aa r X i v : . [ m a t h . DG ] O c t POISSON REDUCTION
CHIARA ESPOSITO
Abstract.
In this paper we develope a theory of reduction for classical sys-tems with Poisson Lie groups symmetries using the notion of momentum mapintroduced by Lu. The local description of Poisson manifolds and Poisson Liegroups and the properties of Lu’s momentum map allow us to define a Poissonreduced space. Introduction
In this paper we prove a generalization of the Marsden-Weinstein reduction tothe general case of an arbitrary Poisson Lie group action on a Poisson manifold.Reduction procedures are known in many different settings. In particular, a re-duction theory is known in the case of Poisson Lie groups acting on symplecticmanifolds [11] and in the case of Lie groups acting on Poisson manifolds [20], [15].An important generalization to the Dirac setting has been studied in [2].The theory of symplectic reduction plays a key role in classical mechanics. Thephase space of a system of n particles is described by a symplectic or more generallyPoisson manifold. Given a symmetry group of dimension k acting on a mechanicalsystem, the dimension of the phase space can be reduced by 2 k . Marsden-Weinsteinreduction formalizes this feature. Recall roughly the notion of Hamiltonian actionsin this setting. Given a Poisson manifold M there are natural Hamiltonian vectorfields { f, ·} on M . Let G be a Lie group acting on M by Φ; the action is Hamiltonianif the vector fields defined by the infinitesimal generator of Φ are Hamiltonian. Moreprecisely, let G be a Lie group acting on a Poisson manifold ( M, π ). The actionΦ : G × M → M is canonical if it preserves the Poisson structure π . Suppose thatthere exists a linear map H : g → C ∞ ( M ) such that the infinitesimal generator Φ X for X ∈ g of the canonical action is induced by H byΦ X = { H X , ·} . A canonical action induced by H is said Hamiltonian if H is a Lie algebra ho-momorphism. We can define a map µ : M → g ∗ , called momentum map, by H X ( m ) = h µ ( m ) , X i for m ∈ M . It is equivariant if the corresponding H is a Liealgebra homomorphism. Given an Hamiltonian action, under certain assumptions,the reduced space has been defined as M//G := µ − ( u ) /G u and it has been provedthat it is a Poisson manifold [16].In this paper we are interested in analyzing the case in which one has an extrastructure on the Lie group, a Poisson structure making it a Poisson Lie group.Poisson Lie groups are very interesting objects in mathematical physics. They maybe regarded as classical limit of quantum groups [6] and they have been studiedas carrier spaces of dynamical systems [10]. It is believed that actions of Poisson Lie groups on Poisson manifolds should be used to understand the “hidden symme-tries” of certain integrable systems [21]. Moreover, the study of classical systemswith Poisson Lie group symmetries may give information about the correspondingquantum group invariant system (an attempt can be found in [7], [8]).The purpose of this paper is to prove that, given a Poisson manifold acted by aPoisson Lie group, under certain conditions, we can also reduce this phase space toanother Poisson manifold.The paper is organized as follows. In Section 2 we recall some basic elementsof Poisson geometry: Poisson manifolds and their local description, Lie bialgebrasand Poisson Lie groups. A nice review of these results can be found in [22] and[19]. The Section 3 is devoted to Poisson actions and associated momentum mapsand we discuss dressing actions and their properties. In Section 4 we present themain result of this paper, the Poisson reduction, and we discuss an example.
Acknowledgments:
I would like to thank my advisor Ryszard Nest and EvaMiranda for many interesting discussions about Poisson reduction and its possibledevelopments. I also wish to thank George M. Napolitano for his help and hisuseful suggestions and Rui L. Fernandes for his comments regarding Dirac reductiontheory.2.
Poisson manifolds, Poisson Lie groups and Lie bialgebras
In this section we introduce the notion of Poisson manifolds and their localdescription, we give some background about Poisson Lie groups and Lie bialgebraswhich will be used in the paper. For more details on this subject, see [11], [6], [19],[22], [23].2.1.
Poisson manifolds and symplectic foliation.
A Poisson structure on asmooth manifold M is a Lie bracket {· , ·} on the space C ∞ ( M ) of smooth functionson M which satisfies the Leibniz rule. This bracket is called Poisson bracket and amanifold M equipped with such a bracket is called Poisson manifold. Therefore, abivector field π on M such that the bracket { f, g } := h π, df ∧ dg i is a Poisson bracket is called Poisson tensor or Poisson bivector field. A Poissontensor can be regarded as a bundle map π ♯ : T ∗ M → T M : h α, π ♯ ( β ) i = π ( α, β ) Definition 2.1.
