Quantum isometries and group dual subgroups

Banica, Teodor; Bhowmick, Jyotishman; De Commer, Kenny

Quantum isometries and group dual subgroups
[Isométries quantiques et duaux de groupes]

Banica, Teodor ; Bhowmick, Jyotishman ; De Commer, Kenny

Annales mathématiques Blaise Pascal, Tome 19 (2012), p. 1-27 / Harvested from Numdam

Access to full text
Full (PDF)
Full (DJVU)

Résumé
Abstract

On étudie les groupes discrets $Λ$ dont les duaux se plongent dans un groupe quantique compact donné, $\hat{Λ} \subset G$ . Dans le cas matriciel $G \subset U_{n}^{+}$ la condition de plongement est équivalente à l’existence d’une application quotient $Γ_{U} \to Λ$ , où $F = {Γ_{U} ∣ U \in U_{n}}$ est une certaine famille de groupes associés à $G$ . On dévéloppe ici un nombre de techniques pour le calcul de $F$ , en partie inspirées pas la classification de Bichon des sous-groupes $\hat{Λ} \subset S_{n}^{+}$ . Ces résultats sont motivés pas la notion de groupe quantique d’isométrie de Goswami, car une variété Riemannienne compacte et connexe ne peut pas avoir des isométries quantiques venant du dual d’un groupe non-abélien.

We study the discrete groups $Λ$ whose duals embed into a given compact quantum group, $\hat{Λ} \subset G$ . In the matrix case $G \subset U_{n}^{+}$ the embedding condition is equivalent to having a quotient map $Γ_{U} \to Λ$ , where $F = {Γ_{U} ∣ U \in U_{n}}$ is a certain family of groups associated to $G$ . We develop here a number of techniques for computing $F$ , partly inspired from Bichon’s classification of group dual subgroups $\hat{Λ} \subset S_{n}^{+}$ . These results are motivated by Goswami’s notion of quantum isometry group, because a compact connected Riemannian manifold cannot have non-abelian group dual isometries.

Publié le : 2012-01-01

MR 2978312 | Zbl 1250.81057

DOI : https://doi.org/10.5802/ambp.303
Classification: 58J42

@article{AMBP_2012__19_1_1_0,
     author = {Banica, Teodor and Bhowmick, Jyotishman and De Commer, Kenny},
     title = {Quantum isometries and group dual subgroups},
     journal = {Annales math\'ematiques Blaise Pascal},
     volume = {19},
     year = {2012},
     pages = {1-27},
     doi = {10.5802/ambp.303},
     zbl = {1250.81057},
     mrnumber = {2978312},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AMBP_2012__19_1_1_0}
}

Banica, Teodor; Bhowmick, Jyotishman; De Commer, Kenny. Quantum isometries and group dual subgroups. Annales mathématiques Blaise Pascal, Tome 19 (2012) pp. 1-27. doi : 10.5802/ambp.303. http://gdmltest.u-ga.fr/item/AMBP_2012__19_1_1_0/

Bibliographie

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

[2] Banica, T.; Bichon, J. Quantum automorphism groups of vertex-transitive graphs of order $\leq 11$ , J. Algebraic Combin., Tome 26 (2007), pp. 83-105 | Article | MR 2335703

[3] Banica, T.; Goswami, D. Quantum isometries and noncommutative spheres, Comm. Math. Phys., Tome 298 (2010), pp. 343-356 | Article | MR 2669439

[4] Banica, T.; Skalski, A. Quantum isometry groups of duals of free powers of cyclic groups (to appear)

[5] Banica, T.; Skalski, A. Quantum symmetry groups of C $^{*}$ -algebras equipped with orthogonal filtrations (to appear)

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

[7] Banica, T.; Vergnioux, R. Invariants of the half-liberated orthogonal group, Ann. Inst. Fourier, Tome 60 (2010), pp. 2137-2164 | Article | Numdam | MR 2791653

[8] Bhowmick, J. Quantum isometry group of the $n$ -tori, Proc. Amer. Math. Soc., Tome 137 (2009), pp. 3155-3161 | Article | MR 2506475

[9] Bhowmick, J.; Goswami, D. Quantum group of orientation preserving Riemannian isometries, J. Funct. Anal., Tome 257 (2009), pp. 2530-2572 | Article | MR 2555012

[10] Bhowmick, J.; Goswami, D. Quantum isometry groups: examples and computations, Comm. Math. Phys., Tome 285 (2009), pp. 421-444 | Article | MR 2461983

[11] Bhowmick, J.; Goswami, D. Quantum isometry groups of the Podlés spheres, J. Funct. Anal., Tome 258 (2010), pp. 2937-2960 | Article | MR 2595730

[12] Bhowmick, J.; Skalski, A. Quantum isometry groups of noncommutative manifolds associated to group $C^{*}$ -algebras, J. Geom. Phys., Tome 60 (2010), pp. 1474-1489 | Article | MR 2661151

[13] Bichon, J. Free wreath product by the quantum permutation group, Alg. Rep. Theory, Tome 7 (2004), pp. 343-362 | Article | MR 2096666

[14] Bichon, J. Algebraic quantum permutation groups, Asian-Eur. J. Math., Tome 1 (2008), pp. 1-13 | Article | MR 2400296

[15] Boca, F. Ergodic actions of compact matrix pseudogroups on C $^{*}$ -algebras, Astérisque, Tome 232 (1995), pp. 93-109 | MR 1372527 | Zbl 0842.46039

