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Quantized Algebras of Functions on Homogeneous Spaces with Poisson Stabilizers

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Abstract

Let G be a simply connected semisimple compact Lie group with standard Poisson structure, K a closed Poisson-Lie subgroup, 0 < q < 1. We study a quantization C(G q /K q ) of the algebra of continuous functions on G/K. Using results of Soibelman and Dijkhuizen-Stokman we classify the irreducible representations of C(G q /K q ) and obtain a composition series for C(G q /K q ). We describe closures of the symplectic leaves of G/K refining the well-known description in the case of flag manifolds in terms of the Bruhat order. We then show that the same rules describe the topology on the spectrum of C(G q /K q ). Next we show that the family of C*-algebras C(G q /K q ), 0 < q ≤ 1, has a canonical structure of a continuous field of C*-algebras and provides a strict deformation quantization of the Poisson algebra \({\mathbb{C}[G/K]}\) . Finally, extending a result of Nagy, we show that C(G q /K q ) is canonically KK-equivalent to C(G/K).

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Authors and Affiliations

  1. Department of Mathematics, University of Oslo, P. O. Box 1053, Blindern, 0316, Oslo, Norway

    Sergey Neshveyev

  2. Faculty of Engineering, Oslo University College, P. O. Box 4, St. Olavs Plass, 0130, Oslo, Norway

    Lars Tuset

Authors
  1. Sergey Neshveyev
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  2. Lars Tuset
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Correspondence to Sergey Neshveyev.

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Communicated by A. Connes

Supported by the Research Council of Norway.

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Neshveyev, S., Tuset, L. Quantized Algebras of Functions on Homogeneous Spaces with Poisson Stabilizers. Commun. Math. Phys. 312, 223–250 (2012). https://doi.org/10.1007/s00220-012-1455-6

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