Liberation theory for noncommutative homogeneous spaces
[Théorie de liberation pour les espaces homogènes non commutatifs]
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 26 (2017) no. 1, pp. 127-156.

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 ×G N )/(G L ×G M-L ×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 ×G N )/(G L ×G M-L ×G N-L ), and the liberation operation for them, with a number of algebraic and probabilistic results.

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Accepté le :
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DOI : 10.5802/afst.1527
Classification : 46L65, 46L54
Mots clés : Liberation theory, Homogeneous space
Banica, Teodor 1

1 Département des Mathématiques, Cergy-Pontoise University, 95000 Cergy-Pontoise, France
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Banica, Teodor. 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. http://www.numdam.org/articles/10.5802/afst.1527/

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

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

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

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

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

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

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

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

[10] Boca, Florin P. 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] Chirvasitu, Alexandru Quantum rigidity of negatively curved manifolds (2015) (https://arxiv.org/abs/1503.07984)

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

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

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

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

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

[18] Köstler, Claus; Speicher, Roland 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] Malacarne, Sara Woronowicz’s Tannaka-Krein duality and free orthogonal quantum groups (2016) (https://arxiv.org/abs/1602.04807)

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

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

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

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

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

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

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

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

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

[30] Voiculescu, Dan V.; Dykema, Kenneth J.; Nica, Alexandru 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] Wang, Shuzhou Free products of compact quantum groups, Commun. Math. Phys., Volume 167 (1995) no. 3, pp. 671-692 | DOI

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

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

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

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

[36] Woronowicz, Stanisław Lech 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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