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

[17] Commer, K. De On projective representations for compact quantum groups, J. Funct. Anal., Tome 260 (2011), pp. 3596-3644 | Article | MR 2781971

[18] Connes, Alain Noncommutative geometry, Academic Press Inc., San Diego, CA (1994) | MR 1303779 | Zbl 0818.46076

[19] Connes, Alain; Marcolli, Matilde Noncommutative geometry, quantum fields and motives, American Mathematical Society, Providence, RI, American Mathematical Society Colloquium Publications, Tome 55 (2008) | MR 2371808

[20] Drinfeld, V. Quantum groups, Proc. ICM Berkeley (1986), pp. 798-820 | MR 934283 | Zbl 0667.16003

[21] Franz, U.; Skalski, A. On idempotent states on quantum groups, J. Algebra, Tome 322 (2009), pp. 1774-1802 | Article | MR 2543634

[22] Goswami, D. Quantum symmetries and quantum isometries of compact metric spaces (arxiv:0811.0095)

[23] Goswami, D. Rigidity of action of compact quantum groups (arxiv:1106.5107)

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

[25] Hajac, P.; Masuda, T. Quantum double-torus, C. R. Acad. Sci. Paris Ser. I Math, Tome 327 (1998), pp. 553-558 | Article | MR 1650603 | Zbl 0973.17013

[26] Huang, H. Faithful compact quantum group actions on connected compact metrizable spaces (arxiv:1202.1175)

[27] J. Bhowmick, B. Das F. D’Andrea; Dabrowski, L. Quantum gauge symmetries in noncommutative geometry (arxiv:1112.3622)

[28] J. Bhowmick, D. Goswami; Skalski, A. Quantum isometry groups of 0-dimensional manifolds, Trans. Amer. Math. Soc., Tome 363 (2011), pp. 901-921 | Article | MR 2728589

[29] J. Bhowmick, F. D’Andrea; Dabrowski, L. Quantum isometries of the finite noncommutative geometry of the standard model, Comm. Math. Phys., Tome 307 (2011), pp. 101-131 | Article | MR 2835874

[30] Jimbo, M. A $q$ -difference analog of $U (g)$ and the Yang-Baxter equation, Lett. Math. Phys., Tome 10 (1985), pp. 63-69 | Article | MR 797001 | Zbl 0587.17004

[31] Jones, V.F.R. Planar algebras I (arxiv:math/9909027)

[32] Kac, G.I.; Paljutkin, V.G. Finite ring groups, Trans. Moscow Math. Soc., Tome 1 (1966), pp. 251-294 | MR 208401 | Zbl 0218.43005

[33] Kawada, Y.; Itô, K. On the probability distribution on a compact group I, Proc. Phys.-Math. Soc. Japan, Tome 22 (1940), pp. 977-998 | MR 3462

[34] Liszka-Dalecki, J.; Tan, P.M. Soł Quantum isometry groups of symmetric groups (to appear)

[35] Pal, A.K. A counterexample on idempotent states on compact quantum groups, Lett. Math. Phys., Tome 37 (1996), pp. 75-77 | Article | MR 1392148 | Zbl 0849.17010

[36] Quaegebeur, J.; Sabbe, M. Isometric coactions of compact quantum groups on compact quantum metric spaces (arxiv:1007.0363)

[37] Raum, S. Isomorphisms and fusion rules of orthogonal free quantum groups and their complexifications (to appear)

[38] Sekine, Y. An example of finite-dimensional Kac algebras of Kac-Paljutkin type, Proc. Amer. Math. Soc., Tome 124 (1996), pp. 1139-1147 | Article | MR 1307564 | Zbl 0845.46031

[39] Speicher, R. Multiplicative functions on the lattice of noncrossing partitions and free convolution, Math. Ann., Tome 298 (1994), pp. 611-628 | Article | MR 1268597 | Zbl 0791.06010

[40] T. Banica, A. Skalski; Tan, P.M. Soł Noncommutative homogeneous spaces: the matrix case, J. Geom. Phys., Tome 62 (2012), pp. 1451-1466 | Article

[41] T. Banica, S. Curran; Speicher, R. Classification results for easy quantum groups, Pacific J. Math., Tome 247 (2010), pp. 1-26 | Article | MR 2718205

[42] Voiculescu, D. V.; Dykema, K. J.; Nica, A. Free random variables, American Mathematical Society, Providence, RI, CRM Monograph Series, Tome 1 (1992) (A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups) | MR 1217253 | Zbl 0795.46049

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

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

[45] Wang, S. Simple compact quantum groups, I, J. Funct. Anal., Tome 256 (2009), pp. 3313-3341 | Article | MR 2504527

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

[47] Woronowicz, S.L. Tannaka-Krein duality for compact matrix pseudogroups. Twisted $S U (N)$ groups, Invent. Math., Tome 93 (1988), pp. 35-76 | Article | MR 943923 | Zbl 0664.58044

[48] Woronowicz, S.L. Compact quantum groups, “Symétries quantiques”, North-Holland (1998), pp. 845-884 | MR 1616348 | Zbl 0997.46045

[49] Yau, S.-T. Open problems in geometry, Proc. Symp. Pure Math., Tome 54 (1993), pp. 1-28 | Article | MR 1216573 | Zbl 0801.53001