Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 42 (2001), No. 2, 407-417.

Quantum Half-Planes via Deformation Quantization^{1}

Do Ngoc Diep Nguyen Viet Hai

Institute of Mathematics, National Center for Natural Sciences and Technology, P. O. Box 631, Bo Ho, 10.000, Hanoi, Vietnam e-mail: dndiep@thevinh.ncst.ac.vn

Haiphong Teacher’s Training College Haiphong City, Vietnam e-mail: hainviet@yahoo.com

Abstract. We give an idea of constructing irreducible unitary representations of Lie groups by using Fedosov deformation quantization in the concrete case of the group Aff(ℝ) of affine transformations of the real line. By an exact computation of the star-product and the operator ℓ̂_{Z}, we show that the resulting representations exhausted all the irreducible representations of this groups.

1. Introduction

Quantization normally means a procedure associating to each classical mechanical system some quantum systems, namely in the Heisenberg model or Schrödinger one. More precisely, the usual formulation of a quantization procedure is a correspondence associating to each symplectic manifold (M, ω) a Hilbert space H of so called quantum states and to each classical observable (i.e. each complex-valued function) f a quantum observable (i.e. a normal operator) Q(f), in such a way that the following relations hold

Q(1) = Id_{H}  (1)
[Q(f), Q(g)] = \frac{ℏ}{i}Q(\{f, g\})  (2)

1This work was supported in part by the Vietnam National Foundation for Fundamental Science Research 0138-4821/93 $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} 2001 Heldermann Verlag

408 Do Ngoc Diep, Nguyen Viet Hai: Quantum Half-Planes via Deformation Quantization

To attack this general problem there are some approaches, such as Feynman path integral quantization, pseudo-differential operator quantization, Weyl quantization, geometric quantization, etc. . . . . Following the geometric quantization procedure, at first one restricts oneself to consider the set of observables to be quantized and secondly interpret the geometric quantization procedure operators, see e.g. [4],

Q(f) := f + \frac{ℏ}{i}∇_{ξ_{f}}

as operators up to the second order approximation in powers of ℏ, satisfying the relation (2). From this point of view the so called Fedosov deformation quantization can be viewed as higher order approximates of operators satisfying the relation (2). The last interpretation is the main idea behind deformation quantization. This deformation quantization essentially differs from the geometric quantization initiated by A. Kirillov, B. Kostant and J.-P. Souriau, see [1], [10].

Many mathematicians attempted to construct quantum objects related with classical ones: The first object created was the so called Podles quantum spheres. Interpreting the classical upper half-plane as the principal affine space of the special linear group SL_{2}(ℝ), one introduces the quantum upper half-plane as some C*-algebra generated by some generators and relations. We are concerned with this upper half-plane from another point of view.

It is well-known that co-adjoint orbits are homogeneous symplectic manifolds with respect to the natural Kirillov form on orbits. A natural question is to associate to these orbits some quantum systems, which could be called quantum co-adjoint orbits. In the most general context, some quantum co-adjoint orbits appeared in [1]–[2]. Still it is difficult to calculate exactly the ⋆-product and the corresponding representations in concrete cases. In this paper we give such a construction for the group Aff(ℝ) of affine transformations of the real line. The main difficulty is the fact that in the concrete case, we should find out explicit formulae. This group has only two nontrivial 2-dimensional orbits which are the upper and lower half-planes. We shall use the same notion of star-product, introduced by M. Flato and A. Lichnerowicz, see [1]. Our main result is the fact that by an exact computation we can find out explicit star-product formula and then by using the Fedosov deformation quantization, the full list of irreducible unitary representations of this group. These results show the effectiveness of the Fedosov quantization, which is not known up to now.

We introduce some notations in §2, in particular, the canonical coordinates are found in Proposition 2.1. The operators ℓ̂_{Z} which define the representation of the Lie algebra aff(ℝ) are found in §3. By exponentiating we obtain the corresponding unitary representation of the Lie group Aff_{0}(ℝ) in Theorem 4.2 of §4.

2. Canonical coordinates on the upper half-planes

Recall that the Lie algebra 𝔤 = aff(ℝ) of affine transformations of the real straight line is described as follows, see for example [4]: The Lie group Aff(ℝ) of affine transformations of type

x ∈ ℝ ↦ ax + b, for some parameters a, b ∈ ℝ, a ≠ 0.

