Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Volume 42 (2001), No. 2, 419-430.
Quantum Co-Adjoint Orbits of the Group of
Affine Transformations of the Complex Line1
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@hn.vnn.vn
Haiphong Teacher’s Training College
Haiphong City, Vietnam
e-mail: hainviet@yahoo.com
Abstract. We construct star-products on the co-adjoint orbit of the Lie group
Aff(C) of affine transformations of the complex line and apply them to obtain the
irreducible unitary representations of this group. These results show the effectiveness of the Fedosov quantization even for groups which are neither nilpotent nor
exponential. Together with the result for the group Aff(R) (see [5]), we thus have
a description of quantum MD co-adjoint orbits.
1. Introduction
The notion of ?-products was a few years ago introduced and played a fundamental role in
the basic problem of quantization, see e.g. references [1, 2, 6, 7, . . . ], as a new approach
to quantization on arbitrary symplectic manifolds. In [5] we have constructed star-products
on upper half-plane, obtained the operator `ˆZ , Z ∈ aff(R) = Lie Aff(R) and proved that the
representation
∂
exp(`ˆZ ) = exp(α + iβes )
∂s
1
This work was supported in part by the National Foundation for Fundamental Science research of Vietnam
and the Alexander von Humboldt Foundation, Germany
c 2001 Heldermann Verlag
0138-4821/93 $ 2.50 420
Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits . . .
of the group Aff 0 (R) coincides with the representation TΩ+ obtained from the orbit method
or Mackey small subgroup method. One of the advantages of this group, with which the computation is rather accessible is the fact that its connected component Aff 0 (R) is exponential.
We could use therefore the canonical coordinates for Kirillov form on the orbits. It is natural
to consider the same problem for the group Aff(C). We can expect that the calculations and
final expressions could be similar to the corresponding ones in real line case, but this group
Aff(C) is no more the exponential, i.e. the exponential map
^
exp : aff(C) → Aff(C)
is no longer a global diffeomorphism and the general theory of D. Arnal and J. Cortet (see
[1], [2]) could not be directly applicable. We overcame these difficulties by a rather different
way which could give new ideas for more general non-exponential groups: To overcome the
main difficulty in applying the deformation quantization to this group, we replace the global
diffeomorphism in Arnal-Cortet’s setting by a local diffeomorphism. With this replacement,
we need to pay attention on the complexity of the symplectic Kirillov form in new coordinates.
We then computed the inverse image of the Kirillov form on appropriate local charts. The
question raised here is how to choose a good local chart in order to make the calculation as
simple as possible. The calculation we propose is realized by using complex analysis on very
simple complex domain.
Our main result consists of an explicit star-product formula (Proposition 3.5) on the
local charts. This means that the functional algebras on co-adjoint orbits admit a suitable
deformation, or in other words, we obtained the quantum co-adjoint orbits of this group as
exact models of new quantum objects, called “quantum punctured complex planes” (C2 \L)q .
Then, by using the Fedosov deformation quantization, it is not hard to obtain the list of all
irreducible unitary representations (Theorem 4.2) of the group Aff(C), although the computation in this case, using the Mackey small subgroup method or the modern orbit method,
is rather delicate. The infinitesimal generators of those exact models of infinite dimensional
irreducible unitary representations, nevertheless, are given by rather simple formulae. We
introduce some preliminary result in §2. The operators `ˆA which define the representation of
the Lie algebra aff(C) are found in §3. In particular, we obtain the unitary representations
^ the universal covering group of the Aff(C), in Theorem 4.3 of §4.
of the Lie group Aff(C),
2. Preliminary results
Recall that the Lie algebra g = aff(C) of affine transformations of the complex line is described
as follows (see [4]). It is well-known that the group Aff(C) is a real 4-dimensional Lie group
which is isomorph to the group of matrices:
a
b
Aff(C) ∼
|a, b ∈ C, a 6= 0
=
0 1
The most easy method is to consider X,Y as complex generators, X = X1 +iX2 and Y = Y1 +
iY2 . Then from the relation [X, Y ] = Y , we get [X1 , Y1 ]−[X2 , Y2 ]+i([X1 Y2 ]+[X2 , Y1 ]) = Y1 +
iY2 . This means that the Lie algebra aff(C) is a real 4-dimensional Lie algebra, having four
Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits . . .
