1Introduction ReductionofSymplecticLieAlgebroidsbyaLieSubalgebroidandaSymmetryLieGroup

Reduction of Symplectic Lie Algebroids

by a Lie Subalgebroid and a Symmetry Lie Group

David IGLESIAS ^†, Juan Carlos MARRERO ^‡, David MART´IN DE DIEGO^†, Eduardo MART´INEZ^§ and Edith PADR ´ON^‡

† Departamento de Matem´aticas, Instituto de Matem´aticas y F´ısica Fundamental, Consejo Superior de Investigaciones Cient´ıficas, Serrano 123, 28006 Madrid, Spain E-mail: [email protected],[email protected]

‡ Departamento de Matem´atica Fundamental, Facultad de Matem´aticas, Universidad de la Laguna, La Laguna, Tenerife, Canary Islands, Spain E-mail: [email protected], [email protected]

§ Departamento de Matem´atica Aplicada, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza, Spain

E-mail: [email protected]

Received November 14, 2006, in final form March 06, 2007; Published online March 16, 2007 Original article is available athttp://www.emis.de/journals/SIGMA/2007/049/

Abstract. We describe the reduction procedure for a symplectic Lie algebroid by a Lie sub- algebroid 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.

Key words: Lie algebroids and subalgebroids; symplectic Lie algebroids; Hamiltonian dyna- mics; reduction procedure

2000 Mathematics Subject Classification: 53D20; 58F05; 70H05

1 Introduction

As it is well known, symplectic manifolds play a fundamental role in the Hamiltonian formulation of Classical Mechanics. In fact, ifM is the configuration space of a classical mechanical system, then the phase space is the cotangent bundleT^∗M, which is endowed with a canonical symplectic structure, the main ingredient to develop Hamiltonian Mechanics. Indeed, given a Hamiltonian functionH, 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 manifoldπ :C→C/F carries a unique symplectic structure ωr such thatπ^∗(ωr) =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 R²ⁿ. On the other hand, an interesting application in Mechanics is the case when we have a Hamiltonian function on the symplectic manifold

?This paper is a contribution to the Proceedings of the Workshop on Geometric Aspects of Integ- rable Systems (July 17–19, 2006, University of Coimbra, Portugal). The full collection is available at http://www.emis.de/journals/SIGMA/Coimbra2006.html

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 al- gebroids. 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-Poincar´e equations for a G-invariant Hamiltonian function H : T^∗Q → R are just the solutions of the Hamilton equations for the reduced Hamiltonianh:T^∗Q/G→Ron the dual vector bundleT^∗Q/Gof 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 byA-tangent bundle toA^∗, which is the subset ofA×T A^∗ given by

T^AA^∗={(b, v)∈A×T A^∗/ρ_A(b) = (T τ_A^∗)(v)},

where ρA is the anchor map of A and τA^∗ : A^∗ → M is the vector bundle projection of the dual bundle A^∗ to A. In this case, T^AA^∗ is a Lie algebroid over A^∗. In fact, it is the pull- back Lie algebroid of A along the bundle projection τ_A^∗ : A^∗ → M in the sense of Higgins and Mackenzie [9]. Moreover, T^AA^∗ 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 τB :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_Be :Be = (B/ker ΩB)/G→N/Gis a symplectic Lie algebroid. In addition, if we have a Hamil- tonian function HM : 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 epimorphismπ_A:A→Ae over πM :M → M, a Lie algebroid structure onf A. An equivalent dual version of this resulte was proved in [3].

2 Hamiltonian Mechanics and symplectic Lie algebroids

2.1 Lie algebroids

LetA be a vector bundle ofrank n over a manifoldM of dimension m andτA:A→M be the vector bundle projection. Denote by Γ(A) theC^∞(M)-module of sections ofτ_A:A→M. ALie algebroid structure ([[·,·]]_A, ρA) onAis a Lie bracket [[·,·]]_Aon the space Γ(A) and a bundle map ρA :A → T M, called the anchor map, such that if we also denote by ρA : Γ(A) → X(M) the homomorphism ofC^∞(M)-modules induced by the anchor map then

[[X, f Y]]A=f[[X, Y]]A+ρA(X)(f)Y,

for X, Y ∈ Γ(A) and f ∈ C^∞(M). The triple (A,[[·,·]]_A, ρ_A) is called a Lie algebroid over M (see [13]).

If (A,[[·,·]]_A, ρA) is a Lie algebroid over M, then the anchor map ρA : Γ(A) → X(M) is a homomorphism between the Lie algebras (Γ(A),[[·,·]]_A) and (X(M),[·,·]).

Trivial examples of Lie algebroids are real Lie algebras of finite dimension and the tangent bundle T M of an arbitrary manifold M. Other examples of Lie algebroids are the following ones:

• The Lie algebroid associated with an inf initesimal action.

