Measured quantum groupoids associated to proper dynamical quantum groups

Abstract

Dynamical quantum groups were introduced by Etingof and Varchenko in connection with the dynamical quantum Yang-Baxter equation, and measured quantum groupoids were introduced by Enock, Lesieur and Vallin in their study of inclusions of type II 1 factors. In this article, we associate to suitable dynamical quantum groups, which are a purely algebraic objects, Hopf C˚-bimodules and measured quantum groupoids on the level of von Neumann algebras. Assuming invariant integrals on the dynamical quantum group, we first construct a fundamental unitary which yields Hopf bimodules on the level of C˚-algebras and von Neumann algebras. Next, we assume properness of the dynamical quantum group and lift the integrals to the operator algebras. In a subsequent article, this construction shall be applied to the dynamical SU q p2q studied by Koelink and Rosengren.

Integration on algebraic quantum groupoids

“…This construction generalizes corresponding results of Kustermans and Van Daele for algebraic quantum groups [9] and of the author for dynamical quantum groups [17], and is detailed in a forthcoming article [13].…”

“…The formulas Thus, every measured multiplier (B, Γ)-Hopf * -algebroid satisfying [17, Section 2.1, condition (A2)] gives rise to a measured multiplier Hopf * -algebroid. Theorem 6.3.2 shows that in the proper case, the assumption µ • C φ C = µ • B ψ B in [17] does not restrict generality.…”

“…To recall the notion of a dynamical quantum group as defined in [17], we need some preliminaries. Let B be a commutative * -algebra with local units and a left action of a discrete group Γ. Denote by e ∈ Γ the unit element.…”

“…Dynamical quantum groups. In [17], we studied integration on dynamical quantum groups in order to construct operator-algebraic completions in the form of measured quantum groupoids. The present results generalize and clarify the results obtained in [17], for example, they explain the deviation of the antipode on the operator-algebraic level from the algebraic antipode observed in [17, Proposition 2.7.13].…”

“…Morphisms are non-degenerate B ⊗ B-linear * -homomorphisms into multipliers that preserve the grading. The product A⊗C of (B, Γ) ev -algebras A and C is the quotient of (1) a map C φ C : A → B ∼ = C, called a left integral in [17], which has to be C-linear, left-invariant with respect to ∆, and to vanish on A γ,γ ′ if γ = e; (2) a map B ψ B : A → B ∼ = B, called a right integral in [17], which has to be B-linear, right-invariant with respect to ∆, and to vanish on A γ,γ ′ if γ ′ = e; (3) a faithful, positive linear functional on B that is quasi-invariant with respect to the action of Γ in a suitable sense and satisfies µ • C φ C = µ • B ψ B . Let us first consider the conditions in (1) and (2).…”