A mapping φ : ( M , π ) → ( M , π ) between two Poisson mani-folds is called a Poisson mapping if ∀ f, g ∈ C ∞ ( M ) one has (1) { f ◦ φ, g ◦ φ } = { f, g } ◦ φ The structure of a Poisson manifold is described by the splitting theorem of AlanWeinstein [23], which shows that locally a Poisson manifold is a direct product of asymplectic manifold with another Poisson manifold whose Poisson tensor vanishesat a point.
Theorem 2.1 (Weinstein) . On a Poisson manifold ( M, π ) , any point m ∈ M has acoordinate neighborhood with coordinates ( q , . . . , q k , p , . . . , p k , y , . . . , y l ) centeredat m , such that (2) π = X i ∂∂q i ∧ ∂∂p i + 12 X i,j φ ij ( y ) ∂∂y i ∧ ∂∂y j φ ij (0) = 0 . OISSON REDUCTION 3
The rank of π at m is k . Since φ depends only on the y i s, this theorem gives adecomposition of the neighborhood of m as a product of two Poisson manifolds: onewith rank k , and the other with rank 0 at m . The term(3) 12 X i,j φ ij ( y ) ∂∂y i ∧ ∂∂y j is called transverse Poisson structure and it is evident that the equations y i = 0determine the symplectic leaf through m .2.2. Lie bialgebras and Poisson Lie groups.Definition 2.2.
A Poisson Lie group ( G, π G ) is a Lie group equipped with a mul-tiplicative Poisson structure π G , i.e. such that the multiplication map G × G → G is a Poisson map. Let G be a Lie group with Lie algebra g . The linearization δ := d e π G : g → g ∧ g of π G at e defines a Lie algebra structure on the dual g ∗ of g and, for this reason,it is called cobracket. The pair ( g , g ∗ ) is called Lie bialgebra. The relation betweenPoisson Lie groups and Lie bialgebras has been proved by Drinfeld [6]: Theorem 2.2. If ( G, π G ) is a Poisson Lie group, then the linearization of π G at e defines a Lie algebra structure on g ∗ such that ( g , g ∗ ) form a Lie bialgebra over g ,called the tangent Lie bialgebra to ( G, π G ) . Conversely, if G is connected and simplyconnected, then every Lie bialgebra ( g , g ∗ ) over g defines a unique multiplicativePoisson structure π G on G such that ( g , g ∗ ) is the tangent Lie bialgebra to thePoisson Lie group ( G, π G ) . From this theorem it follows that there is a unique connected and simply con-nected Poisson Lie group ( G ∗ , π G ∗ ), called the dual of ( G, π G ), associated to theLie bialgebra ( g ∗ , δ ). If G is connected and simply connected, then the dual of G ∗ is G . Example 2.1 ( g = ax + b ) . Consider the Lie algebra g spanned by X and Y withcommutator (4) [ X, Y ] = Y and cobracket given by (5) δ ( X ) = 0 δ ( Y ) = X ∧ Y. The Lie bracket on g ∗ is given by [ X ∗ , Y ∗ ] = Y ∗ . A matrix representation of g is the Lie algebra gl (2 , R ) via X = (cid:18) (cid:19) Y = (cid:18) (cid:19) and X ∗ = (cid:18) (cid:19) Y ∗ = (cid:18) (cid:19) with the metric γ ( a, b ) = tr ( aJbJ ) and J = (cid:18) (cid:19) . CHIARA ESPOSITO
The corresponding Poisson Lie group G and dual G ∗ are subgroups of GL (2 , R ) of matrices with positive determinant are given by (6) G = (cid:26)(cid:18) ξ η (cid:19) : η > (cid:27) G ∗ = (cid:26)(cid:18) s t (cid:19) : s > (cid:27) Poisson actions and Momentum maps
In this section we first introduce the concept of Poisson action of a Poisson Liegroup on a Poisson manifold, which generalizes the canonical action of a Lie groupon a symplectic manifold. We define momentum maps associated to such actionsand finally we consider the particular case of a Poisson Lie group G acting on itsdual G ∗ by dressing transformations. This allows us to study the symplectic leavesof G that are exactly the orbits of the dressing action. These topics can be founde.g. in [11], [12] and [21].From now on we assume that G is connected and simply connected. Definition 3.1.
