Reduction of a symplectic-like Lie algebroid with momentum map and its application to fiberwise linear Poisson structures

A smooth and proper action of a Lie group G on a symplectic manifold (M, Ω) is called a Hamiltonian if G acts by symplectomorphisms and it admits a coadjoint equivariant momentum map $J:M\rightarrow \mathfrak {g}^*$ satisfying the compatibility condition

$$\begin{equation*} \Omega (\xi _M,\cdot )=d\langle J(\cdot ),\xi \rangle \quad \forall \xi \in \mathfrak {g}, \end{equation*}$$

where $\xi _M\in \mathfrak {X}(M)$ is the fundamental vector field corresponding to the Lie algebra element ξ. The Marsden–Weinstein symplectic reduction process introduced in [20] states that if μ is a regular value of J, and G_μ, the stabilizer of μ for the coadjoint representation, acts freely and properly on J⁻¹(μ), then the quotient J⁻¹(μ)/G_μ is a smooth manifold with a naturally induced 'reduced' symplectic form Ω_μ. The identity characterizing Ω_μ is

$$\begin{equation*} \iota _\mu ^*\Omega =\pi _\mu ^*\Omega _\mu , \end{equation*}$$

where ι_μ: J⁻¹(μ)↪M and π_μ: J⁻¹(μ) → J⁻¹(μ)/G_μ are the natural inclusion and projection, respectively.

One can look at the particular and important case when our symplectic structure is not on M, but on its cotangent bundle T*M, and the action of G is the cotangent lift of an action on M. In this case, the lifted action is automatically Hamiltonian with respect to the canonical symplectic form on T*M given in trivializing local coordinates by Ω_c = dxⁱ∧dyⁱ. It can be shown that an equivariant momentum map for this action is given by

$$\begin{equation} J(\alpha _x)(\xi )= \alpha _x(\xi _M(x))\quad \hbox{for all}\quad \alpha _x\in T^*_xM, \xi \in \mathfrak {g}. \end{equation} \tag{ 1.1 }$$

The reduction theory for lifted actions on cotangent bundles was first studied in [26], where only the case of Abelian actions was addressed. The general case was treated in [1] and [12]. The set of results emerging from those and other references is usually known as cotangent bundle reduction. We will expose here the basic lines of this subject and refer to [18, 23] for a more detailed survey. We will assume from now on that the action of G on M is free and proper.

For the case of lifted actions, due to the particularities of the fibered geometry, we can distinguish different situations for the choice of momentum value. These cases are (1) μ = 0, (2) G_μ = G and (3) general values of J. The theory of cotangent bundle reduction establishes the existence of maps from the abstract symplectic reduced spaces to certain cotangent bundles equipped with canonical symplectic forms possibly deformed by a magnetic term. The different possibilities are

• μ = 0. There is a symplectomorphism
  $$\begin{equation*} \phi _0:(J^{-1}(0)/G, \Omega _0)\rightarrow (T^*(M/G),\Omega _c). \end{equation*}$$
• G_μ = G. There is a symplectomorphism
  $$\begin{equation*} \phi _\mu :(J^{-1}(\mu )/G_\mu ,\Omega _\mu )\rightarrow (T^*(M/G_\mu ),\Omega _c-B_\mu ). \end{equation*}$$
• General μ. There is a symplectic embedding
  $$\begin{equation*} \phi _\mu :(J^{-1}(\mu )/G_\mu ,\Omega _\mu )\rightarrow (T^*(M/G_\mu ),\Omega _c-B_\mu ). \end{equation*}$$

In the last two cases, the magnetic term B_μ is the pullback by the cotangent bundle projection of a closed two-form on M/G_μ. This two-form is obtained, for example, via the choice of a principal connection for the fibration M → M/G_μ.

Note also that in the case G_μ = G (which corresponds to values of J for which their coadjoint orbits are trivial), we have M/G_μ = M/G. Therefore, topologically, all the reduced spaces for momentum values μ with trivial coadjoint orbits are equivalent to the same space T*(M/G), and their symplectic forms differ only possibly in the terms B_μ.

This paper develops a generalization of the reduction theory reviewed above for general symplectic manifolds and the particular case of cotangent bundles, to the setup of Lie algebroids. For general Marsden–Weinstein reduction, this happens when one substitutes the tangent bundle TM of a symplectic manifold M by a more general symplectic vector bundle A over M (a symplectic-like Lie algebroid). For cotangent bundle reduction, the generalization consists of substituting TM by a general Lie algebroid A, and T(T*M) by a special construction called the canonical cover (or the prolongation of A over A* following the terminology of [13]) of A*, which happens to be a symplectic-like Lie algebroid. The latter generalization is a particular case of the former, and both cases coincide with Marsden–Weinstein reduction and cotangent bundle reduction, respectively, when A is just the tangent bundle of M. In the remainder of this section, we will give an overview of the new results of this paper.

A Lie algebroid is a natural generalization of the tangent bundle to a manifold. It consists of a vector bundle A → M equipped with a certain geometric structure that allows us to generalize on the one hand, the Lie algebra of vector fields on M to a Lie algebra structure [[ ·, ·]] on the space of sections of A, and on the other, the exterior derivative on differential forms to the derivation d^A of the exterior algebra of multi-sections of A*. The general theory of Lie algebroids is reviewed in section 2. We remark that giving a Lie algebroid structure on the vector bundle A is equivalent to giving a linear Poisson bivector on the dual vector bundle A* of A.

In order to study the reduction process for a Lie algebroid A → M, we introduce in subsection 3.1 the notion of an action by complete lifts on A as an action Φ: G × A → A of a Lie group G by vector bundle automorphisms of a Lie group G on A together with a Lie algebra anti-morphism $\psi : {\mathfrak g}\rightarrow \Gamma (A)$ such that the infinitesimal generator of $\xi \in {\mathfrak g}$ with respect to Φ is just the complete lift of ψ(ξ); or equivalently, an action Φ: G × A* → A* on the dual vector bundle A* by Poisson automorphisms such that the infinitesimal generator of ξ is just the Hamiltonian vector field (with respect to the linear Poisson structure on A*) of the linear function associated with the section ψ(ξ) ∈ Γ(A). The standard example of an action by complete lifts on the Lie algebroid TM is the tangent lift of an action on M.

If Φ: G × A → A is a free and proper action of a connected Lie group G on the Lie algebroid A by complete lifts, then in section 3 we construct an affine action Φ^T: TG × A → A of the tangent Lie group TG such that the orbit space A/TG is a Lie algebroid over the reduced manifold M/G corresponding to the induced action ϕ: G × M → M of G on the base manifold M of the Lie algebroid A. Moreover, we prove that the projection $\widetilde{\pi }:A\rightarrow A/TG$ is a Lie algebroid morphism (see theorem 3.6).

The main idea behind the generalization of symplectic reduction to Lie algebroids consists of realizing that a symplectic manifold can be seen as a Lie algebroid endowed with a symplectic vector space structure on each fiber varying smoothly. Under this point of view, the Lie algebroid is nothing but the tangent bundle of the symplectic manifold, and the symplectic structure on the fibers is the evaluation of the symplectic form to each point. The fact that the symplectic form is closed can then be interpreted as being closed as a differential two-form on the Lie algebroid. This situation can be extended to an arbitrary Lie algebroid, not necessarily the tangent bundle of a symplectic manifold. Therefore, the setup for this paper will be a symplectic-like Lie algebroid, i.e. a Lie algebroid A → M equipped with a non-degenerate smooth 2-section Ω ∈ Γ(∧²A*) satisfying d^AΩ = 0 and an action Φ: G × A → A of a Lie group G by complete lifts on A.

The main result of subsection 3.2 is to obtain a Lie algebroid version of the Marsden–Weinstein reduction for symplectic manifolds. Firstly, we will consider a momentum map $J:M\rightarrow {\mathfrak g}$ for the action ϕ: G × M → M which allows us to define an equivariant map $J^T:A\rightarrow {\mathfrak g}^*\times {\mathfrak g}^*$ for the affine action Φ^T: TG × A → A. Then, in theorem 3.11, we describe the Lie algebroid analog of the Marsden–Weinstein reduction scheme. It states that under a regularity condition involving a value $\mu \in {\mathfrak g}^*$ of J, the quotient A_μ := (J^T)⁻¹(0, μ)/TG_μ is a symplectic-like Lie algebroid over J⁻¹(μ)/G_μ. If Ω is the symplectic-like section on A and $\widetilde{\pi }_\mu :({J^T})^{-1}(0,\mu )\rightarrow A_\mu$ and $\widetilde{\iota }_\mu :({J^T})^{-1}(0,\mu )\rightarrow A$ are the canonical projection and inclusion, respectively, then the reduced symplectic-like section Ω_μ on A_μ is characterized by the condition

$$\begin{equation*} \widetilde{\pi }_\mu ^*\Omega _\mu =\widetilde{\iota }_\mu ^*\Omega . \end{equation*}$$

It is well known that the base manifold of a symplectic-like Lie algebroid has an induced Poisson structure (see [13, 11, 16]). Then, as a consequence of the reduction theorem for symplectic-like Lie algebroids, it is shown in theorem 3.13 that the Poisson structures on the base manifolds of the original and reduced symplectic-like Lie algebroids are related in a similar way. Namely, if { ·, ·} denotes the Poisson structure on M induced by Ω and { ·, ·}_μ is the corresponding structure on J⁻¹(μ)/G_μ induced by the reduced symplectic-like section Ω_μ, then

$$\begin{equation*} \lbrace \tilde{f},\tilde{g}\rbrace _\mu \circ \pi _\mu =\lbrace f,g\rbrace \circ \iota_\mu , \end{equation*}$$

where π_μ: J⁻¹(μ) → J⁻¹(μ)/G_μ and ι_μ: J⁻¹(μ) → M are the canonical projection and the inclusion, respectively, $\tilde{f}, \tilde{g}$ are functions on J⁻¹(μ)/G_μ and f, g are G-invariant extensions to M of ${\tilde{f}}\circ \pi _\mu$ and ${\tilde{g}}\circ \pi _\mu$, respectively. That is, the reduction obtained on the base manifold of the Lie algebroid is just the Marsden–Ratiu reduction for Poisson manifolds [19].

In [2], a theory of reduction for Courant algebroids is presented. A symplectic-like Lie algebroid A induces a Lie bialgebroid and therefore a Courant algebroid on A⊕A* (see [14]). Then, one may apply this Courant reduction process to A⊕A* and could recover, after a long computation, some results described in section 3.2. However, we focus our study in the reduction of the particular case of symplectic-like Lie algebroids which allows us to obtain more explicit results on this type of reduction.

Section 4 studies, within the framework of symplectic-like Lie algebroids, the situation equivalent to cotangent bundle reduction. In this case, the generalization goes as follows. First, the cotangent bundle over a manifold M is replaced by A*, the dual of a Lie algebroid A → M, and then we consider the canonical cover of A* (also known as the prolongation of A over A*), denoted by ${\mathcal T}^AA^*$. This is a natural construction on the dual of a Lie algebroid, which happens to be in a canonical way, a symplectic-like Lie algebroid with the base manifold A*. If A is the tangent bundle of M, then ${\mathcal T}^AA^*$ is just T(T*M). If there is a suitable action of a Lie group G by complete lifts on A, this action can be further lifted to the canonical cover of A*, in a natural way, and this lifted action happens to be a morphism of symplectic-like Lie algebroids. Furthermore, one may define an equivariant momentum map on A* (the base space of ${\mathcal T}^AA^*$) in a similar way as how the classical momentum map (1.1) on T*M is introduced. In general, it is not possible to find an equivariant momentum map for a Poisson action (see, for instance, [8]). However, for the case of the Poisson action Φ*: G × A* → A* associated with an action Φ: G × A → A by complete lifts, an equivariant momentum map is described.

