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Laboratoire de Physique Th´eorique et Hautes ´Energies†,

arXiv:hep-th/9303038v1 4 Mar 1993PAR-LPTHE 93-08UUITP 5/1993hep-th/9303038Symplectic structuresassociated to Lie-Poisson groups.A. Yu.

Alekseev ∗Laboratoire de Physique Th´eorique et Hautes ´Energies†,Paris, France‡.A. Z. Malkin §¶Institute of Theoretical Physics, Uppsala University,Box 803 S-75108, Uppsala, Sweden.23 February 1993AbstractThe Lie-Poisson analogues of the cotangent bundle and coadjoint orbitsof a Lie group are considered.

For the natural Poisson brackets the symplec-tic leaves in these manifolds are classified and the corresponding symplecticforms are described. Thus the construction of the Kirillov symplectic form isgeneralized for Lie-Poisson groups.∗On leave of absence from LOMI, Fontanka 27, St.Petersburg, Russia.†Unit´e Associ´ee au C.N.R.S., URA 280.‡LPTHE, Paris-VI, Tour 16 - 1er ´etage, 4 place Jussieu, F-75252 PARIS CEDEX 05.§Supported in part by a Soros Foundation Grant awarded by the American Physical Society.¶On leave of absence from St.Petersburg University.

Introduction.The method of geometric quantization [1] provides a set of Poisson manifolds as-sociated to each Lie group G. The dual space G∗of the corresponding Lie algebraG plays an important role in this theory. The space G∗carries the Kirillov-KostantPoisson bracket which mimics the Lie commutator in G. Having chosen a basis {εa}in G, we can define structure constants f abc :[εa, εb ] =Xcf abc εc ,(1)where [,] is the Lie commutator in G. 216z On the other hand, we can treat anyelement εa of the basis as a linear function on G∗.

The Kirillov-Kostant Poissonbracket is defined so that it resembles formula (1):{εa, εb } =Xcf abc εc ,(2)The Kirillov-Kostant bracket has two important properties :i. the r.h.s. of (2) is linear in εc,ii.

the group G acts on G∗by means of the coadjoint action and preserves thebracket (2).The Kirillov-Kostant bracket is always degenerate (e. g. at the origin in G∗).According to the general theory of Poisson manifolds [2, 3] the space G∗splits intothe set of symplectic leaves. Usually it is not easy to describe symplectic leavesof a Poisson manifold.

Fortunately an effective description exists in this very case.Symplectic leaves coincide with orbits of the coadjoint action of G in G∗. Kirillovobtained an elegant expression for the symplectic form Ωon the orbit [1]:ΩX(u, v) =< X, [εu, εv] >.

(3)Here < , > is the canonical pairing between G and G∗.The value of the form iscalculated at the point X on the pair of vector fields u and v on the orbit. Theelements εu, εv of the algebra G are defined as follows:u|X= ad∗(εu)X ,(4)where ad∗is the coadjoint action of G on G∗.

The purpose of this paper is to generalizeformula (3) for Lie-Poisson groups.Lie group G equipped with a Poisson bracket {,} is called a Lie-Poisson groupwhen the multiplication in GG × G −→G(5)(g, g′) −→gg′(6)is a Poisson mapping. In other words, the bracket of any two functions f and hsatisfies the following condition:{f, h}(gg′) = {f(gg′), h(gg′)}g + {f(gg′), h(gg′)}g′ .

(7)1

Here we treat f(gg′), h(gg′) as functions of the argument g only in the first term ofthe r.h.s. whereas in the second term they are considered as functions of g′.In the framework of the Poisson theory the natural action of a group on a man-ifold is the Poisson action [4, 5].

It means that the mappingG × M −→M(8)is a Poisson one. In Poisson theory this property replaces property (ii) of Kirillov-Kostant bracket.

There exist direct analogues of the coadjoint orbits for Lie-Poissongroups. Our goal in this paper is to obtain an analogue of formula (3).

However, itis better to begin with Lie-Poisson analogue of the cotangent bundle T ∗G describedin section 2. The symplectic form for this case is obtained in section 3 and then insection 4 the analogue of the Kirillov form appears as a result of reduction.

Section1 is devoted to an exposition of the Kirillov theory. In section 5 some examples areconsidered.When speaking about Lie-Poisson theory the works of Drinfeld [6] , Semenov-Tian-Shansky [5] , Weinstein and Lu [7] must be mentioned.

We follow these paperswhen representing the known results.The theory of Lie-Poisson groups is a quasiclassical version of the theory ofquantum groups. So we often use the attribute “deformed” instead of “Lie-Poisson”.Similarly we call the case when the Poisson bracket on the group is equal to zerothe “classical” one.1Symplectic structures associated to Lie groups.For the purpose of selfconsistency we shall collect in this section some well-knownresults concerning Poisson and symplectic geometry associated to Lie groups.

Themost important part of our brief survey is a theory of coadjoint orbits. Our goalis to rewrite the Kirillov symplectic form so that a generalization can be madestraightforward.Let us fix notations.

The main object of our interest is a Lie group G. Wedenote the corresponding Lie algebra by G. The linear space G is supplied with Liecommutator [,]. If {εa} is a basis in G we can define structure constants f abcin thefollowing way:[εa, εb ] =Xcf abc εc .

(9)The Lie group G has a representation which acts in G. It is called adjoint repre-sentation:εg ≡Ad(g)ε . (10)The corresponding representation of the algebra G is realized by the commutator:ad(ε)η = [ε, η] .

(11)We denote elements of the algebra G by small Greek letters.2

Let us introduce a space G∗dual to the Lie algebra G. There is a canonical pairing< , > between G∗and G and we may construct a basis {la} in G∗dual to the basis{εa} so that< la , εb >= δba . (12)We use small Latin letters for elements of G∗.

Each vector ε from G defines a linearfunction on G∗:Hε(l)=< l, ε >. (13)In particular, a linear function Ha corresponds to an element εa of the basis in G.By duality the group G and its Lie algebra G act in the space G∗via the coadjointrepresentation:< Ad∗(g)l, ε >=< l, Ad(g−1)ε > ,(14)< ad∗(ε)l, η >= −< l, [ε, η] >.

(15)The space G can be considered as a space of left-invariant or right-invariant vectorfields on the group G. Let us define the universal right-invariant one-form θg on Gwhich takes values in G :θg(ε) = −ε . (16)We treat ε in the l.h.s.

of formula (16) as a right-invariant vector field whereasin the r.h.s. as an element of G. Since the one-form θg and the vector field ε areright-invariant the result does not depend on the point g of the group.

θg is knownas Maurer-Cartan form.Similarly, the universal left-invariant one-form µg can be introduced:µg(ε) = ε ,µg = Ad(g−1)θg ,(17)where ε is a left-invariant vector field, Ad acts on values of θg.In the case of matrix group G the invariant forms θg and µg look like follows:θg = dg g−1 ,(18)µg = g−1dg . (19)For any group G there exist two covariant differential operators ∇L and ∇R takingvalues in the space G∗.

These are left and right derivatives:< ∇Lf, ε > (g) = −ddtf(exp(tε)g) ,(20)< ∇Rf, ε > (g) = ddtf(g exp(tε)) . (21)where exp is the exponential map from a Lie algebra to a Lie group.

The simplerelation for left and right derivatives of the same function f holds:∇Rf = −Ad∗(g−1)∇Lf . (22)3

From the very beginning the linear space G∗is not supplied with a natural com-mutator.Nevertheless, we define the commutator [,]∗in G∗and put it equal tozero:[l, m]∗= 0 . (23)The main technical difference of the deformed theory from the classical one is thatthe commutator in G∗is nontrivial.As a consequence, the corresponding groupG∗becomes nonabelian.