Letgbe a real Lie algebra of finite dimension and Φ :g→X(M) an infinitesimal left action of g on a manifoldM, that is, Φ is a R-linear map and

Φ([ξ, η]g) =−[Φ(ξ),Φ(η)], for all ξ, η∈g,

where [·,·]_g is the Lie bracket ong.Then, the trivial vector bundleτ_A:A=M×g→M admits a Lie algebroid structure. The anchor map ρ_A:A→T M of A is given by

ρA(x, ξ) =−Φ(ξ)(x), for (x, ξ)∈M×g=A.

On the other hand, ifξ 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 inA is the constant section on Ainduced by [ξ, η]g. In other words,

[[ξ, η]]_A= [ξ, η]_g.

The resultant Lie algebroid is called theLie algebroid associated with the infinitesimal action Φ.

Let{ξ_α}be a basis ofgand (xⁱ) be a system of local coordinates on an open subsetU ofM such that

[ξ_α, ξ_β]_g =c^γ_αβξ_γ, Φ(ξ_α) = Φⁱ_α ∂

∂xⁱ. If ξe_α:U →M×g is the map defined by

ξeα(x) = (x, ξα), for all x∈U,

then{ξe_α}is a local basis of sections of the action Lie algebroid. In addition, the corresponding local structure functions with respect to (xⁱ) and{ξ˜α}are

C_αβ^γ =c^γ_αβ, ρⁱ_α= Φⁱ_α.

• The Atiyah (gauge) algebroid associated with a principal bundle.

Let πP : P → M be a principal left G-bundle. Denote by Ψ : G×P → P the free action of G on P and by TΨ : 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 τT P|G:T P/G→P/G∼=M given by

(τ_{T P}|G)([v_p]) = [τ_{T P}(v_p)] = [p], for v_p ∈T_pP,

τ_{T P} :T P →P being the canonical projection. A section of the vector bundleτ_{T P}|G:T P/G→ P/G∼=Mmay be identified with a vector field onPwhich isG-invariant. Thus, using that every G-invariant vector field onP isπP-projectable on a vector field onM 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 theAtiyah (gauge) algebroid associated with the principal bundle π_P :P →M (see [11,13]).

LetD:T P →g be a connection in the principal bundleπ_P :P →M and R:T P⊕T P →g be the curvature ofD. 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 thate is the identity element of G, that (xⁱ) 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 D

∂

∂xⁱ|_(x,e)

=D^a_i(x)ξa, R ∂

∂xⁱ|_(x,e), ∂

∂x^j|_(x,e)

=R^a_ij(x)ξa,

forx∈U, then the horizontal lift of the vector field _∂x^∂i is the vector field on U ×Ggiven by ∂

∂xⁱ h

= ∂

∂xⁱ −D^a_iξ_a^L.

Therefore, the vector fields on U ×G {e_i = _∂x^∂i −D_i^aξ^L_a, e_b = ξ_b^L} are G-invariant and they define a local basis{e⁰_i, e⁰_b} of Γ(T P/G).The corresponding local structure functions ofτ_{T P/G}: T P/G→M are

C_ij^k =C_ia^j =−C_ai^j =C_abⁱ = 0, C_ij^a =−R^a_ij, C_ia^c =−C_ai^c =c^c_abD^b_i, C_ab^c =c^c_ab,

ρ^j_i =δ_ij, ρ^a_i =ρⁱ_a=ρ^b_a= 0 (1)

(for more details, see [11]).

An important operator associated with a Lie algebroid (A,[[·,·]]_A, ρ_A) over a manifold M is thedifferential d^A: Γ(∧^kA^∗)→Γ(∧^k+1A^∗) of A which is defined as follows

d^Aµ(X0, . . . , Xk) =

k

X

i=0

(−1)ⁱρA(Xi)(µ(X0, . . . ,Xci, . . . , Xk))

+X

i<j

(−1)^i+jµ([[X_i, X_j]]_A, X₀, . . . ,Xc_i, . . . ,Xc_j, . . . , X_k),

for µ∈Γ(∧^kA^∗) and X₀, . . . , X_k∈Γ(A).It follows that (d^A)² = 0. Note that ifA =T M then d^{T M} is the standard differential exterior for the manifoldM.

On the other hand, if (A,[[·,·]]_A, ρ_A) and (A⁰,[[·,·]]_A⁰, ρ_A⁰) are Lie algebroids over M and M⁰, respectively, then a morphism of vector bundles F :A→A⁰ is aLie algebroid morphism if

d^A(F^∗φ⁰) =F^∗(d^A0φ⁰), for φ⁰ ∈Γ(∧^k(A⁰)^∗). (2)

Note thatF^∗φ⁰ is the section of the vector bundle∧^kA^∗ →M defined by (F^∗φ⁰)_x(a₁, . . . , a_k) =φ⁰_f(x)(F(a₁), . . . , F(a_k)),

forx∈M anda₁, . . . , a_k∈A, wheref :M →M⁰ is the mapping associated with F betweenM and M⁰. We remark that (2) holds if and only if

d^A(g⁰◦f) =F^∗(d^A0g⁰), for g⁰ ∈C^∞(M⁰),

d^A(F^∗α⁰) =F^∗(d^A0α⁰), for α⁰ ∈Γ((A⁰)^∗). (3) This definition of a Lie algebroid morphism is equivalent of the original one given in [9].