It is well-known that this group Aff(ℝ) is a two-dimensional Lie group which is isomorphic to the group of matrices

Aff(ℝ) ≅ \{ ( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) | a, b ∈ ℝ, a ≠ 0\}.

We consider its connected component

G = Aff_{0}(ℝ) = \{( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) | a, b ∈ ℝ, a > 0\}

of identity element. Its Lie algebra

𝔤 = aff(ℝ) ≅ \{( \begin{matrix}α & β \\ 0 & 0 \end{matrix} ) | α, β ∈ ℝ\}

admits a basis of two generators X, Y with the only nonzero Lie bracket [X, Y] = Y, i.e.

𝔤 = aff(ℝ) ≅ {αX + βY | [X, Y] = Y, α, β ∈ ℝ}.

The co-adjoint action of G on 𝔤^{∗} is given (see e.g. [2], [8]) by

⟨K(g)F, Z⟩ = ⟨F, Ad(g^{−1})Z⟩, ∀F ∈ 𝔤^{∗}, g ∈ G and Z ∈ 𝔤.

Denote the co-adjoint orbit of G in 𝔤^{∗}, passing through F by

Ω = K(G)F := {K(g)F | g ∈ G}.

Because the group G = Aff_{0}(ℝ) is exponential (see [4]), for F ∈ 𝔤^{∗} = aff(ℝ)^{∗}, we have

Ω = {K(exp(U)F | U ∈ aff(ℝ)}.

It is easy to deduce that

⟨K(exp U)F, Z⟩ = ⟨F, exp(− ad_{U})Z⟩.

This gives

K(exp U)F = ⟨F, exp(− ad_{U})X⟩X^{∗} + ⟨F, exp(− ad_{U})Y ⟩Y^{∗}.

For a general element U = αX + βY ∈ 𝔤, we have

exp(− ad_{U}) = ∑_{n=0}^{∞} \frac{1}{n!} ( \begin{matrix}0 & 0 \\ β & −α \end{matrix} )^{n} = ( \begin{matrix}1 & 0 \\ L & e^{−α} \end{matrix} ) ,

where L = α + β + \frac{α}{β}(1 − e^{β}). This means that

K(exp U)F = (λ + μL)X^{∗} + (μe^{α})Y^{∗}.

From this formula one deduces from [4] the following description of all co-adjoint orbits of G in 𝔤^{∗}:

• If μ = 0, each point (x = λ, y = 0) on the horizontal axis corresponds to a 0-dimensional

co-adjoint orbit

Ω_{λ} = {λX^{∗}}, λ ∈ ℝ.

• For μ ≠ 0, there are two 2-dimensional co-adjoint orbits: the upper half-plane {(λ, μ) | λ, μ ∈ ℝ, μ > 0} corresponds to the co-adjoint orbit

Ω_{+} := {F = (λ + μL)X^{∗} + (μe^{−α})Y^{∗} | μ > 0},  (3)

and the lower half-plane {(λ, μ) | λ, μ ∈ ℝ, μ < 0} corresponds to the co-adjoint orbit

Ω_{−} := {F = (λ + μL)X^{∗} + (μe^{−α})Y^{∗} | μ < 0}.  (4)