This fact plays a crucial role in the consideration of Lie-Poisson theory. In the classical case the Lie algebra G∗is just abelian and the groupG∗coincides with G∗.The space G∗carries a natural Poisson structure invariant with respect to thecoadjoint action of G on G∗.

Let us remark that the differential of any function onG∗is an element of the dual space , i.e. of the Lie algebra G. It gives us a possibilityto define the following Kirillov-Kostant Poisson bracket:{f, h}(l) =< l, [df(l), dh(l)] >.

(24)In particular, for linear functions Hε the r.h.s. of (24) simplifies:{Hε, Hη} = H[ε,η] ,(25){Ha, Hb} =Xcf abc Hc .

(26)The last formula simulates the commutation relations (1).In general situation the space G∗supplied with Poisson bracket (24) is not asymplectic manifold. The Kirillov-Kostant bracket is degenerate.

For example, inthe simplest case of G = su(2) the space G∗is 3-dimensional. The matrix of Poissonbracket is antisymmetric and degenerates as any antisymmetric matrix in an odd-dimensional space.The relation between symplectic and Poisson theories is the following.AnyPoisson manifold with degenerate Poisson bracket splits into a set of symplecticleaves.

A symplectic leaf is defined so that its tangent space at any point consistsof the values of all hamiltonian vector fields at this point:vh(f) = {h, f} . (27)Each symplectic leaf inherits the Poisson bracket from the manifold.

However, beingrestricted onto the symplectic leaf the Poisson bracket becomesnondegenerate andwe can define the symplectic two-form Ωso that:Ω(vf, vh) = {f, h} . (28)The relation (28) defines Ωcompletely because any tangent vector to the symplecticleaf may be represented as a value of some hamiltonian vector field.If we choose dual bases {ea} and {ea} in tangent and cotangent spaces to thesymplectic leaf we can rewrite the bracket and the symplectic form as follows:{f, h} = −XabP ab < df, ea >< dh, eb > ,(29)4

Ω=XabΩab ea⊗eb = 12XabΩab ea∧εb . (30)Using definition (28) of the form Ωand formulae (29),(30) one can check that thematrix Ωab is inverse to the matrix P ab:XcΩacP cb = δba .

(31)For the particular case of the space G∗with Poisson structure (24), there exists anice description of the symplectic leaves. They coincide with the orbits of coadjointaction (14) of the group G. Starting from any point l0, we can construct an orbitOl0 = {l = Ad∗(g)l0 ,g ∈G} .

(32)Any point of G∗belongs to some coadjoint orbit. The orbit Ol0 can be regarded asa quotient space of the group G over its subgroup Sl0:Ol0 ≈G/Sl0 ,(33)where Sl0 is defined as follows:Sl0 = {g ∈G ,Ad∗(g)l0 = l0} .

(34)In the case of G = SU(2) the coadjoint action is represented by rotations inthe 3-dimensional space G∗. The orbits are spheres and there is one exceptionalzero radius orbit which is just the origin.

The group Sl0 is isomorphic to U(1) andcorresponds to rotations around the axis parallel to l0. For the exceptional orbitSl0 = G and the quotient space G/G is a point.Let us denote by pl0 the projection from G to Ol0:pl0 :g −→lg = Ad∗(g)l0 .

(35)We may investigate the symplectic form Ωon the orbit directly. However, for tech-nical reasons it is more convenient to consider its pull-back ΩGl0 = p∗l0Ωdefined on thegroup G itself.

We reformulate the famous Kirillov’s result in the following form.Let Ol0 be a coadjoint orbit of the group G and pl0 be the projection (35). ThePoisson structure (24) defines a symplectic form ΩonOl0.Theorem 1 The pull-back of Ωalong the projection pl0 is the following:ΩGl0 = 12 < dlg ∧, θg >.

(36)We do not prove formula (36) but the proof of its Lie-Poisson counterpart insection 3 will fill this gap. Let us make only few remarks.

First of all, the form ΩGl0actually is a pull-back of some two-form on the orbit Ol0. Then, ΩGl0 is a closed form:dΩGl0 = 0 .

(37)5

This is a direct consequence of the Jacobi identity for the Poisson bracket (24). Theform ΩGl0 is exact, while the original form Ωbelongs to a nontrivial cohomology class.The left-invariant one-formα =< lg, θg >=< l0, µg >(38)satisfies the equationdα = ΩGl0 .

(39)In physical applications the form α defines an action for a hamiltonian systemon the orbit:S =Zα . (40)Returning to the formula (36) we shall speculate with the definition of G∗.

Inour case G∗= G∗and we may treat lg as an element of G∗. For an abelian groupMaurer-Cartan forms θ and µ coincide with the differential of the group element:θl = µl = dl .

(41)Using (41) we rewrite (36):ΩGl0 = 12 < θl ∧, θg > ,(42)where l is the function of g given by formula (35). Expression (42) admits a straight-forward generalization for Lie-Poisson case.The rest of this section is devoted to the cotangent bundle T ∗G of the group G.Actually, the bundle T ∗G is trivial.

The group G acts on itself by means of rightand left multiplications. Both these actions may be used to trivialize T ∗G.

So wehave two parametrizations ofT ∗G = G × G∗(43)by pairs (g, l) and (g, m) where l and m are elements of g∗. In the left parametrizationG acts on T ∗G as follows:Lh : (g, m) −→(hg, m) ,(44)Rh : (g, m) −→(gh−, Ad∗(h)m) .

(45)In the right parametrization left and right multiplications change roles:Lh : (g, l) −→(hg, Ad∗(h)l) ,(46)Rh : (g, l) −→(gh−, l) . (47)The two coordinates l and m are related:l = Ad∗(g)m .

(48)The cotangent bundle T ∗G carries the canonical symplectic structure ΩT ∗G [2].Using coordinates (g, l, m), we write a formula for ΩT ∗G without the proof:ΩT ∗G = 12(< dm ∧, µg > + < dl ∧, θg >) . (49)6

The symplectic structure on T ∗G is a sort of universal one. We can recover theKirillov two-form (36) for any orbit starting from (49).

More exactly, let us imposein (49) the condition:m = m0 = const . (50)It means that instead of T ∗G we consider a reduced symplectic manifold with thesymplectic structureΩr = 12 < dl, θg > ,(51)where l is subject to constraintl = Ad∗(g)m0 .

(52)Formulae (51), (52) reproduce formulae (35), (36) and we can conclude that thereduction leads to the orbit Om0 of the point m0 in G∗.The aim of this paper is to present Lie-Poisson analogues of formulae (36) and(49). Having finished our sketch of the classical theory, we pass to the deformedcase.2Heisenberg double of Lie bialgebra.One of the ways to introduce deformation leading to Lie-Poisson groups is to considerthe bialgebra structure on G. Following [6], we consider a pair (G, G∗), where we treatG∗as another Lie algebra with the commutator [,]∗.

For a given commutator [,] in Gwe can not choose an arbitrary commutator [,]∗in G∗. The axioms of bialgebra canbe reformulated as follows.

The linear spaceD = G + G∗(53)with the commutator [,]D:[ε, η]D = [ε, η] ,(54)[x, y]D = [x, y]∗,(55)[ε, x]D = ad∗(ε)x −ad∗(x)ε . (56)must be a Lie algebra.

In the last formula (56) ad∗(ε) is the usual ad∗-operatorfor the Lie algebra G acting on G∗. The symbol ad∗(x) corresponds to the coadjointaction of the Lie algebra G∗on its dual space G.The only thing we have to check is the Jacobi identity for the commutator [,]D.If it is satisfied, we call the pair (G, G∗) Lie bialgebra.

Algebra D is called Drinfelddouble. It has the nondegenerate scalar product < , >D :< (ε, x), (η, y) >D=< y, ε > + < x, η > ,(57)where in the r.h.s.

