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

Abstract

Let G be a co-amenable compact quantum group. We show that a right coideal of 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 theory of Poisson boundaries introduced by Izumi for discrete quantum groups and generalize a work of Izumi-Neshveyev-Tuset on SU q (N ) for coamenable 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 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.

One-to-one correspondence between generating functionals and cocycles on quantum groups in presence of symmetry

“…Then the same argument as in Corollary 4.9 in [2] (see also the proofs of Lemma 2.9 (2) in [17] and Proposition 3.5 in [7]) imply that in fact L = L • S • τ z for any z ∈ C. Fixing s ∈ R and applying Theorem 2.8 for η, L and α = τ is ends the proof (recall again that our convention for the scaling group is that of [3], not of [19]). …”

Product type actions ofGq

“…Moreover, we know by [17,Theorem 5.10], [20, Theorem A] and [44,Corollary 4.11] that there exists a G q -equivariant isomorphism from L ∞ (T \G q ) onto Q. In particular, Q is a type I factor by Theorem 3.1.…”

“…In [46], it is conjectured that the classical Poisson boundary H ∞ class ( G, μ) could coincide with the center of the Poisson boundary Z (H ∞ ( G, μ)). (See [17,20,44] for a detail of theory of a Poisson boundary.) In particular, if the fusion rule of G is commutative, the conjecture asks the factoriality of the Poisson boundary.…”

“…We will freely use the notions and the notation introduced in [44]. Let G := (C(G), δ) be a (C * -algebraic) compact quantum group.…”

“…Namely, he has constructed a von Neumann algebra from a random walk on the dual discrete quantum group of a compact quantum group, and shown that is isomorphic to the relative commutant of a fixed point algebra inside an infinite tensor product factor of matrix algebras. Since this pioneering work, the program of realization of Poisson boundaries has been carried out in several papers [17,20,44,49,50]. In particular, it is known that the Poisson boundary of G q is isomorphic to the quantum flag manifold T \G q [17,20,44].…”