421
generators with the only nonzero Lie brackets: [X1 , Y1 ]−[X2 , Y2 ] = Y1 ; [X2 , Y1 ]+[X1 , Y2 ] = Y2
and we can choose another basis denoted again by the same letters such that:
[X1 , Y1 ] = Y1 ; [X1 , Y2 ] = Y2 ; [X2 , Y1 ] = Y2 ; [X2 , Y2 ] = −Y1
Remark 2.1. The exponential map
exp : C −→ C∗ := C\{0}
e∗ ∼
given by z 7→ ez is just the covering map and therefore the universal covering of C∗ : C
= C.
As a consequence one deduces that
^ ∼
Aff(C)
=CnC∼
= {(z, w)|z, w ∈ C}
with the following multiplication law:
(z, w)(z 0 , w0 ) := (z + z 0 , w + ez w0 )
^ in g∗ passing through F ∈ g∗ is denoted by
Remark 2.2. The co-adjoint orbit of Aff(C)
^
^
Ω := K(Aff(C))F
= {K(g)F |g ∈ Aff(C)}.
Then, (see [4]):
1. Each point (α, 0, 0, δ) is 0-dimensional co-adjoint orbit Ω(α,0,0,δ) .
2. The open set β 2 + γ 2 6= 0 is the single 4-dimensional co-adjoint orbit Ω = Ωβ 2 +γ 2 6=0 .
We shall use Ω in the form Ω ∼
= C × C∗ .
Remark 2.3. Set
Hk = {w = q1 + iq2 ∈ C| − ∞ < q1 < +∞; 2kπ < q2 < 2kπ + 2π}; k = 0, ±1, . . .
L = {ρeiϕ ∈ C | 0 < ρ < +∞; ϕ = 0}
and Ck = C\L is a univalent sheet of the Riemann surface of the multi-valued complex
analytic function Ln(w), (k = 0, ±1, . . . ). Then there is a natural diffeomorphism
w ∈ Hk 7−→ ew ∈ Ck with each k = 0, ±1, . . . . Now consider the map:
C × C −→ Ω = C × C∗
(z, w) 7−→ (z, ew ),
with a fixed k ∈ Z. We have a local diffeomorphism
ϕk : C × Hk −→ C × Ck
(z, w) 7−→ (z, ew ).
This diffeomorphism ϕk will be needed in the sequel.
422
Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits . . .
On C × Hk we have the natural symplectic form
1
ωo = [dz ∧ dw + dz ∧ dw],
2
(1)
induced from C2 . Put z = p1 + ip2 , w = q1 + iq2 and (x1 , x2 , x3 , x4 ) = (p1 , q1 , p2 , q2 ) ∈ R4 ,
then
ωo = dp1 ∧ dq1 − dp2 ∧ dq2 .
The corresponding symplectic matrix of ωo is



0
0 −1 0 0
^−1  −1
^  1 0

0
0
 and
=
=
 0
 0 0
0 1 
0 0 −1 0
0
1
0
0
0

0 0
0 0 

0 −1 
1 0
We have therefore the Poisson brackets of functions as follows. With each f, g ∈ C∞ (Ω),
{f, g} = ∧ij
∂f ∂g
∂f ∂g
∂f ∂g
∂f ∂g
∂f ∂g
=
−
−
+
=
i
j
∂x ∂x
∂p1 ∂q1 ∂q1 ∂p1 ∂p2 ∂q2 ∂q2 ∂p2
h ∂f ∂g
∂f ∂g ∂f ∂g
∂f ∂g i
=2
−
+
−
.
∂z ∂w ∂w ∂z ∂z ∂w ∂w ∂z
Proposition 2.4. Fixing the local diffeomorphism ϕk (k ∈ Z), we have:
e in local coor1. For any element A ∈ aff(C), the corresponding Hamiltonian function A
dinates (z, w) of the orbit Ω is of the form
e ◦ ϕk (z, w) = 1 [αz + βew + αz + βew ]
A
2
2. In local coordinates (z, w) of the orbit Ω, the Kirillov form Ω is just the standard form
(1).
