With cedram.org
Annales Mathématiques
Blaise Pascal
Search for an article
Search within the site
Table of contents for this issue | Previous article | Next article
Teodor Banica
Weingarten integration over noncommutative homogeneous spaces
(Intégration de Weingarten sur les espaces homogènes non commutatifs)
Annales mathématiques Blaise Pascal, 24 no. 2 (2017), p. 195-224, doi: 10.5802/ambp.368
Article PDF
Class. Math.: 46L51, 14A22, 60B15
Keywords: Noncommutative manifold, Weingarten integration

Résumé - Abstract

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 $X=G/H\subset \mathbb{C}^N$, with $H\subset G\subset U_N$ 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.


[1] Teodor Banica. The algebraic structure of quantum partial isometries. Infin. Dimens. Anal. Quantum Probab. Relat. Top., 19(1):1-36, 2016. Article
[2] Teodor Banica. Liberation theory for noncommutative homogeneous spaces. Ann. Fac. Sci. Toulouse, Math., 26(1):127-156, 2017. Article
[3] Teodor Banica and Benoît Collins. Integration over compact quantum groups. Publ. Res. Inst. Math. Sci., 43(2):277-302, 2007. Article
[4] Teodor Banica and Debashish Goswami. Quantum isometries and noncommutative spheres. Comm. Math. Phys., 298(2):343-356, 2010. Article
[5] Teodor Banica, Adam Skalski and Piotr Sołtan. Noncommutative homogeneous spaces: the matrix case. J. Geom. Phys., 62(6):1451-1466, 2012. Article
[6] Teodor Banica and Roland Speicher. Liberation of orthogonal Lie groups. Adv. Math., 222(4):1461-1501, 2009. Article
[7] Hari Bercovici and Vittorino Pata. Stable laws and domains of attraction in free probability theory. Ann. Math., 149(3):1023-1060, 1999. Article
[8] Florin P. Boca, Ergodic actions of compact matrix pseudogroups on C$^*$-algebras, Recent advances in operator algebras, Astérisque 232, Société Mathématique de France, 1995, p. 93–109
[9] Benoît Collins and Piotr Śniady. Integration with respect to the Haar measure on the unitary, orthogonal and symplectic group. Comm. Math. Phys., 264(3):773-795, 2006. Article
[10] Kenny De Commer and Makoto Yamashita. Tannaka-Krein duality for compact quantum homogeneous spaces. I. General theory. Theory Appl. Categ., 28:1099-1138, 2013.
[11] Amaury Freslon. On the partition approach to Schur-Weyl duality and free quantum groups. Transform. Groups, 22(3):707-751, 2017. Article
[12] Paweł Kasprzak and Piotr Sołtan. Embeddable quantum homogeneous spaces. J. Math. Anal. Appl., 411(2):574-591, 2014. Article
[13] Piotr Podleś. Symmetries of quantum spaces. Subgroups and quotient spaces of quantum SU(2) and SO(3) groups. Commun. Math. Phys., 170(1):1-20, 1995. Article
[14] Sven Raum and Moritz Weber. The full classification of orthogonal easy quantum groups. Commun. Math. Phys., 341(3):751-779, 2016. Article
[15] Roland Speicher and Moritz Weber. Quantum groups with partial commutation relations. https://arxiv.org/abs/1603.09192, 2016
[16] Pierre Tarrago and Moritz Weber. Unitary easy quantum groups: the free case and the group case. https://arxiv.org/abs/1512.00195, 2015
[17] Shuzhou Wang. Free products of compact quantum groups. Commun. Math. Phys., 167(3):671-692, 1995. Article
[18] Don Weingarten. Asymptotic behavior of group integrals in the limit of infinite rank. J. Math. Phys., 19:999-1001, 1978. Article
[19] Stanisław Lech Woronowicz. Compact matrix pseudogroups. Commun. Math. Phys., 111:613-665, 1987. Article
[20] Stanisław Lech Woronowicz. Tannaka-Krein duality for compact matrix pseudogroups. Twisted SU(N) groups. Invent. Math., 93(1):35-76, 1988. Article