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A Characterization of Right Coideals of Quotient Type and its Application to Classification of Poisson Boundaries

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Abstract

Let \({\mathbb{G}}\) be a co-amenable compact quantum group. We show that a right coideal of \({\mathbb{G}}\) is of quotient type if and only if it is the range of a conditional expectation preserving the Haar state and is globally invariant under the left action of the dual discrete quantum group. We apply this result to the theory of Poisson boundaries introduced by Izumi for discrete quantum groups and generalize a work of Izumi-Neshveyev-Tuset on SU q (N) for co-amenable compact quantum groups with the commutative fusion rules. More precisely, we prove that the Poisson integral is an isomorphism between the Poisson boundary and the right coideal of quotient type by a maximal quantum subgroup of Kac type. In particular, the Poisson boundary and the quantum flag manifold are isomorphic for any q-deformed classical compact Lie group.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo, 153-8914, Japan

    Reiji Tomatsu

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  1. Reiji Tomatsu
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Correspondence to Reiji Tomatsu.

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Communicated by Y. Kawahigashi

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Tomatsu, R. A Characterization of Right Coideals of Quotient Type and its Application to Classification of Poisson Boundaries. Commun. Math. Phys. 275, 271–296 (2007). https://doi.org/10.1007/s00220-007-0267-6

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