. Scand. 122 (2018), 151-160.
• [23] A. Mang and M. Weber, Categories of two-colored pair partitions, part I: Categories indexed by cyclic groups, preprint, 2018.
• [24] A. Mang and M. Weber, Categories of two-colored pair partitions, part II: Categories indexed by semigroups, preprint, 2019.
• [25] S. Raum and M. Weber, The full classification of orthogonal easy quantum groups, Comm. Math. Phys. 341 (2016), 751-779.
• [26] P. Tarrago and M. Weber, Unitary easy quantum groups: the free case and the group case, Int. Math. Res. Notices 2017, 5710-5750.
• [27] S. Wang, Free products of compact quantum groups, Comm. Math. Phys. 167 (1995), 671-692.
• [28] S. Wang, Quantum symmetry groups of finite spaces, Comm. Math. Phys. 195 (1998), 195-211.
• [29] S. L. Woronowicz, Compact matrix pseudogroups, Comm. Math. Phys. 111 (1987), 613-665.
• [30] 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 (2019).