< , > is the canonical pairing of G and G∗. It is easy to see that< G, G >D= 0 ,< G∗, G∗>D= 0 .

(58)7

In other words, G and G∗are isotropic subspaces in D with respect to the form < , >D.We call the form < , >D on the algebra D standard product in D.We shall need two operators P and P ∗acting in D. P is defined as a projectoronto the subspace G:P(x + ε) = ε . (59)The operator P ∗is its conjugate with respect to form (57).

It appears to be aprojector onto the subspace G∗:P ∗(x + ε) = x . (60)The standard product in D enables us to define the canonical isomorphism J :D∗−→D by means of the formula< J(a∗), b >D=< a∗, b > ,(61)where a∗is an element of D∗and b belongs to D. In the r.h.s.

we use the canonicalpairing of D and D∗. The standard product can be defined on the space D∗:< a∗, b∗>D∗=< J(a∗), J(b∗) >D ,(62)where a∗and b∗belong to D∗.

The scalar product < , >D is invariant with respectto the commutator in D:< [a, b], c >D + < b, [a, c] >D= 0 . (63)It is easy to check that the operator J converts ad∗into ad:Jad∗(a)J−1 = ad(a) .

(64)Using the standard scalar product in D, one can construct elements r and r∗inD ⊗D which correspond to the operators P and P ∗:< a⊗b, r >D⊗D=< a, Pb >D ,(65)< a⊗b, r∗>D⊗D= −< a, P ∗b >D. (66)In terms of dual bases {εa} and {la} in G and G∗r =Xaεa ⊗la,r∗= −Xala ⊗εa .

(67)The Lie algebra D may be used to construct the Lie group D.We supposethat D exists (for example, for finite dimensional algebras it is granted by the Lietheorem) and we choose it to be connected. Originally the double is defined as aconnected and simply connected group.

However, we may use any connected groupD corresponding to Lie algebra D. Property (64) can be generalized for Ad and Ad∗:JAd∗(d)J−1 = Ad(d) ,(68)where d is an element of D.8

Let us denote by G and G∗the subgroups in D corresponding to subalgebras Gand G∗in D. In the vicinity of the unit element of D the following two decompositionsare applicable:d = gg∗= h∗h ,(69)where d is an element of D, coordinates g, h belong to the subgroup G, coordinatesg∗, h∗belong to the subgroup G∗.To generalize formula (69), let us consider the set ℑof classes G\D/G∗. We denoteindividual classes by small letters i, j, .

. .. Let us pick up a representative di in eachclass i.

If an element d belongs to the class i, it can be represented in the formd = gdig∗(70)for some g and g∗. In general case the elements g and g∗in decomposition (70) arenot defined uniquely.

If S(di) is a subgroup in GS(di) = {h ∈G ,d−1i hdi ∈G∗} ,(71)we can take a pair (gh, d−1i h−1dig∗) instead of (g, g∗), where h is an arbitrary elementof S(di). We denote T(di) the corresponding subgroup in G∗:T(di) = d−1i S(di)di .

(72)So we have the following stratification of the double D:D =[i∈ℑGdiG∗=[i∈ℑCi . (73)Each cellCi = GdiG∗(74)in this decomposition is isomorphic to the quotient of the direct product G × G∗over S(di), where(g, g∗) ∼(g′, g∗′)if(75)g′ = gh ,g∗′ = d−1i h−1dig∗,h ∈S(di) .

(76)For the inverse element d−1 in the relation (70) we get another stratification ofD in which G and G∗replace each other:D =[i∈ℑG∗d−1i G =[i∈ℑci . (77)Now we turn to the description of the Poisson brackets on the manifold D.Double D admits two natural Poisson structures.

First of them was proposed byDrinfeld [6]. For two functions f and h on D the Drinfeld bracket is equal to{f, h} =< ∇Lf ⊗∇Lh, r > −< ∇Rf ⊗∇Rh, r > ,(78)where < , > is the canonical pairing between D⊗D and D∗⊗D∗.

Poisson bracket (78)defines a structure of a Lie-Poisson group on D. However, the most important forus is the second Poisson structure on D suggested by Semenov-Tian-Shansky [5]:{f, h} = −(< ∇Lf ⊗∇Lh, r > + < ∇Rf ⊗∇Rh, r∗>) . (79)9

The manifold D equipped with bracket (79) is called Heisenberg double or D+.It is a natural analogue of T ∗G in the Lie-Poisson case. When G∗is abelian, G∗= G∗and D+ = T ∗G.

If the double D is a matrix group, we can rewrite the basic formula(79) in the following form:{d1, d2} = −(rd1d2 + d1d2r∗) ,(80)where d1 = d ⊗I , d2 = I ⊗d.The problem which appears immediately in the theory of D+ is the possibledegeneracy of Poisson structure (79) in some points of D. It is important to describethe stratification of D+ into the set of symplectic leaves. The answer is given by thefollowingTheorem 2 Symplectic leaves of D+ are connected components of nonempty inter-sections of left and right stratification cells:Dij = Ci ∩cj = GdiG∗∩G∗d−1j G .

(81)Proof. The tangent space T Sd to the symplectic leaf at the point d coincides with thespace of values of all hamiltonian vector fields at this point.

For concrete calculationslet us choose the left identification of the tangent space to D with D. We can rewritethe Poisson bracket (79) in terms of left derivatives ∇L:{f, h}(d) = −(< ∇Lf ⊗∇Lh, r > + < Ad∗(d−1)∇Lf ⊗Ad∗(d−1)∇Lh, r∗>) == −< ∇Lf ⊗∇Lh, r + Ad(d) ⊗Ad(d) r∗>. (82)Here we use relation (22) between left and right derivatives on a group.A hamiltonian h produces the hamiltonian vector field vh so that the formula< df, vh >= {h, f}(83)holds for any function f. Using (82), (83) we can reconstruct the field vh:vh =< ∇Lh, r + Ad(d) ⊗Ad(d) r∗>2.

(84)Having identified D and D∗by means of the operator J, we can rewrite the r.h.s. of(84) as follows:vh|d = Pdh = (P −Ad(d)P ∗Ad(d−1))J(∇Lh(d)) ,(85)where P acts in D:P = P −Ad(d)P ∗Ad(d−1) .

(86)It is called Poisson operator. Using the fact that the value of ∇Lh at the point d isan arbitrary vector from D∗, we conclude that T Sd coincides with the image of theoperator P:T Sd = ImP .

(87)10

The most simple way to describe the image of P is to use the property:ImP = (KerP∗)⊥. (88)Here conjugation and symbol ⊥correspond to the standard product in D.Theoperator P∗is given by the formulaP∗= P ∗−Ad(d)PAd(d−1) .

(89)Suppose that a vector a = x + ε belongs to KerP∗:P∗(x + ε) = 0 . (90)Let us rewrite the condition (90) in the following form:(Ad(d−1)P ∗−PAd(d−1))(x + ε) = 0 ,(91)or, equivalently,Ad(d−1)x = P(Ad(d−1)x + Ad(d−1)ε) .

(92)Using the propertyP + P ∗= id(93)of the projectors P and P ∗, one can get from (92):P ∗(Ad(d−1)x) = P(Ad(d−1)ε) . (94)The l.h.s.

of (94) is a vector from G∗whereas the r.h.s. belongs to G. So the equation(94) implies that both the l.h.s.

and the r.h.s. are equal to zero.Let V (d) be the subspace in G defined by the following condition:V (d) = {ε ∈G ,Ad(d−1)ε ∈G∗} .

(95)In the same way we define the subspace V ∗(d) in G∗:V ∗(d) = {x ∈G∗,Ad(d−1)x ∈G} . (96)It is not difficult to check that V (d) and V ∗(d) are actually Lie subalgebras in G andG∗.

