Quantum Co-Adjoint Orbits of the Group of Affine Transformations of the Complex Line 1

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1 Beiträge zur lgebra und Geometrie Contributions to lgebra and Geometry Volume 4 (001, No., Quantum Co-djoint Orbits of the Group of ffine Transformations of the Complex Line 1 Do Ngoc Diep Nguyen Viet Hai Institute of Mathematics, National Center for Natural Sciences and Technology, P. O. Box 631, Bo Ho, , Hanoi, Vietnam dndiep@hn.vnn.vn Haiphong Teacher s Training College Haiphong City, Vietnam hainviet@yahoo.com bstract. We construct star-products on the co-adjoint orbit of the Lie group ff(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 ff(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,, 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 ˆl Z, Z aff(r = Lie ff(r and proved that the representation exp(ˆl Z = exp( s + iβes 1 This work was supported in part by the National Foundation for Fundamental Science research of Vietnam and the lexander von Humboldt Foundation, Germany /93 $.50 c 001 Heldermann Verlag

2 40 Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-djoint Orbits... of the group ff 0 (R coincides with the representation T Ω+ 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 ff 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 ff(c. We can expect that the calculations and final expressions could be similar to the corresponding ones in real line case, but this group ff(c is no more the exponential, i.e. the exponential map exp : aff(c Ãff(C is no longer a global diffeomorphism and the general theory of D. rnal and J. Cortet (see [1], [] 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 rnal-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 (C \L q. Then, by using the Fedosov deformation quantization, it is not hard to obtain the list of all irreducible unitary representations (Theorem 4. of the group ff(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. The operators ˆl which define the representation of the Lie algebra aff(c are found in 3. In particular, we obtain the unitary representations of the Lie group Ãff(C, the universal covering group of the ff(c, in Theorem 4.3 of 4.. 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 ff(c is a real 4-dimensional Lie group which is isomorph to the group of matrices: ff(c = {( a b 0 1 } a, b C, a 0 The most easy method is to consider X,Y as complex generators, X = X 1 +ix and Y = Y 1 + iy. Then from the relation [X, Y ] = Y, we get [X 1, Y 1 ] [X, Y ]+i([x 1 Y ]+[X, Y 1 ] = Y 1 + iy. This means that the Lie algebra aff(c is a real 4-dimensional Lie algebra, having four

3 Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-djoint Orbits generators with the only nonzero Lie brackets: [X 1, Y 1 ] [X, Y ] = Y 1 ; [X, Y 1 ]+[X 1, Y ] = Y and we can choose another basis denoted again by the same letters such that: [X 1, Y 1 ] = Y 1 ; [X 1, Y ] = Y ; [X, Y 1 ] = Y ; [X, Y ] = Y 1 Remark.1. The exponential map exp : C C := C\{0} given by z e z is just the covering map and therefore the universal covering of C : C = C. s a consequence one deduces that with the following multiplication law: Ãff(C = C C = {(z, w z, w C} (z, w(z, w := (z + z, w + e z w Remark.. The co-adjoint orbit of Ãff(C in g passing through F g is denoted by Then, (see [4]: Ω := K(Ãff(CF = {K(gF g Ãff(C}. 1. Each point (, 0, 0, δ is 0-dimensional co-adjoint orbit Ω (,0,0,δ.. The open set β + γ 0 is the single 4-dimensional co-adjoint orbit Ω = Ω β +γ 0. We shall use Ω in the form Ω = C C. Remark.3. Set H k = {w = q 1 + iq C < q 1 < + ; kπ < q < kπ + }; k = 0, ±1,... L = {ρe iϕ C 0 < ρ < + ; ϕ = 0} and C k = 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 H k e w C k with each k = 0, ±1,.... Now consider the map: C C Ω = C C (z, w (z, e w, with a fixed k Z. We have a local diffeomorphism ϕ k : C H k C C k (z, w (z, e w. This diffeomorphism ϕ k will be needed in the sequel.

4 4 Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-djoint Orbits... On C H k we have the natural symplectic form ω o = 1 [dz dw + dz dw], (1 induced from C. Put z = p 1 + ip, w = q 1 + iq and (x 1, x, x 3, x 4 = (p 1, q 1, p, q R 4, then ω o = dp 1 dq 1 dp dq. The corresponding symplectic matrix of ω o is = and 1 = We have therefore the Poisson brackets of functions as follows. With each f, g C (Ω, ij f g {f, g} = x i x = f g f g f g + f g = j p 1 q 1 q 1 p 1 p q q p [ f = z g w f g w z + f z g w f w g ]. z à in local coor- Proposition.4. Fixing the local diffeomorphism ϕ k (k Z, we have: 1. For any element aff(c, the corresponding Hamiltonian function dinates (z, w of the orbit Ω is of the form à ϕ k (z, w = 1 [z + βew + z + βe w ]. 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 (ff(c is of the form ( z 0 F = zx + e w Y = e w 0 in local Darboux coordinates (z, w. From this it follows that Ã(F = F, = R tr(f. = ( z βz = R tr e w βe w = 1 [z + βew + z + βe w ]. Using the definition of the Poisson brackets{, }, associated to a symplectic form ω, we have f f {Ã, f} = βew w z βew f z + f w. (

