Liberation theory for noncommutative homogeneous spaces

Teodor Banica

doi:10.5802/afst.1527

Liberation theory for noncommutative homogeneous spaces
[Théorie de liberation pour les espaces homogènes non commutatifs]

Teodor Banica ¹

¹ Département des Mathématiques, Cergy-Pontoise University, 95000 Cergy-Pontoise, France

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 1, pp. 127-156.

Résumé
Abstract

On étudie le problème de liberation, dans le cadre des espaces homogènes. Notre première série de résultats concerne l’axiomatisation et la classification des familles de groupes quantiques compacts $G = (G_{N})$ qui sont « uniformes », dans un sens convenable. On étudie ensuite les espaces quotient du type $X = (G_{M} \times G_{N}) / (G_{L} \times G_{M - L} \times G_{N - L})$ , et l’opération de liberation pour ces espaces, avec des résultats de nature algébrique et probabiliste.

We discuss the liberation question, in the homogeneous space setting. Our first series of results concerns the axiomatization and classification of the families of compact quantum groups $G = (G_{N})$ which are “uniform”, in a suitable sense. We study then the quotient spaces of type $X = (G_{M} \times G_{N}) / (G_{L} \times G_{M - L} \times G_{N - L})$ , and the liberation operation for them, with a number of algebraic and probabilistic results.

Reçu le : 2015-12-13
Accepté le : 2016-03-06
Publié le : 2017-02-06

DOI : 10.5802/afst.1527

Classification : 46L65, 46L54
Mots clés : Liberation theory, Homogeneous space

Affiliations des auteurs :

Teodor Banica ¹

¹ Département des Mathématiques, Cergy-Pontoise University, 95000 Cergy-Pontoise, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Liberation theory for noncommutative homogeneous spaces},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {127--156},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 26},
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     doi = {10.5802/afst.1527},
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Teodor Banica. Liberation theory for noncommutative homogeneous spaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 1, pp. 127-156. doi : 10.5802/afst.1527. https://afst.centre-mersenne.org/articles/10.5802/afst.1527/

Bibliographie
Cité par

[1] Teodor Banica Liberations and twists of real and complex spheres, J. Geom. Phys., Volume 96 (2015), pp. 1-25 | DOI

[2] Teodor Banica The algebraic structure of quantum partial isometries, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Volume 19 (2016) no. 1 (Article ID 1650003, 36 p.) | DOI

[3] Teodor Banica A duality principle for noncommutative cubes and spheres, J. Noncommut. Geom., Volume 10 (2016) no. 3, pp. 1043-1081 | DOI

[4] Teodor Banica; Serban Teodor Belinschi; Mireille Capitaine; Benoît Collins Free Bessel laws, Can. J. Math., Volume 63 (2011) no. 1, pp. 3-37 | DOI

[5] Teodor Banica; Julien Bichon; Benoît Collins The hyperoctahedral quantum group, J. Ramanujan Math. Soc., Volume 22 (2007) no. 4, pp. 345-384

[6] Teodor Banica; Debashish Goswami Quantum isometries and noncommutative spheres, Commun. Math. Phys., Volume 298 (2010) no. 2, pp. 343-356 | DOI

[7] Teodor Banica; Adam Skalski; Piotr Sołtan Noncommutative homogeneous spaces: the matrix case, J. Geom. Phys., Volume 62 (2012) no. 6, pp. 1451-1466 | DOI

[8] Teodor Banica; Roland Speicher Liberation of orthogonal Lie groups, Adv. Math., Volume 222 (2009) no. 4, pp. 1461-1501 | DOI

[9] Hari Bercovici; Vittorino Pata Stable laws and domains of attraction in free probability theory, Ann. Math., Volume 149 (1999) no. 3, pp. 1023-1060 | DOI

[10] Florin P. Boca Ergodic actions of compact matrix pseudogroups on $C^{*}$ -algebras, Recent advances in operator algebras. Collection of talks given in the conference on operator algebras held in Orléans, France in July 1992 (Astérisque), Volume 232, Société Mathématique de France, 1995, pp. 93-109

[11] Alexandru Chirvasitu Quantum rigidity of negatively curved manifolds (2015) (https://arxiv.org/abs/1503.07984)

[12] Benoît Collins; Piotr Śniady Integration with respect to the Haar measure on the unitary, orthogonal and symplectic group, Commun. Math. Phys., Volume 264 (2006) no. 3, pp. 773-795 | DOI