The action
Φ : G × M → M of a Poisson Lie group ( G, π G ) ona Poisson manifold ( M, π ) is called Poisson action if Φ is a Poisson map, where G × M is a Poisson manifold with structure π G ⊕ π . This definition generalizes the notion of canonical action; indeed, if G carries thetrivial Poisson structure π G = 0, the action Φ is Poisson if and only if it preserves π , i.e. if it is canonical. In general, the structure π is not invariant with respectto the action Φ. The easiest examples of Poisson actions are given by the left andright actions of G on itself.For an action Φ : G × M → M we use Φ : g → V ect M : X Φ X to denote theLie algebra anti-homomorphism which defines the infinitesimal generators of thisaction. The proof of the following Theorem can be found in [13]. Theorem 3.1.
The action
Φ : G × M → M is a Poisson action if and only if (7) L Φ X ( π ) = (Φ ∧ Φ) δ ( X ) for any X ∈ g , where L denotes the Lie derivative and δ is the derivative of π G at e . Let Φ : G × M → M be a Poisson action of ( G, π G ) on ( M, π ). Let G ∗ be thedual Poisson Lie group of G and let Φ X be the vector field on M which generatesthe action Φ. In this formalism the definition of momentum map reads (Lu, [11],[12]): Definition 3.2.
A momentum map for the Poisson action
Φ : G × M → M is amap µ : M → G ∗ such that (8) Φ X = π ♯ ( µ ∗ ( θ X )) where θ X is the left invariant 1-form on G ∗ defined by the element X ∈ g = ( T e G ∗ ) ∗ and µ ∗ is the cotangent lift T ∗ G ∗ → T ∗ M . In other words, the momentum map generates the vector field Φ X via the con-struction X ∈ g → θ X ∈ T ∗ G ∗ → α X = µ ∗ ( θ X ) ∈ T ∗ M → π ♯ ( α X ) ∈ T M
It is important to remark that Noether’s theorem still holds in this general context.
OISSON REDUCTION 5
Theorem 3.2.
Let
Φ : G × M → M a Poisson action with momentum map µ : M → G ∗ . If H ∈ C ∞ ( M ) is G -invariant, then µ is an integral of the Hamiltonianvector field associated to H . It is important to point out that in this setting the vector field Φ X is not Hamil-tonian, unless the Poisson structure on G is trivial. In this case G ∗ = g ∗ , thedifferential 1-form θ X is the constant 1-form X on g ∗ , and(9) µ ∗ ( θ X ) = d ( H X ) , where H X ( m ) = h µ ( m ) , X i . This implies that the momentum map is the canonical one and(10) Φ X = π ♯ ( dH X ) = { H X , ·} . In other words, Φ X is the Hamiltonian vector field with Hamiltonian H X ∈ C ∞ ( M ).We observe that, when π G is not trivial, θ X is a Maurer-Cartan form, hence µ ∗ ( θ X )can not be written as a differential of a Hamiltonian function. In the following wegive an example for the infinitesimal generator in this general case.3.1. Dressing Transformations.
One of the most important example of Poissonaction is the dressing action of G on G ∗ . The name “dressing” comes from thetheory of integrable systems and was introduced in this context in [21]. Interestingexamples can be found in [1]. We remark that, given a Poisson Lie group ( G, π G ),the left (right) invariant 1-forms on G ∗ form a Lie algebra with respect to thebracket: [ α, β ] = L π ♯ ( α ) β − L π ♯ ( β ) α − d ( π ( α, β )) . For X ∈ g , let θ X be the left invariant 1-form on G ∗ with value X at e . Let usdefine the vector field on G ∗ (11) l ( X ) = π ♯G ∗ ( θ X ) . The map l : g → T G ∗ : X l ( X ) is a Lie algebra anti-homomorphism. We call l the left infinitesimal dressing action of g on G ∗ ; its linearization at e is the coadjointaction of g on g ∗ . Similarly we can define the right infinitesimal dressing action.Let l ( X ) (resp. r ( X )) a left (resp. right) dressing vector field on G ∗ . If all thedressing vector fields are complete, we can integrate the g -action into an action of G on G ∗ called the dressing action and we say that the dressing actions consist ofdressing transformations. Definition 3.3.
