Reduction of Symplectic Lie Algebroids by a Lie Subalgebroid and a Symmetry Lie Group

We describe the reduction procedure for a symplectic Lie algebroid by a Lie subalgebroid and a symmetry Lie group. Moreover, given an invariant Hamiltonian function we obtain the corresponding reduced Hamiltonian dynamics. Several examples illustrate the generality of the theory.


Introduction
As it is well known, symplectic manifolds play a fundamental role in the Hamiltonian formulation of Classical Mechanics. In fact, if M is the configuration space of a classical mechanical system, then the phase space is the cotangent bundle T * M , which is endowed with a canonical symplectic structure, the main ingredient to develop Hamiltonian Mechanics. Indeed, given a Hamiltonian function H, the integral curves of the corresponding Hamiltonian vector field are determined by the Hamilton equations.
A method to obtain new examples of symplectic manifolds comes from different reduction procedures. One of these procedures is the classical Cartan symplectic reduction process: If (M, ω) is a symplectic manifold and i : C → M is a coisotropic submanifold of M such that its characteristic foliation F = ker(i * ω) is simple, then the quotient manifold π : C → C/F carries a unique symplectic structure ω r such that π * (ω r ) = i * (ω).
A particular situation of it is the well-known Marsden-Weinstein reduction in the presence of a G-equivariant momentum map [18]. In fact, in [7] it has been proved that, under mild assumptions, one can obtain any symplectic manifold as a result of applying a Cartan reduction of the canonical symplectic structure on R 2n . On the other hand, an interesting application in Mechanics is the case when we have a Hamiltonian function on the symplectic manifold satisfying some natural conditions, since one can reduce the Hamiltonian vector field to the reduced symplectic manifold and therefore, obtain a reduced Hamiltonian dynamics.
A category which is closely related to symplectic and Poisson manifolds is that of Lie algebroids. A Lie algebroid is a notion which unifies tangent bundles and Lie algebras, which suggests its relation with Mechanics. In fact, there has been recently a lot of interest in the geometric description of Hamiltonian (and Lagrangian) Mechanics on Lie algebroids (see, for instance, [8,11,19,21]). An important application of this Hamiltonian Mechanics, which also comes from reduction, is the following one: if we consider a principal G-bundle π : Q → M then one can prove that the solutions of the Hamilton-Poincaré equations for a G-invariant Hamiltonian function H : T * Q → R are just the solutions of the Hamilton equations for the reduced Hamiltonian h : T * Q/G → R on the dual vector bundle T * Q/G of the Atiyah algebroid τ T Q/G : T Q/G → R (see [11]). Now, given a Lie algebroid τ A : A → M , the role of the tangent of the cotangent bundle of the configuration manifold is played by A-tangent bundle to A * , which is the subset of A × T A * given by where ρ A is the anchor map of A and τ A * : A * → M is the vector bundle projection of the dual bundle A * to A. In this case, T A A * is a Lie algebroid over A * . In fact, it is the pullback Lie algebroid of A along the bundle projection τ A * : A * → M in the sense of Higgins and Mackenzie [9]. Moreover, T A A * is a symplectic Lie algebroid, that is, it is endowed with a nondegenerate and closed 2-section. Symplectic Lie algebroids are a natural generalization of symplectic manifolds since, the first example of a symplectic Lie algebroid is the tangent bundle of a symplectic manifold.
The main purpose of this paper is to describe a reduction procedure, analogous to Cartan reduction, for a symplectic Lie algebroid in the presence of a Lie subalgebroid and a symmetry Lie group. In addition, for Hamiltonian functions which satisfy some invariance properties, it is described the process to obtain the reduced Hamiltonian dynamics.
The paper is organized as follows. In Section 2, we recall the definition of a symplectic Lie algebroid and describe several examples which will be useful along the paper. Then, in addition, we describe how to obtain Hamilton equations for a symplectic Lie algebroid and a Hamiltonian function on it. Now, consider a symplectic Lie algebroid τ A : A → M with symplectic 2-section Ω A and τ B : B → N a Lie subalgebroid of A. Then, in Section 3 we obtain our main result. Suppose that a Lie group G acts properly and free on B by vector bundle automorphisms. Then, if Ω B is the restriction to B of Ω A , we obtain conditions for which the reduced vector bundle τ e B : B = (B/ ker Ω B )/G → N/G is a symplectic Lie algebroid. In addition, if we have a Hamiltonian function H M : M → R we obtain, under some mild hypotheses, reduced Hamiltonian dynamics.
In the particular case when the Lie algebroid is the tangent bundle of a symplectic manifold, our reduction procedure is just the well known Cartan symplectic reduction in the presence of a symmetry Lie group. This example is shown in Section 4, along with other different interesting examples. A particular application of our results is a "symplectic description" of the Hamiltonian reduction process by stages in the Poisson setting (see Section 4.3) and the reduction of the Lagrange top (see Section 4.4).
In the last part of the paper, we include an Appendix where we describe how to induce, from a Lie algebroid structure on a vector bundle τ A : A → M and a linear epimorphism π A : A → A over π M : M → M , a Lie algebroid structure on A. An equivalent dual version of this result was proved in [3].
If • The Lie algebroid associated with an inf initesimal action.
Let g be a real Lie algebra of finite dimension and Φ : g → X(M ) an infinitesimal left action of g on a manifold M , that is, Φ is a R-linear map and where [·, ·] g is the Lie bracket on g. Then, the trivial vector bundle τ A : A = M × g → M admits a Lie algebroid structure. The anchor map ρ A : A → T M of A is given by On the other hand, if ξ and η are elements of g then ξ and η induce constant sections of A which we will also denote by ξ and η. Moreover, the Lie bracket [[ξ, η]] A of ξ and η in A is the constant section on A induced by [ξ, η] g . In other words, The resultant Lie algebroid is called the Lie algebroid associated with the infinitesimal action Φ.
Let {ξ α } be a basis of g and (x i ) be a system of local coordinates on an open subset U of M such that then { ξ α } is a local basis of sections of the action Lie algebroid. In addition, the corresponding local structure functions with respect to (x i ) and {ξ α } are • The Atiyah (gauge) algebroid associated with a principal bundle.
Let π P : P → M be a principal left G-bundle. Denote by Ψ : G × P → P the free action of G on P and by T Ψ : G × T P → T P the tangent action of G on T P. The space T P/G of orbits of the action is a vector bundle over the manifold M with vector bundle projection τ T P |G : T P/G → P/G ∼ = M given by for v p ∈ T p P, τ T P : T P → P being the canonical projection. A section of the vector bundle τ T P |G : T P/G → P/G ∼ = M may be identified with a vector field on P which is G-invariant. Thus, using that every G-invariant vector field on P is π P -projectable on a vector field on M and that the standard Lie bracket of two G-invariant vector fields is also a G-invariant vector field, we may induce a Lie algebroid structure ([[·, ·]] T P/G , ρ T P/G ) on the vector bundle τ T P |G : T P/G → P/G ∼ = M . The Lie algebroid (T P/G, [[·, ·]] T P/G , ρ T P/G ) is called the Atiyah (gauge) algebroid associated with the principal bundle π P : P → M (see [11,13]). Let D : T P → g be a connection in the principal bundle π P : P → M and R : T P ⊕ T P → g be the curvature of D. We choose a local trivialization of the principal bundle π P : P → M to be U × G, where U is an open subset of M . Suppose that e is the identity element of G, that (x i ) are local coordinates on U and that {ξ a } is a basis of g.
Denote by {ξ L a } the corresponding left-invariant vector fields on G. If for x ∈ U , then the horizontal lift of the vector field ∂ ∂x i is the vector field on U × G given by Therefore, the vector fields on U × G {e i = ∂ ∂x i − D a i ξ L a , e b = ξ L b } are G-invariant and they define a local basis {e ′ i , e ′ b } of Γ(T P/G). The corresponding local structure functions of τ T P/G : (for more details, see [11]). An important operator associated with a Lie algebroid (A, for µ ∈ Γ(∧ k A * ) and X 0 , . . . , X k ∈ Γ(A). It follows that (d A ) 2 = 0. Note that if A = T M then d T M is the standard differential exterior for the manifold M. On Note that F * φ ′ is the section of the vector bundle ∧ k A * → M defined by for x ∈ M and a 1 , . . . , a k ∈ A, where f : M → M ′ is the mapping associated with F between M and M ′ . We remark that (2) holds if and only if This definition of a Lie algebroid morphism is equivalent of the original one given in [9]. If M = M ′ and f = id : M → M then it is easy to prove that F is a Lie algebroid morphism if and only if If F is a Lie algebroid morphism, f is an injective immersion and F is also injective, then the Lie algebroid (A,

