Uniqueness of the Momentum map
Abstract.
We give a detailed discussion of existence and uniqueness of Lu’s momentum map. We introduce the infinitesimal momentum map, and analyze its integrability to the usual momentum map, its existence and its deformations.
Contents
1. Introduction
The classical momentum map for an action of a Lie group on a Poisson manifold provides a mathematical formalization of the notion of conserved quantity associated to symmetries of a dynamical system. The standard definition of momentum map only requires a canonical Lie algebra action and its existence is guaranteed whenever the infinitesimal generators of the Lie algebra action are Hamiltonian vector fields (modulo vanishing of a certain Lie algebra cohomology class). In this paper we focus on a generalization of the momentum map provided by Lu [5], [6].
The detailed construction of this generalized momentum map and its basic properties are recalled in the following section. The basic structure is as follows. Given Poisson Lie group one introduces the dual Poisson Lie group and, under fairly general conditions, carries a Poisson action of (and vice versa). The Lie algebra of is naturally identified with the space of (left)invariant oneforms on :
Given a Poisson manifold with a Poisson action of , a momentum map is a smooth, Poisson map
satisfying
where is the map induced by the action of on . A canonical example of a momentum map is the identity map , in which case coincides with the structure oneform in of the Lie group .
The Poisson structure on gives its Lie algebra a structure of a Lie bialgebra and hence a structure of Gerstenhaber algebra on . On the other hand the Poisson bracket on gives a structure of Lie algebra with bracket which induces a structure of Gerstenhaber algebra on . The map from above lifts to a morphism of Gerstenhaber algebras
which we will call an infinitesimal momentum map (cf. the subsection 3.1 and the proposition 3). The existence of the infinitesimal momentum map has been discussed in [2]. The main subject of this paper is the study of the properties of this infinitesimal momentum map and its relation to the usual momentum map. In particular, we show under which conditions it integrates to the usual momentum map.
The fact that is map of Gerstenhaber algebras reduces to two equations
The second is a MaurerCartan type equation, in fact, in the case when it is precisely the Maurer Cartan equation for the Lie group . In the case when is formal, the second equation admits explicit solution modulo gauge equivalence (cf. Theorem 3.2).
Theorem Suppose that is is a Kähler manifold. The set of gauge equivalence classes of satisfying the equation
(1) 
is in bijective correspondence with the set of the cohomology classes satisfying
(2) 
The following describes conditions under which an infinitesimal momentum map integrates to the usual momentum map (cf. Theorem 3.1 for the details).
Theorem Let be a Poisson manifold and an infinitesimal momentum map. Suppose that and are simply connected and is compact. Then generates an involutive distribution on and a leaf of is a graph of a momentum map if
(3) 
In the section 3.2 we study concrete cases of this globalization question and prove the existence and uniqueness/nonuniqueness of a momentum map associated to a given infinitesimal momentum map for the particular case when the dual Poisson Lie group is abelian, respectively the Heisenberg group. For the second case the result is as follows (cf. Theorem 3.4).
Theorem Let be a Poisson Lie group acting on a Poisson manifold with an infinitesimal momentum map and such that is the Heisenberg group. Let denote the basis of dual to the standard basis of , with central and . Then
(4) 
where is a constant on . The form lifts to a momentum map if and only if . When the set of momentum maps with given is one dimensional with free transitive action of .
Finally, in the last section, we study the question of infinitesimal deformations of a given momentum map. The main result is Theorem 4.1, which describes explicitly the space tangent to the space of momentum maps at a given point. The main result can be formulated as a statement that the space of momentum maps has a structure of flat manifold (in an appropriate topology).
Theorem Infinitesimal deformations of a momentum map are given by smooth maps satisfying the equations
For all ,
(5)  
(6) 
This theorem has the following corollary (cf. Corollary 4.2).
Theorem Suppose that is a compact and semisimple Poisson Lie group with Poisson action on a Poisson manifold and with a momentum map . Any smooth deformation of is given by integrating a Hamiltonian flow on commuting with the action of .
2. Preliminaries: Poisson actions and Momentum maps
In this section we give a brief summary of the notions of Poisson action and momentum map in the Poisson context. We discuss the dressing transformations as an example of Poisson actions that will allow us to introduce the concept of Hamiltonian action.