IfM =M⁰ and f =id:M →M then it is easy to prove thatF is a Lie algebroid morphism if and only if

F[[X, Y]]A= [[F X, F Y]]A⁰, ρA⁰(F X) =ρA(X), forX, Y ∈Γ(A).

If F is a Lie algebroid morphism, f is an injective immersion and F is also injective, then the Lie algebroid (A,[[·,·]]_A, ρA) is a Lie subalgebroid of (A⁰,[[·,·]]_A⁰, ρA⁰).

2.2 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 ∧²A^∗ → M such that Ω_A is nondegenerate and d^AΩA= 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 algebroidA is even. Moreover, if f ∈ C^∞(M) one may introduce the Hamiltonian section H_f^ΩA ∈ Γ(A) of f with respect to ΩA which is characterized by the following condition

i(H^Ω_f^A)ΩA=d^Af. (4)

On the other hand, the map[_ΩA : Γ(A)→Γ(A^∗) given by [_ΩA(X) =i(X)Ω_A, for X∈Γ(A),

is an isomorphism ofC^∞(M)-modules. Thus, one may define a section ΠΩ_A of the vector bundle

∧²A→A as follows ΠΩA(α, β) = ΩA([⁻¹_Ω

A(α), [⁻¹_Ω

A(β)), for α, β ∈Γ(A^∗).

Π_ΩA is atriangular matrix for the Lie algebroidA(Π_ΩA is anA-Poisson bivector field on the Lie algebroidA 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

{·,·}_M :C^∞(M)×C^∞(M)→C^∞(M) which satisfies the following properties:

1. {·,·}_M isR-bilinear and skew-symmetric;

2. It is a derivation in each argument with respect to the standard product of functions, i.e., {f f⁰, g}_M =f{f⁰, g}_M +f⁰{f, g}_M;

3. It satisfies the Jacobi identity, that is,

{f,{g, h}_M}_M +{g,{h, f}_M}_M+{h,{f, g}_M}_M = 0.

In fact, we have that

{f, g}_M = Ω_A(H_f^ΩA,H^Ω_g^A), for f, g∈C^∞(M). (5) Using the Poisson bracket {·,·}_M one may consider the Hamiltonian vector field of a real functionf ∈C^∞(M) as the vector field H^{·,·}_f ^M on M defined by

H^{·,·}_f ^M(g) =−{f, g}_M, for all g∈C^∞(M).

From (4) and (5), we deduce that ρ_A(H^Ω_f^A) = H^{·,·}_f ^M. Moreover, the flow of the vector field H^{·,·}_f ^M on M is covered by a group of 1-parameter automorphisms of A: it is just the (infinitesimal flow) of the A-vector field H^Ω_f^A (see [6]).

A symplectic Lie algebroid may be associated with an arbitrary Lie algebroid as follows (see [11]).

Let (A,[[·,·]]_A, ρA) be a Lie algebroid of ranknover a manifoldM andτA^∗ :A^∗ →M be the vector bundle projection of the dual bundle A^∗ toA. Then, we consider the A-tangent bundle toA^∗ as the subset ofA×T A^∗ given by

T^AA^∗={(b, v)∈A×T A^∗/ρ_A(b) = (T τ_A^∗)(v)}.

T^AA^∗is a vector bundle overA^∗of rank 2nand the vector bundle projectionτ_T^A_A^∗ :T^AA^∗→A^∗ is defined by

τ_T^A_A^∗(b, v) =τT A^∗(v), for (b, v)∈ T^AA^∗.

A section Xe of τ_TAA^∗ : T^AA^∗ → A^∗ is said to be projectable if there exists a section X of τA :A → M and a vector field S ∈ X(A^∗) which is τA^∗-projectable to the vector field ρA(X) and such that ˜X(p) = (X(τ_A^∗(p)), S(p)), for all p ∈ A^∗. For such a projectable section ˜X, we will use the following notation ˜X ≡(X, S). It is easy to prove that one may choose a local basis of projectable sections of the space Γ(T^AA^∗).

The vector bundle τ_TAA^∗ :T^AA^∗ → A^∗ admits a Lie algebroid structure ([[·,·]]_TAA^∗, ρ_TAA^∗).

Indeed, if (X, S) and (Y, T) are projectable sections then

[[(X, S),(Y, T)]]_TAA^∗ = ([[X, Y]]_A,[S, T]), ρ_TAA^∗(X, S) =S.