Proof. 1. Each element F ∈ Ω ⊂ (Aff(C))∗ is of the form
z 0
∗
w ∗
F = zX + e Y = w
e 0
in local Darboux coordinates (z, w). From this it follows that
e ) = hF, Ai = < tr(F.A) =
A(F
1
αz βz
= < tr
= [αz + βew + αz + βew ]
w
w
αe βe
2
2. Using the definition of the Poisson brackets{, }, associated to a symplectic form ω, we
have
e f} = α
{A,
∂f
∂f
∂f
∂f
− βew
− βew
+α
.
∂w
∂z
∂z
∂w
(2)
Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits . . .
423
Let us from now on denote by ξA the Hamiltonian vector field (symplectic gradient) corresponding to the Hamiltonian function Ã, A ∈ aff(C). Now we consider two vector fields:
∂
∂
∂
∂
∂
∂
∂
∂
− β1 ew
− β1 ew
+ α1
; ξB = α2
− β2 ew
− β2 ew
+ α2
,
∂w
∂z
∂z
∂w
∂w
∂z
∂z
∂w
α1 β1
α2 β2
where A =
; B=
∈ aff(C). It is easy to check that
0 0
0 0
ξA = α1
ξA ⊗ ξB = β1 β2 e2w
∂
∂
∂
∂
∂
∂
∂
∂
⊗
+
⊗
+ α1 α2
⊗
+ α1 α2
⊗
+ β1 β2 e2w
∂w ∂w
∂z ∂z
∂z ∂z
∂w ∂w
∂
∂
∂
∂
∂
∂
⊗
+ (α1 β2 − α2 β1 )ew
⊗
+ (α1 β2 − α2 β1 )ew
⊗
+
∂z ∂w
∂z ∂w
∂z ∂w
∂
∂
∂
∂
∂
∂
⊗
.
⊗
+ (β1 β2 − β1 β2 )ew+w
⊗
+ (α1 α2 − α1 α2 )
+(α1 β2 − α2 β1 )ew
∂w ∂w
∂z ∂w
∂z ∂z
Thus,
i
1h
hω, ξA ⊗ ξB i = (α1 β2 − α2 β1 )ew + (α1 β2 − α2 β1 )ew = < tr(F.[A, B]) = hF, [A, B]i.
2
Thus the proposition is proved.
+(α1 β2 − α2 β1 )ew
(k)
3. Computation of operators `ˆA
Proposition 3.1. With A, B ∈ aff(C), the Moyal ?-product satisfies the relation:
^
e ? iB
e − iB
e ? iA
e = i[A,
iA
B].
(3)
Proof. Consider two arbitrary elements A = α1 X + β1 Y ; B = α2 X + β2 Y . Then the
corresponding Hamiltonian functions are:
e = 1 [α1 z + β1 ew + α1 z + β1 ew ]; B
e = 1 [α2 z + β2 ew + α2 z + β2 ew ].
A
2
2
It is easy, then, to see that:
e B)
e = A.
eB
e
P 0 (A,
h
i
e B)
e = {A.
e B}
e = 2 ∂ Ã ∂ B̃ − ∂ Ã ∂ B̃ + ∂ Ã ∂ B̃ − ∂ Ã ∂ B̃
P 1 (A,
∂z ∂w ∂w ∂z
∂z ∂w ∂w ∂z
h
i
1
= (α1 β2 − α2 β1 )ew + (α1 β2 − α2 β1 )ew
2
and P r (Ã, B̃) = 0, ∀r ≥ 2.
Thus,
h
i
e ? iB
e − iB
e ? iA
e = 1 P 1 (iA,
e iB)
e − P 1 (iB,
e iA)
e =
iA
2i
h
i
i
= (α1 β2 − α2 β1 )ew + (α1 β2 − α2 β1 )ew
2
424
Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits . . .
on one hand. On the other hand, because of [A, B] = (α1 β2 − α2 β1 )Y we have
]
i[A,
B] = ihF, [A, B]i =
i
ih
(α1 β2 − α2 β1 )ew + (α1 β2 − α2 β1 )ew
2
The proposition is hence proved.