The kernel of the operator P∗may be represented as a direct sum of V (d) andV ∗(d):KerP∗= V (d) ⊕V ∗(d) . (97)The tangent space T Sd to the symplectic leaf at the point d acquires the formT Sd = (V (d) ⊕V ∗(d))⊥.

(98)The result (98) can be rewritten:T Sd = V (d)⊥∩V ∗(d)⊥= (V (d)⊥∩G∗) ⊕(V ∗(d)⊥∩G) . (99)Here the last expression represents T Sd as a direct sum of its intersections with G andG∗.11

Now we must compare subspace (99) with the tangent space T′d of the intersectionof the stratification cells (theorem 2). Suppose that the point d belongs to the cellDij of the stratification.

We can rewrite the definition of Dij as follows:Dij = GdG∗∩G∗dG = C(d) ∩c(d) . (100)The tangent space to Dij may be represented as an intersection of tangent spacesto left and right cells C(d) and c(d):T′d = Td(C(d)) ∩Td(c(d)) .

(101)For the latter the following formulae are true:Td(C(d)) = G + Ad(d)G∗,(102)Td(c(d)) = G∗+ Ad(d)G . (103)The space Td(C(d)) coincides with V (d)⊥.Indeed, Td(C(d))⊥lies in G becauseTd(C(d))⊥⊂G⊥= G. On the other hand< Td(C(d))⊥, Ad(d)G∗>D= 0 .

(104)Formula (104) implies that Ad(d−1)Td(C(d))⊥⊂G∗⊥= G∗.So Td(C(d))⊥is thesubspace in G which is mapped by Ad(d−1) into G∗. It is the subspace V (d) thatsatisfies these conditions.

So we haveTd(C(d))⊥= V (d) ,Td(C(d)) = V (d)⊥. (105)Similarly,Td(c(d)) = V ∗(d)⊥.

(106)Comparing (99), (101), (105), (106), we conclude that the tangent space T′d tothe cell Dij coincides with the tangent space T Sd to the symplectic leaf. Thus thesymplectic leaf coincides with a connected component of the cell Dij.We have proved theorem 2.

The next question concerns the symplectic structureon the leaves Dij.3Symplectic structure of the Heisenberg double.Each symplectic leaf Dij introduced in the last section carries a nondegenerate Pois-son structure and hence the corresponding symplectic form Ωij can be defined. Towrite down the answer we need several new objects.

Let us denote by Lij the subsetin G × G∗defined as follows:Lij = {(g, g∗) ∈G × G∗,gdig∗∈Dij} . (107)In the same way we construct the subset Mij in G∗× G:Mij = {(h∗, h) ∈G∗× G ,h∗d−1j h ∈Dij} .

(108)12

Finally let Nij be the subset in Lij × Mij:Nij = {[(g, g∗), (h∗, h)] ∈Lij × Mij ,gdig∗= h∗d−1j h} . (109)We can define the projectionpij:Nij −→Dij(110)pij:[(g, g∗), (h∗, h)] −→d = gdig∗= h∗d−1j h(111)and consider the form p∗ijΩij on Nij instead of the original form Ωij on Dij.

It isparallel to the construction of the Kirillov form on the coadjoint orbit (see section1). Parametrizations (107), (108) provide us with the coordinates (g, g∗) and (h∗, h)on Nij.

We can use them to write down the answer:Theorem 3 The symplectic form p∗ijΩij on Nij can be represented as follows:p∗ijΩij = 12(< θh∗∧, θg > + < µg∗∧, µh>) . (112)In the formula (112) θg, θh∗, µh, µg∗are restrictions of the corresponding one-forms from (G × G∗) × (G∗× G) to Nij.

The pairing < , > is applied to values ofMaurer-Cartan forms, which can be treated as elements of G and G∗embedded toD = G + G∗. So we can use < , >D as well as < , >.Proof of theorem 3.The strategy of the proof is quite straightforward.

We consider Poisson bracket(79) on the symplectic leaf Dij. If we use dual bases {ea} and {ea} (a = 1, .

. .

, n=dimD) of right-invariant vector fields and one-forms on D, the formula (79) acquiresthe following form:{f, h}(d) = −< ∇Lf ⊗∇Lh, r + Ad(d) ⊗Ad(d) r∗>== −nXa,b=1< ∇Lf, ea >< ∇Lh, eb >< ea, PJeb >. (113)The last multiplier in (113) is Poisson matrix corresponding to the bracket (79):Pab =< ea, PJeb >.

(114)Here P is the same as in (86). The matrix Pab may be degenerate.

Let us choosevectors {ea,a ∈sij = {1, . .

., nij = dimDij}} so that they form a basis in thespace Td tangent to Dij. Pab is not zero only if both a and b belong to sij.

Thesymplectic form Ωij on the cell Dij can be represented as follows (see section 1):Ωij =nijXa,b=1Ωabea⊗eb ,(115)where the matrix Ωsatisfies the following condition:nijXc=1ΩacPcb = δba . (116)13

So what we need is inverse matrix P−1 for Pab. To make the symbol P−1 meaningfulwe introduce two operators P1 and P2:P1 = (P + Ad(d)P ∗) ,(117)P2 = (P ∗−Ad(d)P) .

(118)P may be decomposed in two ways, using P1 and P2:P = P1P∗2 = −P2P∗1 . (119)Some useful properties of the operators P1 and P2 are collected in the followinglemma.Lemma 1ImP1 = V (d)⊥,ImP2 = V ∗(d)⊥,P(KerP1) = V (d) ,P ∗(KerP2) = V ∗(d) .

(120)Proof. First let us consider the formulaImP1 = (KerP∗1)⊥.

(121)The operator P∗1 looks like follows:P∗1 = P ∗+ PAd(d−1) . (122)The equation for KerP∗1(P ∗+ PAd(d−1))(x + ε) = 0(123)leads immediately to the following restrictions for x and ε:x = 0 ,Ad(d−1)ε ∈G∗.

(124)Comparing (124) with definition (95), we see that KerP∗1 = V (d) and hence ImP1 =V (d)⊥.If a vector x+ε belongs to the kernel of the operator P1, it satisfies the followingequation:(P + Ad(d)P ∗)(x + ε) = 0 . (125)It can be rewritten as a set of conditions for the components x, ε:Ad(d−1)ε ∈G∗,x = −Ad(d−1)ε .

(126)ε again appears to be an element of V (d). This fact may be represented as theequation P(KerP1) = V (d).We omit the proofs of the formulae (120) concerning the operator P2 becausethey are parallel to the proofs given above.The following step is to define inverse operators:P−11:ImP1 −→D/KerP1 ,(127)14

P−12:ImP2 −→D/KerP2 . (128)The solution of the equationP−11,2a = b(129)exists if and only if a ∈ImP1,2 and b is defined up to an arbitrary vector fromKerP1,2.Now we are ready to write down the answer for Ωab:Ωab =< ea, Ωeb >D ,Ω= PP−11−P ∗P−12.

(130)First of all let us check that matrix elements Ωab are well-defined. Vectors eb formthe basis in the space Td = (V (d) ⊕V ∗(d))⊥.

Both P−11and P−12are defined on Tdbecause Td ⊂V (d)⊥= ImP1 and also Td ⊂V ∗(d)⊥= ImP2. So the vector Ωebexists but it is not unique.

It is defined up to an arbitrary vectorδ ∈P(KerP1) + P ∗(KerP2) = V (d) + V ∗(d) . (131)Fortunately the vector ea ∈Td and < ea, δ >= 0 for any δ of the form (131).