Applying the reduction theory of symplectic-like Lie algebroids just developed, we know that the reduction of ${\mathcal T}^AA^*$ at any momentum value is again a symplectic-like Lie algebroid. However, as in the situation of cotangent bundle reduction, it is expected that the extra properties of the symplectic-like Lie algebroid, in this case the prolonged fibered structure, will be recovered in the quotient in some way. This is the content of the results of section 5, for which the obtained new results reduce to the standard cotangent bundle reduction theory in the case that the starting Lie algebroid A → M is the standard Lie algebroid TM. In subsection 5.1, it is shown (theorem 5.1) that if μ = 0, there is a symplectic-like Lie algebroid isomorphism between the reduced symplectic-like Lie algebroid (J^T)⁻¹(0, 0)/TG and ${\mathcal T}^{A_0}A_0^*$. Here A₀ → M/G is a Lie algebroid with total space A/TG. The case G_μ = G is studied in subsection 5.2. There it is shown that (J^T)⁻¹(0, μ)/TG_μ is also isomorphic to ${\mathcal T}^{A_0}A_0^*$, but in this case this isomorphism is canonical between the symplectic-like Lie algebroids if the canonical symplectic-like section on ${\mathcal T}^{A_0}A_0^*$ is modified by the addition of a twisting term which consists of the lift to ${\mathcal T}^{A_0}A_0^*$ of a closed 2-section of A*₀. This is the content of theorem 5.2. Finally, subsection 5.3, in its main result, theorem 5.3 shows that for the most general momentum values, (J^T)⁻¹(0, μ)/TG_μ is canonically embedded as a Lie subalgebroid of ${\mathcal T}^{A_{0,\mu }}A_{0,\mu }^*$, where A_{0, μ} is the Lie algebroid A/TG_μ over M/G_μ, and the prolongation ${\mathcal T}^{A_{0,\mu }}A_{0,\mu }^*$ is equipped with its canonical symplectic-like section minus a magnetic term, just as in the G_μ = G case.

As far as we know, there is similar research being done independently by Martínez [22]. In addition, in the same direction, some similar results in the more general setting of Lie bialgebroids has been discussed in [25].

Let (A, [[ ·, ·]], ρ) be a Lie algebroid over the manifold M and let τ: A → M be the corresponding vector bundle projection. We consider a left action Φ: G × A → A by vector bundle automorphisms of a connected Lie group G on A. Then, Φ induces a linear left action Φ*: G × A* → A* given by

$$\begin{equation*} (\Phi ^*_g)_{|A_x^*}=((\Phi _{g^{-1}})_{|A_{\phi _g(x)}})^*:A_x^*\rightarrow A_{\phi _g(x)}^* \quad \mbox{for } g\in G \mbox{ and } x\in M, \end{equation*}$$

where ϕ: G × M → M is the corresponding action of G on M.

We say that Φ: G × A → A is an action by complete lifts if there is a Lie algebra anti-morphism $\psi :{\mathfrak g}\rightarrow \Gamma (A)$ such that the infinitesimal generator of $\xi \in {\mathfrak g}$ with respect to Φ is just the complete lift of ψ(ξ) to A. Note that this condition implies that ξ_M = ρ(ψ(ξ)), where ξ_M is the infinitesimal generator of the action ϕ: G × M → M with respect to ξ. Moreover, ψ(ξ)^c is a morphic vector field in the sense of [17] and therefore, for all g ∈ G, Φ_g: A → A is a Lie algebroid automorphism. Thus, the induced action Φ*: G × A* → A* of G on A* is Poisson with respect to the corresponding linear Poisson structure on A*. Furthermore, we have the following result.

Proposition 3.1. Let (A, [[ ·, ·]], ρ) be a Lie algebroid on the manifold M, Φ: G × A → A an action by vector bundle automorphisms of a connected Lie group G on A and $\psi :{\mathfrak g}\rightarrow \Gamma (A)$ a Lie algebra anti-morphism. Then, Φ: G × A → A is an action of the Lie group G on A by complete lifts with respect to ψ if and only if Φ*: G × A* → A* is an action on A* by Poisson morphisms such that the infinitesimal generator $\xi _{A^*}$ associated with $\xi \in {\mathfrak g}$ is just the Hamiltonian vector field corresponding to the linear function $\widehat{\psi (\xi )}$ associated with the section ψ(ξ) ∈ Γ(A).

Proof. Denote by $\Pi _{A^*}$ the linear Poisson structure on A*. We will prove that the Hamiltonian vector field

$$\begin{equation*} H_{\widehat{\psi (\xi )}}^{\Pi _{A^*}}=i_{d\widehat{\psi (\xi )}}\Pi _{A^*}\in {\mathfrak X}(A^*) \end{equation*}$$

is just the infinitesimal generator $\xi _{A^*}\in {\mathfrak X}(A^*)$ of ξ with respect to the action Φ*. In fact, if f ∈ C^∞(M) and X ∈ Γ(A), using (2.3), we have that

$$\begin{eqnarray*} H_{\widehat{\psi (\xi )}}^{\Pi _{A^*}}(f\circ \tau _{*})&=&\lbrace f\circ \tau _{*},\widehat{\psi (\xi )}\rbrace _{\Pi _{A^*}}=\rho (\psi (\xi ))(f)\circ \tau _{*}=(\psi (\xi ))^{*c}(f\circ \tau _{*})\\ H_{\widehat{\psi (\xi )}}^{\Pi _{A^*}}(\widehat{X})&=&\lbrace \widehat{X},\widehat{\psi (\xi )}\rbrace _{\Pi _{A^*}}=\widehat{[\! [\psi (\xi ),X]\! ]}= (\psi (\xi ))^{*c}(\widehat{X}). \end{eqnarray*}$$

Here, $\lbrace \cdot ,\cdot \rbrace _{\Pi _{A^*}}$ is the Poisson bracket associated with A*. Thus, $H_{\widehat{\psi (\xi )}}^{\Pi _{A^*}}=(\psi (\xi ))^{*c}$.

On the other hand, the flow of $(\psi (\xi ))^{c}\in {\mathfrak X}(A)$ is $\lbrace \Phi _{\exp (t\xi )}:A\rightarrow A\rbrace _{t\in \mathbb {R}}$ if and only if the flow of $(\psi (\xi ))^{*c}\in {\mathfrak X}( A^*)$ is $\lbrace \Phi ^*_{\exp (-t\xi )}:A^*\rightarrow A^*\rbrace _{t\in \mathbb {R}}$ and, in consequence, we have that the proposition holds.

Examples 3.2. 

• (i)  
  If A = TM and Φ = Tϕ is the tangent lift of the action ϕ: G × M → M, then it is clear that Φ is an action by complete lifts with Lie anti-morphism

  $$\begin{equation*} \psi : {\mathfrak g}\rightarrow {\mathfrak X}(M),\;\;\; \psi (\xi )=\xi _M. \end{equation*}$$
• (ii)  
  Let G be Lie group. If $({\mathfrak g},[\cdot ,\cdot ]_{\mathfrak g})$ is the Lie algebra associated with G, then we have, in a natural way, a Lie algebroid structure on ${\mathfrak g}\times TM\rightarrow M,$ where the Lie bracket and the anchor map are characterized by

  $$\begin{equation*} [(\xi _1,X_1), (\xi _2,X_2)]=([\xi _1,\xi _2]_{\mathfrak g},[X_1,X_2]),\;\;\;\; \rho (\xi ,X)=X \end{equation*}$$

  for all $\xi _1,\xi _2,\xi \in {\mathfrak g}$ and $X_1,X_2,X\in {\mathfrak X}(M).$

Now, consider a free and proper action ϕ: G × M → M of G on the manifold M. We denote by $\Phi :G\times ({\mathfrak g}\times TM)\rightarrow {\mathfrak g}\times TM$ and $\psi :{\mathfrak g}\rightarrow \Gamma ({\mathfrak g}\times TM)\cong C^\infty (M,{\mathfrak g})\times {\mathfrak X}(M)$ the action of G on ${\mathfrak g}\times TM$ and the Lie algebra anti-morphism, respectively, given by

$$\begin{equation*} \Phi _g(\xi ,v_x)=(Ad^G_{g}\xi ,T_x\phi _g(v_x)),\quad \xi \in {\mathfrak g} \mbox{ and } v_x\in T_xM \end{equation*}$$

$$\begin{equation*} \psi (\bar{\xi })=(-\bar{\xi }, \bar{\xi }_M),\quad \bar{\xi }\in {\mathfrak g}, \end{equation*}$$

where $Ad^G:G\times {\mathfrak g}\rightarrow {\mathfrak g}$ is the adjoint action of G on ${\mathfrak g}.$ Note that if $ad^G_{{\bar{\xi }}}:{\mathfrak g}\rightarrow {\mathfrak g}$ denotes the infinitesimal generator of the adjoint action for ${{\bar{\xi }}}\in {\mathfrak g},$ then the infinitesimal generator of $\bar{\xi }\in {\mathfrak g}$ with respect Φ is just $(ad^G_{\bar{\xi }},\bar{\xi }_M^c).$ Thus, Φ is a free and proper action by complete lifts with respect to ψ.

Remark 3.3. Suppose that we have an action Φ: G × A → A of a Lie group G on a Lie algebroid A by complete lifts with respect to $\psi :{\mathfrak g}\rightarrow \Gamma (A)$ such that the corresponding action ϕ: G × M → M on M is free. In such a case, for all x ∈ M, $\psi _x:{\mathfrak g}\rightarrow A_x$ is injective. Indeed, if $\xi ,\xi ^{\prime }\in {\mathfrak g}$ satisfy ψ_x(ξ) = ψ_x(ξ'), we have that ξ_M(x) = ρ(ψ_x(ξ)) = ρ(ψ_x(ξ')) = ξ'_M(x) which implies, using the fact that ϕ is free, that ξ = ξ'.

Next, we will prove that each action of a connected Lie group G over a Lie algebroid A by complete lifts induces an affine action of the Lie group TG over A. Previously, we recalled some facts which were related to the Lie group structure of TG.

If G is a Lie group then TG is also a Lie group. In fact, if · : G × G → G denotes the multiplication of G, then the tangent map T · : TG × TG → TG of · is such that (TG, T · ) is a Lie group. Moreover, TG may be identified with the Cartesian product $G\times {\mathfrak g}$, where $({\mathfrak g},[\cdot ,\cdot ]_{\mathfrak g})$ is the corresponding Lie algebra associated with G. This identification is given by

$$\begin{equation*} TG\rightarrow G\times {\mathfrak g},\quad X_g\in T_gG\rightarrow (g,(T_gl_{g^{-1}})(X_g))\in G\times {\mathfrak g}, \end{equation*}$$

$l_{g^{-1}}:G\rightarrow G$ being the left translation by g⁻¹ on G. The corresponding Lie group structure on $G\times {\mathfrak g}$ is defined as follows:

$$\begin{equation} (g,\xi )\cdot (g^{\prime },\xi ^{\prime })=\big((g\cdot g^{\prime }),\xi ^{\prime } + Ad^G_{(g^{\prime })^{-1}}\xi \big) \end{equation} \tag{ 3.1 }$$

and its associated Lie algebra is ${\mathfrak g}\times {\mathfrak g}$ with the Lie bracket

$$\begin{equation} [(\xi ,\eta ),(\xi ^{\prime },\eta ^{\prime })]_{{\mathfrak g}\times {\mathfrak g}}=([\xi ,\xi ^{\prime }]_{\mathfrak g}, [\xi ,\eta ^{\prime }]_{\mathfrak g}-[\xi ^{\prime },\eta ]_{\mathfrak g}). \end{equation} \tag{ 3.2 }$$

Here $Ad^G:G\times {\mathfrak g}\rightarrow {\mathfrak g}$ denotes the adjoint action of G.

Moreover, if $Coad^{TG}:(G\times {{\mathfrak g}})\times ({{\mathfrak g}}^*\times {\mathfrak g}^*) \rightarrow {\mathfrak g}^* \times {\mathfrak g}^*$ is the left coadjoint action of $TG\cong G\times {\mathfrak g}$ on the dual space of the Lie algebra $({\mathfrak g}\times {\mathfrak g}, [ \cdot ,\cdot ]_{\mathfrak g\times {\mathfrak g}})$, then

$$\begin{equation} Coad ^{TG}_{(g,\xi )} (\mu ^{\prime },\mu ^{\prime \prime })=\big(Coad^G_g\big(\mu ^{\prime }+coad^G_\xi \mu ^{\prime \prime }\big),Coad^G _g\mu ^{\prime \prime }\big) \end{equation} \tag{ 3.3 }$$

for $(g,\xi )\in G\times {\mathfrak g}$ and $(\mu ^{\prime },\mu ^{\prime \prime })\in {\mathfrak g}^*\times {\mathfrak g}^*$, where $Coad ^G:G\times {\mathfrak g}^*\rightarrow {\mathfrak g}^*$ is the left coadjoint action associated with G and $coad^G:{\mathfrak g}\times {\mathfrak g}^*\rightarrow {\mathfrak g}^*$ is the corresponding infinitesimal left coadjoint action.