For each A ∈ aff(C), the corresponding Hamiltonian function is
e = 1 [αz + βew + αz + βew ]
A
2
(k)
and we can consider the operator `A acting on the dense subspace of smooth functions
e i.e. `(k) (f ) = iA
e ? f.
L2 (R2 × (R2 )∗ , dp1 dq1 dp2 dq2 /(2π)2 )∞ by left ?-multiplication by iA,
A
Because of the relation in Proposition 3.1, we have
Corollary 3.2.
(k)
`[A,B]
=
(k)
`A
◦
(k)
`B
−
(k)
`B
◦
(k)
`A
h
:=
(k) (k)
`A , `B
i
(4)
(k)
From this it is easy to see that the correspondence A ∈ aff(C)
7−→ `A = ià ? . is a
representation of the Lie algebra aff(C) on the space C∞ (Ω) [ 2i ] of formal power series, see
[7] for more detail.
Now, let us denote Fz (f ) the partial Fourier transform of the function f from the variable
z = p1 + ip2 to the variable ξ = ξ1 + iξ2 , i.e.
ZZ
1
e−i<(ξz) f (z, w)dp1 dp2 .
Fz (f )(ξ, w) =
2π
2
R
Let us denote by
Fz−1 (f )(z, w)
1
=
2π
ZZ
ei<(ξz) f (ξ, w)dξ1 dξ2
R2
the inverse Fourier transform.
Lemma 3.3. Putting g = g(z, w) = Fz−1 (f )(z, w) we obtain:
r
1. ∂z g = 2i ξg ; ∂zr g = ( 2i ξ) g, r = 2, 3, . . .
r
2. ∂z g = 2i ξg ; ∂zr g = ( 2i ξ) g, r = 2, 3, . . .
3. Fz (zg) = 2i∂ξ Fz (g) = 2i∂ξ f ; Fz (zg) = 2i∂ξ Fz (g) = 2i∂ξ f
4. ∂w g = ∂w (Fz−1 (f )) = Fz −1 (∂w f ); ∂w g = ∂w (Fz−1 (f ) = Fz −1 (∂w f )
Proof. As ∂z = 21 (∂p1 − i∂p2 );
∂z = 12 (∂p1 + i∂p2 ) we obtain 1. and 2.
3. We have Fz (zg) =
ZZ
ZZ
1
1
−i(p1 ξ1 +p2 ξ2 )
=
e
p1 g(z, w)dp1 dp2 + i
e−i(p1 ξ1 +p2 ξ2 ) p2 g(z, w)dp1 dp2 =
2π
2π
= i∂ξ1 Fz (g) + i2 ∂ξ2 Fz (g) = (i∂ξ1 − ∂ξ2 )Fz (g) = 2i∂ξ Fz (g) = 2i∂ξ f
Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits . . .
425
and Fz (zg) =
ZZ
ZZ
1
1
−i(p1 ξ1 +p2 ξ2 )
=
e
p1 g(z, w)dp1 dp2 − i
e−i(p1 ξ1 +p2 ξ2 ) p2 g(z, w)dp1 dp2 = 2i∂ξ f.
2π
2π
4. The proof is straightforward. The Lemma 3.3 is therefore proved.
We also need another lemma which will be used in the sequel.
Lemma 3.4. With g = Fz−1 (f )(z, w), we have:
e g)) = i(α∂ + α∂ξ )f + 1 βew f + 1 βew f.
1. Fz (P 0 (A,
ξ
2
2
e g)) = α∂w f + α∂w f − βew ( i ξ)f − βew ( i ξ)f.
2. Fz (P (A,
2
2
e g)) = (−1)r .2r−1 [βew ( i ξ)r + βew ( i ξ)r ]f
∀r ≥ 2.
3. Fz (P r (A,
2
2
1
e g) = A.g
e = 1 [αzg + βew g + αzg + βew g].