Weconclude that the ambiguity in the definition of the operator Ωdoes not lead to anambiguity for matrix elements Ωab.Now we must check condition (116):δba =nijXc=1ΩacPcb ==nijXc=1< ea, Ωec >< ec, PJ(eb) >=(132)=< ea, ΩPJ(eb) >D.The product ΩP can be easily calculated using (119),(130):ΩP = PP−11 P1P∗2 + P ∗P−12 P2P∗1 == P(P −P ∗Ad(d−1)) + P ∗(P ∗+ PAd(d−1)) =(133)= P + P ∗= I .We must remember that the vector ΩPJ(eb) is defined up to an arbitrary vectorfrom V (d) ⊕V ∗(d) because we used in (133) the “identities”P−11 P1 ≈P−12 P2 ≈id . (134)The ambiguity in (134) does not influence the answer:< ea, ΩPJ(eb) >D=< eb, ea >= δba(135)as it is required by (116).We can rewrite formula (130) in more invariant way:Ωij =< θijd⊗, Ωθijd >D ,(136)15

where θijd is the restriction of the Maurer-Cartan form to the cell Dij. Expression(130) for the operator Ωstill includes inverse operators P−11,2 implying that someequations must be solved.

To this end we consider the pull-back of the form Ωij:p∗ijΩij =< p∗ijθijd⊗, Ωp∗ijθijd >D. (137)There are coordinates (g, g∗) and (h∗, h) on Nij.

The Maurer-Cartan form p∗ijθijd canbe rewritten in two ways:p∗ijθijd = θg + Ad(d)µg∗,(138)p∗ijθijd = θh∗+ Ad(d)µh . (139)Representations (138), (139) allow us to calculate P−11,2p∗ijθijd explicitly:P−11 p∗ijθijd = θg + µg∗,(140)P−12 p∗ijθijd = θh∗−µh .

(141)Let us mention again that solutions (140), (141) are not unique. We can take anypossible value of Ωθijd .

The answer for the form Ωij is independent of this choice.Putting together (130), (137), (140) and (141), we obtain the following formulafor the symplectic form:p∗ijΩij =< (θg + Ad(d)µg∗) ⊗, θg >D −< (θh∗+ Ad(d)µh) ⊗, θh∗>D==< Ad(d)µg∗⊗, θg >D −< Ad(d)µh ⊗, θh∗>D. (142)Actually, the form (142) is antisymmetric.

To make it evident, let us consider theidentity< p∗ijθijd⊗, p∗ijθijd >D==< Ad(d)µg∗⊗, θg >D + < θg ⊗, Ad(d)µg∗>D=(143)=< Ad(d)µh ⊗, θh∗>D + < θh∗⊗, Ad(d)µh >D.Or, equivalently,< Ad(d)µg∗⊗, θg >D −< Ad(d)µh ⊗, θh∗>D== −< θg ⊗, Ad(d)µg∗>D + < θh∗⊗, Ad(d)µh >D. (144)Applying (144) to make (142) manifestly antisymmetric, one gets:p∗ijΩij = 12(< Ad(d)µg∗∧, θg >D + < θh∗∧, Ad(d)µh >D) .

(145)Using representation (111) of d in terms of (g, g∗) and (h∗, h), it is easy to checkthat formula (145) coincides withp∗ijΩij = −12(< µg ∧, Ad(di)θg∗>D + < θh ∧, Ad(dj)µh∗>D) . (146)16

To obtain formula (112) one can use (138),(139):p∗ijθijd = θg + Ad(d)µg∗= θh∗+ Ad(d)µh . (147)Or, equivalently,θg −Ad(d)µh = θh∗−Ad(d)µg∗.

(148)Due to antisymmetry we have< (θg −Ad(d)µh) ∧, (θh∗−Ad(d)µg∗) >D= 0 . (149)Therefore,12(< θh∗∧, θg >D + < µg∗∧, µh >D) == 12(< Ad(d)µg∗∧, θg >D + < θh∗∧, Ad(d)µh >D) = p∗ijΩij ,(150)which coincides with (112).Now we have to check that the r.h.s.

of formula (112) does represent the pull-back of some two-form on Dij. The problem is in the ambiguity of formula (70).Coordinates g and g∗are defined only up to the following change of variables:g′ = gs ,g∗′ = tg∗,(151)wheresdit = di .

(152)Here s is an element of S(di) and t belongs to T(di). The parameter s determines tby means of formula (152).

Similar ambiguity exists in the definition of h and h∗.We can construct an infinitesimal analogue of formula (151). The vector field vε onNijvε = (Ad(g)ε, −Ad(d−1i )ε)(153)does not correspond to any nonzero vector field on Dij.

Here we use coordinates(g, g∗) on Nij and left identification of vector fields on G × G∗and G + G∗. So thefirst term is an element of G and the second one belongs to G∗.

Therefore Ad(g)εbelongs to V (di) (see section 2).Actually we must check two nontrivial statements:i. Form p∗ijΩij is invariant with respect to change of variables (151).

It followsfrom the definition of the Maurer-Cartan forms θ and µ.ii. Tangent vectors (153) belong to the kernel of p∗ijΩij.It is convenient to use expression (146) for p∗ijΩij:p∗ijΩij = −12(< µg ∧, Ad(di)θg∗>D + < θh ∧, Ad(dj)µh∗>D) == −12(ω1 + ω2) ,(154)17

whereω1 =< µg ∧, Ad(di)θg∗>D ,(155)ω2 =< θh ∧, Ad(dj)µh∗>D. (156)We have to consider ω1( .

, vε) and ω2( . , vε).ω1( .

, vε) =< µg, Ad(di)θg∗(vε) >D −< µg(vε), Ad(di)θg∗>D==< µg, Ad(di)Ad(d−1i )ε >D + < Ad(g−1)Ad(g)ε, Ad(di)θg∗>D=(157)=< µg, ε >D + < θg∗, Ad(d−1i )ε >D.Here we use properties (16), (17) of the Maurer-Cartan forms. It is easy to see thatboth terms in the last expression (157) are equal to zero.

First of them< µg, ε >D= 0(158)because both ε and a value of µg belong to G. All the same with the second term:< θg∗, Ad(d−1i )ε >D= 0(159)because for Ad(g)ε ∈V (di) the combination Ad(d−1i )ε belongs to G∗. We remindthat both G and G∗are isotropic subspaces in D.We omit the proof for the second term ω2 in (154) because it is quite parallel tothe one described above.

We conclude that form (112) indeed corresponds to sometwo-form on the symplectic leaf Dij.It is known from general Poisson theory thatdΩ= 0 ,(160)but it is interesting to check that form (112) is closed by direct calculations. Rewrit-ing equation (148) we get:θg −θh∗= Ad(d)µh −Ad(d)µg∗.

(161)Taking the cube of the last equation we get:< θg ∧, θg ∧θg >D −< θh∗∧, θh∗∧θh∗>D ++3 < θg ∧, θh∗∧θh∗>D −3 < θg ∧θg ∧, θh∗>D==< µh ∧, µh ∧µh >D −< µg∗∧, µg∗∧µg∗>D +(162)+3 < µh ∧, µg∗∧µg∗>D −3 < µh ∧µh ∧, µg∗>D.As θg∧θg = 12[θg ∧, θg] and µh∧µh = 12[µh ∧, µh] take values in G, θh∗∧θh∗= 12[θh∗∧, θh∗]and µg∗∧µg∗= 12[µg∗∧, µg∗] take values in G∗we may use the pairing < , >D forthem. Moreover, as both G and G∗are isotropic subspaces in D, we rewrite (162) asfollows:< θg ∧, θh∗∧θh∗>D −< θg ∧θg ∧, θh∗>D −−< µh ∧, µg∗∧µg∗>D + < µh ∧µh ∧, µg∗>D= 0 .

(163)18

We remind that dθg = θg ∧θg and dµg = −µg ∧µg. Thus,dp∗ijΩij = −< dθg ∧, θh∗>D + < θg ∧, dθh∗>D −−< dµh ∧, µg∗>D + < µh ∧, dµg∗>D= 0 .

