Measurable Kac cohomology for bicrossed products

Abstract

Abstract. We study the Kac cohomology for matched pairs of locally compact groups. This cohomology theory arises from the extension theory of locally compact quantum groups. We prove a measurable version of the Kac exact sequence and provide methods to compute the cohomology. We give explicit calculations in several examples using results of Moore and Wigner.

Construction of cocycles for bicrossproduct of Lie groups

“…Then it follows from [5] that a pair (u, v) satisfies (1), (2), and (3) if and only if f u,v is a nonhomogeneous 3-cocycle on K and is equal to 1 whenever one of the elements k 1 , k 2 , and k 3 coincides with e K . Moreover [5], two pairs (u 1 , v 1 ) and (u 2 , v 2 ) satisfy (4) if and only if there exists a differentiable function r :…”

“…The group of equivalence classes of pairs of cocycles is described in [1] for finite groups and in [4] for locally compact groups (see also [5]). The goal of the present paper is to obtain explicit formulas providing a pair of cocycles for the case of Lie groups in terms of a pair of cocycles on the corresponding matched pair of Lie algebras.…”

Measured quantum groupoids associated with matched pairs of locally compact groupoids

“…The aim of this article is to give a large number of examples of measured quantum groupoids as defined by M.Enock and F. Lesieur. We generalize at the same time the notion of matched pair of finite groupoids ( [Val2]) and of locally compact group ( [BS], [VV], [BSV], [BSV1]....), in order to obtain such examples coming from a suitable pseudo multiplicative unitary.…”

“…We study, in the last chapter, two kinds of examples. The first one is pretty natural and comes from matched pairs of groups actions: a very general example of matched pair of groups is the "ax+b" group ([BSV1] Chap 4), and pentagonal transformations lead also to such actions ( [BSV1] Prop 5.1). We prove that, for any matched pair G 1 G 2 of locally compact groups in the sense of [BSV], which acts on a locally compact space X, then X × G 1 , X × G 2 is a matched pair of groupoids.…”

Finding cocycles in the bicrossed product construction for Lie groups

“…The group of equivalence classes of pairs of cocycles was described in [1] for finite groups and in [4] (see also [5]) for locally compact groups. The aim of the present work is to obtain an explicit formula for finding a pair of cocycles in the case where groups are Lie groups by using a pair of cocycles on the corresponding matched pair of Lie algebras.…”