Proof. 1. Applying Lemma 3.3 we obtain P 0 (A,
2
Thus,
e g)) = 1 [αFz (zg) + βew Fz (g) + αFz (zg) + βew Fz (g)] =
Fz (P 0 (A,
2
1
1
1
[2iα∂ξ Fz (g) + 2iα∂ξ Fz (g) + βew Fz (g) + βew Fz (g)] = i(α∂ξ + α∂ξ )f + βew f + βew f.
2
2
2
2.
e g)) = ∧12 ∂p1 A∂
e q1 g + ∧21 ∂q1 A∂
e p1 g + ∧34 ∂p2 A∂
e q2 g + ∧43 ∂q2 A∂
e p2 g =
(P 1 (A,
= α∂w g + α∂w g − βew ∂z g − βew ∂z g.
e g)) =
This implies that: Fz (P 1 (A,
= α∂w Fz (g) + α∂w Fz (g) − βew ∂z Fz (g) − βew ∂z Fz (g) =
i
i
= α∂w f + α∂w f − βew ( ξ)f − βew ( ξ)f.
2
2
3.
e g) = ∧21 ∧21 ∂q1 q1 A∂
e p1 p1 g + ∧21 ∧43 ∂q1 q2 A∂
e p1 p2 g + ∧43 ∧21 ∂q2 q1 A∂
e p2 p1 g +
P 2 (A,
e p2 p2 g = 1 (βew + βew − βew + βew + βew − βew + βew + βew )∂ 2 g +
+ ∧43 ∧43 ∂q2 q2 A∂
z
2
+ (βew + βew + βew − βew − βew + βew + βew + βew )∂z2 g = 2βew ∂z2 g + 2βew ∂z2 g.
This implies also that:
e g)) = 2βew Fz (∂ 2 g) + 2βew Fz (∂ 2 g) = 2βew ( i ξ)2 f + 2βew ( i ξ)2 f.
Fz (P 2 (A,
z
z
2
2
By analogy,
e g) = (−1)3 [4βew ∂z3 g + 4βew ∂z3 g],
P 3 (A,
e g)) = (−1)3 .22 [βew ( i ξ)3 f + βew ( i ξ)3 f ],
Fz (P 3 (A,
2
2
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Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits . . .
and with r ≥ 4
e g) = (−1)r .2r−1 [βew ∂zr g + βew ∂zr g],
P r (A,
e g)) = (−1)r .2r−1 [βew ( i ξ)r + βew ( i ξ)r ]f.
Fz (P r (A,
2
2
Lemma 3.4 is therefore proved.
α β
Proposition 3.5. For each A =
∈ aff(C) and for each compactly supported C∞ 0 0
(k)
(k)
function f ∈ Cc∞ (C × Hk ), we have: `A f := Fz ◦ `A ◦ Fz−1 (f ) =
1
1
1
i
1
= [α( ∂w − ∂ξ )f + α( ∂w − ∂ξ )f + (βew− 2 ξ + βew− 2 ξ )f ].
2
2
2
(5)
Proof. Applying Lemma 3.4, we have:
(k)
e ? Fz−1 (f )) = i
`A (f ) := FZ (iA
X1 1
e Fz−1 (f )) =
( )r Fz P r (A,
r! 2i
r≥0
n
1
1 1
i
1
= i [i(α∂ξ + α∂ξ )f + βew f + βew f ] + ( )[α∂w f + α∂w f − βew ( ξ)f −
2
2
1! 2i
2
2
i
1 −1
i
i
−βew ( ξ)f ] + ( )2 2[βew ( ξ)2 f + βew ( ξ) f ] + · · · +
2
2! 2i
2
2
o
r
r
i
1 −1
i
+ ( )r 2r−1 [βew ( ξ) f + βew ( ξ) f ] + . . .
2
2
r! 2i
nh
1
1 w 1 w 1 w 1
1
1 i
= −(α∂ξ − α∂ξ )f + (α∂w + α∂w )f + i
βe + βe − βe ( ξ) − βew ( ξ) f +
2
2
2
2
2
2
2
h
h
o
2
2i
r
ri
1 1
−1
−1
11
−1
−1
+ . βew ( ξ) + βew ( ξ) f + · · · +
βew ( ξ) + βew ( ξ) f + . . .
