Co-amenability of compact quantum groups

Bédos, Erik; Murphy, Gregory; Tuset, Lars

doi:10.1016/s0393-0440(01)00024-9

Cited by 108 publications

(156 citation statements)

References 13 publications

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“…As the term indicates, this is an amenability property of the representation theory of G. For regular multiplicative unitaries, coamenability has been introduced by Baaj and Skandalis [2], and for compact quantum groups by Banica [4]. See also [8].…”

Section: Compact Quantum Groupsmentioning

confidence: 99%

Connected components of compact matrix quantum groups and finiteness conditions

Cirio

D’Andrea

Pinzari

et al. 2014

Journal of Functional Analysis

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ABSTRACT. We introduce the notion of identity component of a compact quantum group and that of total disconnectedness. As a drawback of the generalized Burnside problem, we note that totally disconnected compact matrix quantum groups may fail to be profinite. We consider the problem of approximating the identity component as well as the maximal normal (in the sense of Wang) connected subgroup by introducing canonical, but possibly transfinite, sequences of subgroups. These sequences have a trivial behaviour in the classical case. We give examples, arising as free products, where the identity component is not normal and the associated sequence has length 1.We give necessary and sufficient conditions for normality of the identity component and finiteness or profiniteness of the quantum component group. Among them, we introduce an ascending chain condition on the representation ring, called Lie property, which characterizes Lie groups in the commutative case and reduces to group Noetherianity of the dual in the cocommutative case. It is weaker than ring Noetherianity but ensures existence of a generating representation. The Lie property and ring Noetherianity are inherited by quotient quantum groups. We show that A u (F ) is not of Lie type. We discuss an example arising from the compact real form of U q (sl 2 ) for q < 0.

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Section: Compact Quantum Groupsmentioning

confidence: 99%

Connected components of compact matrix quantum groups and finiteness conditions

Cirio

D’Andrea

Pinzari

et al. 2014

Journal of Functional Analysis

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“…We only treat a co-amenable G in this paper, and ε extends to the character on C(G). See [3][4][5]43] for details of the amenability.…”

Section: Compact Quantum Groupmentioning

confidence: 99%

Product type actions ofGq

Tomatsu

2015

Advances in Mathematics

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MSC: 46L55 20G42 17B37Keywords: Infinite tensor product type actions Depth 2 subfactors Non-commutative Poisson boundaries Quantum flag manifoldsWe will study a faithful product type action of G q , the q-deformation of a connected semisimple compact Lie group G, and prove that such an action is induced from a minimal action of the maximal torus T of G q . This enables us to classify product type actions of SU q (2) up to conjugacy. We also compute the intrinsic group of G q,Ω , the 2-cocycle deformation of G q that is naturally associated with the quantum flag manifold L ∞ (T \G q ).

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“…Additionally, if G is coamenable, by Corollary 4.6, D(G) is amenable and has the Haagerup property. In particular, SU q (2) is a compact quantum group and coamenable (see [5]). Hence, the Drinfeld double of SU q (2), dual of the quantum Lorentz group in [13], has the Haagerup property.…”

Section: The Last Two Estimates Together Givementioning

confidence: 99%