Symplectic Lie algebroids
Let (A, [[·, ·]] A , ρ A ) be a Lie algebroid over a manifold M . Then A is said to be a symplectic Lie algebroid if there exists a section Ω A of the vector bundle ∧ 2 A * → M such that Ω A is nondegenerate and d A Ω A = 0 (see [11]). The first example of a symplectic Lie algebroid is the tangent bundle of a symplectic manifold.
It is clear that the rank of a symplectic Lie algebroid A is even. Moreover, if f ∈ C ∞ (M ) one may introduce the Hamiltonian section H Ω A f ∈ Γ(A) of f with respect to Ω A which is characterized by the following condition On the other hand, the map ♭ Ω A : Γ(A) → Γ(A * ) given by is an isomorphism of C ∞ (M )-modules. Thus, one may define a section Π Ω A of the vector bundle ∧ 2 A → A as follows Π Ω A is a triangular matrix for the Lie algebroid A (Π Ω A is an A-Poisson bivector field on the Lie algebroid A in the terminology of [2]) and the pair (A, A * ) is a triangular Lie bialgebroid in the sense of Mackenzie and Xu [14]. Therefore, the base space M admits a Poisson structure, that is, a bracket of functions which satisfies the following properties: 1. {·, ·} M is R-bilinear and skew-symmetric; 2. It is a derivation in each argument with respect to the standard product of functions, i.e., 3. It satisfies the Jacobi identity, that is, In fact, we have that (see [6]). A symplectic Lie algebroid may be associated with an arbitrary Lie algebroid as follows (see [11]).
Let (A, [[·, ·]] A , ρ A ) be a Lie algebroid of rank n over a manifold M and τ A * : A * → M be the vector bundle projection of the dual bundle A * to A. Then, we consider the A-tangent bundle to A * as the subset of A × T A * given by T A A * is a vector bundle over A * of rank 2n and the vector bundle projection τ T A A * : A section X of τ T A A * : T A A * → A * is said to be projectable if there exists a section X of τ A : A → M and a vector field S ∈ X(A * ) which is τ A * -projectable to the vector field ρ A (X) and such thatX(p) = (X(τ A * (p)), S(p)), for all p ∈ A * . For such a projectable sectionX, we will use the following notationX ≡ (X, S). It is easy to prove that one may choose a local basis of projectable sections of the space Γ(T A A * ).
The vector bundle τ T A A * : T A A * → A * admits a Lie algebroid structure ([[·, ·]] T A A * , ρ T A A * ). Indeed, if (X, S) and (Y, T ) are projectable sections then is the A-tangent bundle to A * or the prolongation of A over the fibration τ A * : A * → M (for more details, see [11]). Moreover, one may introduce a canonical section Θ T A A * of the vector bundle (T A A * ) * → A * as follows is the canonical symplectic section associated with A. Ω T A A * is a symplectic section for the Lie algebroid T A A * . Therefore, the base space A * admits a Poisson structure {·, ·} A * which is characterized by the following conditions for f, g ∈ C ∞ (M ) and X, Y ∈ Γ(A). Here, Z denotes the linear function on A * induced by a section Z ∈ Γ(A). The Poisson structure {·, ·} A * is called the canonical linear Poisson structure on A * associated with the Lie algebroid A. Next, suppose that (x i ) are local coordinates on an open subset U of M and that {e α } is a local basis of Γ(A) on U . Denote by (x i , y α ) the corresponding local coordinates on A * and by ρ i α , C γ αβ the local structure functions of A with respect to the coordinates (x i ) and to the basis {e α }. Then, we may consider the local sections {X α , P α } of the vector bundle τ T A A * : T A A * → A * given by We have that {X α , P α } is a local basis of Γ(T A A * ) and (for more details, see [11]). Example 1.