2 2!
2
2
2 k!
2
2
h 1
i
1
1
1
i
i
= α( ∂w − ∂ξ ) + α( ∂w − ∂ξ ) + βew e− 2 ξ + βew e− 2 ξ f
2
2
2
2
h 1
i
1
1
1
i
= α( ∂w − ∂ξ ) + α( ∂w − ∂ξ ) + (βew− 2 ξ + βew− 2 ξ ) f
2
2
2
The proposition is therefore proved.
Remark 3.6. Set u=w − 12 ξ;v = w + 12 ξ we obtain
∂f
∂f
i
(k)
`ˆA (f ) = α
+α
+ (βeu + βeu )f |(u,v) ,
∂u
∂u 2
∂
∂
i
(k)
`ˆA = α
+α
+ (βeu + βeu ),
∂u
∂u 2
which provides a (local) representation of the Lie algebra aff(C).
i.e.
(6)
Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits . . .
427
^
4. The irreducible representations of Aff
(C)
(k)
Since `ˆA is a representation of the Lie algebra aff(C), we have:
(k)
exp(`ˆA ) = exp α
i
∂
∂
+α
+ (βeu + βeu )
∂u
∂u 2
is just the corresponding representation of the corresponding connected and simply con^ Let us first recall the well-known list of all the irreducible unitary
nected Lie group Aff(C).
representations of the group of affine transformations of the complex line, see [4] for more
details.
^
Theorem 4.1. Up to unitary equivalence, every irreducible unitary representation of Aff(C)
is unitarily equivalent to one of the following one-to-another nonequivalent irreducible unitary
representations:
1. The unitary characters of the group, i.e. the one-dimensional unitary representation
Uλ , λ ∈ C, acting in C following the formula
^ λ ∈ C.
Uλ (z, w) = ei<(zλ) , ∀(z, w) ∈ Aff(C),
2. The infinite dimensional irreducible representations Tθ , θ ∈ S1 , acting on the Hilbert
space L2 (R × S1 ) following the formula:
h
i
=(x + z) ]) f (x ⊕ z),
Tθ (z, w)f (x) = exp i(<(wx) + 2πθ[
2π
(7)
^ x ∈ R × S1 = C\{0}; f ∈ L2 (R × S1 );
where (z, w) ∈ Aff(C);
x ⊕ z = <(x + z) + 2πi{
=(x + z)
}.
2π
In this section we will prove the following important theorem which is of interest for us both
in theory and practice.
(k)
^ is the irreducible unitary
Theorem 4.2. The representation exp(`ˆA ) of the group Aff(C)
^ associated to Ω by the orbit method, i.e.
representation Tθ of the group Aff(C)
(k)
exp(`ˆA )f (x) = [Tθ (exp A)f ](x),
α β
2
1
where f ∈ L (R × S ); A =
∈ aff(C); θ ∈ S1 ; k = 0, ±1, . . .
0 0
Proof. Putting x = eu ∈ C\{0} = R × S1 and recall that
a b
α β
= exp(A) = exp
,
0 1
0 0
428
Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits . . .
we can rewrite (7) as following:
eα − 1 u
=eu+α u⊕α
[Tθ (exp A)f ](e ) = exp i(<(
βe ) + 2πθ[
]) f (e ),
α
2π
u
where
=(u + α)
=(u + α)
} = u + α − 2πi[
].
2π
2π
Therefore, for the one-parameter subgroup exp tA, t ∈ R, we have the action formula:
u ⊕ α = <(u + α) + 2πi{
u
etα − 1 u
=eu+tα u⊕tα
Tθ (exp tA)f (e ) = exp i(<
βe + 2πθ[
]) f (e
).