We shall work from now on for the fixed co-adjoint orbit Ω_{+}. The case of the co-adjoint orbit Ω_{−} can be treated similarly. First we study the geometry of this orbit and introduce some canonical coordinates in it. It is well-known from the orbit method [8] that the Lie algebra 𝔤 = aff(ℝ) can be realized as the complete right-invariant Hamiltonian vector fields on co-adjoint orbits Ω ≅ G_{F} \ G with flat (co-adjoint) action of the Lie group G = Aff_{0}(ℝ). On the orbit Ω_{+} we choose a fix point F = Y^{∗}. It is well-known from the orbit method that we can choose an arbitrary point F on Ω. It is easy to see that the stabilizer of this (and therefore of any) point is trivial, G_{F} = {e}. We identify therefore G with G_{Y^{∗}} \ G. There is a natural diffeomorphism Id_{ℝ} × exp(.) from the standard symplectic space ℝ^{2} with symplectic 2-form dp ∧ dq in the canonical Darboux (p, q)-coordinates, onto the upper halfplane ℍ_{+} ≅ ℝ⋊ℝ_{+} with coordinates (p, e^{q}), which is, from the above coordinate description, also diffeomorphic to the co-adjoint orbit Ω_{+}. We can therefore use (p, q) as the standard canonical Darboux coordinates in Ω_{Y^{∗}}. There are also non-canonical Darboux coordinates (x, y) = (p, e^{q}) on Ω_{Y^{∗}}. We show now that in these coordinates (x, y), the Kirillov form looks like ω_{Y^{∗}}(x, y) = \frac{1}{y}dx ∧ dy, but in the canonical Darboux coordinates (p, q), the Kirillov form is just the standard symplectic form dp ∧ dq. This means that there are symplectomorphisms between the standard symplectic space (ℝ^{2}, dp∧dq), the upper half-plane (ℍ_{+}, \frac{1}{y}dx∧dy) and the co-adjoint orbit (Ω_{Y^{∗}}, ω_{Y^{∗}}). Each element Z ∈ 𝔤 can be considered as a linear functional Z̃ on co-adjoint orbits, as subsets of 𝔤^{∗}, Z̃(F) := ⟨F, Z⟩. It is well-known that this linear function is just the Hamiltonian function associated with the Hamiltonian vector field ξ_{Z}, which represents Z ∈ 𝔤 following the formula

(ξ_{Z}f)(x) := \frac{d}{dt}f(x exp(tZ))|_{t=0}, ∀f ∈ C^{∞}(Ω_{+}).

The Kirillov form ω_{F} is defined by the formula

ω_{F} (ξ_{Z}, ξ_{T}) = ⟨F, [Z, T]⟩, ∀Z, T ∈ 𝔤 = aff(ℝ).  (5)

This form defines the symplectic structure and the Poisson brackets on the co-adjoint orbit Ω_{+}. For the derivative along the direction ξ_{Z} and the Poisson bracket we have relation ξ_{Z}(f) = {Z̃, f}, ∀f ∈ C^{∞}(Ω_{+}). It is well-known in differential geometry that the correspondence Z ↦ ξ_{Z}, Z ∈ 𝔤 defines a representation of our Lie algebra by vector fields on co-adjoint orbits. If the action of G on Ω_{+} is flat [4], we have the second Lie algebra homomorphism from strictly Hamiltonian right-invariant vector fields into the Lie algebra of smooth functions on the orbit with respect to the associated Poisson brackets.

Denote by ψ the indicated symplectomorphism from ℝ^{2} onto Ω_{+}

(p, q) ∈ ℝ^{2} ↦ ψ(p, q) := (p, e^{q}) ∈ Ω_{+}

Proposition 2.1. 1. The Hamiltonian function f_{Z} = Z̃ in canonical coordinates (p, q) of the orbit Ω_{+} is of the form

Z̃ ∘ ψ(p, q) = αp + βe^{q}, if Z = ( \begin{matrix}α & β \\ 0 & 0 \end{matrix} ) .

2. In the canonical coordinates (p, q) of the orbit Ω_{+}, the Kirillov form ω_{Y^{∗}} is just the standard form ω = dp ∧ dq.

Proof. 1. Each element F ∈ (aff(ℝ))^{∗} is of the form F = xX^{∗} + yY^{∗}. This means that the value of the function f_{Z} = Z̃ on the element Z = αX + βY is

Z̃(F) = ⟨F, Z⟩ = ⟨xX^{∗} + yY^{∗}, αX + βY ⟩ = αx + βy.

It follows therefore that

Z̃ ∘ ψ(p, q) = αp + βe^{q}, if Z = ( \begin{matrix}α & β \\ 0 & 0 \end{matrix} ) .  (6)

2. In canonical Darboux coordinates (p, q), F = pX^{∗} + e^{q}Y^{∗} ∈ Ω_{+}, and for Z = ( \begin{matrix}α_{1} & β_{1} \\ 0 & 0 \end{matrix} ),

T = ( \begin{matrix}α_{2} & β_{2} \\ 0 & 0 \end{matrix} ), we have

⟨F, [Z, T]⟩ = ⟨pX^{∗} + e^{q}Y^{∗}, (α_{1}β_{2} − α_{2}β_{1})Y ⟩ = (α_{1}β_{2} − α_{2}β_{1})e^{q},

i.e.