The following proposition describes how ψ works with respect to the action Φ.

Proposition 3.4. Let Φ: G × A → A be an action of a connected Lie group G on the Lie algebroid A by complete lifts with respect to $\psi :{\mathfrak g}\rightarrow \Gamma (A)$. Then,

$$\begin{equation} \Phi _g(\psi (Ad^G_{g^{-1}}\xi )(x))=\psi (\xi )(\phi _g(x)) \end{equation} \tag{ 3.4 }$$

for all $\xi \in {\mathfrak g},$ g ∈ G and x ∈ M.

Proof. We organize the proof in two steps.

First step. Suppose that G is a connected and simply connected Lie group. Consider the map

$$\begin{equation*} \psi ^{cv}:{\mathfrak g}\times {\mathfrak g}\rightarrow {\mathfrak X}(A),\quad\psi ^{cv}(\xi ,\eta )=(\psi (\xi ))^c + (\psi (\eta ))^v. \end{equation*}$$

Using (2.6), we may prove easily that ψ^cv is an infinitesimal action of TG over A, that is, ψ^cv is $\mathbb {R}$-linear and

$$\begin{equation*} \psi ^{cv}([(\xi ,\eta ),(\xi ^{\prime },\eta ^{\prime })]_{{\mathfrak g}\times {\mathfrak g}})=[\psi ^{cv}(\xi ,\eta ), \psi ^{cv}(\xi ^{\prime },\eta ^{\prime })]. \end{equation*}$$

Then, since the vector field $\psi ^{cv}(\xi ,\eta )\in {\mathfrak X}(A)$ is complete, from Palais theorem (see [24]), there is a unique action Φ^T: TG × A → A from $TG\cong G\times {\mathfrak g}$ such that for all $(\xi ,\eta )\in {\mathfrak g}\times {\mathfrak g}$

$$\begin{equation*} (\xi ,\eta )_A=\psi ^{cv}(\xi ,\eta )=(\psi (\xi ))^c + (\psi (\eta ))^v. \end{equation*}$$

Here $(\xi ,\eta )_A\in {\mathfrak X}(A)$ is the infinitesimal generator of (ξ, η) with respect to the action Φ^T.

Now, suppose that g = exp _G(η). Then, we have that

$$\begin{equation} \Phi _g\big(\psi \big(Ad^G_{g^{-1}}\xi \big)(x)\big)=\Phi ^T\big((g,0_{\mathfrak g}),\psi \big(Ad^G_{g^{-1}}\xi \big)(x)\big) \end{equation} \tag{ 3.5 }$$

with $0_{\mathfrak g}$ being the zero of ${\mathfrak g}.$ In fact, one can prove that

$$\begin{equation} \pi :\mathbb {R}\rightarrow G\times {\mathfrak g},\quad s\mapsto \pi (s)= (\exp _G(s\eta ),0_{\mathfrak g}) \end{equation} \tag{ 3.6 }$$

is a one-parameter subgroup and $\frac{{\rm d}\pi }{{\rm d}s}_{|s=0}=(\eta ,0_{\mathfrak g})$. So, $\Phi ^T((\exp _G(s\eta ),0_{\mathfrak g}),\psi (Ad^G_{g^{-1}}\xi (x))$ is just the integral curve of $\phi ^T(\eta , 0_{\mathfrak g})=(\psi (\eta ))^c$ at the point $\psi (Ad^G_{g^{-1}}\xi )(x)\in A_x.$ In consequence,

$$\begin{equation*} \Phi ^T\big((\exp _G(s\eta ),0_{\mathfrak g}),\psi \big(Ad^G_{g^{-1}}\xi \big)(x)\big)=\Phi \big({\exp _G(s\eta )},\psi \big(Ad^G_{g^{-1}}\xi \big)(x)\big). \end{equation*}$$

In particular, when s = 1, we obtain (3.5).

Furthermore,

$$\begin{equation} \psi \big(Ad^G_{g^{-1}}\xi \big)(x)=\Phi ^T\big(\big(e,Ad^G_{g^{-1}}\xi \big),0_x\big), \end{equation} \tag{ 3.7 }$$

where 0_x is the zero of A_x and e is the identity element of G.

In fact, in order to prove (3.7), we consider the one-parameter subgroup

$$\begin{equation} \pi ^{\prime }:\mathbb {R}\rightarrow G\times {\mathfrak g},\quad s\mapsto \pi ^{\prime }(s)=\big(e,sAd^G_{g^{-1}}\xi \big). \end{equation} \tag{ 3.8 }$$

Then, $\frac{{\rm d}\pi ^{\prime }}{{\rm d}s}_{|s=0}=(0_{\mathfrak g},Ad^G_{g^{-1}}\xi )\in {\mathfrak g}\times {\mathfrak g}$ and, therefore, $\Phi ^T((e,sAd^G_{g^{-1}}\xi ),0_x))$ is just the integral curve of $\psi ^{cv} (0_{\mathfrak g},Ad^G_{g^{-1}}\xi )=(\psi (Ad^G_{g^{-1}}\xi ))^v\in {\mathfrak X}(A)$ at the point 0_x ∈ A_x, i.e.

$$\begin{equation*} \Phi ^T\big(\big(e,sAd^G_{g^{-1}}\xi \big),0_x\big)=s\psi \big(Ad^G_{g^{-1}}\xi \big)(x). \end{equation*}$$

In particular, if s = 1 we obtain (3.7).

Now, from (3.1), (3.5) and (3.7), we deduce that

$$\begin{equation*} \begin{array}{r@{\,}c@{\,}l}\Phi _g\big(\psi \big(Ad^G_{g^{-1}}\xi \big)(x)\big)&=&\Phi ^T\big((g,0_{\mathfrak g}),\Phi ^T\big(\big(e,Ad^G_{g^{-1}}\xi \big),0_x\big)\big)\\\ms &=& \Phi ^T\big((g,0_{\mathfrak g})\cdot \big(e,Ad^G_{g^{-1}}\xi \big),0_x\big)\\\ms &=&\Phi ^T((e,\xi )\cdot (g,0_{\mathfrak g}),0_x)= \Phi ^T((e,\xi ),\Phi ^T((g,0_{\mathfrak g}),0_{x})). \end{array} \end{equation*}$$

On the other hand, using (3.5) (with $\xi =0_{\mathfrak g}$), it follows that $\Phi ^T ((g,0_{\mathfrak g}),0_x)=0_{\phi _g(x)}$. In addition, from (3.7) (with g = e), we obtain that $\Phi ^T ((e,\xi ),0_{\phi _g(x)})=\psi (\xi )(\phi _g(x))$. This proves (3.4) for g = exp _G(η). Finally, using that G is connected, we conclude that (3.4) holds for all g ∈ G.

Second step. Now, we suppose that G is a connected Lie group with Lie algebra ${\mathfrak g}.$ Denote by $\widetilde{G}$ the universal covering of G and by $\widetilde{\mathfrak g}$ its corresponding Lie algebra. Then, the covering projection $p:\widetilde{G}\rightarrow G$ is a local isomorphism of Lie groups and the map

$$\begin{equation*} \widetilde{\Phi }:\widetilde{G}\times A\rightarrow A,\quad \widetilde{\Phi }(\widetilde{g},a_x)=\Phi (p(\widetilde{g}),a_x) \end{equation*}$$

is an action of $\widetilde{G}$ over A by complete lifts with respect to the Lie algebra anti-morphism $\widetilde{\psi }=\psi \circ T_ep:\widetilde{\mathfrak g}\rightarrow \Gamma (A).$ So, for all g ∈ G, x ∈ M and $\xi \in {\mathfrak g}$, there are $\widetilde{g}\in \widetilde{G}$ and $\widetilde{\xi }\in \widetilde{\mathfrak g}$ such that

$$\begin{equation*} p(\widetilde{g})=g \mbox{ and } (T_{\widetilde{e}}p)(\widetilde{\xi })=\xi . \end{equation*}$$

Here, $\widetilde{e}$ is the identity element of $\widetilde{G}$. Therefore, using the first step

$$\begin{equation*} \widetilde{\Phi }_{\widetilde{g}}(\widetilde{\psi }(Ad^{\widetilde{G}}_{\widetilde{g}^{-1}}\widetilde{\xi })(x))= {\psi }({\xi })({\phi }_{{g}}(x)). \end{equation*}$$

Finally, since $(T_{\widetilde{e}}p)(Ad^{\widetilde{G}}_{\widetilde{g}^{-1}}\widetilde{\xi })=Ad^G_{g^{-1}}\xi$, we obtain (3.4).

As we previously claimed, from an action of G on A by complete lifts, we can define an affine action of TG on A as it is described in the following theorem.

Theorem 3.5. Let Φ: G × A → A be an action of the connected Lie group G by complete lifts on the Lie algebroid A with respect to $\psi :{\mathfrak g}\rightarrow \Gamma (A).$ Then,

$$\begin{equation} \Phi ^T:(G\times {\mathfrak g})\times A\rightarrow A,\quad \Phi ^T((g,\xi ),a_x)=\Phi _g(a_x + \psi (\xi )(x)) \end{equation} \tag{ 3.9 }$$

defines an affine action of $TG\cong G\times {\mathfrak g}$ over A. Moreover, if $(\xi ,\eta )\in {\mathfrak g}\times {\mathfrak g},$ its infinitesimal generator (ξ, η)_A with respect to the action Φ^T is

$$\begin{equation} (\xi ,\eta )_A=(\psi (\xi ))^c + (\psi (\eta ))^v. \end{equation} \tag{ 3.10 }$$

Proof. Equation (3.4) allows us to prove that $\Phi ^T:(G\times {\mathfrak g})\times A\rightarrow A$ is an affine action of $TG\cong G\times {\mathfrak g}$ over A. In fact,

$$\begin{equation*} \fl\begin{array}{r@{\,}c@{\,}l}\Phi ^T((g,\xi )\cdot (h,\eta ),a_x)&=&\Phi ^T\big(\big(g\cdot h,\eta + Ad^G_{h^{-1}}\xi \big),a_x\big)=\Phi _{gh}(a_x) + \Phi _{gh}\big(\psi \big(\eta + Ad^G_{h^{-1}}\xi \big)(x)\big)\\\ms &=&\Phi _g(\Phi _h(a_x) + \Phi _h\big(\psi (\eta )(x) + \psi \big(Ad^G_{h^{-1}}\xi \big)(x))\big)\\\ms &=& \Phi _g(\Phi _h(a_x) + \Phi _h(\psi (\eta )(x))) + \Phi _g(\psi (\xi )(\phi _h(x)))\\\ms &=& \Phi ^T((g,\xi ),\Phi ^T((h,\eta ),a_x))\end{array} \end{equation*}$$

and

$$\begin{equation*} \Phi ^T((e,0_{\mathfrak g}),a_x)=\Phi _e(a_x) + \Phi _e(\psi (0_{\mathfrak g})(x))=a_x. \end{equation*}$$

Moreover, using the one-parameter subgroups defined in (3.6) and (3.8), one may conclude easily that the infinitesimal generator (ξ, η)_A of $(\xi ,\eta )\in {\mathfrak g}\times {\mathfrak g}$ with respect to Φ^T is

$$\begin{equation*} (\xi ,\eta )_A=(\xi ,0_{\mathfrak g})_A + (0_{\mathfrak g},\eta )_A=(\psi (\xi ))^c + (\psi (\eta ))^v. \end{equation*}$$

Note that, under the same hypotheses as in theorem 3.5, the action Φ: G × A → A is free and proper if and only if the corresponding action ϕ: G × M → M on M is free and proper. Moreover, if Φ: G × A → A is free and proper, then so is Φ^T: TG × A → A.

On the other hand, we recall that the space of orbits N/H of a free and proper action of a Lie group H on a manifold N is a quotient differentiable manifold and the canonical projection π: N → N/H is a surjective submersion (see [1]). With this, we prove a preliminary reduction result.

Theorem 3.6. Let (A, [[ ·, ·]], ρ) be a Lie algebroid on M and Φ: G × A → A a free and proper action of a connected Lie group G on A by complete lifts with respect to the Lie algebra anti-morphism $\psi :{\mathfrak g}\rightarrow \Gamma (A).$ Then, A/TG is a Lie algebroid over M/G and the projection $\widetilde{\pi }:A\rightarrow A/TG$ is a Lie algebroid epimorphism.