5 Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-djoint Orbits Let us from now on denote by ξ the Hamiltonian vector field (symplectic gradient corresponding to the Hamiltonian function Ã, aff(c. Now we consider two vector fields: ξ = 1 w β 1e w z β 1e w z + 1 w ; ξ B = w β e w z β e w z + w, ( ( 1 β where = 1 β ; B = aff(c. It is easy to check that ξ ξ B = β 1 β e w z z + 1 w w + β 1β e w z z + 1 w w + +( 1 β β 1 e w z w + ( 1β β 1 e w z w + ( 1β β 1 e w z w + +( 1 β β 1 e w z w + (β 1β β 1 β e w+w z z + ( 1 1 w w. Thus, ω, ξ ξ B = 1 ] [( 1 β β 1 e w + ( 1 β β 1 e w = R tr(f.[, B] = F, [, B]. Thus the proposition is proved. 3. Computation of operators ˆl (k Proposition 3.1. With, B aff(c, the Moyal -product satisfies the relation: iã i B i B iã = i [, B]. (3 Proof. Consider two arbitrary elements = 1 X + β 1 Y ; B = X + β Y. Then the corresponding Hamiltonian functions are: à = 1 [ 1z + β 1 e w + 1 z + β 1 e w ]; B = 1 [ z + β e w + z + β e w ]. It is easy, then, to see that: P 0 (Ã, B = Ã. B P 1 (Ã, B = {Ã. B} [ à = B z w à B w z + à B z w à w = 1 ] [( 1 β β 1 e w + ( 1 β β 1 e w and P r (Ã, B = 0, r. Thus, iã i B i B iã = 1 [P 1 i (iã, i B P 1 (i B, ] iã = = i ] [( 1 β β 1 e w + ( 1 β β 1 e w B ] z

6 44 Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-djoint Orbits... on one hand. On the other hand, because of [, B] = ( 1 β β 1 Y we have i[ã, B] = i F, [, B] = i ] [( 1 β β 1 e w + ( 1 β β 1 e w The proposition is hence proved. For each aff(c, the corresponding Hamiltonian function is à = 1 [z + βew + z + βe w ] and we can consider the operator l (k acting on the dense subspace of smooth functions L (R (R, dp 1 dq 1 dp dq /( by left -multiplication by iã, i.e. l(k (f = iã f. Because of the relation in Proposition 3.1, we have Corollary 3.. [ ] l (k [,B] = l(k l(k B l(k B l(k := l (k, l(k B (4 From this it is easy to see that the correspondence aff(c l (k = iã. is a representation of the Lie algebra aff(c on the space C (Ω [ [ i ]] of formal power series, see [7] for more detail. Now, let us denote F z (f the partial Fourier transform of the function f from the variable z = p 1 + ip to the variable ξ = ξ 1 + iξ, i.e. F z (f(ξ, w = 1 Let us denote by Fz 1 (f(z, w = 1 the inverse Fourier transform. R e ir(ξz f(z, wdp 1 dp. R e ir(ξz f(ξ, wdξ 1 dξ Lemma 3.3. Putting g = g(z, w = Fz 1 (f(z, w we obtain: 1. z g = i ξg ; r zg = ( i ξr g, r =, 3,.... z g = i ξg ; r zg = ( i ξr g, r =, 3, F z (zg = i ξ F z (g = i ξ f ; F z (zg = i ξ F z (g = i ξ f 4. w g = w (Fz 1 (f = F 1 z ( w f; w g = w (Fz 1 (f = F 1 z ( w f Proof. s z = 1 ( p 1 i p ; z = 1 ( p 1 + i p we obtain 1. and. 3. We have F z (zg = = 1 e i(p 1ξ 1 +p ξ p 1 g(z, wdp 1 dp + i 1 e i(p 1ξ 1 +p ξ p g(z, wdp 1 dp = = i ξ1 F z (g + i ξ F z (g = (i ξ1 ξ F z (g = i ξ F z (g = i ξ f