ω_{F} (ξ_{Z}, ξ_{T}) = (α_{1}β_{2} − α_{2}β_{1})e^{q}.  (7)

Let us consider two vector fields

ξ_{Z} = α_{1} \frac{∂}{∂q} − β_{1}e^{q} \frac{∂}{∂p} and ξ_{T} = α_{2} \frac{∂}{∂q} − β_{2}e^{q} \frac{∂}{∂p}.

We have

ξ_{Z} ⊗ ξ_{T} = α_{1}α_{2} \frac{∂}{∂q} ⊗ \frac{∂}{∂q} + (α_{1}β_{2} − α_{2}β_{1})e^{q} \frac{∂}{∂p} ⊗ \frac{∂}{∂q} + β_{1}β_{2}e^{2q} \frac{∂}{∂p} ⊗ \frac{∂}{∂p}.  (8)

From (7) and (8) we conclude that in the canonical coordinates the Kirillov form is just the standard symplectic form ω = dp ∧ dq.

3. Computation of generators ℓ̂_{Z}

Let us denote by Λ the 2-tensor associated with the Kirillov standard form ω = dp ∧ dq in canonical Darboux coordinates. We use also the multi-index notation. Let us consider the well-known Moyal ⋆-product of two smooth functions u, v ∈ C^{∞}(ℝ^{2}), defined by

u ⋆ v = u.v + ∑_{r≥1} \frac{1}{r!}(\frac{1}{2i})^{r}P^{r}(u, v),

where

P^{r}(u, v) := Λ^{i_{1}j_{1}}Λ^{i_{2}j_{2}} … Λ^{i_{r}j_{r}}∂_{i_{1}i_{2}...i_{r}}u∂_{j_{1}j_{2}...j_{r}}v,

with

∂_{i_{1}i_{2}...i_{r}} := \frac{∂^{r}}{∂x^{i_{1}} … ∂x^{i_{r}}} , x := (p, q) = (p_{1}, …, p_{n}, q^{1}, …, q^{n})

as multi-index notation. It is well-known that this series converges in the Schwartz distribution spaces S(ℝ^{n}). We apply this to the special case n = 1. In our case we have only x = (x^{1}, x^{2}) = (p, q).

Proposition 3.1. In the above mentioned canonical Darboux coordinates (p, q) on the orbit Ω_{+}, the Moyal ⋆-product satisfies the relation

iZ̃ ⋆ iT̃ − iT̃ ⋆ iZ̃ = i{ \atop [Z, T]}, ∀Z, T ∈ aff(ℝ).

Proof. Consider the elements Z = α_{1}X + β_{1}Y and T = α_{2}X + β_{2}Y. Then as noted above the corresponding Hamiltonian functions are Z̃ = α_{1}p + β_{1}e^{q} and T̃ = α_{2}p + β_{2}e^{q}. It is easy then to see that P^{0}(Z̃, T̃) = Z̃.T̃,

P^{1}(Z̃, T̃) = {Z̃, T̃} = ∂_{p}Z̃∂_{q}T̃ − ∂_{q}Z̃∂_{p}T̃ = (α_{1}β_{2} − α_{2}β_{1})e^{q},

P^{2}(Z̃, T̃) = Λ^{12}Λ^{12}∂_{pp}Z̃∂_{qq}T̃ + Λ^{12}Λ^{21}∂_{pq}Z̃∂_{qp}T̃ + Λ^{21}Λ^{12}∂_{qp}Z̃∂_{pq}T̃ + Λ^{21}Λ^{21}∂_{qq}Z̃∂_{pp}T̃ = 0.

By analogy we have P^{k}(Z̃, T̃) = 0, ∀k ≥ 2. Thus,

iZ̃ ⋆ iT̃ − iT̃ ⋆ iZ̃ = \frac{1}{2i}[P^{1}(iZ̃, iT̃) − P^{1}(iT̃, iZ̃)] = i(α_{1}β_{2} − α_{2}β_{1})e^{q},

on one hand.

