Invariants of the half-liberated orthogonal group
Annales de l'Institut Fourier, Volume 60 (2010) no. 6, pp. 2137-2164.

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.

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.

DOI: 10.5802/aif.2579
Classification: 20G42, 16W30, 46L65
Keywords: Quantum group, maximal torus, root system
Mot clés : groupe quantique, tore maximal, système de racines
Banica, Teodor 1; Vergnioux, Roland 2

1 Université de Toulouse 3 Département de Mathématiques 118, route de Narbonne 31062 Toulouse (France)
2 Université de Caen Département de Mathématiques BP 5186 14032 Caen Cedex (France)
@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{\textquoteright}institut Fourier},
     volume = {60},
     number = {6},
     year = {2010},
     doi = {10.5802/aif.2579},
     mrnumber = {2791653},
     zbl = {1277.46040},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2579/}
}
TY  - JOUR
AU  - Banica, Teodor
AU  - Vergnioux, Roland
TI  - Invariants of the half-liberated orthogonal group
JO  - Annales de l'Institut Fourier
PY  - 2010
SP  - 2137
EP  - 2164
VL  - 60
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.2579/
DO  - 10.5802/aif.2579
LA  - en
ID  - AIF_2010__60_6_2137_0
ER  - 
%0 Journal Article
%A Banica, Teodor
%A Vergnioux, Roland
%T Invariants of the half-liberated orthogonal group
%J Annales de l'Institut Fourier
%D 2010
%P 2137-2164
%V 60
%N 6
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.2579/
%R 10.5802/aif.2579
%G en
%F AIF_2010__60_6_2137_0
Banica, Teodor; Vergnioux, Roland. Invariants of the half-liberated orthogonal group. Annales de l'Institut Fourier, Volume 60 (2010) no. 6, pp. 2137-2164. doi : 10.5802/aif.2579. https://aif.centre-mersenne.org/articles/10.5802/aif.2579/

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

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

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

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

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

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

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

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

[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 | DOI | MR | Zbl

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

[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 | DOI | MR | Zbl

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

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

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

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

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

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

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

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

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

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

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

Cited by Sources: