Unitaires multiplicatifs en dimension finie et leurs sous-objets

Baaj, Saad; Blanchard, Etienne; Skandalis, Georges

doi:10.5802/aif.1719

Cited by 18 publications

(20 citation statements)

References 16 publications

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“…That pre-subgroups give rise to idempotent states was not emphasized in [1], but can easily be seen from [1, Proposition 3.5(a)]. Here we prove that, conversely, every idempotent state comes from a pre-subgroup, cf.…”

Section: Introductionmentioning

confidence: 86%

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

Franz¹,

Skalski²

2009

Comptes Rendus. Mathématique

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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. To cite this article: U. Franz, A. Skalski, C. R. Acad. Sci. Paris, Ser. I 347 (2009). RésuméUne nouvelle caractérisation des états idempotents sur des groupes quantiques finis ou compacts. 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. Pour citer cet article : U. Franz, A. Skalski, C. R. Acad. Sci. Paris, Ser. I 347 (2009). Kawada et Itô [8] ont montré que toute mesure idempotente sur un groupe compact est induite par la mesure de Haar d'un de ses sous-groupes compacts, voir aussi [6]. Depuis Pal [11] nous savons que l'analogue de ce théorème pour les groupes quantiques est fausse. Recemment nous avons donné de nouvelles examples d'états idempotents sur des groupes quantiques finis qui ne sont pas induits par des états de Haar de sous-groupes quantiques, voir [4]. Nous en avons également donné une caractérisation en termes de sous-hypergroupes quantiques.

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Section: Introductionmentioning

confidence: 86%

“…La première ressemble au résultat classique de Kawada et Itô, mais il fallait remplacer les sous-groupes quantiques par les pré-sous-groupes [1]. La deuxième, en terme de sous-algèbres coïdeales, découle ensuite d'un résultat de Baaj, Blanchard et Skandalis.…”

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

Franz¹,

Skalski²

2009

Comptes Rendus. Mathématique

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“…The next corollary is the infinite-dimensional version of [8, Theorem 3.2] and also the generalization of [8, Theorem 4.1] and [1]. Corollary 6.2.…”

Section: Proofmentioning

confidence: 93%

Coideals, Quantum Subgroups and Idempotent States

Kasprzak

Khosravi

2017

Q J Math

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We establish a one to one correspondence between idempotent states on a locally compact quantum group G and integrable coideals in the von Neumann algebra L ∞ (G) that are preserved by the scaling group. In particular we show that there is a one to one correspondence between idempotent states on G and ψ G -expected left-invariant von Neumann subalgebras of L ∞ (G). We characterize idempotent states of Haar type as those corresponding to integrable normal coideals preserved by the scaling group. We also establish a one to one correspondence between open subgroups of G and central idempotent states on the dual G. Finally we characterize coideals corresponding to open quantum subgroups of G as those that are normal and admit an atom. As a byproduct of this study we get a number of universal lifting results for Podleś condition, normality and regularity and we generalize a number of results known before to hold under the coamenability assumption.2010 Mathematics Subject Classification. Primary: 46L65 Secondary: 43A05, 46L30, 60B15.

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“…We recall that the dual object of a discrete multiplier Hopf C*algebra is, in the setting provided by [6,4], a compact Hopf C*-algebra. There is a Fourier transform, see [24, pg.…”

Section: K-theory Of Discrete Multiplier Hopf C*-algebrasmentioning

confidence: 99%

Isomorphisms and automorphisms of discrete multiplier Hopf C*-algebras

Kucerovsky

2014

Positivity

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Abstract. We construct Hopf algebra isomorphisms of discrete multiplier Hopf C*-algebras, and Hopf AF C*-algebras (generalized quantum UHF algebras), from K-theoretical data. Some of the intermediate results are of independent interest, such as a result that Jordan maps of Hopf algebras intertwine antipodes, and the applications to automorphisms of Hopf algebras. IntroductionClassification is a recurring theme in mathematics, and probably the most successful approach to classifying C*-algebras has been to use Ktheory, often augumented by some additional information, as a classifying functor [10]. We consider the case of C*-algebras with Hopf algebra structure, and we find that in many cases there is a product structure on the K-theory group. We then address the problem of constructing Hopf algebra maps from algebra maps respecting the product structure on the K-theory group, with applications to constructing automorphisms and isomorphisms of Hopf algebras. Theorems 2.10, 3.2 and 2.11 allow constructing Hopf algebra (co-anti) automorphisms or isomorphisms from purely K-theoretical data. These results are used to study bi-inner Hopf *-automorphisms, which are the Hopf *-automorphisms of a Hopf C*-algebra that are inner as algebra automorphisms, both in the dual algebra and in the given algebra. We extend the results of [21]. We also develop techniques of independent interest, involving Jordan bi-algebra maps, linear positivity preserving maps, and other related concepts. In the last two sections, we give an example where we apply the techniques to the case of a Hopf AF C*-algebra, which includes an interesting special case that we call the quantum UHF case. With a compact Hopf AF C*-algebra can be associated two K-theory 1991 Mathematics Subject Classification. Primary 47L80, 16T05; Secondary 47L50, 16T20 .

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Unitaires multiplicatifs en dimension finie et leurs sous-objets

Cited by 18 publications

References 16 publications

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

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

Coideals, Quantum Subgroups and Idempotent States

Isomorphisms and automorphisms of discrete multiplier Hopf C*-algebras

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