(T^AA^∗,[[·,·]]_TAA^∗, ρ_T^A_A^∗) 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_TAA^∗ of the vector bundle (T^AA^∗)^∗→A^∗ as follows

Θ_TAA^∗(γ)(b, v) =γ(b),

for γ ∈ A^∗ and (b, v) ∈ T_γ^AA^∗. Θ_TAA^∗ is called the Liouville section associated with A and Ω_TAA^∗ =−d^TAA∗Θ_TAA^∗ is thecanonical symplectic section associated withA. Ω_TAA^∗ is a sym- plectic section for the Lie algebroidT^AA^∗.

Therefore, the base space A^∗ admits a Poisson structure {·,·}_A^∗ which is characterized by the following conditions

{f ◦τA^∗, g◦τA^∗}_A^∗ = 0, {f◦τA^∗,X}b _A^∗=ρA(X)(f)◦τA^∗,

{X,b Yb}_A^∗ =−[[X, Y\]]A,

for f, g ∈ C^∞(M) and X, Y ∈ Γ(A). Here, Zb denotes the linear function on A^∗ induced by a sectionZ ∈Γ(A).The Poisson structure{·,·}_A^∗ is called thecanonical linear Poisson structure on A^∗ associated with the Lie algebroid A.

Next, suppose that (xⁱ) 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 (xⁱ, y_α) the corresponding local coordinates on A^∗ and by ρⁱ_α, C_αβ^γ the local structure functions of A with respect to the coordinates (xⁱ) and to the basis {e_α}. Then, we may consider the local sections {X_α,P^α} of the vector bundle τ_TAA^∗ :T^AA^∗→A^∗ given by

X_α=

e_α◦τ_A^∗, ρⁱ_α ∂

∂xⁱ

, P^α =

0, ∂

∂y_α

. We have that {X_α,P^α} is a local basis of Γ(T^AA^∗) and

[[X_α,X_β]]_TAA^∗=C_αβ^γ X_γ, [[X_α,P^β]]_TAA^∗ = [[P^α,P^β]]_TAA^∗= 0, ρ_TAA^∗(X_α) =ρⁱ_α ∂

∂xⁱ, ρ_TAA^∗(P^α) = ∂

∂y_α, Θ_TAA^∗=y_αX^α, Ω_TAA^∗ =X^α∧ P_α+1

2C_αβ^γ y_γX^α∧ X^β,

{X^α,P_α} being the dual basis of {X_α,P^α}. Moreover, if H, H⁰ : A^∗ → R are real functions on A^∗ it follows that

{H, H⁰}_A^∗ =−∂H

∂yα

∂H⁰

∂y_βC_αβ^γ yγ+ ∂H

∂xⁱ

∂H⁰

∂yα

− ∂H

∂yα

∂H⁰

∂xⁱ

ρⁱ_α, H^Ω_H^TAA∗ = ∂H

∂yα

X_α−

ρⁱ_α∂H

∂xⁱ +C_αβ^γ yγ

∂H

∂y_β

P^α, (6)

H^{·,·}_H ^A∗ = ∂H

∂yα

ρⁱ_α ∂

∂xⁱ −

ρⁱ_α∂H

∂xⁱ +C_αβ^γ y_γ∂H

∂yβ

∂

∂yα

, (for more details, see [11]).

Example 1.

1. If A is the standard Lie algebroid T M then T^AA^∗ = T(T^∗M), Ω_TAA^∗ is the canonical symplectic structure onA^∗ =T^∗M and{·,·}_T^∗_M is the canonical Poisson bracket onT^∗M induced by Ω_TAA^∗.

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 Section2.1), then the Lie algebroid T^AA^∗ →A^∗ may be identified with the trivial vector bundle

(M ×g^∗)×(g×g^∗)→M×g^∗

and, under this identification, the canonical symplectic section Ω_T^A_A^∗ is given by Ω_T^A_A^∗(x, α)((ξ, β),(ξ⁰, β⁰)) =β⁰(ξ)−β(ξ⁰) +α[ξ, ξ⁰]g,

for (x, α)∈A^∗ =M ×g^∗ and (ξ, β),(ξ⁰, β⁰)∈g×g^∗, where [·,·]_g is the Lie bracket on g.

The anchor map ρ_TAA^∗ : (M ×g^∗)×(g×g^∗) → T(M ×g^∗) ∼= T M ×(g^∗ ×g^∗) of the A-tangent bundle to A^∗ is

ρ_T^A_A^∗((x, α),(ξ, β)) = (−Φ(ξ)(x), α, β)

and the Lie bracket of two constant sections (ξ, β) and (ξ⁰, β⁰) is just the constant section ([ξ, ξ⁰]_g,0).

Note that in the particular case whenM is a single point (that is,Ais the real algebra g) then the linear Poisson bracket{·,·}_g^∗ong^∗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 overA^∗ ∼=T^∗P/Gand the Lie algebroid T^AA^∗ 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 algebroide T(T^∗P)/G → T^∗P/G. Ω is juste the canonical symplectic section of T^AA^∗ ∼= 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 πP : P → M, the space A^∗ ∼=T^∗P/G may be identified with the Whitney sumW =T^∗M⊕_M eg^∗,whereeg^∗ is the coadjoint bundle associated with the principal bundleπ_P :P →M.In addition, under the above identification, the Poisson bracket on A^∗ ∼=T^∗P/Gis just the so-called Weinstein space Poisson bracket (for more details, see [20]).

2.3 Hamilton equations and symplectic Lie algebroids

Let Abe a Lie algebroid over a manifold M and ΩA be a symplectic section ofA. Then, as we know, Ω_A induces a Poisson bracket {·,·}_M on M.

A Hamiltonian function forAis a realC^∞-functionH:M →RonM. IfH is a Hamiltonian function one may consider the Hamiltonian section H^Ω_H^A of H with respect to Ω_A and the Hamiltonian vector field H_H^{·,·}M of H with respect to the Poisson bracket {·,·}_M on M.

The solutions of the Hamilton equations forH in Aare just the integral curves of the vector fieldH^{·,·}_H ^M.

Now, suppose that our symplectic Lie algebroid is theA-tangent bundle toA^∗,T^AA^∗,where Ais an arbitrary Lie algebroid. As we know, the base space ofT^AA^∗isA^∗ and the corresponding Poisson bracket{·,·}_A^∗ onA^∗ is just the canonical linear Poisson bracket onA^∗ associated with the Lie algebroidA. Thus, ifH :A^∗ →Ris a Hamiltonian function forT^AA^∗,we have that the solutions of the Hamilton equations for H in T^AA^∗ (or simply, the solutions of the Hamilton equations for H) are the integral curves of the Hamiltonian vector fieldH^{·,·}_H ^A∗ (see [11]).

If (xⁱ) is a system of local coordinates on an open subset U of M and {e_α} is a local ba- sis of Γ(A) on U, we may consider the corresponding system of local coordinates (xⁱ, y_α) on τ_A⁻¹∗(U) ⊆ A^∗. In addition, using (6), we deduce that a curve t → (xⁱ(t), y_α(t)) on τ_A⁻¹∗(U) is a solution of the Hamilton equations forH if and only if

dxⁱ

dt =ρⁱ_α∂H

∂xⁱ, dy_α dt =−

C_αβ^γ yγ

∂H

∂yβ

+ρⁱ_α∂H

∂xⁱ

, (7)

(see [11]).

Example 2.

1. Let Abe the Lie algebroid associated with an infinitesimal action Φ :g→X(M) of a Lie algebrag on the manifoldM. IfH :A^∗ =M×g^∗ →Ris a Hamiltonian function,{ξ_α} is

a basis of gand (xⁱ) is a system of local coordinates onM such that [ξ_α, ξ_β]_g =c^γ_αβξ_γ, Φ(ξ_α) = Φⁱ_α ∂

∂xⁱ,

then the curve t→(xⁱ(t), yα(t)) onA^∗ =M×g^∗ is a solution of the Hamilton equations forH if and only if

dxⁱ

dt = Φⁱ_α∂H

∂xⁱ, dyα

dt =−

c^γ_αβy_γ∂H

∂y_β + Φⁱ_α∂H

∂xⁱ

.

Note that ifMis a single point (that is,Ais the Lie algebrag) then the above equations are just the (minus) Lie–Poisson equations on g^∗ for the Hamiltonian function H :g^∗ →R.

2. Let πP : P → M be a principal G-bundle, D : T P → g be a principal connection and R:T P ⊕T P →gbe the curvature ofD. 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 (xⁱ) are local coordinates on U and that {ξ_a} is a basis of g such that [ξa, ξb]g = c^c_abξc. Denote by D_i^a (respectively, R^a_ij) the components of D (respectively, R) with respect to the coordinates (xⁱ) and to the basis {ξ_a} and by (xⁱ, 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 andc:t→(xⁱ(t), pi(t),p¯a(t)) is a curve onA^∗∼=T^∗P/Gthen, using (1) and (7), we conclude that c is a solution of the Hamilton equations forh if and only if

dxⁱ dt = ∂h

∂pi

, dp_i

dt =−∂h

∂xⁱ +R^a_ijp¯_a∂h

∂pj

−c^c_abD_i^bp¯_c ∂h

∂p¯a

, d¯pa

dt =c^c_abD_i^bp¯c

∂h

∂p_i −c^c_abp¯c

∂h

∂p¯_b. (8)

These equations are justthe Hamilton–Poincar´e equations associated with the G-invariant Hamiltonian functionH=h◦π_T^∗_P, π_T^∗_P :T^∗P →T^∗P/Gbeing the canonical projection (see [11]).

Note that in the particular case whenG is the trivial Lie group, equations (8) reduce to dxⁱ

dt = ∂h

∂pi

, dp_i

dt =−∂h

∂xⁱ,

which are the classical Hamilton equations for the Hamiltonian functionh:T^∗P∼=T^∗M→R.

3 Reduction of the Hamiltonian dynamics on a symplectic Lie algebroid by a Lie subalgebroid and a symmetry Lie group

3.1 Reduction of the symplectic Lie algebroid

Let A be a Lie algebroid over a manifoldM and Ω_A be a symplectic section of A.In addition, suppose thatBis a Lie subalgebroid over the submanifoldN ofMand denote byi_B :B →Athe corresponding monomorphism of Lie algebroids. The following commutative diagram illustrates the above situation

N i_N

- M τ_B

?

τ_A

?

B iB

- A

ΩA induces, in a natural way, a section ΩB of the real vector bundle∧²B^∗ →N. In fact, Ω_B=i^∗_BΩ_A.

Since i_B is a Lie algebroid morphism, it follows that

d^BΩB = 0. (9)

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 pointx ofN, we will denote by ker ΩB(x) the vector subspace ofBx defined by ker ΩB(x) ={a_x ∈Bx/i(ax)ΩB(x) = 0} ⊆Bx.

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 submanifoldN.

Moreover, we may consider the vector subbundle ker ΩB of B whose fiber over the point x∈N is the vector space ker Ω_B(x). In addition, we have that:

Lemma 1. ker ΩB is a Lie subalgebroid over N of B.

Proof . It is sufficient to prove that

X, Y ∈Γ(ker Ω_B)⇒[[X, Y]]_B ∈Γ(ker Ω_B), (10)

where [[·,·]]_B is the Lie bracket on Γ(B).

Now, using (9), it follows that (10) holds.

Next, we will suppose that there isa proper and free actionΨ :G×B →B of a Lie groupG on B by vector bundle automorphisms. Then, the following conditions are satisfied:

C1) N is the total space of a principal G-bundle over Ne with principal bundle projection π_N :N →Ne ∼=N/Gand we will denote by ψ:G×N →N the corresponding free action of Gon N.

C2) Ψ_g :B →B is a vector bundle isomorphism over ψ_g :N →N, for allg∈G.

The following commutative diagram illustrates the above situation

N ψg

- N τ_B

?

τ_B

?

B Ψg

- B

@

@

@

@ R

Ne ∼=N/G

πN

πN

In such a case, the action Ψ of the Lie group Gon the presymplectic Lie algebroid (B,Ω_B) is said to be presymplectic if

Ψ^∗_gΩB= ΩB, for all g∈G. (11)

Now, we will prove the following result.

Theorem 1. Let Ψ :G×B →B be a presymplectic action of the Lie group G on the presym- plectic Lie algebroid(B,Ω_B). Then, the action ofGonB induces an action ofGon the quotient vector bundle B/ker Ω_B such that:

a) The space of orbits Be = (B/ker ΩB)/G of this action is a vector bundle over Ne =N/G and the diagram

N π_N

- Ne =N/G τ_B

?

τBe

?

B eπB

-Be= (B/ker Ω_B)/G

defines an epimorphism of vector bundles, where eπB :B →Be is the canonical projection.

b) There exists a unique section Ω

Be of ∧²Be^∗→Ne such that eπ_B^∗Ω

Be = Ω_B. c) Ω

Be is nondegenerate.

Proof . If g∈Gand x∈M then Ψ_g:B_x→B_ψg(x) is a linear isomorphism and Ψg(ker ΩB(x)) = ker ΩB(ψg(x)).

Thus, Ψ induces an action Ψ :e G×(B/ker ΩB) → (B/ker ΩB) of G on B/ker ΩB and Ψeg

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.e

On the other hand, the vector bundle projectionτ_{B/ker ΩB} :B/ker ΩB→N is also equivariant with respect to the actionΨ.e Consequently, it induces a smooth mapτ

Be :Be= (B/ker Ω_B)/G→ Ne =N/Gsuch that the following diagram is commutative

Be= (B/ker ΩB)/G τ

Be

- Ne =N/G π_{B/ker ΩB}

?

π_N

? B/ker Ω_B τ_{B/ker ΩB}

- N

Now, ifx ∈N then, using that the actionψ is free, we deduce that the map π_{Bx/ker ΩB(x)} : B_x/ker Ω_B(x)→τ⁻¹

Be (π_N(x)) is bijective. Thus, one may introduce a vector space structure on τ⁻¹

Be (π_N(x)) in such a way that the mapπ_{Bx/ker ΩB(x)} :B_x/ker Ω_B(x)→τ⁻¹

Be (π_N(x)) is a linear isomorphism. Furthermore, if π_N(y) =π_N(x) then π_{By/ker ΩB(y)}=π_{Bx/ker ΩB(x)}◦Ψe⁻¹_g .

Therefore, Be is a vector bundle over Ne with vector bundle projection τ

Be : Be → Ne and if x ∈ N then the fiber of Be over πN(x) is isomorphic to the vector space Bx/ker ΩB(x). This proves a).