[13] Stephen Curran; Roland Speicher Quantum invariant families of matrices in free probability, J. Funct. Anal., Volume 261 (2011) no. 4, pp. 897-933 | DOI

[14] Kenny De Commer; Makoto Yamashita Tannaka–Krein duality for compact quantum homogeneous spaces. I. General theory, Theory Appl. Categ., Volume 28 (2013), pp. 1099-1138 (electronic only)

[15] Amaury Freslon On the partition approach to Schur-Weyl duality and free quantum groups (2014) (https://arxiv.org/abs/1409.1346v1)

[16] Debashish Goswami; Soumalya Joardar Rigidity of action of compact quantum groups on compact, connected manifolds (2013) (https://arxiv.org/abs/1309.1294v1)

[17] Igor Klep; Victor Vinnikov; Jurij Volčič Null- and Positivstellensätze for rationally resolvable ideals (2015) (https://arxiv.org/abs/1504.08004)

[18] Claus Köstler; Roland Speicher A noncommutative de Finetti theorem: invariance under quantum permutations is equivalent to freeness with amalgamation, Commun. Math. Phys., Volume 291 (2009) no. 2, pp. 473-490 | DOI

[19] Sara Malacarne Woronowicz’s Tannaka-Krein duality and free orthogonal quantum groups (2016) (https://arxiv.org/abs/1602.04807)

[20] Sergey Neshveyev; Lars Tuset Compact quantum groups and their representation categories, Cours Spécialisés (Paris), 20, Société Mathématique de France, 2013, iv+169 pages

[21] Alexandru Nica; Roland Speicher Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, 335, Cambridge University Press, 2006, xv+417 pages

[22] Piotr Podleś Quantum spheres., Lett. Math. Phys., Volume 14 (1987), pp. 193-202 | DOI

[23] Piotr Podleś Symmetries of quantum spaces. Subgroups and quotient spaces of quantum $S U (2)$ and $S O (3)$ groups, Commun. Math. Phys., Volume 170 (1995) no. 1, pp. 1-20 | DOI

[24] Sven Raum Isomorphisms and fusion rules of orthogonal free quantum groups and their complexifications, Proc. Am. Math. Soc., Volume 140 (2012) no. 9, pp. 3207-3218 | DOI

[25] Sven Raum; Moritz Weber The full classification of orthogonal easy quantum groups, Commun. Math. Phys., Volume 341 (2016) no. 3, pp. 751-779 | DOI

[26] Piotr Sołtan On actions of compact quantum groups., Ill. J. Math., Volume 55 (2011) no. 3, pp. 953-962

[27] Roland Speicher Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Ann., Volume 298 (1994) no. 4, pp. 611-628 | DOI

[28] Pierre Tarrago; Moritz Weber The classification of tensor categories of two-colored noncrossing partitions (2015) (https://arxiv.org/abs/1509.00988)

[29] Pierre Tarrago; Moritz Weber Unitary Easy Quantum Groups: the free case and the group case (2015) (https://arxiv.org/abs/1512.00195)

[30] Dan V. Voiculescu; Kenneth J. Dykema; Alexandru Nica Free random variables. A noncommutative probability approach to free products with applications to random matrices, operator algebras and harmonic analysis on free groups, CRM Monograph Series, 1, American Mathematical Society, 1992, v+70 pages

[31] Shuzhou Wang Free products of compact quantum groups, Commun. Math. Phys., Volume 167 (1995) no. 3, pp. 671-692 | DOI

[32] Shuzhou Wang Quantum symmetry groups of finite spaces, Commun. Math. Phys., Volume 195 (1998) no. 1, pp. 195-211 | DOI

[33] Moritz Weber On the classification of easy quantum groups, Adv. Math., Volume 245 (2013), pp. 500-533 | DOI

[34] Don Weingarten Asymptotic behavior of group integrals in the limit of infinite rank, J. Math. Phys., Volume 19 (1978), pp. 999-1001 | DOI

[35] Stanisław Lech Woronowicz Compact matrix pseudogroups., Commun. Math. Phys., Volume 111 (1987), pp. 613-665 | DOI

[36] Stanisław Lech Woronowicz Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math., Volume 93 (1988) no. 1, pp. 35-76 | DOI

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