Weingarten integration over noncommutative homogeneous spaces
[Intégration de Weingarten sur les espaces homogènes non commutatifs]
Annales Mathématiques Blaise Pascal, Tome 24 (2017) no. 2, pp. 195-224.

On présente une extension de la formule d’intégration de Weingarten, pour les espaces homogènes non commutatifs, vérifiant des hypothèses « d’aisance » adéquates. Les espaces qu’on considère sont des variétés algebriques non commutatives, généralisant les espaces du type X=G/H N , avec HGU N étant des sous-groupes du groupe unitaire, vérifiant certaines conditions d’uniformité. On traite d’abord les questions d’axiomatisation, ensuite on établit la formule de Weingarten, et on finit avec quelques conséquences probabilistes.

We discuss an extension of the Weingarten formula, to the case of noncommutative homogeneous spaces, under suitable “easiness” assumptions. The spaces that we consider are noncommutative algebraic manifolds, generalizing the spaces of type X=G/H N , with HGU N being subgroups of the unitary group, subject to certain uniformity conditions. We discuss various axiomatization issues, then we establish the Weingarten formula, and we derive some probabilistic consequences.

Publié le : 2017-11-20
DOI : https://doi.org/10.5802/ambp.368
Classification : 46L51,  14A22,  60B15
Mots clés: Variété non commutative, Integration de Weingarten
@article{AMBP_2017__24_2_195_0,
     author = {Teodor Banica},
     title = {Weingarten integration over noncommutative homogeneous spaces},
     journal = {Annales Math\'ematiques Blaise Pascal},
     pages = {195--224},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {24},
     number = {2},
     year = {2017},
     doi = {10.5802/ambp.368},
     language = {en},
     url = {ambp.centre-mersenne.org/item/AMBP_2017__24_2_195_0/}
}
Teodor Banica. Weingarten integration over noncommutative homogeneous spaces. Annales Mathématiques Blaise Pascal, Tome 24 (2017) no. 2, pp. 195-224. doi : 10.5802/ambp.368. https://ambp.centre-mersenne.org/item/AMBP_2017__24_2_195_0/

[1] Teodor Banica The algebraic structure of quantum partial isometries, Infin. Dimens. Anal. Quantum Probab. Relat. Top., Volume 19 (2016) no. 1, pp. 1-36 | Article | Zbl 1354.46064

[2] Teodor Banica Liberation theory for noncommutative homogeneous spaces, Ann. Fac. Sci. Toulouse, Math., Volume 26 (2017) no. 1, pp. 127-156 | Article | Zbl 06774844

[3] Teodor Banica; Benoît Collins Integration over compact quantum groups, Publ. Res. Inst. Math. Sci., Volume 43 (2007) no. 2, pp. 277-302 | Article | Zbl 1129.46058

[4] Teodor Banica; Debashish Goswami Quantum isometries and noncommutative spheres, Comm. Math. Phys., Volume 298 (2010) no. 2, pp. 343-356 | Article | Zbl 1204.58004

[5] Teodor Banica; Adam Skalski; Piotr Sołtan Noncommutative homogeneous spaces: the matrix case, J. Geom. Phys., Volume 62 (2012) no. 6, pp. 1451-1466 | Article | Zbl 1256.46038

[6] Teodor Banica; Roland Speicher Liberation of orthogonal Lie groups, Adv. Math., Volume 222 (2009) no. 4, pp. 1461-1501 | Article | Zbl 1247.46046

[7] Hari Bercovici; Vittorino Pata Stable laws and domains of attraction in free probability theory, Ann. Math., Volume 149 (1999) no. 3, pp. 1023-1060 | Article | Zbl 0945.46046

[8] Florin P. Boca Ergodic actions of compact matrix pseudogroups on C * -algebras, Recent advances in operator algebras (Astérisque) Volume 232, Société Mathématique de France, 1995, pp. 93-109 | Zbl 0842.46039

[9] Benoît Collins; Piotr Śniady Integration with respect to the Haar measure on the unitary, orthogonal and symplectic group, Comm. Math. Phys., Volume 264 (2006) no. 3, pp. 773-795 | Article | Zbl 1108.60004

[10] 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 | Zbl 1337.46045

[11] Amaury Freslon On the partition approach to Schur-Weyl duality and free quantum groups, Transform. Groups, Volume 22 (2017) no. 3, pp. 707-751 | Article | Zbl 06793985

[12] Paweł Kasprzak; Piotr Sołtan Embeddable quantum homogeneous spaces, J. Math. Anal. Appl., Volume 411 (2014) no. 2, pp. 574-591 | Article | Zbl 1337.46047

[13] Piotr Podleś 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 | Article | Zbl 0853.46074

[14] Sven Raum; Moritz Weber The full classification of orthogonal easy quantum groups, Commun. Math. Phys., Volume 341 (2016) no. 3, pp. 751-779 | Article | Zbl 1356.46061

[15] Roland Speicher; Moritz Weber Quantum groups with partial commutation relations (2016) (https://arxiv.org/abs/1603.09192)

[16] Pierre Tarrago; Moritz Weber Unitary easy quantum groups: the free case and the group case (2015) (https://arxiv.org/abs/1512.00195)

[17] Shuzhou Wang Free products of compact quantum groups, Commun. Math. Phys., Volume 167 (1995) no. 3, pp. 671-692 | Article | Zbl 0838.46057

[18] Don Weingarten Asymptotic behavior of group integrals in the limit of infinite rank, J. Math. Phys., Volume 19 (1978), pp. 999-1001 | Article | Zbl 0388.28013

[19] Stanisław Lech Woronowicz Compact matrix pseudogroups, Commun. Math. Phys., Volume 111 (1987), pp. 613-665 | Article | Zbl 0627.58034

[20] Stanisław Lech Woronowicz Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups, Invent. Math., Volume 93 (1988) no. 1, pp. 35-76 | Article | Zbl 0664.58044