Proof. Since Φ: G × A → A is an action by Lie algebroid automorphisms A/G is a Lie algebroid over M/G with vector bundle projection τ/G: A/G → M/G (see [10, 15]). The space of sections of this vector bundle may be identified with that of G-invariant sections Γ(A)^G of A. Under this identification the bracket and the anchor map of the Lie algebroid structure on A/G is just

$$\begin{equation*} [\! [X, Y]\! ]_{A/G}= [\! [X, Y]\! ], \quad\rho _{A/G}(X({\pi }(x)))=T_x{\pi }(\rho (X(x)) \end{equation*}$$

for all X, Y ∈ Γ(A)^G and x ∈ M, where π: M → M/G is the quotient projection corresponding to the induced action ϕ: G × M → M.

On the other hand, using that ϕ: G × M → M is free, we have that $\psi _x:{\mathfrak g}\rightarrow A_x$ is injective (see remark 3.3). Thus, for all x ∈ M, we have that

$$\begin{equation*} \dim \lbrace \psi _x(\xi )/\xi \in {\mathfrak g}\rbrace =\dim {\mathfrak g}\mbox{ for all }x\in M. \end{equation*}$$

Therefore, since $\psi :{\mathfrak g}\rightarrow \Gamma (A)$ is a Lie algebra anti-morphism, we deduce that

$$\begin{equation*} \psi ({\mathfrak g})=\bigcup _{x\in M}\lbrace \psi _x(\xi )/\xi \in {\mathfrak g}\rbrace \end{equation*}$$

is a Lie subalgebroid of A over M. Moreover, from proposition 3.4, we have that the Lie group G acts by Lie algebroid automorphisms on $\psi ({\mathfrak g})$. So, one may induce a Lie algebroid structure on the quotient vector bundle $\psi ({\mathfrak g} )/G$ such that $\psi ({\mathfrak g} )/G$ is a Lie subalgebroid of A/G. Now, we will show that it is also an ideal.

If X ∈ Γ(A) is G-invariant then X ○ ϕ_g = Φ_g ○ X for all g ∈ G. Thus, the flow ϒ_t(ξ): A → A of the vector field (ψ(ξ))^c and the flow φ_t(ξ): M → M of (ρ○ψ)(ξ) satisfy the following property:

$$\begin{equation*} \Upsilon _t(\xi )\circ X=X\circ \varphi _t(\xi )\quad \mbox{ for all }t\in \mathbb {R}. \end{equation*}$$

Equivalently,

$$\begin{equation*} \widehat{X}\circ \Upsilon _t(\xi )^*=\widehat{X,} \end{equation*}$$

where ϒ_t(ξ)* is the dual morphism of ϒ_t(ξ): A → A and $\widehat{X}$ is the linear function associated with the section X.

Therefore, we have that

$$\begin{equation*} \frac{{\rm d}}{{\rm d}t}_{|t=0}(\widehat{X}\circ \Upsilon _t(\xi )^*)=0. \end{equation*}$$

Since ϒ_t(ξ)* is the flow of the vector field ψ(ξ)*^c, the previous equation is equivalent to the relation

$$\begin{equation} [\! [\psi (\xi ),X]\! ]=0\quad\mbox{for all }\xi \in {\mathfrak g}. \end{equation} \tag{ 3.11 }$$

On the other hand, let Y be a G-invariant section of $\psi ({\mathfrak g})$ and {ξ_i} a basis of ${\mathfrak g}$. Then,

$$\begin{equation*} Y=\sum Y^i \psi (\xi _i), \end{equation*}$$

with Yⁱ real functions on A. Moreover, using proposition 3.4 and the fact that ψ_x is injective, we have that

$$\begin{equation} Y^i\circ \phi _g = (Ad^G)_j^i(g)Y^j, \end{equation} \tag{ 3.12 }$$

where Ad^G_gξ_i = (Ad^G)^j_i(g)ξ_j. Hence, from (3.11),

$$\begin{equation*} [\! [X,Y]\! ]= \sum \rho (X)(Y^i)\psi (\xi _i). \end{equation*}$$

Now, using that X and Y are G-invariant sections, we obtain that [[X, Y]] is a G-invariant section and, as a consequence, [[X, Y]] is a G-invariant section of the vector bundle $\psi ({\mathfrak g} )\rightarrow M$. Thus, $\psi ({\mathfrak g} )/G$ is indeed an ideal of A/G of constant rank (since the action Φ is free) and therefore, the quotient vector bundle $(A/G)/(\psi ({\mathfrak g})/G)$ admits a Lie algebroid structure over M/G.

Finally, we have that this vector bundle is isomorphic to A/TG and, thus, a Lie algebroid structure on A/TG is induced in such a way that this isomorphism is a Lie algebroid isomorphism. In fact, using (3.4), one may prove that Φ induces a free and proper action $\bar{\Phi }:G\times A/\psi ({\mathfrak g})\rightarrow A/\psi ({\mathfrak g})$ on $A/\psi ({\mathfrak g})$ such that the projection $\widetilde{\Psi }_1:A\rightarrow A/\psi ({\mathfrak g})$ is equivariant, with respect to the action Φ^T and $\bar{\Phi }$. So, $\widetilde{\Psi }_1$ induces a smooth map $\Psi _1:A/TG\rightarrow (A/\psi ({\mathfrak g}))\big /G$. Moreover, one easily proves that Ψ₁ is a one-to-one correspondence. On the other hand, the map

$$\begin{equation*} \Psi _2:(A/\psi ({\mathfrak g}))/G\rightarrow (A/G) / (\psi ({\mathfrak g})/G),\quad [\widetilde{\Psi }_1(a)]\rightarrow \widetilde{\Psi }_2([a]) \end{equation*}$$

is bijective, where $\widetilde{\Psi }_2:A/G\rightarrow (A/G)/( \psi ({\mathfrak g})/G)$ is the corresponding quotient map. Consequently, the reduced vector bundle A/TG → M/G is isomorphic to the vector bundles

$$\begin{equation} (A/\psi ({\mathfrak g}))/G\rightarrow M/G\quad\mbox{and}\quad (A/G)/ (\psi ({\mathfrak g})/G)\rightarrow M/G \end{equation} \tag{ 3.13 }$$

Note that, using the above isomorphisms, the space of sections of the vector bundle A/TG → M/G may be identified with the quotient space

$$\begin{equation*} \Gamma (A)^G/\Gamma (\psi ({\mathfrak g}))^G, \end{equation*}$$

where Γ(A)^G (respectively, $\Gamma ( \psi ({\mathfrak g} ))^{G}$) is the space of G-invariant sections on A (respectively, $\psi ({\mathfrak g})$). Under this identification the Lie algebroid structure ([[ ·, ·]]_A/TG, ρ_A/TG) on A/TG is characterized by

$$\begin{equation} \begin{array}{l}[\! [[X],[Y]]\! ]_{A/TG} = [[\! [X, Y]\! ]],\\\ms \rho _{A/TG }([X])\circ {\pi }=T{\pi } \circ \rho (X)\quad \mbox{ for } X, Y\in \Gamma (A)^{G}. \end{array} \end{equation} \tag{ 3.14 }$$

This implies that the canonical projection $\widetilde{\pi }:A\rightarrow A/TG$ is a Lie algebroid epimorphism.

Examples 3.7. 

• (i)  
  In the case when A = TM and Φ = Tϕ is the tangent lift of the action ϕ: G × M → M, the reduced Lie algebroid TM/TG from the previous theorem is isomorphic to T(M/G) with its standard Lie algebroid structure.
• (ii)  
  For the case $A={\mathfrak g}\times TM$ from (ii) in examples 3.2, we obtain that the reduced Lie algebroid $({\mathfrak g}\times TM)/TG$ with respect to the action $\Phi ^T: (G\times {\mathfrak g})\times ({\mathfrak g}\times TM)\rightarrow {\mathfrak g}\times TM$ given by

  $$\begin{equation*} \Phi ^T((g,\bar{\xi }), (\xi ,v_x))=\big(Ad^G_{g}(\xi - \bar{\xi }), T_x\phi _g(v_x+\bar{\xi }_M(x))\big) \end{equation*}$$

  can be identified with the Atiyah Lie algebroid TM/G induced by the principal bundle π: M → M/G.

We recall the construction of this last Lie algebroid. Firstly, we denote by τ: TM → M the projection of TM on M which is equivariant with respect to the tangent lift action Tϕ: G × TM → TM and ϕ: G × M → M. The sections of the induced vector bundle τ/G: TM/G → M/G can be identified with the G-invariant vector fields on M. Moreover, the set of G-invariant vector fields is closed with respect to the Lie bracket of vector fields. Using this fact, one can define the Lie algebroid structure ([[ ·, ·]]_TM/G, ρ_TM/G) on TM/G by

$$\begin{equation*} [\! [X,Y]\! ]_{TM/G}=[X,Y],\quad \rho _{TM/G}(X(x))=T_x\pi (X(x)) \end{equation*}$$

for X, Y G-invariant vector fields of M and x ∈ M. The corresponding Lie algebroid is known as Atiyah Lie algebroid (see [13]).

Now, we have the following vector bundle epimorphism:

$$\begin{equation} F:({\mathfrak g}\times TM)/TG\rightarrow TM/G,\quad F([(\xi ,v_x)]_{TG})=[v_x+\xi _M]_G. \end{equation} \tag{ 3.15 }$$

Note that F is well defined. Indeed, for all $\xi ,\bar{\xi }\in {\mathfrak g}$, g ∈ G and v_x ∈ T_xM, one has that

$$\begin{equation*} \fl\begin{array}{r@{\,}c@{\,}l}F\big(\big[Ad_g^G(\xi - \bar{\xi }), T_x\phi _g(v_x + \bar{\xi }_M(x))\big]_{TG}\big)&= &\big[T_x\phi _g(v_x + \bar{\xi }_M(x))+\big(Ad_g^G(\xi - \bar{\xi })\big)_M(\phi _g(x))\big]_G\\\ms &=&[T_x\phi _g(v_x + \bar{\xi }_M(x))+T_x\phi _g(\xi _M(x) - \bar{\xi }_M(x))]_G\\\ms &=&[v_x+\xi _M(x)]_G. \end{array} \end{equation*}$$

Moreover, since

$$\begin{equation} [(\xi ,v_x)]_{TG}=[(0,v_x+\xi _M(x))]_{TG},\quad \mbox{for all } v_x\in T_xM \mbox{ and } \xi \in {\mathfrak g}, \end{equation} \tag{ 3.16 }$$

we deduce that F is a vector bundle isomorphism.

On the other hand, using (3.16), we obtain that if {X_i} is a local basis of G-invariant vector fields on M, then {[(0, X_i)]_TG} is a local base of $\Gamma (({\mathfrak g}\times TM)/TG).$ This fact allows us to prove that F is a Lie algebroid isomorphism.

A Lie algebroid (A, [[ ·, ·]], ρ) on the manifold M is symplectic-like if there is a nondegenerate 2-section Ω ∈ Γ(∧²A*) on A* which is closed, i.e. d^AΩ = 0. In such a case, for each function $f:M\rightarrow {\mathbb R}$ on M we have the Hamiltonian section on A which is characterized by

$$\begin{equation*} i_{{\mathcal H}^\Omega _f}\Omega =d^Af. \end{equation*}$$

The base space M of a symplectic-like Lie algebroid A is a Poisson manifold, where the Poisson bracket on M is given by

$$\begin{equation} \lbrace f,g\rbrace =\rho ({\mathcal H}^\Omega _g)(f)\quad f,g\in C^\infty (M) \end{equation} \tag{ 3.17 }$$

(see [13, 11, 16]).