Recall that a Poisson Lie group is a Lie group equipped with a multiplicative Poisson structure . From the Drinfeld theorem [1], given a Poisson Lie group , the linearization of at defines a Lie algebra structure on such that form a Lie bialgebra over . For this reason, in the following we always assume that is connected and simply connected.
Definition 2.1.
The action of on is called Poisson action if the map is Poisson, where is a Poisson product with structure
Given an action , we denote by the Lie algebra antihomomorphism from to which defines the infinitesimal generator of this action.
Proposition 1.
Assume that is a connected Poisson Lie group. Then the action is a Poisson action if and only if
(7) 
for any , where is the derivative of at .
The proof of this Proposition can be found in [7]. Motivated by this fact, we introduce the following definition.
Definition 2.2.
A Lie algebra action is called an infinitesimal Poisson action of the Lie bialgebra on if it satisfies eq. (7).
Definition 2.3.
A momentum map for the Poisson action is a map such that
(8) 
where is the left invariant 1form on defined by the element and is the cotangent lift .
2.1. Dressing Transformations
One of the most important example of Poisson action is the dressing action of on . Consider a Poisson Lie group , its dual and its double , with Lie algebras , and , respectively.
Let the vector field on defined by
(9) 
for each . Here is the left invariant 1form on defined by . The map is a Lie algebra antihomomorphism. Using the MaurerCartan equation for :
(10) 
The action is an infinitesimal Poisson action of the Lie bialgebra on the Poisson Lie group , called left infinitesimal dressing action (see, for example, [5]). Similarly, the right infinitesimal dressing action of on is defined by where is the right invariant 1form on .
Let (resp. ) a left (resp. right) dressing vector field on . If all the dressing vector fields are complete, we can integrate the action into a Poisson action on called the dressing action and we say that the dressing actions consist of dressing transformations. The orbits of the dressing actions are precisely the symplectic leaves in (see [9], [5]).
The momentum map for the dressing action of on is the opposite of the identity map from to itself.
Definition 2.4.
A multiplicative Poisson tensor on is complete if each left (equiv. right) dressing vector field is complete on .
It has been proved in [5] that a Poisson Lie group is complete if and only if its dual Poisson Lie group is complete. Assume that is a complete Poisson Lie group. We denote respectively the left (resp. right) dressing action of on its dual by (resp. ).
Definition 2.5.
A momentum map for a left (resp. right) Poisson action is called Gequivariant if it is such with respect to the left dressing action of on , that is, (resp. )
A momentum map is equivariant if and only if it is a Poisson map, i.e. . Given this generalization of the concept of equivariance introduced for Lie group actions, it is natural to call Hamiltonian action a Poisson action induced by an equivariant momentum map.
3. The infinitesimal momentum map
In this section we study the conditions for the existence and the uniqueness of the momentum map. In particular, we give a new definition of the momentum map, called infinitesimal, in terms of oneforms and we study the conditions under which the infinitesimal momentum map determines a momentum map in the usual sense. We describe the theory of reconstruction of the momentum map from the infinitesimal one in two explicit cases. Finally, we provide the conditions which ensure the uniqueness of the momentum map.
3.1. The structure of a momentum map
Recall that, for the Poisson Lie group we identify with the space of left invariant 1forms on ; this space is closed under the bracket defined by and the induced bracket on , by the above identification, coincides with the original Lie bracket on (see [10]).
Proposition 2.
Let be two left invariant 1forms on , such that , then
(11) 
and
(12) 
Proof.
Let us consider and element and the correspondent left invariant vector field . Recall that given a Poisson manifold, the Poisson structure always induces a Lie bracket on the space of oneform on the manifold (see [8]) by
(13) 
Using this explicit formula for we can see that
(14) 
This proves that is a left invariant 1form. In particular, since , eq. (11) is proved^{1}^{1}1This relation has already been claimed in [4]. Moreover, we have
(15) 
since . From , eq. (12) follows. ∎
As a direct consequence, recalling that the pullback and the differential commute and using the equivariance of the momentum map, we have the following proposition:
Proposition 3.
Given a Poisson action with equivariant momentum map , the forms satisfy the following identities:
(16)  
(17) 
This motivates the following Definition.
Definition 3.1.
Let be a Poisson manifold and a Poisson Lie group. An infinitesimal momentum map is a morphism of Gerstenhaber algebras
(18) 
The following theorem describes the conditions in which an infinitesimal momentum map determines a momentum map in the usual sense.
Theorem 3.1.
Let be a Poisson manifold and a linear map which satisfies the conditions (16)(17). Then:

The set generate an involutive distribution on .

If is connected and simply connected, the leaves of coincide with the graphs of the maps satisfying and acts freely and transitively on the space of leaves by left multiplication on the second factor.

The vector fields define a homomorphism from to . If they integrate to the action (e.g. when is compact and simply connected), then is a Poisson action and is its momentum map if and only if the functions
(19) satisfy
(20) for all .
Proof.

Using the eqs. (10) and (17), the valued form on satisfies as a consequence, from the Frobenius theorem, it defines a distribution on . Let be any of its leaves and let , denote the projection onto the first (resp. second) factor in . Since the linear span of , at any point coincides with , the restriction of the projection to is an immersion. Finally, since , is a covering map.

Under the hypothesis that is simply connected, is a diffeomorphism and
is a smooth map whose graph coincides with . It is immediate, that . Moreover, since ’s are left invariant it follows immediately that the action of on the space of leaves by left multiplication of the second factor is free and transitive.

Suppose that the condition (20) is satisfied. Then
and coincides with the set of zero’s of . Hence, is a Poisson map and, in particular
i.e. it is a equivariant map.
∎
The first of the equations in the Definition (3), the equation
(21) 
can be solved explicitly in the case when is a Kähler manifold. Before stating the result, we need to introduce the concept of gauge equivalence of solutions of (21):
Definition 3.2.
Two solutions and of eq. (21) are said to be gauge equivalent, if there exists a smooth function such that
(22) 
Theorem 3.2.
Suppose that is is a Kähler manifold. The set of gauge equivalence classes of satisfying the equation
(23) 
is in bijective correspondence with the set of the cohomology classes satisfying
(24) 
Proof.
Since is is a Kähler manifold, is a formal CDGA (commutative differential graded algebra) [3]. As a consequence,
(25) 
is a formal DGLA and, in particular, there exists a bijection between the equivalence classes of Maurer Cartan elements of and Maurer Cartan elements of .
A MaurerCartan element in is an element in satisfying
(26) 
and the claim is proved. ∎
3.2. The reconstruction problem
In this section we discuss the conditions under which the distribution defined in Theorem 3.1 admits a leaf satisfying eq. (20). In particular, we analyze the case where the structure on is trivial and the Heisenberg group case. In the following we keep the assumption that is connected and simply connected.
3.2.1. The abelian case
Suppose that is abelian. Then, the forms satisfy , hence (since ), for some .
Let us denote by the linear functions . Then and the leaves of the distribution coincide with the level sets (on ) of the functions
(27) 
Furthermore, we have
(28) 
In this case, the basic identity (13) reduces to
(29) 
hence
(30) 
for some constants . By the Jacobi identity, the constants define a class . Suppose that this class vanishes (for example if semisimple). Then, there exists a such that . Hence, given a leaf ,
(31) 
if and only if is given by
(32) 
In other words, the space of leaves of which give a momentum map coincides with the affine space modeled on (which again vanishes when is semisimple). This proves the following theorem.
Theorem 3.3.
Suppose that is a connected and simply connected Lie group with trivial Poisson structure and is compact. Then an infinitesimal momentum map is a map such that
(33) 
The element is a two cocycle on with values in . The infinitesimal momentum map is generated by a momentum map if this cocycle vanishes and, in this case, is unique.
3.2.2. the Heisenberg group case
Suppose now that is the Heisenberg group. Let be a basis for , where is central and . Let be the dual basis of . The cocycle on is given by
(34) 
then
(35) 
There are essentially two possibilities for the Lie bialgebra structure on , which give the following two possibilities for the Lie algebra structure on . Either
(36) 
or
(37) 
The result below will turn out to be independent of the choice (the computations will be done using the second choice, which corresponds to , with acting by rotation on ). Below we use the notation
(38) 
Applying the Cartan formula and the identity to the basic equation (13), we get
(39) 
In our case it gives the following equations
(40) 
which are also satisfied after replacing with . Let denote the ideal generating our distribution . Then, from above,
(41) 
and
(42) 
Here, as before, is a leaf of . Using the relation (12), we get
(43) 
In particular, is invariant under left translations. Since is zero at the identity, we get
(44) 
Since , the function is leafwise constant. Using the definition (19) and the equation (44) it follows that also is leafwise constant. Hence we have
Lemma 3.1.
is constant on and necessary condition for existence of the momentum map is .
By assuming that , given a leaf by eq. (41),
(45) 
for some constants and . Setting to , we get
(46) 
The final result is as follows.
Theorem 3.4.
Let be a Poisson Lie group acting on a Poisson manifold with an infinitesimal momentum map and such that is the Heisenberg group. Let denote the basis of dual to the standard basis of , with central and . Then
(47) 
where is a constant on . The form lifts to a momentum map if and only if . When the set of momentum maps with given is one dimensional with free transitive action of .
4. Infinitesimal deformations of a momentum map
In the following we study infinitesimal deformations of a given momentum map.
Let be a Poisson manifold with a Poisson action of a Poisson Lie group generated by the momentum map . In the following we denote by the exponential map. Suppose that , , is a differentiable path of momentum maps for this action. We can assume that is of the form
(48) 
for some differentiable maps and .
Theorem 4.1.
In the notation above the following identities hold.
For all ,
(49)  
(50) 
Proof.
Let us compute
First note that
(51) 
where and are the right and left multiplication, respectively. Calculating the derivative we get:
(52) 
and
(53) 
The differential of the exponential map is a map from the cotangent bundle of to the cotangent bundle of . It can be trivialized as . Furthermore, hence the map is given by . We get
(54) 
and finally
(55) 
Since is independent of , and we get the identity (50).
Corollary 4.2.
Suppose that is a Poisson manifold with a Poisson action of a compact semisimple Poisson Lie group . Then any infinitesimal deformation of a momentum map as above is generated by a one parameter family of gauge transformations.
Proof.
Since the relation (49) implies that and is compact semisimple, is a Lie coboundary, i. e. there exists a function
such that
(59) 
In particular, it is easy to check that , where . Now observe that
(60) 
hence
(61) 
Substituting the equations 59 and 61 in (50) we get
(62) 
In other words the Hamiltonian vector field associated to commutes with the group action and is tangent to the derivative of at as claimed. ∎
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