(164)Now it is interesting to consider the classical limit of our theory to recover thestandard answer for T ∗G. There is no deformation parameter in bracket (79) but itmay be introduced by hand:{f, h}γ = γ{f, h} .

(165)For the new bracket (165) we have the symplectic form:Ωγij = 1γ Ωij . (166)The classical limit γ →0 makes sense only for the main cell corresponding todi = dj = I.

The idea is to parametrize a vicinity of the unit element in the groupG∗by means of the exponential map:g∗= exp(γm) ,(167)h∗= exp(γl) ,(168)where m and l belong to G∗. Coordinates m and l are adjusted in such a way thatthey have finite values after the limit procedure.

When γ tends to zero, the formulad = gg∗= h∗h(169)leads to the following relations:g = h ,l = Ad∗(g)m . (170)Expanding the form Ωγ into the series in γ we keep only the constant term (singular-ity γ−1 disappears from the answer because the corresponding two-form is identicallyequal to zero).

The answer is the following:Ωγ = 12(< dm ∧, µg > + < dl ∧, θg >)(171)and it recovers classical answer (49) (see section 1). Deriving formula (171), we usethe expansions for the Maurer-Cartan forms on G∗:θg∗= γdm + O(γ2) ,(172)µh∗= γdl + O(γ2) .

(173)We have considered general properties of the symplectic structure on the Heisen-berg double D+ and now we turn to the theory of orbits for Lie-Poisson groups.19

4Theory of orbits.In this section we describe reductions of the Heisenberg double D+ which lead toLie-Poisson analogues of coadjoint orbits. We consider quotient spaces of the doubleD over its subgroups G and G∗: FR =D/G ,F ∗R=D/G∗,FL =G\D ,F ∗L =G∗\D.

They inherit Poisson bracket from the double D+. Indeed, let us pickup FR as an example.

Functions on FR may be regarded as functions on D invariantwith respect to right action of G :f(dg) = f(d) . (174)The right derivative ∇Rf is orthogonal to G for functions on FR:< ∇Rf, G >= 0 .

(175)For a pair of invariant functions f and h the second term in the formula (79) vanishesbecause r∗∈G∗⊗G. The first term is an invariant function because the left derivative∇L preserves the condition (174).

So we conclude that the Poisson bracket{f, h} = −< ∇Lf ⊗∇Lh, r >(176)is well-defined on invariant functions and hence it can be treated as a Poisson bracketon FR. The purpose of this section is to study the stratification of the space FR intosymplectic leaves and describe the corresponding symplectic forms on them.

Onecan consider FL, F ∗R, F ∗L in the same way.Using stratification (77) of the double D we can obtain the stratification of thespace FR:FR =[jG∗/T−j =[jG∗j . (177)Each stratification cell G∗j is just an orbit of the natural action of G∗on the quotientspace FR = D/G by the left multiplication.

We denote the orbit of the class of unityin D by G∗0. It is a quotient of G∗over discrete subgroup E=G∗∩G, G∗0 =G/E.We have factorized the double D over the right action of the group G. However,the same group acts on the quotient space by the left multiplications:g:dG −→gdG .

(178)Here the class dG is mapped into the class gdG. In the vicinity of the unit elementon the maximum cell GG∗∩G∗G the action (178) looks like follows:gg∗= g∗′(g, g∗)g′(g, g∗) .

(179)The element g∗′(g, g∗) is a result of the left action of the element g on the pointg∗∈G∗⊂FR. In the classical limit, when g∗and g∗′ are very close to the identity,formula (179) transforms into the coadjoint action of G on G∗:g∗= I + γl + .

. .,(180)20

g∗′ = I + γl′ + . .

.,(181)l′ = Ad∗(g)l . (182)For historical reasons transformations (179) are called dressing transformations.

Wedenote them AD∗to remind their relation to the coadjoint action:g∗′(g, g∗) = AD∗(g)g∗. (183)As we have mentioned, the transformation AD∗is defined on the space FR glob-ally.

For some values of g and g∗in (183) the element g∗′ does not exist and theresult of the action of g on g∗belongs to some other cell G∗j of stratification (177).So we have a correct definition of the AD∗-orbit in the Lie-Poisson case. The ques-tion is whether they coincide with symplectic leaves or not.

In general the answeris negative. Characterizing the situation we shall systematically omit the proofsconcerning standard Poisson theory [2, 3].A powerful tool for studying symplectic leaves is a dual pair.

By definition apair of Poisson mappings of symplectic manifold S to different Poisson manifoldsPL and PR:Sւց(184)PLPRis called a dual pair, if Poisson bracket of any function on S lifted from PR vanisheswhen the second function is lifted from PL and in this case only. Symplectic leaves inPR can be obtained in the following way.

Take a point in PL, consider its preimagein S and project it into PR. Connected components of the image of this projectionare symplectic leaves in PR.As an example let us consider the following pair of Poisson mappings:D+ւց(185)FLFR .This pair is not a dual pair because D+ is not a symplectic manifold.

However, thepair (185) is related to a family of dual pairs:Dijւց(186)FLFR .Here we use symplectic leaves Dij instead of D+. One can prove that pair of map-pings (186) is a dual pair by direct calculation with bracket (79).

Choosing dualpairs with different indices ij, we cover all space FR and find all the symplectic leavesin this space.21

Let us apply the general prescription to the dual pair (186). We pick up a classGx ∈ImLDij ⊂Ti\G∗⊂FL.

Its preimage in Dij is an intersection Kij(x) = Gx∩Dij.Projecting Kij(x) into FR, we get a symplectic leaf:AD∗(G)xG ∩ImRDij . (187)Let us remark that ImRDij is an intersection G∗j ∩(∪g∗∈G∗AD∗(G)dig∗G).

It impliesthat we may use G∗j instead of ImRDij in the formula (187). So all the symplecticleaves in FR are intersections of orbits of dressing transformations AD∗and orbitsG∗j of the action of G∗in FR.

To get all the leaves we have to use all the cells Dij inD. The orbits of AD∗-action in FR appear to have a complicated structure.

Eachorbit Op0 = AD∗(G) p0 ( p0 ∈FR) may be represented as a sum of its cells:Op0 =[j(AD∗(G) p0 ∩G∗j) =[jOjp0 . (188)Each cell of stratification (188) is a symplectic leaf in FR.Now we turn to the description of symplectic forms on the leaves (188).

Asusually, it is convenient to use coordinates on the orbit and on the group G at thesame time. Formulagh∗0d−1j G = h∗d−1j G(189)for the action of AD∗on the point h∗0T−j ∈G∗j provides us with the projection fromthe subsetGj(h∗0) = {g ∈G ,gh∗0d−1j∈G∗d−1j G}(190)to the cell Ojh∗0 of the orbit:pj:Gj(h∗0) −→Ojh∗0 ,(191)pj:g −→h∗T−j ,(192)where h∗is the same as in (189).

Instead of the symplectic form Ωj on the cell Ojh∗0we shall consider its pull-back p∗j(h∗0)Ωj defined on Gj(h∗0). It is easy to obtain theanswer, using formula (112) for the symplectic form on Dij.

We put the parameterof the symplectic leaf g∗=g∗0 =const. It kills the second term and the rest gives usthe following answer:p∗j(h∗0)Ωj = 12 < θh∗∧, θg >.

(193)There is no manifest dependence on dj in (193), but one must remember that gtakes values in the very special subset of G (190). The dependence is hidden there.Anyway, the final result of our investigation is quite elegant.Each orbit of thedressing transformations in FR splits into the sum of symplectic leaves (188) andthe symplectic form on each leaf can be represented in the uniformed way (193).As in section 3 one can check independently that two-form (193) is really a pull-back of some closed form on Ojh∗0.