7 Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-djoint Orbits and F z (zg = = 1 e i(p 1ξ 1 +p ξ p 1 g(z, wdp 1 dp i 1 e i(p 1ξ 1 +p ξ p g(z, wdp 1 dp = i ξ f. 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: 1. F z (P 0 (Ã, g = i( ξ + ξf + 1 βew f + 1 βew f.. F z (P 1 (Ã, g = wf + w f βe w ( i ξf βew ( i ξf. 3. F z (P r (Ã, g = ( 1r. r 1 [βe w ( i ξr + βe w ( i ξr ]f r. Proof. 1. pplying Lemma 3.3 we obtain P 0 (Ã, g = Ã.g = 1 [zg + βew g + zg + βe w g]. Thus, F z (P 0 (Ã, g = 1 [F z(zg + βe w F z (g + F z (zg + βe w F z (g] = 1 [i ξ F z(g + i ξ F z (g + βe w F z (g + βe w F z (g] = i( ξ + ξ f + 1 βew f + 1 βew f.. (P 1 (Ã, g = 1 p1 à q1 g + 1 q1 à p1 g + 34 p à q g + 43 q à p g = 1 This implies that: F z (P (Ã, g = = w g + w g βe w z g βe w z g. = w F z (g + w F z (g βe w z F z (g βe w z F z (g = = w f + w f βe w ( i ξf βew ( i ξf. 3. P (Ã, g = 1 1 q1 q 1 à p1 p 1 g q1 q à p1 p g q q 1 à p p 1 g q q à p p g = 1 [ (βe w + βe w βe w + βe w + βe w βe w + βe w + βe w zg + + (βe w + βe w + βe w βe w βe w + βe w + βe w + βe w zg ] = βe w zg + βe w zg. This implies also that: F z (P (Ã, g = βew F z ( zg + βe w F z ( zg = βe w ( i ξ f + βe w ( i ξ f. By analogy, P 3 (Ã, g = ( 13 [4βe w 3 zg + 4βe w 3 zg], F z (P 3 (Ã, g = ( 13. [βe w ( i ξ3 f + βe w ( i ξ3 f],

8 46 Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-djoint Orbits... and with r 4 P r (Ã, g = ( 1r. r 1 [βe w r zg + βe w r zg], F z (P r (Ã, g = ( 1r. r 1 [βe w ( i ξr + βe w ( i ξr ]f. Lemma 3.4 is therefore proved. ( β Proposition 3.5. For each = aff(c and for each compactly supported C function f Cc (C H k, we have: l (k f := F z l (k F z 1 (f = = [( 1 w ξ f + ( 1 w ξ f + i (βew 1 ξ + βe w 1 ξ f]. (5 Proof. pplying Lemma 3.4, we have: l (k 1 r! ( 1 ( i r F z P r 1 (Ã, Fz (f = (f := F Z(ià F z 1 (f = i r 0 { = i [i( ξ + ξ f + 1 βew f + 1 βew f] + 1 1! ( 1 i [ wf + w f βe w ( i ξf βe w ( i ξf] + 1! ( 1 i [βe w ( i ξ f + βe w ( i ξ f] r! ( 1 i r r 1 [βe w ( i r ξ f + βe w ( i } rf] ξ +... = ( ξ ξ f + 1 {[ 1 ( w + w f + i βew + 1 βew 1 βew ( 1 ξ 1 βew ( 1 ] ξ f ! [ βe w ( 1 ξ + βe w ( 1 = = ]f ξ [ βe w ( 1 r k! ξ + βe w ( 1 r] ξ [( 1 w ξ + ( 1 w ξ + i βew e 1 ξ + i ] βew e 1 ξ f [ ( 1 w ξ + ( 1 w ξ + i ] 1 (βew ξ + βe w 1 ξ f } f +... The proposition is therefore proved. Remark 3.6. Set u=w 1ξ;v = w + 1 ξ we obtain ˆl (k (f = f u + f u + i (βeu + βe u f (u,v, i.e. (6 ˆl (k = u + u + i (βeu + βe u, which provides a (local representation of the Lie algebra aff(c.

