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 which are “uniform”, in a suitable sense. We study then the quotient spaces of type , and the liberation operation for them, with a number of algebraic and probabilistic results.
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 qui sont « uniformes », dans un sens convenable. On étudie ensuite les espaces quotient du type , et l’opération de liberation pour ces espaces, avec des résultats de nature algébrique et probabiliste.
Accepted:
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Mots-clés : Liberation theory, Homogeneous space
Teodor Banica 1

@article{AFST_2017_6_26_1_127_0, author = {Teodor Banica}, 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}, number = {1}, year = {2017}, doi = {10.5802/afst.1527}, language = {en}, url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1527/} }
TY - JOUR AU - Teodor Banica TI - Liberation theory for noncommutative homogeneous spaces JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2017 SP - 127 EP - 156 VL - 26 IS - 1 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1527/ DO - 10.5802/afst.1527 LA - en ID - AFST_2017_6_26_1_127_0 ER -
%0 Journal Article %A Teodor Banica %T Liberation theory for noncommutative homogeneous spaces %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2017 %P 127-156 %V 26 %N 1 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1527/ %R 10.5802/afst.1527 %G en %F AFST_2017_6_26_1_127_0
Teodor Banica. Liberation theory for noncommutative homogeneous spaces. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 26 (2017) no. 1, pp. 127-156. doi : 10.5802/afst.1527. https://afst.centre-mersenne.org/articles/10.5802/afst.1527/
[1] Liberations and twists of real and complex spheres, J. Geom. Phys., Volume 96 (2015), pp. 1-25 | DOI
[2] 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] A duality principle for noncommutative cubes and spheres, J. Noncommut. Geom., Volume 10 (2016) no. 3, pp. 1043-1081 | DOI
[4] Free Bessel laws, Can. J. Math., Volume 63 (2011) no. 1, pp. 3-37 | DOI
[5] The hyperoctahedral quantum group, J. Ramanujan Math. Soc., Volume 22 (2007) no. 4, pp. 345-384
[6] Quantum isometries and noncommutative spheres, Commun. Math. Phys., Volume 298 (2010) no. 2, pp. 343-356 | DOI
[7] Noncommutative homogeneous spaces: the matrix case, J. Geom. Phys., Volume 62 (2012) no. 6, pp. 1451-1466 | DOI
[8] Liberation of orthogonal Lie groups, Adv. Math., Volume 222 (2009) no. 4, pp. 1461-1501 | DOI
[9] Stable laws and domains of attraction in free probability theory, Ann. Math., Volume 149 (1999) no. 3, pp. 1023-1060 | DOI
[10] Ergodic actions of compact matrix pseudogroups on -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] Quantum rigidity of negatively curved manifolds (2015) (https://arxiv.org/abs/1503.07984)
[12] 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] Quantum invariant families of matrices in free probability, J. Funct. Anal., Volume 261 (2011) no. 4, pp. 897-933 | DOI
[14] Tannaka–Krein duality for compact quantum homogeneous spaces. I. General theory, Theory Appl. Categ., Volume 28 (2013), pp. 1099-1138 (electronic only)
[15] On the partition approach to Schur-Weyl duality and free quantum groups (2014) (https://arxiv.org/abs/1409.1346v1)
[16] Rigidity of action of compact quantum groups on compact, connected manifolds (2013) (https://arxiv.org/abs/1309.1294v1)
[17] Null- and Positivstellensätze for rationally resolvable ideals (2015) (https://arxiv.org/abs/1504.08004)
[18] 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] Woronowicz’s Tannaka-Krein duality and free orthogonal quantum groups (2016) (https://arxiv.org/abs/1602.04807)
[20] Compact quantum groups and their representation categories, Cours Spécialisés (Paris), 20, Société Mathématique de France, 2013, iv+169 pages
[21] Lectures on the combinatorics of free probability, London Mathematical Society Lecture Note Series, 335, Cambridge University Press, 2006, xv+417 pages
[22] Quantum spheres., Lett. Math. Phys., Volume 14 (1987), pp. 193-202 | DOI
[23] Symmetries of quantum spaces. Subgroups and quotient spaces of quantum and groups, Commun. Math. Phys., Volume 170 (1995) no. 1, pp. 1-20 | DOI
[24] 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] The full classification of orthogonal easy quantum groups, Commun. Math. Phys., Volume 341 (2016) no. 3, pp. 751-779 | DOI
[26] On actions of compact quantum groups., Ill. J. Math., Volume 55 (2011) no. 3, pp. 953-962
[27] Multiplicative functions on the lattice of non-crossing partitions and free convolution, Math. Ann., Volume 298 (1994) no. 4, pp. 611-628 | DOI
[28] The classification of tensor categories of two-colored noncrossing partitions (2015) (https://arxiv.org/abs/1509.00988)
[29] Unitary Easy Quantum Groups: the free case and the group case (2015) (https://arxiv.org/abs/1512.00195)
[30] 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] Free products of compact quantum groups, Commun. Math. Phys., Volume 167 (1995) no. 3, pp. 671-692 | DOI
[32] Quantum symmetry groups of finite spaces, Commun. Math. Phys., Volume 195 (1998) no. 1, pp. 195-211 | DOI
[33] On the classification of easy quantum groups, Adv. Math., Volume 245 (2013), pp. 500-533 | DOI
[34] Asymptotic behavior of group integrals in the limit of infinite rank, J. Math. Phys., Volume 19 (1978), pp. 999-1001 | DOI
[35] Compact matrix pseudogroups., Commun. Math. Phys., Volume 111 (1987), pp. 613-665 | DOI
[36] 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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