On the other hand, because [Z, T] = ZT − T Z = (α_{1}β_{2} − α_{2}β_{1})Y, we have

i{ \atop [Z, T]} = i(α_{1}β_{2} − α_{2}β_{1})e^{q} = iZ̃ ⋆ iT̃ − iT̃ ⋆ iZ̃.

The proposition is hence proved.

Consequently, to each adapted chart ψ in the sense of [2], we associate a G-covariant ⋆-product.

Proposition 3.2. (See [6].) Let ⋆ be a formal differentiable ⋆-product on C^{∞}(M, ℝ), which is covariant under G. Then there exists a representation τ of G in Aut N[[ν]] such that

τ(g)(u ⋆ v) = τ(g)u ⋆ τ(g)v.

Let us denote by F_{p}u the partial Fourier transform of the function u from the variable p to the variable η, i.e.

F_{p}(u)(η, q) := \frac{1}{\sqrt{2π}} ∫_{ℝ} e^{−ipη}u(p, q)dp.

Let us denote by F_{p}^{−1}(u) the inverse Fourier transform.

Lemma 3.3. We have 1. ∂_{p}F_{p}^{−1}(u) = iF_{p}^{−1}(η.u),

2. F_{p}(p.v) = i∂_{η}F_{p}(v),

3. P^{k}(Z̃, F_{p}^{−1}(u)) = (−1)^{k}βe^{q} \frac{∂^{k}F_{p}^{−1}(u)}{∂^{k}p}, with k ≥ 2.

Proof. The first two formulas are well-known from theory of Fourier transforms. We reproduce them to locate notation.

1. ∂_{p}F_{p}^{−1}(u) = ∂_{p}(\frac{1}{\sqrt{2π}} ∫_{ℝ} e^{ipη}u(η, q)dη) = \frac{1}{\sqrt{2π}} ∫_{ℝ} iηe^{ipη}u(η, q)dη = iF_{p}^{−1}(η.u).

2. i∂_{η}F_{p}(v) = i∂_{η}(\frac{1}{\sqrt{2π}} ∫_{ℝ} e^{−ipη}v(p, q)dp = i\frac{1}{\sqrt{2π}} ∫_{ℝ} −ipe^{−ipη}v(p, q)dp = = \frac{1}{\sqrt{2π}} ∫_{ℝ} e^{−ipη}pv(p, q)dp = F_{p}(p.v).

3. Remark that Λ = ( \begin{matrix}0 & −1 \\ 1 & 0 \end{matrix} ) in the standard symplectic Darboux coordinates (p, q) on

the orbit Ω_{+} and we have had Z̃ = αp + βe^{q}, then

P^{2}(Z̃, F_{p}^{−1}(u)) = Λ^{12}Λ^{12}∂_{pp}Z̃∂_{qq}F_{p}^{−1}(u)) + Λ^{12}Λ^{21}∂_{pq}Z̃∂_{qp}F_{p}^{−1}(u)) +
Λ^{21}Λ^{12}∂_{qp}Z̃∂_{pq}F_{p}^{−1}(u)) + Λ^{21}Λ^{21}∂_{qq}Z̃∂_{pp}F_{p}^{−1}(u)) = (−1)^{2}βe^{q}∂_{pp}^{2}F_{p}^{−1}(u)).

By analogy we obtain

P^{k}(Z̃, F_{p}^{−1}(u)) = (−1)^{k}βe^{q}∂_{p...p}^{k}F_{p}^{−1}(u)), ∀k ≥ 3.

The lemma is therefore proved.

For each Z ∈ aff(ℝ), the corresponding Hamiltonian function is Z̃ = αp + βe^{q} and we can consider the operator ℓ_{Z} acting on dense subspace L^{2}(ℝ^{2}, \frac{dpdq}{2π})^{∞} of smooth functions by left ⋆- multiplication by iZ̃, i.e. ℓ_{Z}(u) = iZ̃ ⋆ u. It is then extended to the whole space L^{2}(ℝ^{2}, \frac{dpdq}{2π}). It is easy to see that, because of the relation in Proposition 3.1, the correspondence Z ∈ aff(ℝ) ↦ ℓ_{Z} = iZ̃ ⋆ . is a representation of the Lie algebra aff(ℝ) on the space N[[\frac{i}{2}]] of formal power series in the parameter ν = \frac{i}{2} with coefficients in N = C^{∞}(M, ℝ), see e.g. [6] for more detail.