Note that if f ∈ C^∞(M), then the Hamiltonian vector field of f with respect to the Poisson structure on M is $\rho ({\mathcal H}_f^\Omega )$. Thus, the solutions of Hamilton's equations for f are the integral curves of the vector field $\rho ({\mathcal H}_f^\Omega ).$

Now, in the rest of this section, we suppose that (A, [[ ·, ·]], ρ, Ω) is a symplectic-like Lie algebroid over M, that Φ: G × A → A is an action of a connected Lie group G on A by complete lifts with respect to the Lie algebra anti-morphism $\psi :{\mathfrak g}\rightarrow \Gamma (A)$ and that $J:M\rightarrow {\mathfrak g}^*$ is an equivariant smooth map, i.e.

$$\begin{equation*} Coad^G_g(J(x))=J(\phi _g(x)),\quad \forall x\in M,\quad\forall g\in G. \end{equation*}$$

The action Φ is said to be a Hamiltonian action with the momentum map $J:M\rightarrow {\mathfrak g}^*$ if

$$\begin{equation} \Phi _g^*(\Omega )=\Omega \mbox{ and } i_{\psi (\xi )}\Omega =d^A {J}_\xi , \mbox{ for all } g\in G \mbox{ and } \xi \in {\mathfrak g}, \end{equation} \tag{ 3.18 }$$

where J_ξ is the real function on M given by

$$\begin{equation} {J}_\xi (x)=J(x)(\xi ) \quad\mbox{ for } x\in M. \end{equation} \tag{ 3.19 }$$

Note that the previous condition implies that

$$\begin{equation} {\mathcal L}_{\psi (\xi )}^A\Omega =0. \end{equation} \tag{ 3.20 }$$

Now, let $J^T:A\rightarrow ({\mathfrak g}\times {\mathfrak g})^*\cong {\mathfrak g}^*\times {\mathfrak g}^*$ be the map given by

$$\begin{equation} J^T(a)=((TJ\circ \rho )(a) , J(\tau (a))). \end{equation} \tag{ 3.21 }$$

Then, we have

Lemma 3.8. The map $J^T:A\rightarrow {\mathfrak g}^*\times {\mathfrak g}^*$ is equivariant for the action Φ^T: TG × A → A.

Proof. Let $(g,\xi )\in G\times {\mathfrak g}\cong TG$ and a ∈ A_x. Since J is equivariant, we have that

$$\begin{equation} T_xJ (\rho ( \psi (\xi )(x)))=coad^G_\xi (J(x)) \quad \mbox{for }\xi \in {\mathfrak g}\mbox{ and } x\in M. \end{equation} \tag{ 3.22 }$$

Moreover, using that Φ_g is a Lie algebroid morphism over ϕ_g and that J is equivariant, we obtain

$$\begin{equation} TJ\circ \rho \circ \Phi _g = TJ \circ T\phi _g\circ \rho = Coad^G _g\circ TJ\circ \rho . \end{equation} \tag{ 3.23 }$$

As a consequence, from (3.3), (3.21), (3.22) and (3.23), we have that

$$\begin{eqnarray*} \fl Coad ^{TG}_{(g,\xi )} (J^T(a))&=& Coad ^{TG}_{(g,\xi )} (TJ (\rho (a)) ,J(\tau (a)))\\\ms &=&(Coad^G_g(TJ\circ \rho (a))+Coad^G_g\big(coad^G_\xi J(\tau (a))\big),Coad^G_g (J(\tau (a))))\\\ms &=& \big(Coad^G_g(TJ\circ \rho (a))+Coad^G_g\big(coad^G_\xi J(\tau (a))\big),J (\phi _g(\tau (a)))\big)\\\ms &=& \big((TJ \circ \rho )( \Phi _g(a) )+ (TJ\circ \rho )(\Phi _g(\psi (\xi )(\tau (a)))), J \big(\tau \big(\Phi ^T _{(g,\xi )}(a)\big)\big) \big)\\\ms &=& \big((TJ\circ \rho )\big(\Phi ^T_{(g,\xi )}(a)\big),J\big(\tau \big(\Phi ^T_{(g,\xi )}(a)\big)\big)\big)\\\ms &=& J^T (\Phi ^T_{(g,\xi )}(a)). \end{eqnarray*}$$

Hence, J^T is equivariant with respect to Φ^T.

Proposition 3.9. Let (A, [[ ·, ·]], ρ, Ω) be a symplectic-like Lie algebroid over M, Φ: G × A → A a Hamiltonian action of a connected Lie group G on A with Lie algebra anti-morphism $\psi :{\mathfrak g} \rightarrow \Gamma (A)$ and equivariant momentum map $J:M\rightarrow {\mathfrak g}^*$. Let $\mu \in {\mathfrak g}^*$ be a regular value of J such that $T_x J\circ \rho : A_x \rightarrow T_{\mu }{\mathfrak g}^*$ has a constant rank for all x ∈ J⁻¹(μ). Then,

• (i)  
  (J^T)⁻¹(0, μ) is a Lie subalgebroid of A over J⁻¹(μ);
• (ii)  
  the restriction $\psi _\mu :{\mathfrak g}_\mu \rightarrow \Gamma (A)$ of ψ to the isotropy algebra ${\mathfrak g}_\mu$ of μ with respect to the coadjoint action takes values in Γ((J^T)⁻¹(0, μ));
• (iii)  
  the isotropy Lie group G_μ of μ with respect to the coadjoint action acts on (J^T)⁻¹(0, μ) by complete lifts with respect to $\psi _\mu :{\mathfrak g}_\mu \rightarrow \Gamma ((J^T)^{-1} (0,\mu ));$
• (iv)  
  the action of G_μ on the Lie subalgebroid (J^T)⁻¹(0, μ) induces an affine action Φ^T_μ: TG_μ × (J^T)⁻¹(0, 0) → (J^T)⁻¹(0, 0).

Proof. 

• (i)  
  Note that since μ is a regular value of J, J⁻¹(μ) is a regular submanifold of M. In fact, (0, μ) is a regular value for $J^T:A\rightarrow \mathfrak g^*\times \mathfrak g^*\cong T{\mathfrak g}^*.$ Thus, using that $T_xJ\circ \rho _x:A_x\rightarrow T_\mu {\mathfrak g}^*$ has a constant rank for all x ∈ J⁻¹(μ), we deduce that

  $$\begin{equation*} (J^T)^{-1} (0,\mu )=\lbrace a\in A/TJ(\rho (a))=0,\ J(\tau (a))=\mu \rbrace \end{equation*}$$

  is a vector subbundle of A on J⁻¹(μ) of rank $n-\dim G,$ where n = rank A. On the other hand, from (3.21) we have that J^T is a Lie algebroid morphism; then, it is straightforward to see that the restriction $\tau _{|(J^T)^{-1} (0,\mu )}: (J^T)^{-1} (0,\mu )\rightarrow J^{-1}(\mu )$ of τ: A → M to (J^T)⁻¹(0, μ) is a Lie subalgebroid of (A, [[ ·, ·]], ρ).
• (ii)  
  If x ∈ J⁻¹(μ) and

  $$\begin{equation*} \xi \in {\mathfrak g}_\mu =\big\lbrace \eta \in {\mathfrak g}/coad^G_\eta \mu =0\big\rbrace , \end{equation*}$$

  then, since J is an equivariant map, we have that

  $$\begin{equation*} T_xJ(\rho _x (\psi (\xi )(x))) = T_xJ(\xi _M(x))= coad^G _\xi (\mu ) = 0. \end{equation*}$$

  Thus, the restriction of ψ(ξ) to J⁻¹(μ) is a section of the vector bundle (J^T)⁻¹(0, μ) → J⁻¹(μ).
• (iii)  
  Using the equivariance of J and the fact that Φ_g is a Lie algebroid automorphism, for any g ∈ G we have that the action Φ: G × A → A induces an action Φ_μ of G_μ on (J^T)⁻¹(0, μ). In fact, if g ∈ G_μ, a ∈ (J^T)⁻¹(0, μ) and x ∈ J⁻¹(μ), then

  $$\begin{equation*} J(\phi _g(x))=Coad_g^G(J(x))=Coad_g^G\mu =\mu \end{equation*}$$

  and

  $$\begin{eqnarray*} (TJ\circ \rho )(\Phi _g(a))&=& T(J\circ \phi _g)(\rho (a))= T(Coad^G _{g}\circ J)(\rho (a))= 0. \end{eqnarray*}$$

  Moreover, from (ii), we have that Φ_μ is an action by complete lifts with respect to the Lie algebra anti-morphism ψ_μ.
• (iv)  
  It is a direct consequence of (iii) and theorem 3.5.

Let $\mu \in {\mathfrak g}^*$ be a regular value of $J:M\rightarrow {\mathfrak g}^*$ such that $T_x J\circ \rho _x : A_x \rightarrow T_{\mu }{\mathfrak g}^*$ has a constant rank for all x ∈ J⁻¹(μ). Suppose that the corresponding action ϕ_μ: G_μ × J⁻¹(μ) → J⁻¹(μ) is free and proper. Then, using theorem 3.6 and proposition 3.9, we obtain that A_μ = (J^T)⁻¹(0, μ)/TG_μ is a Lie algebroid over J⁻¹(μ)/G_μ. In the following result, we will prove that A_μ is a symplectic-like Lie algebroid. For this purpose, we will need the following properties.

Lemma 3.10. Let (A, [[ ·, ·]], ρ, Ω) be a symplectic-like Lie algebroid over the manifold M and Φ: G × A → A a Hamiltonian action of a connected Lie group G on A with an equivariant momentum map $J:M\rightarrow {\mathfrak g}^*$ and associated Lie algebra anti-morphism $\psi :{\mathfrak g}\rightarrow \Gamma (A).$ If $\mu \in {\mathfrak g}^*$, then for any x ∈ M,

• (i)  
  $(\psi _\mu )_x({\mathfrak g}_\mu )=\psi _x({\mathfrak g})\cap \ker (T_xJ\circ \rho _x);$
• (ii)  
  $\ker (T_xJ\circ \rho _x)=(\psi _x({\mathfrak g}))^\perp =\lbrace a_x\in A_x/\Omega _x(a_x,b_x)=0, \forall b_x\in \psi _x({\mathfrak g})\rbrace .$

Proof. 

• (i)  
  It is an immediate consequence of the fact that J is equivariant.
• (ii)  
  If a_x ∈ A_x, using (2.2) and (3.19), we deduce that

  $$\begin{equation*} \Omega (a_x,\psi (\xi )(x))=-(i_{\psi (\xi )}\Omega )(a_x)=-(d^A {J}_\xi )(a_x)=-(T_xJ(\rho _x(a_x)))({\xi }) \end{equation*}$$

  for all $\xi \in {\mathfrak g}.$ Thus, one concludes immediately (ii) from this relation.

The following result may be seen as the analog of Marsden–Weinstein reduction theorem for symplectic-like Lie algebroids.

Theorem 3.11. Reduction theorem of symplectic-like Lie algebroids. Let (A, [[ ·, ·]], ρ, Ω) be a symplectic-like Lie algebroid and Φ: G × A → A a Hamiltonian action of a connected Lie group G on A with an equivariant momentum map $J:M\rightarrow {\mathfrak g}^*$ and associated Lie algebra anti-morphism $\psi :{\mathfrak g}\rightarrow \Gamma (A)$. Suppose that $\mu \in {\mathfrak g}^*$ is a regular value of $J:M\rightarrow {\mathfrak g}^*$ such that $T_x J\circ \rho _x : A_x \rightarrow T_{\mu }{\mathfrak g}^*$ has a constant rank for all x ∈ J⁻¹(μ) and the restricted action ϕ_μ: G_μ × J⁻¹(μ) → J⁻¹(μ) is free and proper. Then, A_μ = (J^T)⁻¹(0, μ)/TG_μ is a symplectic-like Lie algebroid over J⁻¹(μ)/G_μ with a symplectic-like section Ω_μ characterized by the condition

$$\begin{equation*} \widetilde{\pi }_\mu ^*\Omega _\mu =\widetilde{\iota }_\mu ^*\Omega , \end{equation*}$$

where $\widetilde{\pi }_\mu :(J^T)^{-1} (0,\mu )\rightarrow A_\mu$ is the canonical projection and $\;\widetilde{\iota }_\mu :(J^T)^{-1}(0,\mu )\rightarrow A$ is the canonical inclusion.

Proof. Since A_μ is a Lie algebroid over J⁻¹(μ)/G_μ, one needs to prove that this algebroid is symplectic-like.

Let $\widetilde{\Omega }_\mu =\widetilde{\iota }_\mu ^*\Omega$ be the 2-cocycle on the Lie subalgebroid (J^T)⁻¹(0, μ) → J⁻¹(μ) induced by Ω. We will prove that $\widetilde{\Omega }_\mu$ induces a symplectic-like 2-section Ω_μ over A_μ.