We suggest this proposition as an exercise for aninterested reader.22

We have classified symplectic leaves in the quotient space FR = D/G and inparticular in its maximum cell G∗0 = G∗/E. In this content the idea to find symplecticleaves in the group G∗itself arises naturally.

To this end let us consider the followingsequence of projections G∗U →G∗→G∗0, where G∗U is a universal covering groupof the group G∗. The group G∗U is a Lie-Poisson group.

The Poisson bracket onthe group G∗U is defined uniquely by the Lie commutator in G [6]. The coveringG∗U →G∗0 appears to be a Poisson mapping.

Using this property one can checkthat G∗is a Lie-Poisson group and the corresponding Poisson bracket makes bothprojections G∗U →G∗and G∗→G∗0 Poisson mappings. It implies that symplecticleaves in G∗U and in G∗are connected components of preimages of symplectic leavesin G∗0.

Corresponding symplectic forms can be obtained by pull-back from (193).On the other hand, the formula (193) gives an expression for symplectic forms onthe leaves in G∗U and G∗, if we treat h∗as an element of one of these groups and g asan element of GU, universal covering group of G. Then we define the action of G∗Uon G∗0 by the formula (189) (g is a projection to G of some element gU ∈GU) andlift the action of GU from G∗0 to G∗U or G∗. It is always possible by the definition ofthe universal covering group.

We can identify symplectic leaves in G∗U or G∗withorbits of the action of GU, which we have just defined.It is remarkable that in the deformed case the groups G and G∗may be consideredon the same footing. Formula (193) defines symplectic structure on the orbit of G∗-action in D/G∗as well as on the orbit of G-action in D/G.

The only thing we haveto change is the relation between g and h∗:h∗g0diG∗= gdiG∗. (194)To consider the classical limit we can introduce a deformation parameter intothe formula (193):p∗j(h∗0)Ωγj = 12γ < θh∗∧, θg >.

(195)In this way one can recover the classical Kirillov form (36) as we did it for T ∗G insection 3.5Examples.In this section we shall consider two concrete examples to clarify constructions de-scribed in sections 2–4.1. The first example concerns the Borel subalgebra B+ of semisimple Lie algebraG.

The algebra B+ consists of Cartan subalgebra H ⊂G and nilpotent subalgebraN+ generated by the Chevalley generators corresponding to positive roots. In thesimplest case G = sl(n)B+ is just an algebra of traceless upper triangular matrices.We may define the projection p : B+ →H.

Let us call p(ε) ∈H a diagonal part of εand denote it εd.The dual space B∗+ can be identified with another Borel subalgebra B−⊂G, whereB−= H+N −includes the nilpotent subalgebra N −corresponding to negative roots.23

The canonical pairing of B+ and B−is given by the Killing form K(x, y) ≡Tr(xy)on G:< x, ε >= K(x, ε) + K(xd, εd) . (196)The natural commutator on B∗+ = B−defines a structure of bialgebra on B+.

Thedouble D is isomorphic to the direct sum of G and H:D(B+) ≃G ⊕H ,(197)Isomorphism (197) looks like follows:(x, ε) −→(x + ε, xd −εd) . (198)The first component of the r.h.s.

in (198) belongs to G and satisfies the correspondingcommutation relations, while the second component is an element of H. Elementsof D, satisfying the conditionsx = xd ,ε = εd ,xd + εd = 0 ,(199)belong to the center of D.The group D in this case is a product of semisimple Lie group G and its Cartansubgroup H:D = G × H . (200)The groups B+ and B−, corresponding to the algebras B+ and B−, can be embeddedinto D as follows:B+ −→(B+, (B+)d) ,(201)B−−→(B−, (B−)−1d ) ,(202)where (B+)d, (B−)d are diagonal parts of the matrices B+, B−.

The decomposition(73) in this case may be described more precisely:D =[i∈WB+WiB−,(203)where W is Weyl group of G and the pair Wi = (wi, I) consists of the element wifrom W and the unit element I in H. For nontrivial wi spaces V (Wi), V ∗(Wi) (95),(96) are nonempty.For the algebras B+ and B−we can use matrix notations (18), (19) for the Maurer-Cartan forms. For example,θB+ = (dB+B−1+ , db+b−1+ ) ,(204)µB−= (B−1−dB−, −b−1−db−) .

(205)Here b+ and b−are diagonal parts of B+ and B−correspondingly. The invariantpairing < , >D acquires the form:< (g1, h1), (g2, h2) >D= Tr(g1g2 −h1h2) .

(206)24

Now we can rewrite form (112) on the cell Dij in this particular case:d = (B+wiB−, (B+)d(B−)−1d ) = (B′−w−1j B′+, (B′−)−1d (B′+)d)(207)p∗ijΩij = 12Tr(dB′−B′−1−∧dB+B−1+ + db′−b′−1−∧db+b−1+ ++B−1−dB−∧B′−1+ dB′+ + b−1−db−∧b′−1+ db+) . (208)We have the symplectic structure on D+ and it is interesting to specialize Poissonbracket (79) for this case.

We use tensor notations and write down the Poissonbracket for matrix elements of d and h, (d, h) ∈D:{d1, d2} = −(r+d1d2 + d1d2r−) ,(209){d1, h2} = −(ρd1h2 + d1h2ρ) ,(210){h1, h2} = 0 . (211)Here r+ and r−are the standard classical r-matrices, corresponding to the Liealgebra G:r+ = 12Xhi ⊗hi +Xα∈∆+eα ⊗e−α ,(212)r−= −12Xhi ⊗hi −Xα∈∆+e−α ⊗eα ,(213)and ρ is the diagonal part of r+ :ρ = 12Xhi ⊗hi .

(214)As a result of general consideration we have obtained the symplectic structurecorresponding to nontrivial Poisson bracket (209)–(211). At this point we leave thefirst example and pass to the next one.2.

Now we take a semisimple Lie algebra G as an object of the deformation. Itis the most popular and interesting example.

The dual space G∗may be realized asa subspace in B+⊕B−:G∗= {(x, y) ∈B+⊕B−,xd + yd = 0} . (215)The pairing between G and G∗is the following:< (x, y), z >= Tr{(x −y)z}(216)and the Lie algebra structure on G∗is inherited from B+⊕B−.

It is easy to provethat the algebra double is isomorphic to the direct sum of two copies of G [6]:D ≃G ⊕G(217)25

{x, (y, z)} −→(x + y, x + z) = (d, d′) ,(218)< (d1, d′1), (d2, d′2) >D= Tr(d1d2 −d′1d′2) ,(219)where x ∈G , (y, z) ∈G∗. Therefore, the group double D is a product of two copiesof G:D = G × G .

(220)The subgroups G and G∗can be realized in D as follows:G = {(g, g) ∈D} ,(221)G∗= {(L+, L−) ∈D ,(L+)d(L−)d = I} . (222)Any pair (X, Y ) ∈D can be decomposed into the product of the elements fromG∗and G by means of the same Weyl group W:X = L+wig ,(223)Y = L−g .

(224)Here (L+, L−) ∈G∗, g ∈G and wi is an element of the Weyl group W. So we havethe following decomposition:D =[i∈WG∗WiG ,(225)where Wi = (wi, I).In this example we do not consider the symplectic structure on D+ and passdirectly to the description of orbits. The space FR = D/G can be decomposed as ingeneral case:FR =[i∈W(G∗/T−i) ,(226)where T−i is the subgroup of B+, generated by the positive roots, which transforminto the negative ones by the element wi of the Weyl group:T−i = {t ∈B+ ,td = I ,w−1i twi ∈B−} .