We study now the convergence of the formal power series. In order to do this, we look at the ⋆-product of iZ̃ as the ⋆-product of symbols and define the differential operators corresponding to iZ̃. It is easy to see that the resulting correspondence is a representation of 𝔤 by pseudo-differential operators.

Proposition 3.4. For each Z ∈ aff(ℝ) and for each compactly supported C^{∞}-function u ∈ C_{c}^{∞}(ℝ^{2}), we have

ℓ̂_{Z}(u) := F_{p} ∘ ℓ_{Z} ∘ F_{p}^{−1}(u) = α(\frac{1}{2}∂_{q} − ∂_{η})u + iβe^{q− \frac{η}{2}} u.

Proof. For each Z ∈ 𝔤 = aff(ℝ), we have

ℓ̂_{Z}(u) := F_{p} ∘ ℓ_{Z} ∘ F_{p}^{−1}(u) = F_{p}(iZ̃ ⋆ F_{p}^{−1}(u)) = iF_{p}(∑_{r≥0} (\frac{1}{2i})^{r} P^{r}(Z̃, F_{p}^{−1}(u)).

Remark that

P^{1}(Z̃, F_{p}^{−1}(u)) = {Z̃, F_{p}^{−1}(u)} = α∂_{q}F_{p}^{−1}(u) − βe^{q}∂_{p}F_{p}^{−1}(u)

and applying Lemma 3.3, we obtain:

iF_{p}( ∑_{r≥0} \frac{1}{r!}(\frac{1}{2i})^{r}P^{r}(Z̃, F_{p}^{−1}(u)) =

= iF_{p}[(αp + βe^{q})F_{p}^{−1}(u) + \frac{1}{2i}α∂_{q}F_{p}^{−1}(u) − \frac{1}{2i}βe^{q}∂_{p}F_{p}^{−1}(u)+
+\frac{1}{2!} (\frac{−1}{2i})^{2} βe^{q}∂_{p}^{2}F_{p}^{−1}(u) + … + \frac{1}{n!} (\frac{−1}{2i})^{n} βe^{q}∂_{p}^{n}F_{p}^{−1}(u) + …]
= i[αi∂_{η}u + βe^{q}u + \frac{1}{2i}α∂_{q}u − \frac{1}{2i}βe^{q}F_{p}(iF_{p}^{−1}(η.u))+
+\frac{1}{2!} (−\frac{1}{2i})^{2} βe^{q}F_{p}(i^{2}F_{p}^{−1}(η^{2}.u)) + … + \frac{1}{n!} (−\frac{1}{2i})^{n} βe^{q}F_{p}(i^{n}F_{p}^{−1}(η^{n}.u)) + …]
= i[iα∂_{η}u + \frac{1}{2i}α∂_{q}u + βe^{q}u − βe^{q} \frac{η}{2}u+
+\frac{1}{2!}βe^{q} (\frac{η}{2})^{2} u + … + \frac{1}{n!}(−1)^{n}βe^{q} (\frac{η}{2})^{n} u + …]
= α(\frac{1}{2}∂_{q} − ∂_{η})u + iβe^{q}[1 − \frac{η}{2} + \frac{1}{2!} (\frac{η}{2})^{2} + … + (−1)^{n} \frac{1}{n!} (\frac{η}{2})^{n} + …]
= α(\frac{1}{2}∂_{q} − ∂_{η})u + iβe^{q− \frac{η}{2}} u.

The proposition is therefore proved.

Remark 3.5. Set s = q − \frac{η}{2}, t = q + \frac{η}{2}, we have

ℓ̂_{Z}(u) = α\frac{∂u}{∂s} + iβe^{s}u, i.e. ℓ̂_{Z} = α\frac{∂}{∂s} + iβe^{s},  (9)

which provides a representation of the Lie algebra aff(ℝ).

4. The associate irreducible unitary representations

Our aim in this section is to exponentiate the obtained representation ℓ̂_{Z} of the Lie algebra aff(ℝ) to the corresponding representation of the Lie group Aff_{0}(ℝ). We shall prove that the result is exactly the irreducible unitary representation T_{Ω_{+}} obtained from the orbit method or Mackey small subgroup method applied to the group Aff(ℝ). Let us recall first the wellknown list of all the irreducible unitary representations of the group of affine transformation of the real line.