Suppose that X_μ, Y_μ ∈ Γ(A_μ). Then, we may choose two sections $\widetilde{X}_\mu ,\widetilde{Y}_\mu \in \Gamma ((J^T)^{-1}(0,\mu ))$ such that the following diagram is commutative:

We will see that $\widetilde{\Omega }_\mu (\widetilde{X}_\mu ,\widetilde{Y}_\mu )$ is a G_μ-invariant function (or, equivalently, a π_μ-basic function).

Denote by $([\! [\cdot ,\cdot ]\! ]_{(J^T)^{-1} (0,\mu )},\rho _{(J^T)^{-1} (0,\mu )})$ the Lie algebroid structure on (J^T)⁻¹(0, μ) → J⁻¹(μ).

As we know, the vertical bundle of π_μ is generated by the vector fields on J⁻¹(μ) of the form $\rho _{(J^T)^{-1}(0,\mu )}(\psi _\mu (\xi )),$ with $\xi \in {\mathfrak g}_\mu .$

Now, we have that

$$\begin{eqnarray*} &&\fl\big(\rho _{(J^T)^{-1}(0,\mu )}(\psi _\mu (\xi ))\big)(\widetilde{\Omega }_\mu (\widetilde{X}_\mu ,\widetilde{Y}_\mu ))=\big({\mathcal L}_{\psi _\mu (\xi )}^{(J^T)^{-1} (0,\mu )}\widetilde{\Omega }_\mu \big)(\widetilde{X}_\mu ,\widetilde{Y}_\mu )\nonumber\\ &&+\, \widetilde{\Omega }_\mu \big([\! [\psi _\mu (\xi ),\widetilde{X}_\mu ]\! ]_{(J^T)^{-1} (0,\mu )},\widetilde{Y}_\mu \big)+ \widetilde{\Omega }_\mu \big(\widetilde{X}_\mu ,[\! [\psi _\mu (\xi ),\widetilde{Y}_\mu ]\! ]_{(J^T)^{-1} (0,\mu )}\big). \end{eqnarray*}$$

On the other hand, using that $\widetilde{X}_\mu$ and $\widetilde{Y}_\mu$ are G_μ-invariant, we deduce that

$$\begin{equation*} [\! [\psi _\mu (\xi ),\widetilde{X}_\mu ]\! ]_{(J^T)^{-1} (0,\mu )}=[\! [\psi _\mu (\xi ),\widetilde{Y}_\mu ]\! ]_{(J^T)^{-1} (0,\mu )}=0 \end{equation*}$$

(see (3.11)). In addition, from (3.20), it follows that

$$\begin{equation*} {\mathcal L}_{\psi _\mu (\xi )}^{(J^T)^{-1} (0,\mu )}\widetilde{\Omega }_\mu =0, \end{equation*}$$

which implies that

$$\begin{equation*} \rho _{(J^{T})^{-1}(0,\mu )}(\psi _\mu (\xi ))(\widetilde{\Omega }_\mu (\widetilde{X}_\mu ,\widetilde{Y}_\mu ))=0. \end{equation*}$$

Thus, for all X_μ, Y_μ ∈ Γ(A_μ) there is a function Ω_μ(X_μ, Y_μ) on J⁻¹(μ)/G_μ such that

$$\begin{equation*} \Omega _\mu (X_\mu ,Y_\mu )\circ \pi _\mu =\widetilde{\Omega }_\mu (\widetilde{X}_\mu ,\widetilde{Y}_\mu ). \end{equation*}$$

Note that the function Ω_μ(X_μ, Y_μ) does not depend on the chosen sections $\widetilde{X}_\mu ,\widetilde{Y}_\mu \in \Gamma ((J^T)^{-1} (0,\mu ))$ which project on $\widetilde{X}_\mu$ and $\widetilde{Y}_\mu ,$ respectively. In fact, from lemma 3.10, we have that

$$\begin{equation*} \ker (\widetilde{\Omega }_\mu (x))=(\ker \widetilde{\pi }_\mu )_{|(J^{T})_x^{-1}(0,\mu )}=(\psi _\mu )_x({\mathfrak g}_\mu ) \mbox{ for all } x\in J^{-1}(\mu ). \end{equation*}$$

Therefore, the map

$$\begin{equation*} \Gamma (A_\mu )\times \Gamma (A_\mu )\rightarrow C^\infty (J^{-1}(\mu )/G_\mu ),\quad ({X}_\mu , {Y}_\mu )\mapsto {\Omega }_\mu ({X}_\mu , {Y}_\mu ) \end{equation*}$$

defines a section Ω_μ of the vector bundle Λ²A*_μ → J⁻¹(μ)/G_μ and it is clear that

$$\begin{equation*} \widetilde{\pi }_\mu ^*{\Omega _\mu }=\widetilde{\iota }_\mu ^*\Omega =\widetilde{\Omega }_\mu . \end{equation*}$$

This implies that

$$\begin{equation*} \widetilde{\pi }_\mu ^*(d^{A_\mu }{\Omega _\mu })=d^{(J^T)^{-1} (0,\mu )}(\widetilde{\pi }_\mu ^*{\Omega }_\mu )=d^{(J^T)^{-1} (0,\mu )}\widetilde{\iota }_\mu ^*\Omega =\widetilde{\iota }_\mu ^*(d^A\Omega )=0 \end{equation*}$$

and since $\widetilde{\pi }_\mu :(J^T)^{-1} (0,\mu )\rightarrow A_\mu =(J^T)^{-1} (0,\mu )/TG_\mu$ is an epimorphism of vector bundles, we conclude that $d^{A_\mu }{\Omega }_\mu =0.$

Finally, we will prove that Ω_μ is non-degenerate. In fact, if x ∈ J⁻¹(μ) and $\widetilde{v}_x\in \ker (T_xJ\circ \rho _x)$ is such that

$$\begin{equation*} {\Omega }_\mu ({\pi _\mu (x)})(\widetilde{\pi }_\mu (\widetilde{v}_x),u_{\pi _\mu (x)}))=0,\quad \forall u_{\pi _\mu (x)}\in (A_\mu )_{\pi _\mu (x)}, \end{equation*}$$

then

$$\begin{equation*} \widetilde{\Omega }(x)(\widetilde{v}_x,\widetilde{u}_x)=0 \quad\mbox{for all }\widetilde{u}_x\in \ker (T_xJ\circ \rho _x). \end{equation*}$$

Consequently (see lemma 3.10)

$$\begin{equation*} \widetilde{v}_x\in (\ker (T_xJ\circ \rho _x))^\perp =\psi _x({\mathfrak g}) \end{equation*}$$

and thus,

$$\begin{equation*} \widetilde{v}_x\in (\psi _\mu )_x({\mathfrak g}_\mu ) \end{equation*}$$

which implies that $\widetilde{\pi }_\mu (\widetilde{v}_x)=0$.

Remark 3.12. In the particular case when M is a symplectic manifold and A is the standard symplectic-like Lie algebroid TM → M, then theorem 3.11 reproduces the classical Marsden–Weinstein reduction result for the symplectic manifold M.

Since A_μ → J⁻¹(μ)/G_μ is a symplectic-like Lie algebroid, the base space J⁻¹(μ)/G_μ is a Poisson manifold. In fact, we will prove that J⁻¹(μ)/G_μ is the reduced Poisson manifold obtained from the reduction process of Marsden–Ratiu [19].

Theorem 3.13. Under the hypotheses of theorem 3.11, if { ·, ·}_μ is the Poisson bracket on J⁻¹(μ)/G_μ, we have that

$$\begin{equation} \lbrace \tilde{f},\tilde{g}\rbrace _\mu \circ \pi _\mu =\lbrace f,g\rbrace \circ i_\mu \end{equation} \tag{ 3.24 }$$

for $\tilde{f},\tilde{g}\in C^\infty ( J^{-1}(\mu )/G_\mu )$, where i_μ: J⁻¹(μ) → M is the canonical inclusion and f, g ∈ C^∞(M) are arbitrary G-invariant extensions of $\tilde{f}\circ \pi _\mu$ and $\tilde{g}\circ \pi _\mu ,$ respectively.

Proof. From theorem 3.11, we deduce that $(A_\mu ,[\! [\cdot ,\cdot ]\! ]_{A_\mu },\rho _{A_\mu },\Omega _\mu )$ is a symplectic-like Lie algebroid on J⁻¹(μ)/G_μ. Then, one can define a Poisson structure on J⁻¹(μ)/G_μ as in (3.17). We will prove that the associated Poisson bracket { · · }_μ satisfies (3.24).

If $\tilde{f}, \tilde{g}:J^{-1}(\mu )/G_\mu \rightarrow {\mathbb R}$ are two real functions on J⁻¹(μ)/G_μ and ${f}:M\rightarrow \mathbb {R}$, $g:M\rightarrow \mathbb {R}$ are arbitrary G-invariant extensions of $\tilde{f}\circ \pi _\mu$ and $\tilde{g}\circ \pi _\mu ,$ respectively, for any $\xi \in {\mathfrak g}$ satisfying ρ(ψ(ξ))(f) = ρ(ψ(ξ))(g) = 0, we have that

$$\begin{equation*} d^Af(\psi (\xi ))=d^Ag(\psi (\xi ))=0, \end{equation*}$$

or, equivalently,

$$\begin{equation*} \Omega ({\mathcal H}_f^\Omega ,\psi (\xi ))=\Omega ({\mathcal H}_g^\Omega ,\psi (\xi ))=0. \end{equation*}$$

Therefore, ${\mathcal H}_f^\Omega (x),{\mathcal H}_g^\Omega (x)\in \psi _x({\mathfrak g})^\perp =(J^T)^{-1}_x(0,\mu )$ for all x ∈ J⁻¹(μ).

On the other hand, if x ∈ J⁻¹(μ) and a_x ∈ (J^T)⁻¹_x(0, μ), then, using theorem 3.11, the fact that $(\widetilde{\pi }_\mu ,\pi _\mu )$ is a Lie algebroid epimorphism and that (J^T)⁻¹(0, μ) → J⁻¹(μ) is a Lie subalgebroid of A, one deduces that

$$\begin{equation*} \begin{array}{r@{\,}c@{\,}l}\Omega _\mu \big(\widetilde{\pi }_\mu \big({\mathcal H}_f^\Omega (x)\big),\widetilde{\pi }_\mu (a_x)\big)\big) &=&\Omega \big({\mathcal H}_f^{\Omega }(x),a_x\big)=(d^Af)(a_x)\\\ms &=& \big(d^{(J^T)^{-1}(0,\mu )}(\widetilde{f}\circ \pi _\mu )\big)(a_x)=(d^{A_\mu }\widetilde{f})(\widetilde{\pi }_\mu (a_x))\\\ms &=& \Omega _\mu \big({\mathcal H}_{\widetilde{f}}^{\Omega _\mu }({\pi _\mu (x)}),\widetilde{\pi }_\mu (a_x)\big). \end{array} \end{equation*}$$

So since Ω_μ is non-degenerate,

$$\begin{equation*} \widetilde{\pi }_\mu \big({\mathcal H}_f^\Omega ({x})\big)={\mathcal H}_{\tilde{f}}^{\Omega _\mu }({\pi _\mu (x)}). \end{equation*}$$

Thus, using again that $(\widetilde{\pi }_\mu ,\pi _\mu )$ is an epimorphism of Lie algebroids, we conclude that

$$\begin{equation*} T_x\pi _\mu \big(\rho _{(J^T)^{-1}(0,\mu )}\big({\mathcal H}_f^\Omega (x)\big)=\rho _{A_\mu }\big({\mathcal H}_{\widetilde{f}}^{\Omega _\mu }({\pi _\mu (x)})\big),\quad\forall x\in J^{-1}(\mu ). \end{equation*}$$

Therefore, if x ∈ J⁻¹(μ)

$$\begin{equation*} \begin{array}{r@{\,}c@{\,}l}\lbrace \widetilde{f},\widetilde{g}\rbrace _\mu (\pi _\mu (x))&=&-\big(\rho _{A_\mu }\big({\mathcal H}_{\widetilde{f}}^{\Omega _\mu }({\pi _\mu (x)})\big)\big)\widetilde{g} \\\ms &=&-\big(T_x\pi _\mu \big(\rho _{(J^T)^{-1}(0,\mu )}\big({\mathcal H}_f^\Omega (x)\big)\big)\big)\widetilde{g}\\\ms &=& -\big(\rho _{(J^T)^{-1}(0,\mu )}\big({\mathcal H}_f^\Omega (x)\big)\big)(\widetilde{g}\circ \pi _\mu )\\\ms &=&-\big(\rho \big({\mathcal H}_f^\Omega (x)\big)g=\lbrace f,g\rbrace (x). \end{array} \end{equation*}$$