α
2π
By a direct computation:
∂
[Tθ (exp tA)f ](eu ) =
∂t
(8)
=(u+tα)
∂ i etα − 1 u etα − 1 u
=eu+tα =
exp (
βe +
] + f (eu+tα−2πi[ 2π ] )
βe ) + 2πθi[
∂t
2
α
α
2π
tα
tα
u+tα =(u+tα)
=e
∂
i e −1 u e −1 u
+ exp (
βe +
βe ) + 2πθi[
]
f (eu+tα−2πi[ 2π ] ) =
2
α
α
2π
∂t
tα
=eu+tα u⊕tα ∂f
i
e −1 u
βe ) + 2πθi[
] αe
= (βeu+tα + βeu+tα) Tθ (exp tA)f (eu ) + exp i(<(
α
2π
∂u
2
on one hand. On the other hand, we have:
(k)
`ˆA ([Tθ (exp tA)f ](eu ) =
(9)
i
∂
∂
[Tθ (exp tA)f ](eu ) + α
[Tθ (exp tA)f ](eu ) + (βeu + βeu ) Tθ (exp tA)f ](eu ) =
∂u
∂u
2
tα
tα
i e −1 u
e −1 u
=eu+tα u⊕tα
=α (
βe ) exp i(<(
βe ) + 2πθ[
] f (e
)+
2
α
α
2π
etα − 1 u
=eu+tα u⊕tα ∂f
+α exp i(<(
βe ) + 2πθ[
] e
+
α
2π
∂u
i etα − 1 u
etα − 1 u
=eu+tα u⊕tα
+α (
βe ) exp i(<(
βe ) + 2πθ[
] f (e
)+
2
α
α
2π
i
i
+ (βeu + βeu )[Tθ (exp tA)f ](eu ) = (βeu+tα + βeu+tα )[Tθ (exp tA)f ](eu )+
2
2
tα
e −1 u
=eu+tα u⊕αt ∂f
+ exp i(<(
βe ) + 2πθ[
] αe
α
2π
∂u
(8) and (9) imply that
=α
∂
(k)
[Tθ (exp tA)f ](x) = `ˆA [Tθ (exp tA)f ](x)
∂t
∀x ∈ R × S1 .
Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits . . .
429
Furthermore,
Tθ (exp tA)f ](eu )|t=0 = exp(2πi[
=u
=eu
]θ)f (eu−2πi[ 2π ] ) = f (eu ).
2π
(k)
This means that: exp(`ˆA )f (x) and [Tθ (exp tA)f ](x) together are the solution of the Cauchy
problem
∂
(k)
u(t, x) = `ˆA u(t, x);
∂t
u(0, x) = id.
(k)
The operator `ˆA is sufficiently well-behaved, so that the Cauchy problem has a unique
solution. From this uniqueness we deduce that
(k)
exp(`ˆA )f (x) ≡ [Tθ (exp tA)f ](x)
The theorem is hence proved.
∀x ∈ R × S1 .
Remark 4.3. We say that a real Lie algebra g is in the class MD iff every K-orbit is of
dimension 0 or dim g. Furthermore, it was proven in [4, Theorem 4.4] that, up to isomorphism,
every Lie algebra of class MD is one of the following:
1. commutative Lie algebras,
2. the Lie algebra aff(R) of affine transformations of the real line,
3. the Lie algebra of affine transformations of the complex line.
Thus, by calculation for the group of affine transformations of the real line Aff(R)) in [5]
and here for the group of affine transformations of the complex line Aff(C) we obtained a
description of the quantum MD co-adjoint orbits.
References
[1] Arnal, D.; Cortet, J. C.: ?-product and representations of nilpotent Lie groups. J. Geom.
Phys. 2 (1985)(2), 86–116.
[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. Chapman
& Hall/CRC. Press #LM 2003, 1999.
[5] Do Ngoc Diep; Nguyen Viet Hai: Quantum half-plane via Deformation Quantization.
Math. QA/9905002, 2 May 1999.
[6] Fedosov, B.: Deformation Quantization and Index Theory. Akademie Verlag, Berlin
1996.
[7] Gutt, S.: Deformation quantization. ICTP Workshop on Representation Theory of Lie
groups. SMR 686/14, 1993.
430
Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-Adjoint Orbits . . .
[8] Gel’fand, 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.
[9] Kirillov, A. A.: Elements of the Theory of Representations. Springer Verlag, Berlin New York - Heidelberg 1976.
Received October 10, 1999; revised version December 12, 2000