Narrow Results

'ax + b' is the group of affine transformations of the real line R. In quantum version ab = q²ba, where q² = e^{-i ℏ} is a number of modulus 1. The main problem of constructing quantum deformation of this group on the C*-level consists in non-selfadjointness of Δ(b) = a ⊗ b + b ⊗ I. This problem is overcome by introducing (in addition to a and b) a new generator β commuting with a and anticommuting with b. β (or more precisely β ⊗ β) is used to select a suitable selfadjoint extension of a ⊗ b + b ⊗ I. Furthermore we have to assume that , where k = 0,1,2, ·. In this case, q is a root of 1.

To construct the group, we write an explicit formula for the Kac–Takesaki operator W. It is shown that W is a manageable multiplicative unitary in the sense of [3,19]. Then using the general theory we construct a C*-algebra A and a comultiplication Δ ∈ Mor (A,A ⊗ A). A should be interpreted as the algebra of all continuous functions vanishing at infinity on quantum 'ax + b'-group. The group structure is encoded by Δ. The existence of coinverse also follows from the general theory [19].

The notion of left (respectively right) regular object of a tensor C*-category equipped with a faithful tensor functor into the category of Hilbert spaces is introduced. If such a category has a left (respectively right) regular object, it can be interpreted as a category of corepresentations (respectively representations) of some multiplicative unitary. A regular object is an object of the category which is at the same time left and right regular in a coherent way. A category with a regular object is endowed with an associated standard braided symmetry.

Conjugation is discussed in the context of multiplicative unitaries and their associated Hopf C*-algebras. It is shown that the conjugate of a left regular object is a right regular object in the same category. Furthermore the representation category of a locally compact quantum group has a conjugation. The associated multiplicative unitary is a regular object in that category.

We present versions of several classical results on harmonic functions and Poisson boundaries in the setting of locally compact quantum groups. In particular, the Choquet–Deny theorem holds for compact quantum groups; also, the result of Kaimanovich–Vershik and Rosenblatt, which characterizes group amenability in terms of harmonic functions, admits a noncommutative analogue in the separable case. We also explore the relation between classical and quantum Poisson boundaries by investigating the spectrum of the quantum group. We apply this machinery to find a concrete realization of the Poisson boundaries of the compact quantum group SU_q(2) arising from measures on its spectrum.

Given an extension $0\to V\to G\to Q\to 1$ of locally compact groups, with $V$ abelian, and a compatible essentially bijective $1$-cocycle $\eta :Q\to \widehat{V}$, we define a dual unitary $2$-cocycle on $G$ and show that the associated deformation of $Ĝ$ is a cocycle bicrossed product defined by a matched pair of subgroups of $Q⋉\widehat{V}$. We also discuss an interpretation of our construction from the point of view of Kac cohomology for matched pairs. Our setup generalizes that of Etingof and Gelaki for finite groups and its extension due to Ben David and Ginosar, as well as our earlier work on locally compact groups satisfying the dual orbit condition. In particular, we get a locally compact quantum group from every involutive nondegenerate set-theoretical solution of the Yang–Baxter equation, or more generally, from every brace structure. On the technical side, the key new points are constructions of an irreducible projective representation of $G$ on $L^2(Q)$ and a unitary quantization map $L^2(G)\to HS(L^2(Q))$ of Kohn–Nirenberg type.