A multiplicative Poisson tensor π G on G is complete if each left(equiv. right) dressing vector field is complete on G . From the definition of dressing action follows (the proof can be found in [21])that the orbits of the right or left dressing action of G ∗ (resp. G ) are the symplecticleaves of G (resp. G ∗ ).It can be proved (see [11]) that if π G is complete, both left and right dressingactions are Poisson actions with momentum map given by the identity.Assume that G is a complete Poisson Lie group. We denote respectively the left(resp. right) dressing action of G on its dual G ∗ by g l g (resp. g r g ). Definition 3.4.
A momentum map µ : M → G ∗ for a left (resp. right) Poissonaction Φ is called G-equivariant if it is such with respect to the left dressing actionof G on G ∗ , that is, µ ◦ Φ g = λ g ◦ µ (resp. µ ◦ Φ g = ρ g ◦ µ ) CHIARA ESPOSITO
It is important to remark that a momentum map is G -equivariant if and only ifit is a Poisson map, i.e. µ ∗ π = π G ∗ . Definition 3.5.
An action
Φ : G × M → M of a Poisson Lie group ( G, π G ) on aPoisson manifold ( M, π ) is said Hamiltonian if it is a Poisson action generated byan equivariant momentum map. Poisson Reduction
In this section we present the main result of this paper. We show that, givena Hamiltonian action Φ, as defined above, we can define a reduced manifold interms of momentum map and prove that it is a Poisson manifold. The approachused is a generalization of the orbit reduction [14] in symplectic geometry. Recallthat, under certain conditions, the orbit space of Φ is a smooth manifold and itcarries a Poisson structure. First, we give an alternate proof of this claim. Then,we consider a generic orbit O u of the dressing action of G on G ∗ , for u ∈ G ∗ ,and we prove that the set µ − ( O u ) /G is a regular quotient manifold with Poissonstructure induced by the Poisson structure on M . Similarly to the symplectic case,this reduced space is isomorphic to the space µ − ( u ) /G u which will be regarded asthe Poisson reduced space.4.1. Poisson structure on
M/G . Consider a Hamiltonian action of a connectedand simply connected Poisson Lie group (
G, π G ) on a Poisson manifold ( M, π ). Itis known that, if the action is proper and free, the orbit space
M/G is a smoothmanifold, it carries a Poisson structure such that the natural projection M → M/G is a Poisson map (a proof of this result can be found in [21]). In this section wegive an alternate proof of this result, by introducing an explicit formulation for theinfinitesimal generator of the Hamiltonian action, in terms of local coordinates.As discussed in the previous section, a Hamiltonian action is a Poisson actioninduced by an equivariant momentum map µ : M → G ∗ by formula (8). In otherwords, the map α : g → Ω ( M ) : X α X = µ ∗ ( θ X )is a Lie algebra homomorphism such thatΦ X = π ♯ ( α X )The dual map of α defines a g ∗ -valued 1-form on M , still denoted by α , satisfyingMaurer-Cartan equation (as proved in [11]) dα + 12 [ α, α ] g ∗ = 0 . In particular, { α X : X ∈ g } defines a foliation F on M . Lemma 4.1.
The space of G -invariant functions on M is closed under Poissonbracket. Hence π defines a Poisson structure on M/G
Proof.
Let H i , i = 1 , . . . n be local coordinates on M such that F = Ker { dH , . . . , dH n } OISSON REDUCTION 7
Then(12) α X = X i c i ( X ) dH i and(13) Φ X [ f ] = π ♯ ( α X ) = X i c i ( X ) { H j , f } M . This implies that a function f ∈ C ∞ ( M ) is G -invariant (Φ X [ f ] = 0) if and only if { H i , f } = 0 for any i . If f, g are G -invariant functions on M , we have { H i , f } = { H i , g } = 0 for any i . Then, using the Jacobi identity we get { H i , { f, g }} = 0.Since G is connected, the result follows. (cid:3) Poisson reduced space.
Assume that G is connected, simply connected andcomplete. In order to define a reduced space and to prove that it is a Poissonmanifold we consider a generic orbit O u of the dressing orbit of G on G ∗ passingthrough u ∈ G ∗ . First, we prove the following: Lemma 4.2.
