QUANTUM 'ax + b' GROUP

Abstract

'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 [Formula: see text], 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 Classical Limit of Representation Theory of the Quantum Plane

“…This coincides with Woronowicz's construction of the quantum 'ax + b' group [27] using the theory of multiplicative unitaries, restricted to the semigroup setting with B > 0, so that we don't run into the difficulty of the self-adjointness of the coproduct. The multiplicative unitary involved produces the corepresentation of the quantum plane desired, and the corepresentation obtained in this way is shown to have a classical limit towards the unitary representation for the classical group.…”

“…This function and its many variants are being studied [7,19,25] and applied to vast amount of different areas, for example the construction of the 'ax + b' quantum group by Woronowicz et.al. [17,27], the harmonic analysis of the non-compact quantum group U q (sl(2, R)) and its modular double [1,15,16], the q-deformed Toda chains [12] and hyperbolic knot invariants [10]. One of the important properties of this function is its invariance under the duality b ↔ b −1 that provides the basis for the definition of the modular double of U q (sl(2, R)) first introduced by Faddeev [3], and also related, for example, to the self-duality of Liouville theory [15] that has no classical counterpart.…”

The Haagerup property for locally compact quantum groups

“…The locally compact quantum groups quantum E .2/ (see [78]), its dual O E .2/ (see [72,78]), quantum az C b (see [80]) and quantum ax C b (see [81]) have the Haagerup property. Indeed, they are all coamenable, as follows for example from [67, Theorem 3.14] 1) , and the two last examples are self-dual, up to 'reversing the group operation', i.e.…”

On low-dimensional locally compact quantum groups

“…quantum "ax+b"-group considered by S.L. Woronowicz and S. Zakrzevski [66] and by A. Van Daele [55] are defined by the relations…”

“…Kac and the second author [21] and on the other hand by M. Enock and J.-M. Schwartz (for a survey see [12]). However, this theory was not general enough to cover important new examples constructed starting from the eighties [3], [24], [25], [31], [41], [42], [52], [56], [60] - [66], which motivated essential efforts to get a generalization that would cover these examples and that would be as elegant and symmetric as the theory of Kac algebras. Important steps in this direction were made by S. Baaj and G. Skandalis [4], S.L.…”