If A is the standard Lie algebroid T M then T
2. If A is the Lie algebroid associated with an infinitesimal action Φ : g → X(M ) of a Lie algebra g on a manifold M (see Section 2.1), then the Lie algebroid T A A * → A * may be identified with the trivial vector bundle and, under this identification, the canonical symplectic section Ω T A A * is given by for (x, α) ∈ A * = M × g * and (ξ, β), (ξ ′ , β ′ ) ∈ g × g * , where [·, ·] g is the Lie bracket on g.
Note that in the particular case when M is a single point (that is, A is the real algebra g) then the linear Poisson bracket {·, ·} g * on g * is just the (minus) Lie-Poisson bracket induced by the Lie algebra g.
3. Let π P : P → M be a principal G-bundle and A = T P/G be the Atiyah algebroid associated with π P : P → M . Then, the cotangent bundle T * P to P is the total space of a principal G-bundle over A * ∼ = T * P/G and the Lie algebroid T A A * may be identified with the Atiyah algebroid T (T * P )/G → T * P/G associated with this principal G-bundle (see [11]). Moreover, the canonical symplectic structure on T * P is G-invariant and it induces a symplectic section Ω on the Atiyah algebroid T (T * P )/G → T * P/G. Ω is just the canonical symplectic section of T A A * ∼ = T (T * P )/G → A * ∼ = T * P/G. Finally, the linear Poisson bracket on A * ∼ = T * P/G is characterized by the following property: if on T * P we consider the Poisson bracket induced by the canonical symplectic structure then the canonical projection π T * P : T * P → T * P/G is a Poisson morphism.
We remark that, using a connection in the principal G-bundle π P : P → M , the space A * ∼ = T * P/G may be identified with the Whitney sum W = T * M ⊕ M g * , where g * is the coadjoint bundle associated with the principal bundle π P : P → M. In addition, under the above identification, the Poisson bracket on A * ∼ = T * P/G is just the so-called Weinstein space Poisson bracket (for more details, see [20]). (see [11]). If (x i ) is a system of local coordinates on an open subset U of M and {e α } is a local basis of Γ(A) on U , we may consider the corresponding system of local coordinates (

is a solution of the Hamilton equations for H if and only if
(see [11]).
1. Let A be the Lie algebroid associated with an infinitesimal action Φ : g → X(M ) of a Lie algebra g on the manifold M . If H :

solution of the Hamilton equations for H if and only if
Note that if M is a single point (that is, A is the Lie algebra g) then the above equations are just the (minus) Lie-Poisson equations on g * for the Hamiltonian function H : g * → R.
2. Let π P : P → M be a principal G-bundle, D : T P → g be a principal connection and R : T P ⊕ T P → g be the curvature of D. We choose a local trivialization of the principal bundle π P : the components of D (respectively, R) with respect to the coordinates (x i ) and to the basis {ξ a } and by (x i , y α = p i ,p a ) the corresponding local coordinates on A * ∼ = T * P/G (see Section 2.1). If h : T * P/G → R is a Hamiltonian function and c : t → (x i (t), p i (t),p a (t)) is a curve on A * ∼ = T * P/G then, using (1) and (7), we conclude that c is a solution of the Hamilton equations for h if and only if These equations are just the Hamilton-Poincaré equations associated with the G-invariant Hamiltonian function H = h • π T * P , π T * P : T * P → T * P/G being the canonical projection (see [11]).
Note that in the particular case when G is the trivial Lie group, equations (8) reduce to which are the classical Hamilton equations for the Hamiltonian function h : 3 Reduction of the Hamiltonian dynamics on a symplectic Lie algebroid by a Lie subalgebroid and a symmetry Lie group

