Invariants of the half-liberated orthogonal group  [ Invariants du groupe orthogonal semi-libéré ]
Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 2137-2164.

Le groupe orthogonal semi-libéré O n * est un groupe quantique intermédiaire entre le groupe orthogonal O n et sa version libre O n + . Nous discutons ici ses propriétés algébriques de base, et nous classifions ses représentations irréductibles. Cette classification est établie grâce à une mise en relation avec le groupe U n et des méthodes inspirées de la théorie des algèbres de Lie. Un groupe discret non abélien joue le rôle de réseau des poids. Nous utilisons ces résultats pour montrer que le groupe quantique discret dual est à croissance polynomiale.

The half-liberated orthogonal group O n * appears as intermediate quantum group between the orthogonal group O n , and its free version O n + . We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by using a certain twisting-type relation between O n * and U n , a non abelian discrete group playing the role of weight lattice, and a number of methods inspired from the theory of Lie algebras. We use these results for showing that the dual discrete quantum group has polynomial growth.

DOI : https://doi.org/10.5802/aif.2579
Classification : 20G42,  16W30,  46L65
Mots clés : groupe quantique, tore maximal, système de racines
@article{AIF_2010__60_6_2137_0,
     author = {Banica, Teodor and Vergnioux, Roland},
     title = {Invariants of the half-liberated orthogonal group},
     journal = {Annales de l'Institut Fourier},
     pages = {2137--2164},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {6},
     year = {2010},
     doi = {10.5802/aif.2579},
     mrnumber = {2791653},
     zbl = {1277.46040},
     language = {en},
     url = {www.numdam.org/item/AIF_2010__60_6_2137_0/}
}
Banica, Teodor; Vergnioux, Roland. Invariants of the half-liberated orthogonal group. Annales de l'Institut Fourier, Tome 60 (2010) no. 6, pp. 2137-2164. doi : 10.5802/aif.2579. http://www.numdam.org/item/AIF_2010__60_6_2137_0/

[1] Banica, T. Le groupe quantique compact libre U(n), Comm. Math. Phys., Volume 190 (1997), pp. 143-172 | Article | MR 1484551 | Zbl 0906.17009

[2] Banica, T. Symmetries of a generic coaction, Math. Ann., Volume 314 (1999), pp. 763-780 | Article | MR 1709109 | Zbl 0928.46038

[3] Banica, T.; Bichon, J. Quantum groups acting on 4 points, J. Reine Angew. Math., Volume 626 (2009), pp. 74-114 | MR 2492990 | Zbl 1187.46058

[4] Banica, T.; Bichon, J.; Collins, B. The hyperoctahedral quantum group, J. Ramanujan Math. Soc., Volume 22 (2007), pp. 345-384 | MR 2376808 | Zbl 1185.46046

[5] Banica, T.; Collins, B. Integration over compact quantum groups, Publ. Res. Inst. Math. Sci., Volume 43 (2007), pp. 277-302 | Article | MR 2341011 | Zbl 1129.46058

[6] Banica, T.; Speicher, R. Liberation of orthogonal Lie groups, Adv. Math., Volume 222 (2009), pp. 1461-1501 | Article | MR 2554941 | Zbl pre05614878

[7] Banica, T.; Vergnioux, R. Fusion rules for quantum reflection groups, J. Noncommut. Geom., Volume 3 (2009), pp. 327-359 | Article | MR 2511633 | Zbl pre05578949

[8] Banica, T.; Vergnioux, R. Growth estimates for discrete quantum groups, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Volume 12 (2009), pp. 321-340 | Article | MR 2541400 | Zbl 1189.46059

[9] Bhowmick, J.; Goswami, D.; Skalski, A. Quantum isometry groups of 0-dimensional manifolds (arxiv:0807.4288)

[10] Bichon, J.; De Rijdt, A.; Vaes, S. Ergodic coactions with large multiplicity and monoidal equivalence of quantum groups, Comm. Math. Phys., Volume 262 (2006), pp. 703-728 | Article | MR 2202309 | Zbl 1122.46046

[11] Brauer, R. On algebras which are connected with the semisimple continuous groups, Ann. of Math., Volume 38 (1937), pp. 857-872 | Article | JFM 63.0873.02 | MR 1503378 | Zbl 0017.39105

[12] Collins, B.; Śniady, P. Integration with respect to the Haar measure on the unitary, orthogonal and symplectic group, Comm. Math. Phys., Volume 264 (2006), pp. 773-795 | Article | MR 2217291 | Zbl 1108.60004

[13] Drinfeld, V. G. Quantum groups, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) (1987), pp. 798-820 | MR 934283

[14] Goswami, D. Quantum group of isometries in classical and noncommutative geometry, Comm. Math. Phys., Volume 285 (2009), pp. 141-160 | Article | MR 2453592 | Zbl 1228.81188

[15] Köstler, C.; Speicher, R. A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation, Comm. Math. Phys., Volume 291 (2009), pp. 473-490 | Article | MR 2530168 | Zbl 1183.81099

[16] Pinzari, C.; Roberts, J. Ergodic actions of compact quantum groups from solutions of the conjugate equations (arxiv:0808.3326)

[17] Vaes, S.; Vergnioux, R. The boundary of universal discrete quantum groups, exactness and factoriality, Duke Math. J., Volume 140 (2007), pp. 35-84 | Article | MR 2355067 | Zbl 1129.46062

[18] Vergnioux, R. Orientation of quantum Cayley trees and applications, J. Reine Angew. Math., Volume 580 (2005), pp. 101-138 | Article | MR 2130588 | Zbl 1079.46048

[19] Vergnioux, R. The property of rapid decay for discrete quantum groups, J. Operator Theory, Volume 57 (2007), pp. 303-324 | MR 2329000 | Zbl 1120.58004

[20] Wang, S. Free products of compact quantum groups, Comm. Math. Phys., Volume 167 (1995), pp. 671-692 | Article | MR 1316765 | Zbl 0838.46057

[21] Wang, S. Quantum symmetry groups of finite spaces, Comm. Math. Phys., Volume 195 (1998), pp. 195-211 | Article | MR 1637425 | Zbl 1013.17008

[22] Wenzl, H. On the structure of Brauer’s centralizer algebras, Ann. of Math., Volume 128 (1988), pp. 173-193 | Article | MR 951511 | Zbl 0656.20040

[23] Woronowicz, S. L. Compact matrix pseudogroups, Comm. Math. Phys., Volume 111 (1987), pp. 613-665 | Article | MR 901157 | Zbl 0627.58034

[24] Woronowicz, S. L. Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math., Volume 93 (1988), pp. 35-76 | Article | EuDML 143589 | MR 943923 | Zbl 0664.58044

[25] Woronowicz, S. L. Differential calculus on compact matrix pseudogroups (quantum groups), Comm. Math. Phys., Volume 122 (1989), pp. 125-170 | Article | MR 994499 | Zbl 0751.58042