Theorem 4.1. [7] Every irreducible unitary representation of the group Aff(ℝ) of all the affine transformations of the real line, up to unitary equivalence, is equivalent to one of the pairwise nonequivalent representations:

• the infinite-dimensional representation S, realized in the space L^{2}(ℝ^{∗}, \frac{dy}{|y|}), where ℝ^{∗} = ℝ \ {0} and defined by the formula

(S(g)f)(y) := e^{iby}f(ay), where g = ( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) ,

• the representation U_{λ}^{ε}, where ε = 0, 1, λ ∈ ℝ, realized in the 1-dimensional Hilbert space ℂ^{1} and given by the formula

U_{λ}^{ε}(g) = |a|^{iλ}(sgn a)^{ε}.

Let us consider now the connected component G = Aff_{0}(ℝ). The irreducible unitary representations can be obtained easily from the orbit method machinery.

Theorem 4.2. The representation exp(ℓ̂_{Z}) of the group G = Aff_{0}(ℝ) is exactly the irreducible unitary representation T_{Ω_{+}} of G = Aff_{0}(ℝ) associated following the orbit method construction, to the orbit Ω_{+}, which is the upper half-plane ℍ ≅ ℝ ⋊ ℝ^{∗}, i.e.

(exp(ℓ̂_{Z})f)(y) = (T_{Ω_{+}}(g)f)(y) = e^{iby}f(ay), ∀f ∈ L^{2}(ℝ^{∗}, \frac{dy}{|y|}),

where g = exp Z = ( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) .

Proof. We choose an admissible Lie sub-algebra 𝔥 = ⟨X⟩. Let us denote by H the corresponding analytic subgroup of G with Lie algebra 𝔥. The corresponding representation is Ind_{H}^{G} χ_{F} = Ind_{H}^{G} χ_{Y^{∗}}. The homogeneous space H \ G is homeomorphic to ℝ^{∗} = ℝ \ {0} with the quasi-invariant measure \frac{dy}{|y|}. The corresponding representation T_{Ω_{+}} is given exactly by the same formula as the representation S in Theorem 4.1. More precisely, for the element

Z = ( \begin{matrix}α & β \\ 0 & 0 \end{matrix} ) ∈ 𝔤 = aff(ℝ),

exp Z = exp ( \begin{matrix}α & β \\ 0 & 0 \end{matrix} ) = ( \begin{matrix}a & b \\ 0 & 0 \end{matrix} ) = \{ \begin{matrix}( \begin{matrix}e^{α} & \frac{β}{α}(e^{α} − 1) \\ 0 & 0 \end{matrix} ) & if α ≠ 0 \\ ( \begin{matrix}1 & β \\ 0 & 1 \end{matrix} ) & if α = 0 \end{matrix}

It is reasonable to simplify the notation, to consider the second case. Remark that because y = e^{q} is the natural but non-canonical coordinate in ℝ^{∗} ≅ H \ G we can write the induced representation obtained from the orbit method construction as

T_{Ω_{+}}(exp Z)f(e^{s}) = exp(i\frac{β}{α}(e^{α} − 1)e^{s})f(e^{α+s}).  (10)

Therefore for the one-parameter subgroup exp(tZ), t ∈ ℝ, the action is given by the formula

T_{Ω_{+}}(exp tZ)f(e^{s}) = exp(i\frac{β}{α}(e^{tα} − 1)e^{s})f(e^{tα+s}).