Reduction of the symplectic Lie algebroid
Let A be a Lie algebroid over a manifold M and Ω A be a symplectic section of A. In addition, suppose that B is a Lie subalgebroid over the submanifold N of M and denote by i B : B → A the corresponding monomorphism of Lie algebroids. The following commutative diagram illustrates the above situation Ω A induces, in a natural way, a section Ω B of the real vector bundle ∧ 2 B * → N . In fact, Since i B is a Lie algebroid morphism, it follows that However, Ω B is not, in general, nondegenerate. In other words, Ω B is a presymplectic section of the Lie subalgebroid τ B : B → N . For every point x of N , we will denote by ker Ω B (x) the vector subspace of B x defined by In what follows, we will assume that the dimension of the subspace ker Ω B (x) is constant, for all x ∈ N. Thus, the space B/ ker Ω B is a quotient vector bundle over the submanifold N . Moreover, we may consider the vector subbundle ker Ω B of B whose fiber over the point x ∈ N is the vector space ker Ω B (x). In addition, we have that: Proof . It is sufficient to prove that where [[·, ·]] B is the Lie bracket on Γ(B). Now, using (9), it follows that (10) holds.
Next, we will suppose that there is a proper and free action Ψ : G × B → B of a Lie group G on B by vector bundle automorphisms. Then, the following conditions are satisfied: C1) N is the total space of a principal G-bundle over N with principal bundle projection π N : N → N ∼ = N/G and we will denote by ψ : G × N → N the corresponding free action of G on N .
C2) Ψ g : B → B is a vector bundle isomorphism over ψ g : N → N , for all g ∈ G.
The following commutative diagram illustrates the above situation In such a case, the action Ψ of the Lie group G on the presymplectic Lie algebroid (B, Ω B ) is said to be presymplectic if Now, we will prove the following result.
Thus, Ψ induces an action Ψ : G × (B/ ker Ω B ) → (B/ ker Ω B ) of G on B/ ker Ω B and Ψ g is a vector bundle isomorphism, for all g ∈ G. Moreover, the canonical projection π B : B → B/ ker Ω B is equivariant with respect to the actions Ψ and Ψ.
On the other hand, the vector bundle projection τ B/ ker Ω B : B/ ker Ω B → N is also equivariant with respect to the action Ψ. Consequently, it induces a smooth map τ e B : B = (B/ ker Ω B )/G → N = N/G such that the following diagram is commutative Now, if x ∈ N then, using that the action ψ is free, we deduce that the map π Bx/ ker Ω B (x) : (π N (x)) is bijective. Thus, one may introduce a vector space structure on isomorphism. Furthermore, if π N (y) = π N (x) then π By/ ker Ω B (y) = π Bx/ ker Ω B (x) • Ψ −1 g . Therefore, B is a vector bundle over N with vector bundle projection τ e B : B → N and if x ∈ N then the fiber of B over π N (x) is isomorphic to the vector space B x / ker Ω B (x). This proves a).
On the other hand, it is clear that the section Ω B induces a section Ω (B/ ker Ω B ) of the vector bundle ∧ 2 (B/ ker Ω B ) → N which is characterized by the condition Then, using (11) and (12), we deduce that the section Ω (B/ ker Ω B ) is G-invariant, that is, Thus, there exists a unique section Ω e B of the vector bundle ∧ 2 B * → N such that This proves b) (note that π B = π B/ ker Ω B • π B ). Now, using (12) and the fact ker π B = ker Ω B , it follows that the section Ω (B/ ker Ω B ) is nondegenerate. Therefore, from (13), we conclude that Ω e B is also nondegenerate (note that if Next, we will describe the space of sections of B = (B/ ker Ω B )/G. For this purpose, we will use some results contained in the Appendix of this paper.
Let Γ(B) p e π B be the space of π B -projectable sections of the vector bundle τ B : B → N. As we know (see the Appendix), a section X of τ B : B → N is said to be π B -projectable if there exists a section π B (X) of the vector bundle τ e On the other hand, it is easy to prove that the section π B • X is π (B/ ker Ω B ) -projectable if and only if it is G-invariant. In other words, (π B • X) is π (B/ ker Ω B ) -projectable if and only if for and therefore, the above facts, imply that Consequently, using some results of the Appendix (see (A.2) in the Appendix), we deduce that as C ∞ (Ñ )-modules. Moreover, we may prove the following result Proof . The first part of the Theorem follows from (14), (15) and Theorem A.1 in the Appendix.
On the other hand, if the conditions i) and ii) in the Theorem hold then, using Theorem 1 and the fact that π B is a Lie algebroid epimorphism, we obtain that Finally, we deduce the following corollary.
Corollary 1. Let Ψ : G × B → B be a presymplectic action of the Lie group G on the presymplectic Lie algebroid (B, Ω B ) and suppose that: ii) If X ∈ Γ(B) p e π B and Y ∈ Γ(ker Ω B ), we have that Then, the reduced vector bundle τ e B : B → N admits a unique Lie algebroid structure such that π B : B → B is a Lie algebroid epimorphism. Moreover, the nondegenerate section Ω e B induces a symplectic structure on the Lie algebroid τ e where Z g , W g ∈ Γ(ker Ω B ) and ψ g : N → N is the diffeomorphism associated with Ψ g : B → B. Thus, using the fact that Ψ g is a Lie algebroid isomorphism, it follows that Therefore, from condition ii) in the corollary, we deduce that