(227)The dressing transformations act on the space FR as follows:gL+wi = Lg+wigh ,(228)gL−= Lg−h ,(229)where (Lg+, Lg−) is the result of the dressing action AD∗(g) and ig is the index of thecell, where it lies. By the general theory the symplectic leaves in FR are intersectionsof the cells (G∗/T−i) and the orbits of the dressing transformations.

The analogue(193) of the Kirillov two-form can be rewritten in the following form:p∗jΩj = 12Tr(dL+L−1+ −dL−L−1−) ∧dgg−1 . (230)26

It is convenient to define the matrixL = L+wiL−1−. (231)It transforms under the action of the transformations (228), (229) in a simple way:Lg = Lg+wig(Lg−)−1 = gLg−1 .

(232)Being an element of G, the matrix L defines a mapping from FR to G by means ofthe formula (231). On each orbit of the conjugations (232) we can find a matrix Lof canonical form.

Let us denote it by L0:Lg = gL0g−1 = Lg+wig(Lg−)−1 . (233)Using two different parametrizations of the same matrix L, we can rewrite (230):p∗jΩj = 12Tr{g−1dgL0 ∧g−1dgL−10+ L−1+ dL+wj ∧L−1−dL−w−1j } .

(234)Formula (234) was obtained for wi = I in the paper [8] as a by-product of theinvestigations of WZ model. The first term in (234) is rather universal.

It dependsneither on the choice of the Borel subalgebra in the definition of the deformationnor on the cell of FR. On the contrary, the second term keeps the information aboutthe particular choice of (B+, B−) pair and it depends on the element wi of the Weylgroup characterizing the cell of the orbit.It is instructive to write down the Poisson bracket for the matrix elements of L.Using the classical r-matrices r+, r−(212), (213) and tensor notations, we have [5]:{L1, L2} = r+L1L2 + L1L2r−−L1r+L2 −L2r−L1 .

(235)Let us remind that the same symplectic form (230) corresponds to another Pois-son structure{g1, g2} = r+g1g2 −g1g2r+ = r−g1g2 −g1g2r−,(236)if instead of conditions (228), (229) we impose the following set of constraints onL+, L−and g:L+gwi = gLwiLL′+ ,(237)L−g = gLL′−. (238)6Discussion.In this section we formulate several problems related to the symplectic structuresdescribed in the paper.

The first of them concerns the quantum version of the pre-sented formalism. In the classical case the Kirillov symplectic form appears in thecontent of the theory of geometric quantization.

Roughly speaking, some coadjointorbits of the group G equipped with the Kirillov form correspond to irreducible27

representations of the Lie algebra G. The cotangent bundle T ∗G with its canoni-cal symplectic structure corresponds to the regular representation of G. Actually,we may restrict ourselves to the latter case because all the particular irreduciblerepresentations can be obtained from the regular one by means of the reductionprocedure. For Lie-Poisson groups the problem is not so simple even for D+.

Afterthe quantization the Poisson algebra (80) becomes the quantum algebra of functionson D+. Its basic relations can be written in the following form:d1d2 = Rd2d1R∗,(239)where we use tensor notations, R and R∗are quantum R-matrices corresponding tothe classical counterparts r and r∗.

The result we expect as an outcome of geometricquantization is an irreducible representation of the algebra (239) corresponding toa symplectic leaf in D+. It is easy to find such a representation for the main cellD00 = GG∗∩G∗G.

Algebra (239) Funkq(D+) acts in the space Funkq(G). It isan analogue of the standard regular representation in the space of functions on thegroup G. The algebra Funkq(G) is defined by the basic relations [9]Rg1g2 = g2g1R .

(240)On the cell D00 we can decompose the element d as a productd = gh∗= g∗h(241)of elements from G and G∗. Matrix elements of G act on the space Funkq(G) bymeans of multiplication and matrix elements of G∗generalize differential operators.The regular representation in Funkq(G) was considered in [10], where the quantumanalogue of the Fourier transformation was constructed.We expect that representations corresponding to other symplectic leaves Dij canbe found and presented in a similar form.

This would give a good basis for thegeometric quantization in the direct meaning of the word, i.e. establishing of thecorrespondence between the orbits and the quantum group representations.ForG=SU(n) this correspondence has been described in paper [11] by means of quan-tization of orbits of the dressing transformations.

It is a simple case because forG = SU(n) D = GG∗= G∗G and orbits are symplectic leaves. It should be men-tioned that this correspondence appears in a natural way in the course of investiga-tions of the quantum groups representation theory for the deformation parameterq being a root of unity.

If qN = 1, there exists an irreducible representation of thedeformed universal enveloping algebra Uq(G) corresponding to any orbit of dressingtransformations [12].Another problem which we would like to mention is a possible application of themachinery of sections 3 and 4 to physics. Having the closed form Ω, we can solve atleast locally the equationdα = Ω.

(242)The one-form α may be treated as a lagrangian of some mechanical system so thatthe action looks like follows:S0 =Zα . (243)28

If we add an appropriate hamiltonian H, we get a system with the actionS =Z(α −Hdt) . (244)Symplectic structures described in sections 3 and 4 provide a wide class of dynamicalsystems (244).

For the classical groups one obtains many interesting examples inthis way. Among them one finds the WZNW model and the gravitational WZ model[13].

Realizing the same idea for the Lie-Poisson case, one can hope to constructintegrable deformations of these systems.Acknowledgements.We are grateful to L.D.Faddeev, A.G.Reiman and K.Gawedzki for stimulating dis-cussions. We would like to thank M.A.Semenov-Tian-Shansky for the guidance inthe theory of Lie-Poisson groups.

The work of A.A. was supported by the joint pro-gram of CNRS (France) and Steklov Mathematical Institute (Russia). He thanksProf.

P.K.Mitter for perfect conditions in Paris. We are grateful to Prof. A.Niemifor hospitality in Uppsala where this work was completed.References[1] A.A.Kirillov, Elements of the theory of representations, Springer-Verlag, 1976.

[2] V.I.Arnold, Mathematical methods of classical mechanics, Springer-Verlag,1980. [3] A.Weinstein, The local structure of Poisson manifolds, J.Diff.Geom., v.18, n.3,1983, pp.523-557.

[4] V.G.Drinfeld, private communications. [5] M.A.Semenov-Tian-Shansky, Dressing transformations and Poisson-Lie groupactions, Publ.

RIMS, Kyoto University 21, no.6, 1985, p.1237. [6] V.G.Drinfeld, Quantum groups, in Proc.

ICM, MSRI, Berkeley, 1986, p.798. [7] J.H.Lu, A.Weinstein, Poisson-Lie groups, dressing transformations and Bruhatdecompositions, J.Diff.Geom., v.31, 1990, p.501.

[8] K.Gawedzki, F.Falceto, On quantum group symmetries of conformal field theo-ries, preprint IHES/P/91/59, September 1991. [9] L.D.Faddeev, N.Yu.Reshetikhin, L.Takhtajan, Quantization of Lie groups andLie algebras, Algebra i Analiz, v.1, 1989, p.178 (in Russian), English version inLeningrad Math.Journ, v.1.29

[10] M.A.Semenov-Tian-Shansky, Poisson-Lie groups, quantum duality principleand the twisted quantum double, Theor.Math.Phys. v.93, no.2, 1992, pp.302-329 (in Russian).

[11] B.Jurˇco, P.ˇStoviˇcek, Quantum dressing orbits on compact groups, preprintLMU-TPW-1992-3, 1992. [12] C.De Concini, V.G.Kac, Representations of quantum groups at roots of 1, Col-loque Dixmier, Progress in Math., Birkh¨auser, 1990, pp.471-506.

[13] A.Alekseev, S.Shatashvili, Path integral quantization of the coadjoint orbits ofthe Virasoro group and 2-D gravity, Nucl.Phys. B 323, 1989, pp.719-733.30

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