On the other hand, it is clear that the section Ω_B induces a section Ω_{(B/ker ΩB)} of the vector bundle ∧²(B/ker ΩB)→N which is characterized by the condition

π_B^∗(Ω_{(B/ker ΩB)}) = ΩB. (12)

Then, using (11) and (12), we deduce that the section Ω_{(B/ker ΩB)} isG-invariant, that is, Ψe^∗_g(Ω_{(B/ker ΩB)}) = Ω_{(B/ker ΩB)}, for all g∈G.

Thus, there exists a unique section Ω

Be of the vector bundle ∧²Be^∗→Ne such that π_B/^∗ _{ker Ω}

B(Ω

Be) = Ω_{(B/ker ΩB)}. (13)

This proves b) (note thatπe_B=π_{B/ker ΩB}◦π_B).

Now, using (12) and the fact kerπ_B = ker Ω_B, it follows that the section Ω_{(B/ker ΩB)} is nondegenerate. Therefore, from (13), we conclude that Ω

Be is also nondegenerate (note that if x∈M thenπ_{B/ker ΩB} :B_x/ker Ω_B(x)→Be_πN(x) is a linear isomorphism).

Next, we will describe the space of sections ofBe= (B/ker Ω_B)/G. For this purpose, we will use some results contained in the Appendix of this paper.

Let Γ(B)^p

eπ_B be the space ofπeB-projectable sections of the vector bundleτB:B →N.As we know (see the Appendix), a section Xof τ_B:B →N is said to beeπ_B-projectable if there exists a section eπ_B(X) of the vector bundle τ

Be :Be→ Ne such that πe_B◦X =eπ_B(X)◦π_N.Thus, it is clear that X is eπB-projectable if and only ifπB◦X is aπ_{(B/ker ΩB)}-projectable section.

On the other hand, it is easy to prove that the section π_B◦X is π_{(B/ker ΩB)}-projectable if and only if it is G-invariant. In other words, (π_B◦X) isπ_{(B/ker ΩB)}-projectable if and only if for everyg∈Gthere existsYg∈Γ(kerπB) such that Ψg◦X= (X+Yg)◦ψg,whereψ:G×N →N is the corresponding free action ofG onN. Note that

kereπ_B = kerπ_B= ker Ω_B (14)

and therefore, the above facts, imply that Γ(B)^p

πeB ={X∈Γ(B)/∀g∈G, ∃Y_g ∈Γ(ker Ω_B) and Ψ_g◦X = (X+Y_g)◦ψ_g}. (15) Consequently, using some results of the Appendix (see (A.2) in the Appendix), we deduce that

Γ(B)e ∼= {X ∈Γ(B)/∀g∈G, ∃Y_g∈Γ(ker Ω_B) and Ψ_g◦X= (X+Y_g)◦ψ_g} Γ(ker ΩB)

asC^∞( ˜N)-modules.

Moreover, we may prove the following result

Theorem 2 (The reduced symplectic Lie algebroid). LetΨ :G×B →B be a presympletic action of the Lie group G on the Lie algebroid (B,ΩB). Then, the reduced vector bundle τ

Be : Be= (B/ker Ω_B)/G→Ne =N/Gadmits a unique Lie algebroid structure such thateπ_B :B→Be is a Lie algebroid epimorphism if and only if the following conditions hold:

i) The space Γ(B)^p

πeB is a Lie subalgebra of the Lie algebra(Γ(B),[[·,·]]_B).

ii) Γ(ker ΩB) is an ideal of this Lie subalgebra.

Furthermore, if the conditionsi) and ii) hold, we get that there exists a short exact sequence of Lie algebroids

0→ker Ω_B →B→Be→0 and that the nondegenerate section Ω

Be induces a symplectic structure on the Lie algebroid τ

Be : Be→N .e

Proof . The first part of the Theorem follows from (14), (15) and TheoremA.1in the Appendix.

On the other hand, if the conditionsi) and ii) in the Theorem hold then, using Theorem1 and the fact that eπ_B is a Lie algebroid epimorphism, we obtain that

πe_B^∗(d^BeΩ

Be) = 0, which implies that d^BeΩ

Be = 0.

Note that if G is the trivial group, the condition ii) of Theorem 2 is satisfied and if only if Γ(ker ΩB) is an ideal of Γ(B).

Finally, we deduce the following corollary.

Corollary 1. Let Ψ :G×B→B be a presymplectic action of the Lie group Gon the presym- plectic Lie algebroid (B,Ω_B) and suppose that:

i) Ψ_g :B →B is a Lie algebroid isomorphism, for all g∈G.

ii) If X ∈Γ(B)^p

πeB and Y ∈Γ(ker Ω_B), we have that [[X, Y]]_B ∈Γ(ker Ω_B).