Reduction of the Hamiltonian dynamics
Let A be a Lie algebroid over a manifold M and Ω A be a symplectic section of A. In addition, suppose that B is a Lie subalgebroid of A over the submanifold N of M and that G is a Lie group such that one may construct the symplectic reduction τ e B : B = (B/ ker Ω B )/G → N = N/G of A by B and G as in Section 3.1 (see Theorem 2).
We will also assume that B and N are subsets of A and M , respectively (that is, the corresponding immersions i B : B → A and i N : N → M are the canonical inclusions), and that N is a closed submanifold.
Thus, using that π B is a Lie algebroid epimorphism and the fact that H e N • π N = H N , it follows that This implies that

Examples and applications 4.1 Cartan symplectic reduction in the presence of a symmetry Lie group
Let M be a symplectic manifold with symplectic 2-form Ω T M . Suppose that N is a submanifold of M , that G is a Lie group and that N is the total space of a principal G-bundle over N = N/G. We will denote ψ : G × N → N the free action of G on N , by π N : N → N = N/G the principal bundle projection and by Ω T N the 2-form on N given by Here, i N : N → M is the canonical inclusion.
Proposition 1. If the vertical bundle to π N is the kernel of the 2-form Ω T N , that is, then there exists a unique symplectic 2- Proof . We will prove the result using Theorem 1 and Corollary 1. In fact, we apply Theorem 1 to the followings elements: • On the other hand, it is well-known that the Lie bracket of a π N -projectable vector field and a π N -vertical vector field is a π N -vertical vector field. Therefore, we may use Corollary 1 and we deduce that the symplectic vector bundle τ g T N : T N → N = N/G admits a unique Lie algebroid structure such that the canonical projection π T N : T N → T N is a Lie algebroid epimorphism. In addition, Ω g T N is a symplectic section of the Lie algebroid τ g T N : T N → N = N/G. Finally, under the identification between the vector bundles T N and T (N/G), one may see that the resultant Lie algebroid structure on the vector bundle τ T (N/G) : T (N/G) → N/G is the just standard Lie algebroid structure. Remark 1. The reduction process described in Proposition 1 is just the classical Cartan symplectic reduction process (see, for example, [20]) for the particular case when the kernel of the presymplectic 2-form on the submanifold N of the original symplectic manifold is just the union of the tangent spaces of the G-orbits associated with a principal G-bundle with total space N. Now, assume that the submanifold N is closed. Then, using Theorem 3, we may prove the following result:

Symplectic reduction of symplectic Lie algebroids by Lie subalgebroids
As in Section 2, we will assume that the dimension of the subspace ker Ω B (x) is constant, for all x ∈ N. Then, one may consider the vector bundle τ ker Ω B : ker Ω B → N which is a Lie subalgebroid of the Lie algebroid τ B : B → N. Proof . It follows using Theorem 2 (in this case, G is the trivial Lie group G = {e}).
Next, we will consider the particular case when the manifold M is a single point.
Proof . From Corollary 3 we deduce the result.
The symplectic Lie algebra ( h, Ω e h ) is called the symplectic reduction of the Lie algebra (g, Ω g ) by the Lie subalgebra h.
This reduction process of symplectic Lie algebras plays an important role in the description of symplectic Lie groups (see, for instance, [5]).

