A new characterisation of idempotent states on finite and compact quantum groups

Uwe Franz; Adam Skalski

doi:10.1016/j.crma.2009.06.015

Algebra/Functional Analysis

A new characterisation of idempotent states on finite and compact quantum groups
[Une nouvelle caractérisation des états idempotents sur des groupes quantiques finis ou compacts]

Présenté par : Gilles Pisier

Uwe Franz ¹ ; Adam Skalski ²

¹ Département de mathématiques de Besançon, Université de Franche-Comté, 16, route de Gray, 25030 Besançon, France
² Department of Mathematics and Statistics, Lancaster University, Lancaster, United Kingdom

Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 991-996.

Abstract (VO)
Résumé

We show that idempotent states on finite quantum groups correspond to pre-subgroups in the sense of Baaj, Blanchard, and Skandalis. It follows that the lattices formed by the idempotent states on a finite quantum group and by its coidalgebras are isomorphic. We show, furthermore, that these lattices are also isomorphic for compact quantum groups, if one restricts to expected coidalgebras.

Nous donnons une caractérisation des états idempotents sur un groupe quantique fini en termes des pré-sous-groupes introduits par Baaj, Blanchard, et Skandalis, et en déduisons un isomorphisme entre le réseau des états idempotents et le réseau des sous-algèbres coïdéales d'un groupe quantique fini. Cet isomorphisme s'étend aux groupes quantiques compacts, si on le restreind au sous-algèbres coïdéales expectées.

Reçu le : 2009-06-02
Accepté le : 2009-06-25
Publié le : 2009-07-22

DOI : 10.1016/j.crma.2009.06.015

Affiliations des auteurs :

Uwe Franz ¹ ; Adam Skalski ²

¹ Département de mathématiques de Besançon, Université de Franche-Comté, 16, route de Gray, 25030 Besançon, France
² Department of Mathematics and Statistics, Lancaster University, Lancaster, United Kingdom

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Uwe Franz; Adam Skalski. A new characterisation of idempotent states on finite and compact quantum groups. Comptes Rendus. Mathématique, Volume 347 (2009) no. 17-18, pp. 991-996. doi : 10.1016/j.crma.2009.06.015. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2009.06.015/

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Paweł Kasprzak; Fatemeh Khosravi; Piotr M. Sołtan Kawada-Itô-Kelley theorem for quantum semigroups, Journal of Mathematical Analysis and Applications, Volume 483 (2020) no. 2, p. 123594 | DOI:10.1016/j.jmaa.2019.123594
Haonan Zhang Infinitely divisible states on finite quantum groups, Mathematische Zeitschrift, Volume 294 (2020) no. 1-2, p. 571 | DOI:10.1007/s00209-019-02288-8
Haonan Zhang Idempotent states on Sekine quantum groups, Communications in Algebra, Volume 47 (2019) no. 10, p. 4095 | DOI:10.1080/00927872.2019.1579335
Alexandru Chirvasitu Relative Fourier transforms and expectations on coideal subalgebras, Journal of Algebra, Volume 516 (2018), p. 271 | DOI:10.1016/j.jalgebra.2018.08.033
Paweł Kasprzak; Fatemeh Khosravi Coideals, Quantum Subgroups and Idempotent States, The Quarterly Journal of Mathematics (2017) | DOI:10.1093/qmath/haw051
Uwe Franz; Hun Hee Lee; Adam Skalski Integration over the quantum diagonal subgroup and associated Fourier-like algebras, International Journal of Mathematics, Volume 27 (2016) no. 09, p. 1650073 | DOI:10.1142/s0129167x16500737
Pekka Salmi; Adam Skalski Idempotent States on Locally Compact Quantum Groups II, The Quarterly Journal of Mathematics (2016) | DOI:10.1093/qmath/haw045
Paweł Kasprzak; Piotr M. Sołtan Embeddable quantum homogeneous spaces, Journal of Mathematical Analysis and Applications, Volume 411 (2014) no. 2, p. 574 | DOI:10.1016/j.jmaa.2013.07.084
MEHRDAD KALANTAR; MATTHIAS NEUFANG; ZHONG-JIN RUAN POISSON BOUNDARIES OVER LOCALLY COMPACT QUANTUM GROUPS, International Journal of Mathematics, Volume 24 (2013) no. 03, p. 1350023 | DOI:10.1142/s0129167x13500237
Uwe Franz; Adam Skalski; Reiji Tomatsu On square roots of the Haar state on compact quantum groups, Journal of Pure and Applied Algebra, Volume 216 (2012) no. 10, p. 2079 | DOI:10.1016/j.jpaa.2012.01.020
Pekka Salmi Compact quantum subgroups and left invariant C⁎-subalgebras of locally compact quantum groups, Journal of Functional Analysis, Volume 261 (2011) no. 1, p. 1 | DOI:10.1016/j.jfa.2011.03.003

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