Then, the reduced vector bundle τ

Be :Be → Ne admits a unique Lie algebroid structure such that πeB :B → Be is a Lie algebroid epimorphism. Moreover, the nondegenerate section Ω

Be induces a symplectic structure on the Lie algebroid τ

Be :Be →N.e Proof . If X, Y ∈Γ(B)^p

eπB then

Ψ_g◦X= (X+Z_g)◦ψ_g, Ψ_g◦Y = (Y +W_g)◦ψ_g, for all g∈G,

whereZ_g, W_g ∈Γ(ker Ω_B) andψ_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 Ψ_g◦[[X, Y]]_B◦ψ_g⁻¹ = [[X+Z_g, Y +W_g]]_B, for all g∈G.

Therefore, from conditionii) in the corollary, we deduce that Ψg◦[[X, Y]]B◦ψ_g⁻¹−[[X, Y]]B∈Γ(ker ΩB), for all g∈G, which implies that [[X, Y]]B ∈ Γ(B)^p

eπB. This proves that Γ(B)^p

πeB is a Lie subalgebra of the Lie

algebra (Γ(B),[[·,·]]_B).

3.2 Reduction of the Hamiltonian dynamics

Let A be a Lie algebroid over a manifoldM and ΩA be a symplectic section ofA. In addition, suppose thatBis a Lie subalgebroid ofAover the submanifoldN ofM and thatGis a Lie group such that one may construct the symplectic reduction τ

Be : Be = (B/ker Ω_B)/G → Ne = N/G of A byB and Gas in Section 3.1(see Theorem 2).

We will also assume thatB and N are subsets of A and M, respectively (that is, the corre- sponding immersions i_B :B →A and i_N :N →M are the canonical inclusions), and thatN is a closed submanifold.

Theorem 3 (The reduction of the Hamiltonian dynamics). LetHM :M →Rbe a Hami- tonian function for the symplectic Lie algebroid A such that:

i) The restriction HN of HM to N isG-invariant and

ii) If H^Ω_H^A

M is the Hamiltonian section of HM with respect to the symplectic section ΩA, we have that H^Ω_H^A

M(N)⊆B.

Then:

a) HN induces a real function H

Ne :Ne →R such that H

Ne◦πN =HN. b) The restriction ofH^Ω_H^A

M toN iseπ_B-projectable over the Hamiltonian section of the function HNe with respect to the reduced symplectic structure Ω

Be and

c) If γ : I → M is a solution of the Hamilton equations for H_M in the symplectic Lie algebroid(A,ΩA) such thatγ(t0)∈N, for somet0 ∈I, thenγ(I)⊆N andπN◦γ :I →Ne is a solution of the Hamilton equations for H

Ne in the symplectic Lie algebroid (B,e Ω

Be).

Proof . a) Using that the functionHN isG-invariant, we deducea).

b) If x ∈ N and a_x ∈ B_x then, since eπ_B^∗Ω

Be = Ω_B (see Theorem 2), H^Ω_H^A

M(N) ⊆ B and τ_B :B →N is a Lie subalgebroid of A, we have that

(i(eπB(H^Ω_H^A

M(x)))Ω

Be(πN(x))(πeB(ax))) = (d^BHN)(x)(ax).

Thus, using thatπe_B is a Lie algebroid epimorphism and the fact thatH

Ne◦π_N =H_N,it follows that

(i(eπB(H^Ω_H^A

M(x)))Ω

Be(πN(x)))(eπB(ax))) = (d^BeH

Ne)(πN(x))(πeB(ax))

= (i(H^Ω_H^Be

Ne(π_N(x)))Ω

Be(π_N(x)))(eπ_B(a_x))).

This implies that πe_B(H^Ω_H^A

M(x)) =H^Ω_H^Be

Ne(π_N(x)).

c) Usingb), we deduce that the vector fieldρ_B((H^Ω_H^A

M)_|N) isπ_N-projectable on the vector field ρBe(H^Ω_H^Be

Ne) (it is a consequence of the equalityρ

Be◦eπ_B =T π_N◦ρ_B). This provesc).Note thatN is closed and that the integral curves ofρ_B((H^Ω_H^A

M)_|N) (respectively,ρ

Be(H^Ω_H^Be

Ne)) are the solutions of the Hamilton equations for H_M inA (respectively, for H

Ne inB) with initial condition ine N

(respectively, in Ne).

4 Examples and applications

4.1 Cartan symplectic reduction in the presence of a symmetry Lie group LetM be a symplectic manifold with symplectic 2-form ΩT M. Suppose thatN is a submanifold ofM, thatGis a Lie group and thatN is the total space of a principalG-bundle overNe =N/G.

We will denote ψ:G×N →N the free action of Gon N, byπ_N :N →Ne =N/Gthe principal bundle projection and by ΩT N the 2-form onN given by

Ω_{T N} =i^∗_NΩ_{T M}.

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,

V πN = ker ΩT N, (16)

then there exists a unique symplectic2-formΩ_T(N/G)onNe =N/Gsuch thatπ^∗_NΩ_T(N/G) = ΩT N.