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

By Reiji Tomatsu. 2006

80 citations recorded by the source corpus.

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 SUq(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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/f07df62680dd022f0a32fac5d25c09e1c452ed04

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Examples of factors which have no Cartan subalgebras (2012, methodology): “We use similar notation to [31], which has a good survey of the modular theory on compact quantum groups.”
The boundary of universal discrete quantum groups, exactness, and factoriality (2005, methodology): “A recent preprint of Tomatsu [ 21 ] provides a computation for the Poisson boundary of the dual of an arbitrary q-deformed compact Lie group.”