Reduction of a symplectic Lie algebroid by a symplectic Lie subalgebroid and a symmetry Lie group
We will assume that Ω B is nondegenerate. Thus, Ω B is a symplectic section on the Lie subalgebroid τ B : B → N and ker Ω B (x) = {0}, for all x ∈ N.
Now, if g ∈ G and X, Y ∈ Γ(A) G then, since Ψ g is a Lie algebroid morphism, we deduce that i.e., [[X, Y ]] A ∈ Γ(A) G . Thus, Γ(A) G is a Lie subalgebra of (Γ(A), [[·, ·]] A ) and therefore, using Theorem A.1 (see Appendix), it follows that the quotient vector bundle A/G admits a unique Lie algebroid structure such that π A is a Lie algebroid morphism.
Next, we will apply Corollary 5 to the following example.
Example 3. Let A be a Lie algebroid over a manifold M and G be a Lie group as in Remark 2. Note that the action Ψ of G on A is principal and it induces an action Ψ * of G on A * which is also principal. In fact, Ψ * g = Ψ t g −1 , where Ψ t g −1 is the dual map of Ψ g −1 : A → A. The space A * /G of orbits of this action is a quotient vector bundle over M/G which is isomorphic to the dual vector bundle to τ A/G : A/G → M/G. Moreover, the canonical projection π A * : Now, we consider the action T * Ψ of G on the A-tangent bundle to A * , T A A * , given by for g ∈ G and (a x , X αx ) ∈ T A αx A * . Since Ψ g and T Ψ t g −1 are Lie algebroid isomorphisms, we have that T * g Ψ is also a Lie algebroid isomorphism. Therefore, from Remark 2, we have that the space In addition, if Θ T A A * ∈ Γ((T A A * ) * ) is the Liouville section associated with A, it is easy to prove that Thus, if Ω T A A * ∈ Γ(∧ 2 (T A A * ) * ) is the canonical symplectic section associated with A, it follows that This implies that T * Ψ is a symplectic action of G on T A A * . Consequently, using Corollary 5, we deduce that the quotient vector bundle τ T A A * /G : T A A * /G → A * /G admits a unique Lie algebroid structure such that the canonical projection π T A A * : T A A * → T A A * /G is a Lie algebroid epimorphism. Moreover, there exists a unique symplectic structure Ω ( Next, we will prove that the symplectic Lie algebroid (T A A * /G, Ω (T A A * )/G ) is isomorphic to the A/G-tangent bundle to A * /G. For this purpose, we will introduce the map (π A , T π A * ) : then a x = 0 and, therefore, . On the other hand, using that π A (respectively, T π A * ) is a Lie algebroid morphism between the Lie algebroids τ A : A → M and τ A/G : A/G → M/G (respectively, τ T A * : T A * → A * and τ T (A * /G) : T (A * /G) → A * /G) we deduce that (π A , T π A * ) is also a Lie algebroid morphism M.
Moreover, if Θ T A/G (A * /G) ∈ Γ(T A/G (A * /G)) * is the Liouville section associated with the Lie algebroid τ A/G : A/G → M/G, we have that where Ω T A/G (A * /G) is the canonical symplectic section associated with the Lie algebroid τ A/G : Finally, using the above results, we obtain that (π A , T π A * ) is a Lie algebroid isomorphism and, in addition, In conclusion, we have proved that the symplectic Lie algebroids ((T A A * )/G, Ω (T A A * )/G ) and For this purpose, we will consider the Atiyah algebroids τ T Q/G 2 : T Q/G 2 → Q 2 = Q/G 2 and τ T Q/G 1 : T Q/G 1 → Q 1 = Q/G 1 associated with the principal bundles π 2 Q : Q → Q 2 = Q/G 2 and π 1 Q : Q → Q 1 = Q/G 1 . We also consider the corresponding dual vector bundles τ T * Q/G 2 : T * Q/G 2 → Q 2 = Q/G 2 and τ T * Q/G 1 : Then, the original Hamiltonian system (T * Q, H) may be considered as a Hamiltonian system, with Hamiltonian function H, in the standard symplectic Lie algebroid Note that this symplectic Lie algebroid is isomorphic to the T Q-tangent bundle to T * Q.
On the other hand, the reduced Poisson Hamiltonian system (T * Q/G 1 , H 1 ) may be considered as a Hamiltonian system, with Hamiltonian function H 1 : T * Q/G 1 → R in the symplectic Lie algebroid Now, in order to give the symplectic description of the two steps of the reduction process, we proceed as follows: First step: The action of G 2 on Q induces a symplectic action of G 2 on the symplectic Lie algebroid τ T T Q (T * Q) : T T Q (T * Q) ∼ = T (T * Q) → T * Q. Furthermore (see Example 3), the symplectic reduction of this Lie algebroid by G 2 is just the T Q/G 2 -tangent bundle to T * Q/G 2 , that is, T T Q/G 2 (T * Q/G 2 ) (note that the standard Lie bracket of two G 2 -invariant vector fields on Q is again a G 2 -invariant vector field). Therefore, the Hamiltonian system, in the symplectic Lie algebroid τ T (T * Q) : T (T * Q) → T * Q, with Hamiltonian function H : T * Q → R may be reduced to a Hamiltonian system, in the symplectic Lie algebroid τ T T Q/G 2 (T * Q/G 2 ) : T T Q/G 2 (T * Q/G 2 ) → T * Q/G 2 , with Hamiltonian function H 2 : T * Q/G 2 → R.
Second step: The action of the Lie group G 1 on Q induces a Lie algebroid action of the quotient Lie group G 1 /G 2 on the Atiyah algebroid τ T Q/G 2 : T Q/G 2 → Q 2 = Q/G 2 . Thus (see Remark 2), the quotient vector bundle (T Q/G 2 )/(G 1 /G 2 ) → (Q/G 2 )/(G 1 /G 2 ) admits a quotient Lie algebroid structure. If follows that this Lie algebroid is isomorphic to the Atiyah algebroid τ T Q/G 1 : T Q/G 1 → Q 1 = Q/G 1 . On the other hand, the Lie algebroid action of G 1 /G 2 on the Atiyah algebroid τ T Q/G 2 : T Q/G 2 → Q 2 = Q/G 2 induces a symplectic action of G 1 /G 2 on the symplectic Lie algebroid T T Q/G 2 (T * Q/G 2 ) → T * Q/G 2 which is also a Lie algebroid action (see Example 3). Moreover (see again Example 3), the symplectic reduction of τ T T Q/G 2 (T * Q/G 2 ) : Therefore, the Hamiltonian system in the symplectic Lie algebroid τ T T Q/G 2 (T * Q/G 2 ) : T T Q/G 2 (T * Q/G 2 ) → T * Q/G 2 , with Hamiltonian function H 2 : T * Q/G 2 → R, may be reduced to the Hamiltonian system in the symplectic Lie algebroid τ T T Q/G 1 (T * Q/G 1 ) : T T Q/G 1 (T * Q/G 1 ) → T * Q/G 1 , with Hamiltonian function

