QUANTUM ‘az + b’ GROUP ON COMPLEX PLANE

‘az + b’ is the group of affine transformations of complex plane C. The coefficients a, b ∈ C. In quantum version a, b are normal operators such that ab = q2ba, where q is the deformation parameter. We shall assume that q is a root of unity. More precisely q = e 2πi N , where N is an even natural number. 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, 18]. Then using the general theory we construct a C∗-algebra A and a comultiplication ∆ ∈ Mor(A, A⊗A). A should be interpreted as the algebra of all continuous functions vanishing at infinity on quantum ‘az+b’group. The group structure of is encoded by ∆. The existence of coinverse also follows from the general theory [18]. In the appendix, we briefly discuss the case of real q. 0. Introduction. The group ‘az + b’ considere in this paper is the group of affine transformations of the complex plane C. The group will be denoted by G. The ∗-algebra A of polynomial functions on G is generated by three normal commuting elements a, a−1, b subject to the one relation: a−1a = I. The comultiplication ∆ encoding the group structure is the ∗-algebra homomorphism from A into A⊗A such that ∆(a) = a⊗ a, ∆(b) = a⊗ b+ b⊗ I. (0.1) One can easily verify that (A,∆) is a Hopf ∗-algebra. In particular counit e and coinverse κ are given by the formulae: e(a) = 1, κ(a) = a−1, e(b) = 0, κ(b) = −a−1b. (0.2) Now we perform quantum deformation of G. The quantum ‘az + b’-group on the level of Hopf ∗-algebra is an object with no problems. The deformation parameter q is a complex number of modulus 1. Let Ao be the ∗-algebra generated by three elements a, a−1, b subject to the following relations: a−1a = aa−1 = 1, aa∗ = a∗a, bb∗ = b∗b, ab = qba, ab∗ = b∗a. (0.3) The comultiplication ∆ : Ao → Ao ⊗ Ao is the ∗-algebra homomorphism acting on generators in the way described in (0.1). In what follows, q is a root of unity of the form