Let
Φ : G × M → M be a free and Hamiltonian action of a compactPoisson Lie group ( G, π G ) on a Poisson manifold ( M, π ) . Then: (i) O u is closed and the Poisson structure π G ∗ does not depend on the transver-sal coordinates on O u . (ii) µ − ( O u ) /G is a smooth manifold.Proof. (i) If G is compact, any G -action is automatically proper. This impliesthat, given u ∈ G ∗ the generic orbit O u of the dressing action is closed.From section (3.1) we know that O u is the symplectic leaf through u . Usingthe local description of Poisson manifolds introduced in Theorem (2.1) itis evident that π G ∗ restricted to O u does not depend on the transversalcoordinates y i .(ii) If the action Φ is free, the momentum map µ : M → G ∗ is a submersion ontosome open subset of G ∗ . This implies that µ − ( u ) is a closed submanifoldof M . As µ is equivariant, it follows that µ − ( u ) is G -invariant. Freeand proper actions of G on M restrict to free and proper G -actions on G -invariant submanifolds. In particular, the action of G on µ − ( u ) is stillproper, then G · µ − ( u ) is closed. Using the equivariance we have that G · µ − ( u ) = µ − ( O u ), which is still G -invariant. The action of G on µ − ( O u )is proper and free, so we can conclude that the orbit space µ − ( O u ) /G isa smooth manifold. (cid:3) We aim to prove that the manifold
N/G := µ − ( O u ) /G carries a Poisson struc-ture. In the previous Lemma we stated that π G ∗ restricted to O u does not dependon the transversal coordinates y i ’s; if x i are local coordinates along N = µ − ( O u )and H i are pullback of the transversal coordinates y i ’s by(14) H i := y i ◦ µ we can easily deduce that the Poisson structure π on M involves derivatives in H i only in the combination ∂ x i ∧ ∂ H i CHIARA ESPOSITO
This is evident because the differential d µ between T M | N /T N and T G ∗ /T O u is abijective map. Moreover, since { y i , y j } vanishes on the orbit O u , { H i , H j } vanisheson the preimage N and dH i ’s are in the span of { α X : X ∈ g } .Now we introduce the ideal I generated by H i and prove some properties. Lemma 4.3.
Let I = { f ∈ C ∞ ( M ) : f | N = 0 } . (i) I is defined in an open G -invariant neighborhood U of N . (ii) I is closed under Poisson bracket.Proof. (i) The coordinates H i are locally defined but we can show that I isglobally defined. Considering a different neighborhood on the orbit of G ∗ wehave transversal coordinates y ′ i and their pullback to M will be H ′ i = y ′ i ◦ µ .The coordinates H ′ i are defined in a different open neighborhood V of N ,but we can see that the ideal I generated by H i coincides with I ′ generatedby H ′ i on the intersection of U and V , then it is globally defined.(ii) Since µ is a Poisson map we have: { H i , H j } M = { y i ◦ µ , y j ◦ µ } M = { y i , y j } G ∗ ◦ µ . Hence the ideal I is closed under Poisson brackets. (cid:3) Motivated by this Lemma we use the following identification C ∞ ( N/G ) ≃ ( C ∞ ( U ) / I ) G . Lemma 4.4.
Suppose that
N/G is an embedded submanifold of the smooth manifold
M/G , then (15) ( C ∞ ( U ) / I ) G ≃ ( C ∞ ( U ) G + I ) / I Proof.
Let f be a smooth function on U satisfying [ f ] ∈ ( C ∞ ( U ) / I ) G . As theequivalence class [ f ] is G -invariant, we have(16) f ( G · m ) = f ( m ) + i ( m ) , where i ∈ I and G · m is a generic orbit of the Hamiltonian action of G on M . It isclear that f | N is G -invariant and hence it defines a smooth function ¯ f ∈ C ∞ ( N/G ).Since
N/G is a k -dimensional embedded submanifold of the n -dimensional smoothmanifold M/G , the inclusion map ι : N/G → M/G has local coordinates represen-tation:(17) ( x , . . . , x k ) ( x , . . . , x k , c k +1 , . . . , c n )where c i are constants. Hence we can extend ¯ f to a smooth function φ on M/G bysetting ¯ f ( x , . . . , x k ) = φ ( x , . . . , x k , , . . . , f of φ by pr : M → M/G is G -invariant and satisfies(18) ˜ f − f | N = 0 , hence ˜ f − f ∈ I . (cid:3) Using these results we can prove the following:
Theorem 4.1.
Let
Φ : G × M → M be a free Hamiltonian action of a compactPoisson Lie group ( G, π G ) on a Poisson manifold ( M, π ) with momentum map µ : M → G ∗ . The orbit space N/G has a Poisson structure induced by π . OISSON REDUCTION 9
Proof.