A Lagrange top
In Sections 4.2 and 4.3, we discussed the reduction of a symplectic Lie algebroid A by a Lie subalgebroid B and a symmetry Lie group G for the particular case when G is the trivial group and for the particular case when B is symplectic, respectively. Next, we will consider an example such that the Lie subalgebroid B is not symplectic and the symmetry Lie group G is not trivial: a Lagrange top.
A Lagrange top is a classical mechanical system which consists of a symmetric rigid body with a fixed point moving in a gravitational field (see, for instance, [4,15]).
A Lagrange top may be described as a Hamiltonian system on the A-tangent bundle to A * , where A is an action Lie algebroid over the sphere S 2 in R 3 The Lie algebroid structure on A is defined as follows.
Let SO(3) be the special orthogonal group of order 3. As we know, the Lie algebra so(3) of SO(3) may identified with R 3 and, under this identification, the Lie bracket [·, ·] so(3) is just the cross product on R 3 .
We have the standard left action of SO(3) on S 2 and the corresponding infinitesimal left action Φ : so(3) ∼ = R 3 → X(S 2 ) given by A is the Lie algebroid over S 2 associated with the infinitesimal left action Φ. Thus, A = The dual vector bundle A * to A is the trivial vector bundle τ e A * : A * ∼ = S 2 × R 3 → S 2 and the Hamiltonian function H : for x = (x, y, z) ∈ S 2 and π = (π 1 , π 2 , π 3 ) ∈ R 3 , where (I, I, J) are the (positive) eigenvalues of the inertia tensor, m is the mass, g is the gravity and l is the distance from the fixed point of the body to the centre of mass. The A-tangent bundle to A * , T e A A * , is isomorphic to the trivial vector bundle over

Under the canonical identification T A
Now, we consider the Lie subgroup G of SO(3) of rotations about the z-axis. The elements of G are of the form It is clear that G is isomorphic to S 1 = R/(2πZ).
Note that the standard action of G ∼ = S 1 on S 2 is not free (the north and the south poles are fixed points). Therefore, we must restrict this action to the open subset M of S 2 M = S 2 − {(0, 0, 1), (0, 0, −1)}.
If Ω B is the restriction of Ω T A A * to the Lie subalgebroid B then dim(ker Ω B (([θ], t), π)) = 1, for all (([θ], t), π) ∈ N and the section s of τ B : B → N defined by is a global basis of Γ(ker Ω B ). Moreover, the restriction of the symplectic action T * Ψ to the Lie algebroid τ B : B → N is a presymplectic action. In fact, the action of G on N is given by for [θ ′ ] ∈ G ∼ = S 1 and (([θ], t), π) ∈ N , and the action of G on B is We have that  (see Theorem 2). On the other hand, using Theorem 3, we deduce that the original Hamiltonian system, with Hamiltonian function H : A * → R in the symplectic Lie algebroid τ T A A * : T A A * → A * , may be reduced to a Hamiltonian system, with Hamiltonian function H : N → R, in the symplectic Lie algebroid τ e B : B → N .
Next, we will give an explicit description of the reduced symplectic Lie algebroid and the reduced Hamiltonian system on it.
In fact, one may prove that the Lie algebroid τ e B : B → N is isomorphic to the trivial vector bundle τ R 3 ×R 4 : R 3 × R 4 → R 3 with basis R 3 and fiber R 4 and, under this identification, the anchor map ρ e B is given by