By a direct computation, we obtain

\frac{∂}{∂t}T_{Ω_{+}}(exp tZ)f(e^{s}) =  (11)
= i\frac{β}{α}e^{s}αe^{tα} exp(i\frac{β}{α}(e^{tα} − 1)e^{s})f(e^{ts+s}) + exp(i\frac{β}{α}(e^{tα} − 1)e^{s})αe^{tα+s}.∂_{s}f
= e^{tα+s} exp(i\frac{β}{α}(e^{tα} − 1)e^{s})[iβf(e^{tα+s}) + α.∂_{s}f],

on one hand. On the other hand, we have

ℓ̂_{Z}T_{Ω_{+}}(exp tz)f(e^{s}) =  (12)
= (iβe^{s} + α.∂_{s})[exp(i\frac{β}{α}(e^{tα} − 1)e^{s})f(e^{tα+s})] = iβe^{s} exp(i\frac{β}{α}(e^{tα} − 1)e^{s})f(e^{tα+s}) +
+ α[i\frac{β}{α}(e^{tα} − 1)e^{s} exp(i\frac{β}{α}(e^{tα} − 1)e^{s})f(e^{tα+s}) + exp(i\frac{β}{α}(e^{tα} − 1)e^{s})e^{tα+s}.∂_{s}f]
= e^{tα+s} exp(i\frac{β}{α}(e^{tα} − 1)e^{s})[iβf(e^{tα+s}) + α.∂_{s}f].

From (11) and (12) follows that

\frac{∂}{∂t}T_{Ω_{+}}(exp(tZ))f(y) = ℓ̂_{Z}T_{Ω_{+}}(exp(tZ))f(y).

Obviously, T_{Ω_{+}}(exp(tZ))f(y)|_{t=0} = f(y). This means that T_{Ω_{+}}(exp(tZ))f(y) is the unique solution of the Cauchy problem

\{ \begin{matrix}\frac{∂}{∂t}U(t, y) & = ℓ̂_{Z}U(t, y) \\ U(0, y) & = Id \end{matrix}

This gives exp(ℓ̂_{Z})f(y) ≡ T_{Ω_{+}}(exp Z)f(y). The proof of the theorem is therefore achieved. By analogy, we have also

Theorem 4.3. The representation exp(ℓ̂_{Z}) of the group G = Aff_{0}(ℝ) is exactly the irreducible unitary representation T_{Ω_{−}} of G = Aff_{0}(ℝ) associated following the orbit method construction, to the orbit Ω_{−}, which is the lower half-plane ℍ ≅ ℝ ⋊ ℝ^{∗}, i.e

(exp(ℓ̂_{Z})f)(y) = (T_{Ω_{−}}(g)f)(y) = e^{iby}f(ay), ∀f ∈ L^{2}(ℝ^{∗}, \frac{dy}{|y|}),

where g = exp Z = ( \begin{matrix}a & b \\ 0 & 1 \end{matrix} ) .

Remark 4.4.

1. We have demonstrated how all the irreducible unitary representations of the connected group of affine transformations could be obtained from deformation quantization. It is reasonable to refer to the algebras of functions on co-adjoint orbits with this ⋆-product as quantum ones.

2. In a forthcoming work, we shall do the same calculation for the group of affine transformations of the complex straight line ℂ. This achieves the description of quantum \overline{MD} co-adjoint orbits, see [4] for definition of \overline{MD} Lie algebras.

References

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[2] Arnal, D.; Cortet, J. C.: Représentations ⋆ des groupes exponentiels. J. Funct. Anal. 92 (1990), 103–135.

[3] Arnold, V. I.: Mathematical Methods of Classical Mechanics. Springer Verlag, Berlin - New York - Heidelberg 1984.

[4] Do Ngoc Diep: Noncommutative Geometry Methods for Group C*-Algebras. CRC Press, #LM 2003, 1999.

[5] Fedosov, B.: Deformation quantization and index theory. Akademie Verlag, Berlin 1996.

[6] Gutt, S.: Deformation quantization. ICTP Workshop on Representation Theory of Lie groups. SMR 686/14, 1993.

[7] Gelfand, I. M.; Naimark, M. A.: Unitary representations of the group of affine transformations of the straight line. Dokl. AN SSSR 55 (1947)(7), 571–574.

[8] Kirillov, A. A.: Elements of the theory of representation. Springer Verlag, Berlin - New York - Heidelberg 1976.

[9] Kirillov, A. A.: Unitary representations of nilpotent Lie groups. Russian Math. Survey, 1962, 17–52.

[10] Kostant, B.: On certain unitary representations which arise from a quantization theory. Lecture Notes in Math. 170 (1970), 237–253.

[11] Meise, R.; Vogt, D.: Introduction to Functional Analysis. Clarendon Press, Oxford 1997.

Received October 10, 1999; revised version December 12, 2000