First we prove that the Poisson bracket of M induces a well defined Poissonbracket on ( C ∞ ( U ) G + I ) / I . In fact, for any f + i ∈ C ∞ ( U ) G / I and j ∈ I thePoisson bracket { f + i, j } still belongs to the ideal I . Since the ideal I is closedunder Poisson brackets, { i, j } belongs to I . The function j , by definition on theideal I , can be written as a linear combination of H i , so { f, j } = P i a i { f, H i } . ByLemma 4.1, we have { f, H i } = 0, hence { f + i, j } ∈ I as stated. Finally, usingthe isomorphism proved in the Lemma (4.4) and the identification C ∞ ( N/G ) ≃ ( C ∞ ( U ) / I ) G , the claim is proved. (cid:3) Finally, we observe that there is a natural isomorphism(19) µ − ( u ) /G u ≃ µ − ( O u ) /G. We refer to µ − ( u ) /G u as the Poisson reduced space.5. An example
In this section we discuss a concrete example of Poisson reduction. Considerthe Lie bialgebra g = ax + b discussed in Example (2.1). The Poisson tensor onthe dual Poisson Lie group G ∗ is given, in the coordinates ( s, t ) introduced in thematrix representation, by(20) π G ∗ = st∂ s ∧ ∂ t . It is clear that ( s, t ) are global coordinates on G ∗ . First, we need to study theorbits of the dressing action. Remember that the dressing orbits O u through apoint u ∈ G ∗ are the same as the symplectic leaves, hence it is clear that they aredetermined by the equation t = 0. The symplectic foliation of the manifold G ∗ inthis case is given by two open orbits, determined by the conditions t > t < t = 0 and a ∈ R + .Consider a Hamiltonian action Φ : G × M → M of G on a generic Poissonmanifold M induced by the equivariant momentum map µ : M → G ∗ . Its pullback(21) µ ∗ : C ∞ ( G ∗ ) −→ C ∞ ( M )maps the coordinates s and t on G ∗ to x ( u ) = s ( µ ( u )) y ( u ) = t ( µ ( u )) . It is important to underline that we have no information on the dimension of M ,so x and y are just a pair of the possible coordinates. Nevertheless, since µ is aPoisson map, we have(22) { x, y } = xy on M . The infinitesimal generators of the action Φ can be written in terms of thesecoordinates ( x, y ) as(23) Φ( X ) = x { y, ·} Φ( Y ) = x { x − , ·} . In the following, we discuss the Poisson reduction case by case, by consideringthe different dressing orbits studied above.
Case 1: ( t > O u determined by the condition t > s and t are both positive, we can put(24) x = e p , y = e q . Since { x, y } = xy we have(25) { p, q } = 1 . For this reason the preimage of the dressing orbit can be split as N = R × M and C ∞ ( N ) is given explicitly by the set of functions generated by y − . Theinfinitesimal generators are given by(26) Φ( X ) = e p { e q , ·} Φ( Y ) = e p { e − p , ·} which is the action of G on the plane. Hence the Poisson reduction in this case isgiven by(27) ( C ∞ ( M )[ y − ]) G . Case 2: ( t < y = − e q .Case 3: ( t = 0). The orbit O u is given by fixed points on the line t = 0, then wechoose the point s = 1. Consider the ideal I = h x − , y i of functions vanishing on N . It is easy to check that it is G -invariant, hence the Poisson reduction in thiscase is simply given by(28) ( C ∞ ( M ) / I ) G . Questions and future directions.
The theory of Poisson reduction can befurther developed, as it has been obtained under the assumption that the orbit space
M/G is a smooth manifold. This result could be proved under weaker hypothesis,for instance requiring that
M/G is an orbifold.As stated in the introduction, the idea of momentum map and Poisson reductioncan be also used for the study of symmetries in quantum mechanics. In particular,the approach of deformation quantization would provide a relation between classicaland quantum symmetries. A notion of quantum momentum map has been definedin [7], [8] and it can be used to define the quantization of the Poisson reduction.At classical level, Poisson reduction could be generalized to actions of Dirac Liegroups [17] on Dirac manifolds [4]. Finally, a possible development of this theory isits integration to symplectic groupoids by means of the theories on the integrabilityof Poisson brackets [5] and Poisson Lie group actions [9].
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