Annales Mathématiques
Blaise Pascal
no. 2
Weingarten integration over noncommutative homogeneous spaces
[Intégration de Weingarten sur les espaces homogènes non commutatifs]
Teodor Banica1
1 Department of Mathematics Cergy-Pontoise University 95000 Cergy-Pontoise, France
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⊂ℂ N , avec H⊂G⊂U 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⊂ℂ N , with H⊂G⊂U 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 :
DOI : 10.5802/ambp.368
Classification : 46L51, 14A22, 60B15
Keywords: Noncommutative manifold, Weingarten integration
Mot clés : Variété non commutative, Integration de Weingarten
Teodor Banica 1

1 Department of Mathematics Cergy-Pontoise University 95000 Cergy-Pontoise, France
Licence : CC-BY 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@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 = {https://ambp.centre-mersenne.org/articles/10.5802/ambp.368/}
}
TY  - JOUR
AU  - Teodor Banica
TI  - Weingarten integration over noncommutative homogeneous spaces
JO  - Annales mathématiques Blaise Pascal
PY  - 2017
SP  - 195
EP  - 224
VL  - 24
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - https://ambp.centre-mersenne.org/articles/10.5802/ambp.368/
DO  - 10.5802/ambp.368
LA  - en
ID  - AMBP_2017__24_2_195_0
ER  - 
%0 Journal Article
%A Teodor Banica
%T Weingarten integration over noncommutative homogeneous spaces
%J Annales mathématiques Blaise Pascal
%D 2017
%P 195-224
%V 24
%N 2
%I Annales mathématiques Blaise Pascal
%U https://ambp.centre-mersenne.org/articles/10.5802/ambp.368/
%R 10.5802/ambp.368
%G en
%F 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/articles/10.5802/ambp.368/

[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 | DOI | Zbl

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

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

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

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

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

[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 | DOI | Zbl

[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

[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 | DOI | Zbl

[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

[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 | DOI | Zbl

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

[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 | DOI | Zbl

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

[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 | DOI | Zbl

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

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

[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 | DOI | Zbl

Cité par Sources :