The vector bundle ${\mathcal T}^AA^*\rightarrow A^*$

Let (A, [[ ·, ·]], ρ) be a Lie algebroid of rank n over a manifold M of dimension m with τ: A → M the associated vector bundle projection and let F: M' → M be a smooth map from a manifold M' to M. If x' ∈ M', we consider the vector subspace

$$\begin{equation*} ({\mathcal T}^AM^{\prime })_{x^{\prime }}=\lbrace (a,v)\in A_{F(x^{\prime })}\times T_{x^{\prime }}M^{\prime }/\rho (a)=T_{x^{\prime }}F(v)\rbrace \end{equation*}$$

of $A_{F(x^{\prime })}\times T_{x^{\prime }}M^{\prime }$ of dimension $n+m^{\prime }-\dim (\rho (A_{F(x^{\prime })}) + T_{x^{\prime }}F(T_{x^{\prime }}M^{\prime }))$, where m' is the dimension of M'. If we suppose that $\dim (\rho (A_{F(x^{\prime })}) + T_{x^{\prime }}F(T_{x^{\prime }}M^{\prime }))$ is constant over F(M') (for instance, if F is a submersion), then ${\mathcal T}^AM^{\prime }$ is a vector bundle over M' which is called the prolongation of A over F (see [9, 13]). In this case, a section ${\mathcal X}$ of ${\mathcal T}^AM^{\prime }\rightarrow M^{\prime }$ is said to be projectable if there exist a section X of A and a vector field V on M', F-projectable over ρ(X), satisfying

$$\begin{equation*} {\mathcal Z}(m^{\prime })=(X(F(m^{\prime })),V(m^{\prime }))\quad \mbox{for all } m^{\prime }\in M^{\prime }. \end{equation*}$$

The section ${\mathcal Z}$ will be denoted by ${\mathcal Z}=(X,V).$ Note that one may choose a local basis $\lbrace {\mathcal Z}_I\rbrace$ of $\Gamma ({\mathcal T}^AA^*)$ such that, for all I, ${\mathcal Z}_I$ is a projectable section.

On the other hand, a section $\widetilde{\gamma }$ of the dual vector bundle $({\mathcal T}^AM^{\prime })^*\rightarrow M^{\prime }$ is said to be projectable if there exist a section α of A* and a 1-form β of M' such that

$$\begin{equation*} \widetilde{\gamma }(X,V)=\alpha (X)\circ F + \beta (V),\quad \mbox{ for $(X,V)$ a projectable section of ${\mathcal T}^AM^{\prime }$}. \end{equation*}$$

In such a case, we will use $\widetilde{\gamma }=(\alpha ,\beta ).$ Note that one may choose a local basis $\lbrace {\mathcal Z}^I\rbrace$ of $\Gamma (({\mathcal T}^AA^*)^*)$ such that, for all I, ${\mathcal Z}^I$ is a projectable section.

A particular case is when the function F is the dual bundle projection τ_*: A* → M of the Lie algebroid A. Then, the prolongation $\tau _{{\mathcal T}^AA^*}:{\mathcal T}^AA^*\rightarrow A^*$ of A over τ_*: A* → M is called the A-tangent bundle of A*. In such a case, $\dim (\rho (A_{x}) + T_{\alpha _x}\tau _{*}(T_{\alpha _x}A^*))$ is just the dimension of M for all α_x ∈ A*_x, and the rank of ${\mathcal T}^AA^*$ is 2n.

A basis of local sections of the vector bundle $\tau _{{\mathcal T}^AA^*}:{\mathcal T}^AA^*\rightarrow A^*$ is defined as follows. If (xⁱ) are local coordinates on an open subset U of M, {e_I} is a basis of sections of the vector bundle τ⁻¹(U) → U and (xⁱ, y_I) are the corresponding local coordinates on A*, then $\lbrace {\mathcal X}_I, {\mathcal Y}^I\rbrace$ is a local basis of $\Gamma ({\mathcal T}^AA^*)$, where ${\mathcal X}_{I}$ and ${\mathcal Y}^{I}$ are the projectable sections defined by

$$\begin{equation} {\mathcal X}_{I} = \left(e_{I}, \rho _{I}^i \displaystyle \frac{\partial }{\partial x^i}\right), \quad{\mathcal Y}^{I} = \left(0, \displaystyle \frac{\partial }{\partial y_{I}}\right). \end{equation} \tag{ 4.1 }$$

The symplectic-like Lie algebroid structure on ${\mathcal T}^AA^*\rightarrow A^*$

The vector bundle ${\mathcal T}^AA^*$ admits a Lie algebroid structure $([\! [\cdot ,\cdot ]\! ]_{{\mathcal T}^AA^*},\rho _{{\mathcal T}^AA^*})$ which is characterized by the following conditions:

$$\begin{equation*} [\! [(X,V),(X^{\prime },V^{\prime })]\! ]_{{\mathcal T}^AA^*}=([\! [X,Y]\! ],[V,V^{\prime }]),\quad \rho _{{\mathcal T}^AA^*}(X,V)=V \end{equation*}$$

for (X, V), (X', V') projectable sections of ${{\mathcal T}^AA^*}.$

If $d^{{\mathcal T}^AA^*}$ is the differential associated with this Lie algebroid structure, then

$$\begin{equation} \begin{array}{@{}l@{}}d^{{\mathcal T}^AA^*}f(X_1,V_1)= {\rm d}f(V_1) \\\ms d^{{\mathcal T}^AA^*}(\alpha ,\beta )((X_1,V_1), (X_2,V_2))= d^A\alpha (X_1,X_2)\circ \tau _*+ {\rm d}\beta ( V_1,V_2), \end{array} \end{equation} \tag{ 4.2 }$$

where $f:A^*\rightarrow {\mathbb R}$ is a smooth function, $(\alpha ,\beta )\in \Gamma (({\mathcal T}^AA^*)^*)$ and $(X_i,V_i)\in \Gamma ({\mathcal T}^AA^*)$ are projectable sections of $({\mathcal T}^AA^*)^*$ and ${\mathcal T}^AA^*$, respectively.

The canonical section λ_A of the dual bundle to ${\mathcal T}^A A^*$ (which is called the Liouville section associated with the Lie algebroid A) may be defined as follows:

$$\begin{equation} \lambda _A(a,v_\alpha )=\alpha (a)\quad\mbox{for } \alpha \in A^*\quad\mbox{and } (a,v_\alpha )\in {{\mathcal T}_\alpha ^AA^*}. \end{equation} \tag{ 4.3 }$$

The section Ω_A of $\wedge ^2({{\mathcal T}^AA^*})^*\rightarrow A^*$ given by

$$\begin{equation*} \Omega _A=-d^{{{\mathcal T}^AA^*}}\lambda _A \end{equation*}$$

is nondegenerate and $d^{{\mathcal T}^A A^*}\Omega _A=0.$ Thus, Ω_A is a symplectic-like section of the Lie algebroid ${{\mathcal T}^AA^*}\rightarrow A^*$ which is called the canonical symplectic-like section associated with the Lie algebroid A. The Poisson structure on the base space A* induced by this symplectic-like section is just the linear Poisson structure on A* associated with the Lie algebroid A (see [13]). For this reason, ${\mathcal T}^AA^*$ may be considered as the canonical cover of the fiberwise linear Poisson structure on A*. If $\lbrace {\mathcal X}_{I}, {\mathcal Y}^{I}\rbrace$ is the local basis of sections of ${\mathcal T}^AA^*$ described in (4.1), the local expressions of λ_A and Ω_A are

$$\begin{equation*} \lambda _{A} = y_{I}{\mathcal X}^I,\quad\Omega _{A} = {\mathcal X}^I \wedge {\mathcal Y}_{I} + {\textstyle\frac{1}{2}} C_{IJ}^K y_K {\mathcal X}^I \wedge {\mathcal X}^J \end{equation*}$$

where $\lbrace {\mathcal X}^I, {\mathcal Y}_{I}\rbrace$ is the dual basis of $\lbrace {\mathcal X}_{I}, {\mathcal Y}^{I}\rbrace$ and C^K_IJ are the local structure functions of the bracket [[ ·, ·]] (for more details, see [13]).

Examples 4.1. 

• (i)  
  Note that if A is the standard Lie algebroid TM, then the symplectic-like Lie algebroid ${\mathcal T}^AA^*\rightarrow A^*$ may be identified with the standard Lie algebroid T(T*M) → T*M and, under this identification, Ω_A is the canonical symplectic Ω_M structure of T*M.
• (ii)  
  For the case $A={\mathfrak g}\times TM$ from (ii) in examples 3.2, we have that ${\mathcal T}^AA^*\rightarrow A^*$ can be identified with ${\mathcal T}^{\mathfrak g}{\mathfrak g}^*\oplus {\mathcal T}^{TM}({T^*M})\rightarrow {\mathfrak g}^*\times T^*M$, i.e.

  $$\begin{equation*} ({\mathfrak g}\times T{\mathfrak g}^*)\oplus T(T^*M)\rightarrow {\mathfrak g}^*\times T^*M. \end{equation*}$$

  Under this identification, the symplectic-like structure $\Omega _{{\mathfrak g}\times TM}$ on ${\mathcal T}^{{\mathfrak g}\times TM}({\mathfrak g}^*\times T^*M)$ is just $\Omega _{\mathfrak g} \oplus \Omega _{M},$ where Ω_M is the standard symplectic 2-form on T*M and $\Omega _{{\mathfrak g}}$ is the symplectic-like structure on ${\mathfrak g}\times T{\mathfrak g}^*\rightarrow {\mathfrak g}^*$ characterized by

  $$\begin{equation*} (\Omega _{\mathfrak g})_{\eta _{0}}((\xi ,\eta ), (\xi ^{\prime },\eta ^{\prime }))=\eta ^{\prime }(\xi )-\eta (\xi ^{\prime }) + \eta _0{[\xi ,\xi ^{\prime }]_{\mathfrak g}} \end{equation*}$$

  for all $\eta _0,\eta ,\eta ^{\prime }\in {\mathfrak g}^*$ and $\xi ,\xi ^{\prime }\in {\mathfrak g}^*.$

The action of a Lie group G on ${\mathcal T}^AA^*\rightarrow A^*$ by complete lifts

Now, suppose that Φ: G × A → A is a free and proper action of a connected Lie group G by complete lifts with respect to the Lie algebra anti-morphism $\psi :{\mathfrak g}\rightarrow \Gamma (A).$ Denote by ϕ: G × M → M the corresponding action on M and by Φ*: G × A* → A* the left dual action on A*. In what follows, we will describe a free and proper canonical action by complete lifts on ${\mathcal T}^AA^*$ induced by Φ.

Proposition 4.2. Let Φ: G × A → A be a free and proper action of a connected Lie group G on the Lie algebroid A by complete lifts with respect to $\psi :{\mathfrak g}\rightarrow \Gamma (A).$ Then the map $(\Phi ,T\Phi ^*):G\times {\mathcal T}^AA^*\rightarrow {\mathcal T}^AA^*$ given by

$$\begin{equation} \fl(\Phi ,T\Phi ^*)(g,(a_x,v_{\alpha _x}))=(\Phi _g(a_x),(T_{\alpha _x}\Phi _{g}^*)(v_{\alpha _x})),\quad \alpha _x\in A_x^*\quad \mbox{and } (a_x,v_{\alpha _x})\in {\mathcal T}_{\alpha _x}^AA^* \end{equation} \tag{ 4.4 }$$

defines a free and proper left canonical action of G on the symplectic-like Lie algebroid ${\mathcal T}^AA^*$ by complete lifts with respect to the Lie algebra anti-morphism $\psi ^T:{\mathfrak g}\rightarrow \Gamma ({\mathcal T}^AA^*)$ defined by

$$\begin{equation} \psi ^T(\xi )=(\psi (\xi ), \xi _{A^*})\quad\mbox{for }\xi \in {\mathfrak g}, \end{equation} \tag{ 4.5 }$$

where $\xi _{A^*}$ is the infinitesimal generator of ξ with respect to Φ*.