9 Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-djoint Orbits The irreducible representations of ff(c Since ˆl (k is a representation of the Lie algebra aff(c, we have: exp(ˆl (k = exp( u + u + i (βeu + βe u is just the corresponding representation of the corresponding connected and simply connected Lie group Ãff(C. Let us first recall the well-known list of all the irreducible unitary 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 Ãff(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 U λ, λ C, acting in C following the formula U λ (z, w = e ir(zλ, (z, w Ãff(C, λ C.. The infinite dimensional irreducible representations T θ, θ S 1, acting on the Hilbert space L (R S 1 following the formula: [ ] ( I(x + z T θ (z, wf (x = exp i(r(wx + θ[ ] f(x z, (7 where (z, w Ãff(C; x R S1 = C\{0}; f L (R S 1 ; x z = R(x + z + i{ I(x + z }. In this section we will prove the following important theorem which is of interest for us both in theory and practice. Theorem 4.. The representation exp(ˆl (k of the group Ãff(C is the irreducible unitary representation T θ of the group Ãff(C associated to Ω by the orbit method, i.e. exp(ˆl (k f(x = [T θ(exp f](x, ( β where f L (R S 1 ; = aff(c; θ S ; k = 0, ±1,... Proof. Putting x = e u C\{0} = R S 1 and recall that ( a b = exp( = exp 0 1 ( β 0 0,

10 48 Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-djoint Orbits... we can rewrite (7 as following: ( [T θ (exp f](e u = exp i(r( e 1 βeu + θ[ Ieu+ ] f(e u, where I(u + I(u + u = R(u + + i{ } = u + i[ ]. Therefore, for the one-parameter subgroup exp t, t R, we have the action formula: By a direct computation: [ Tθ (exp tf ] (e u = exp(i(r et 1 βe u + θ[ Ieu+t ] f(e u t. = ( ( i 1 exp t (et + exp ( i 1 (et βe u + et 1 βe u + θi[ Ieu+t = i (βeu+t + βe u+t[ T θ (exp tf ] ( (e u + exp i(r( et 1 on one hand. On the other hand, we have: ( [Tθ (exp tf](e u = (8 t βe u + et 1 βe u + θi[ Ieu+t ] I(u+t u+t i[ + f(e ] ] I(u+t t f(eu+t i[ ] = βe u + θi[ Ieu+t ] u t f e u ˆl (k ([T θ(exp tf](e u = (9 = ( [Tθ (exp tf](e u + ( [Tθ (exp tf](e u + i u u (βeu + βe u [ T θ (exp tf](e u ] = = i 1 ( (et βe u exp i(r( et 1 βe u + θ[ Ieu+t ] f(e u t + ( + exp i(r( et 1 βe u + θ[ Ieu+t ] u t f e u + + i 1 ( (et βe u exp i(r( et 1 βe u + θ[ Ieu+t ] f(e u t + + i (βeu + βe u [T θ (exp tf](e u = i (βeu+t + βe u+t [T θ (exp tf](e u + (8 and (9 imply that ( + exp i(r( et 1 t [T θ(exp tf](x = βe u + θ[ Ieu+t ] u t f e u (k( ˆl [Tθ (exp tf](x x R S 1.

11 Furthermore, Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-djoint Orbits T θ (exp tf](e u t=0 = exp(i[ Ieu Iu ]θf(eu i[ ] = f(e u. This means that: exp(ˆl (k f(x and [T θ(exp tf](x together are the solution of the Cauchy problem { (k u(t, x = ˆl t u(t, x; u(0, x = id. ˆl (k The operator solution. From this uniqueness we deduce that is sufficiently well-behaved, so that the Cauchy problem has a unique exp(ˆl (k f(x [T θ(exp tf](x x R S 1. The theorem is hence proved. 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,. 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 ff(r in [5] and here for the group of affine transformations of the complex line ff(c we obtained a description of the quantum MD co-adjoint orbits. References [1] rnal, D.; Cortet, J. C.: -product and representations of nilpotent Lie groups. J. Geom. Phys. (1985(, [] rnal, D.; Cortet, J. C.: Représentations des groupes exponentiels. J. Funct. nal. 9 (1990, [3] rnold, V. I.: Mathematical Methods of Classical Mechanics. Springer Verlag, Berlin - New York - Heidelberg [4] Do Ngoc Diep: Noncommutative Geometry Methods for Group C*-lgebras. Chapman & Hall/CRC. Press #LM 003, [5] Do Ngoc Diep; Nguyen Viet Hai: Quantum half-plane via Deformation Quantization. Math. Q/990500, May [6] Fedosov, B.: Deformation Quantization and Index Theory. kademie Verlag, Berlin [7] Gutt, S.: Deformation quantization. ICTP Workshop on Representation Theory of Lie groups. SMR 686/14, 1993.

12 430 Do Ngoc Diep, Nguyen Viet Hai: Quantum Co-djoint Orbits... [8] Gel fand, I. M.; Naimark, M..: Unitary representations of the group of affine transformations of the straight line. Dokl. N SSSR 55 (1947(7, [9] Kirillov,..: Elements of the Theory of Representations. Springer Verlag, Berlin - New York - Heidelberg Received October 10, 1999; revised version December 1, 000