Conclusions and future work
In this paper we have generalized the Cartan symplectic reduction by a submanifold in the presence of a symmetry Lie group to the Lie algebroid setting. More precisely, we develop a procedure to reduce a symplectic Lie algebroid to a certain quotient symplectic Lie algebroid, using a Lie subalgebroid and a symmetry Lie group. In addition, under some mild assumptions we are able to reduce the Hamiltonian dynamics. Several examples illustrate our theory. As we already mentioned in the introduction, a particular example of reduction of a symplectic manifold is the well known Marsden-Weinstein reduction of a symmetric Hamiltonian dynamical system (M, Ω, H) in the presence of a G-equivariant momentum map J : M → g * , where g is the Lie algebra of the Lie group G which acts on M by Φ : G × M → M . In this situation, the submanifold is the inverse image J −1 (µ) of a regular value µ ∈ g * of J, and the Lie group acting on J −1 (µ) is the isotropy group of µ. It would be interesting to generalize this procedure to the setting of symplectic Lie algebroids. In this case, we would have to define a proper notion of a momentum map for this framework. This appropiate moment map would allow us to get the right submanifold in order to apply the reduction procedure we have developed in this paper. Moreover, under these hypotheses, we should be able to reduce also the Hamiltonian dynamics. These topics are the subject of a forthcoming paper (see [10]).
A natural generalization of symplectic manifolds is that of Poisson manifolds. This type of manifolds play also a prominent role in the Hamiltonian description of Mechanics, in particular for systems with constraints or with symmetry. For Poisson manifolds, one can also develop a reduction procedure (see [17]). On the other hand, as we mentioned in Section 2.2, a symplectic section on a Lie algebroid A can be seen as a particular example of a triangular matrix for A, that is, a Poisson structure in the Lie algebroid setting. Therefore, it should be interesting to generalize our reduction constructions to Poisson structures on Lie algebroids and, more generally, to obtain a reduction procedure for Lie bialgebroids (we remark that triangular matrices on Lie algebroids are a particular example of Lie bialgebroids, the so-called triangular Lie bialgebroids).
On the other hand, recently, in [1] it has been described a reduction procedure for Courant algebroids and Dirac structures. We recall that there is a relation between Courant algebroids and Lie bialgebroids: there is a one-to-one correspondence between Lie bialgebroids and pairs of complementary Dirac structures for a Courant algebroid (see [12]). Since symplectic Lie algebroids are a particular example of this situation, it would be interesting to compare both approaches. This is the subject of a forthcoming paper.

Appendix
Suppose that τ A : A → M and τ e A : A → M are real vector bundles and that π A : A → A is a vector bundle epimorphism over the surjective submersion π M : M → M .
Thus, one may consider the vector subbundle ker π A of A whose fiber over the point x ∈ M is the kernel of the linear epimorphism (π A ) |Ax : A x → A π M (x) .
On the other hand, a section X of the vector bundle τ A : A → M is said to be π A -projectable if there exists a section π A (X) of τ e A : A → M such that We will denote by Γ(A) p π A the set of π A -projectable sections. Γ(A) p π A is a C ∞ ( M )-module and it is clear that Γ(ker π A ) is a C ∞ ( M )-submodule of Γ(A) p π A . Note that if X ∈ Γ(A) p π A then there exists a unique section π A (X) of the vector bundle such that (A.1) holds.
Thus, we may define a map Π A : Γ(A) p π A → Γ( A) and it follows that Π A is an epimorphism of C ∞ ( M )-modules and that ker Π A = Γ(ker π A ).
Therefore, we have that the C ∞ ( M )-modules Γ( A) and Γ(A) p π A Γ(ker π A ) are isomorphic, that is, Moreover, one may prove that following result. Proof . Assume that ([[·, ·]] e A , ρ e A ) is a Lie algebroid structure on the vector bundle τ e A : A → M such that π A is a Lie algebroid epimorphism.
If X ∈ Γ(A) p π A then, using that we directly conclude that the vector field ρ A (X) on M is π M -projectable on the vector field ρ e A (π A (X)) on M . Thus, if Y ∈ Γ(A) p π A and α ∈ Γ( A * ), we have that for all x ∈ M . Therefore, since (π * A (d e A α)) = d A (π * A α) it follows that Consequently, we deduce the following result This proves (i) and (ii). Conversely, suppose that conditions (i) and (ii) hold. Then, we will define a Lie bracket [[·, ·]] e A on the space Γ( A) as follows. If X and Y are sections of τ e A : A → M , we can choose X, Y ∈ Γ(A) p π A such that π A (X) = X, π A (Y ) = Y . Next, will see that Z ∈ Γ(ker π A ) ⇒ ρ A (Z) is a π M -vertical vector field. (A. 4) In fact, if f ∈ C ∞ ( M ) we have that On the other hand, which implies that ρ A (Z)( f • π M ) = 0 and, thus, the vector field ρ A (Z) is vertical with respect to the map π M . Now, let X be a section of τ e A : A → M and X be a section of τ A : A → M such that π A (X) = X. If f ∈ C ∞ ( M ) it follows that and, since we obtain that ρ A (X)( f • π M ) is a basic function with respect to the map π M . Therefore, the vector field ρ A (X) is π M -projectable to a vector field ρ e A ( X) on M . Using (A.4), we deduce that the map ρ e A : Γ( A) → X( M ) is well-defined. In fact, ρ e A is a homomorphism of C ∞ ( M )-modules and if we also denote by ρ e A : A → T M the corresponding bundle map then In addition, since ([[·, ·]] A , ρ A ) is a Lie algebroid structure on A, we conclude that the pair ([[·, ·]] e A , ρ e A ) is also a Lie algebroid structure on A. On the other hand, from (A.5), it follows that for f ∈ C ∞ ( M ).