Proof. Note that the map (Φ, TΦ*) is well defined. In fact, since Φ is an action by complete lifts, then Φ_g: A → A is a Lie algebroid isomorphism for all g ∈ G. So, using (2.4) with k = 0, we have that

$$\begin{equation} \rho (\Phi _g(a_x))=T_x\phi _g(\rho (a_x))\quad \mbox{ for all $g\in G,$ $x\in M$ and $a_x\in A_x.$} \end{equation} \tag{ 4.6 }$$

Moreover,

$$\begin{equation} T_{\Phi ^*_{g}(\alpha _x)}\tau _{*}(T_{\alpha _x}\Phi ^*_{g}(v_{\alpha _x}))=T_{\alpha _x}(\phi _g\circ \tau _{*})(v_{\alpha _x})=T_x\phi _g(\rho (a_x)) \end{equation} \tag{ 4.7 }$$

for all $v_{\alpha _x}\in T_{\alpha _x}A^*.$ Thus, from (4.6) and (4.7), we deduce that

$$\begin{equation*} T_{\Phi ^*_{g}(\alpha _x)}\tau _{*}(T_{\alpha _x}\Phi ^*_{g}(v_{\alpha _x}))=\rho (\Phi _g(a_x)) \end{equation*}$$

for all $(a_x,v_{\alpha _x})\in {\mathcal T}_{\alpha _x}^AA^*$, that is,

$$\begin{equation*} (\Phi ,T^*\Phi )(g,(a_x,v_{\alpha _x}))\in {\mathcal T}^A_{\Phi ^*_g(\alpha _x)}A^*. \end{equation*}$$

Obviously, (Φ, TΦ*) is a free and proper action. We will now show that this action on ${\mathcal T}^AA^*$ is by complete lifts. Firstly, note that the map ${\psi }^T:{\mathfrak g}\rightarrow \Gamma ({\mathcal T}^AA^*)$ is well defined. In fact, since ρ(ψ(ξ)) is just the infinitesimal generator ξ_M of ξ with respect to the action ϕ: G × M → M and the projection τ_*: A* → M is equivariant, we have that

$$\begin{equation*} \rho (\psi (\xi ))=T\tau _*(\xi _{A^*})\quad \mbox{ for all }\xi \in {\mathfrak g}. \end{equation*}$$

On the other hand, the infinitesimal generator $\xi _{{\mathcal T}^AA^*}\in {\mathfrak X}({{\mathcal T}^AA^*})$ of $\xi \in {\mathfrak g}$ with respect to the action (Φ, TΦ*) is the pair (ξ_A, ξ^c_A*), where ξ_A is the infinitesimal generator of $\xi \in {\mathfrak g}$ with respect to Φ and $\xi ^c_{A^*}$ is the complete lift of $\xi _{A^*}.$ Moreover, the complete lift of ψ^T(ξ) with respect to the Lie algebroid ${\mathcal T}^AA^*$ is just $(\psi (\xi )^c,\xi _{A^*}^c).$ This is a consequence of the fact that $(\psi (\xi )^c,\xi _{A^*}^c)\in {\mathfrak X}({\mathcal T}^AA^*)$ is $\tau _{{\mathcal T}^AA^*}$-projectable on $\rho _{{\mathcal T}^AA^*} (\psi (\xi ), \xi _{A^*})=\xi _{A^*}$ and that, from (4.2), we deduce that

$$\begin{equation*} {\mathcal L}^{{\mathcal T}^AA^*}_{(\psi (\xi ),\xi _{A^*})}(\alpha ,\beta )=\big({\mathcal L}^{A}_{\psi (\xi )}\alpha ,{\mathcal L}_{\xi _{A^*}}\beta \big) \end{equation*}$$

for every projectable section (α, β) on $\Gamma (({\mathcal T}^AA^*)^*)$. Here ${\mathcal L}$ is the standard Lie derivative.

Therefore, (Φ, TΦ*) is an action by complete lifts and consequently by automorphisms of Lie algebroids. Finally, a direct computation, using (4.3), proves that the action (Φ, TΦ*) preserves the Liouville section λ_A, i.e

$$\begin{equation*} (\Phi ,T\Phi ^*)_g^*\lambda _A=\lambda _A \quad \mbox{ for all $g\in G.$} \end{equation*}$$

Thus, using (2.4) and the fact that (Φ, TΦ*)_g is an automorphism of Lie algebroids, we conclude that (Φ, TΦ*)_g preserves the canonical symplectic-like section Ω_A of ${\mathcal T}^AA^*.$

The momentum map for the canonical action of G on the Lie algebroid ${\mathcal T}^AA^*\rightarrow A^*$

Denote by $J_{A^*}:A^*\rightarrow {\mathfrak g}^*$ the map given by

$$\begin{equation} J_{A^*}(\alpha _x)(\xi )=\alpha _x(\psi (\xi )(x))\quad\mbox{with } x\in M,\quad\alpha _x\in A_x^*\quad\mbox{and }\xi \in {\mathfrak g}. \end{equation} \tag{ 4.8 }$$

Then, we have the following result.

Proposition 4.3. The map $J_{A^*}:A^*\rightarrow {\mathfrak g}^*$ is an equivariant momentum map for the Poisson action Φ*: G × A* → A*.

Proof. From proposition 3.1, we have that if $\Pi _{A^*}$ is the linear Poisson structure on A* and $\widehat{\psi (\xi )}$ is the linear function associated with the section ψ(ξ) ∈ Γ(A), for each $\xi \in {\mathfrak g},$ the Hamiltonian vector field

$$\begin{equation*} H_{\widehat{\psi (\xi )}}^{\Pi _{A^*}}=i_{d\widehat{\psi (\xi )}}\Pi _{A^*}\in {\mathfrak X}(A^*) \end{equation*}$$

is just the infinitesimal generator $\xi _{A^*}\in {\mathfrak X}(A^*)$ of ξ with respect to the action Φ*. Note that the function $(J_{A^*})_\xi :A^*\rightarrow {\mathbb R}$ given by

$$\begin{equation*} (J_{A^*})_\xi (\alpha _x)=(J_{A^*}(\alpha _x))(\xi ) \end{equation*}$$

is just $\widehat{{\psi }(\xi )}.$ Thus, $J_{A^*}$ is a momentum map for the Poisson action Φ*: G × A* → A*.

Now, we will prove that $J_{A^*}$ is equivariant, i.e.

$$\begin{equation*} J_{A^*}\circ \Phi _{g}^*=Coad_{g}^G\circ J_{A^*}. \end{equation*}$$

Indeed, if x ∈ M and α_x ∈ A*, then, from proposition 3.4, we have that

$$\begin{equation*} \begin{array}{r@{\,}c@{\,}l}J_{A^*}(\Phi _{g}^*(\alpha _x))(\xi )&=&(\Phi ^*_{g}(\alpha _x))(\psi (\xi )(\phi _g(x)))=\alpha _x(\Phi _{g^{-1}}(\psi (\xi )(\phi _g(x)))\\\ms &=&\alpha _x\big(\psi \big(Ad^G_{g^{-1}}\xi \big)(x)\big)=J_{A^*}(\alpha _x)\big(Ad^G_{g^{-1}}\xi \big)\\\ms &=&\big(\big(Coad^G_{g}\big)(J_{A^*}(\alpha _x)\big)(\xi ) \end{array} \end{equation*}$$

for all $\xi \in {\mathfrak g}.$

Now, using lemma 3.8, we have that the map

$$\begin{equation} J_{A^*}^T:{\mathcal T}^AA^*\rightarrow {\mathfrak g}^*\times {\mathfrak g}^*,\quad J_{A^*}^T(a_x,v_{\alpha _x})= (T_{\alpha _x}J_{A^*}(v_{\alpha _x}), J_{A^*}(\alpha _x)) \end{equation} \tag{ 4.9 }$$

is equivariant with respect to the action $(\Phi ,T\Phi ^*)^T:TG\times {\mathcal T}^AA^*\rightarrow {\mathcal T}^AA^*.$

From the injectivity of ψ_x (see remark 3.3), it follows that the restriction of $J_{A^*}:A^*\rightarrow {\mathfrak g}^*$ to A*_x is a linear epimorphism and therefore, for all α_x ∈ A*_x the restriction of the tangent map $T_{\alpha _x}J_{A^*}:T_{\alpha _x}A^*\rightarrow T_{J_{A^*}(\alpha _x)}{\mathfrak g}^*\cong {\mathfrak g}^*$ to $T_{\alpha _x}A_x^*$ is surjective. Thus, all the elements of ${\mathfrak g}^*$ are regular values of $J_{A^*}$ and

$$\begin{equation*} T_{\alpha _x}J_{A^*}\circ (\rho _{{\mathcal T}^{A}A^*)_{\alpha _x}}:{{\mathcal T}_{\alpha _x}^{A}A^*}\rightarrow {\mathfrak g}^* \end{equation*}$$

is surjective for all α_x ∈ A*_x. Note that

$$\begin{equation*} T_{\alpha _x}A^*_x=\ker T_{\alpha _x}\tau _{*}\subseteq (\rho _{{\mathcal T}^AA^*})_{\alpha _x}({\mathcal T}_{\alpha _x}^AA^*). \end{equation*}$$

In conclusion, if $\mu \in {\mathfrak g}^*,$ then $J_{A^*}^{-1}(\mu )$ is a regular submanifold of A* and $(J^T_{A^*})^{-1}(0,\mu )$ is a Lie subalgebroid of ${\mathcal T}^{A}A^*$ over $J_{A^*}^{-1}(\mu )$ (see proposition 3.9). In fact, $(J^T_{A^*})^{-1}(0,\mu )$ is just the prolongation

$$\begin{equation*} {\mathcal T}^A(J_{A^*}^{-1}(\mu )) \end{equation*}$$

of the Lie algebroid A over the restriction $(\tau _*)_{|(J_{A^*})^{-1}(\mu )}:J_{A^*}^{-1}(\mu )\rightarrow M$ of τ_*: A* → M to the submanifold $J_{A^*}^{-1}(\mu )$. Note that $J_{A^*}^{-1}(\mu )$ is an affine subbundle of A* over M and that $(\tau _*)_{|J_{A^*}^{-1}(\mu )}:J_{A^*}^{-1}(\mu )\rightarrow M$ is the projection.

Note that, under this assumption, the isotropy group G_μ is just G.

We will prove that the reduced symplectic-like Lie algebroid

$$\begin{equation*} ({\mathcal T}^A{A^*})_0={\mathcal T}^AJ_{A^*}^{-1}(0)\,/\,TG\rightarrow J_{A^*}^{-1}(0)/G \end{equation*}$$

is the canonical cover of a fiberwise linear Poisson structure on the dual A*₀ of a certain Lie algebroid A₀ over M/G.

Description of the Lie algebroid A₀. The Lie algebroid A₀ over M/G is the space of orbits A/TG of the affine action of TG on A (see theorem 3.6). As we know (see the proof of theorem 3.6), if $\widetilde{\pi }: A\rightarrow A_0=A/TG$ and π: M → M/G are the canonical projections and $([\! [\cdot ,\cdot ]\! ]_{A_0},\rho _{A_0})$ is the Lie algebroid structure on A₀, then

$$\begin{equation} [\! [X_0,Y_0]\! ]_{A_0}\circ \pi =\widetilde{\pi }([\! [X,Y]\! ]),\quad \rho _{A_0}(X_0)=T\pi (\rho (X)) \end{equation} \tag{ 5.1 }$$

for X₀, Y₀ ∈ Γ(A₀) and X, Y ∈ Γ(A) satisfying

$$\begin{equation*} X_0\circ \pi =\widetilde{\pi }\circ X,\quad Y_0\circ\pi=\widetilde{\pi }\circ Y. \end{equation*}$$

Note that with this structure, $\widetilde{\pi }:A\rightarrow A_0$ is an epimorphism of Lie algebroids.

Now, we will prove that the vector bundle A₀ = A/TG → M/G is isomorphic to

$$\begin{equation*} \big(J_{A^*}^{-1}(0)/G\big)^*\rightarrow M/G. \end{equation*}$$

In fact, one may easily test that the submanifold $J_{A^*}^{-1}(0)$ is just the annihilator $(\psi ({\mathfrak g}))^0$ of $\psi ({\mathfrak g})$. Therefore, the restriction $\tau _{*}^0=\tau _{*|J_{A^*}^{-1}(0)}:J_{A^*}^{-1}(0)\rightarrow M$ of τ_*: A* → M to this submanifold is a vector bundle over M. Moreover, a direct computation proves that this vector bundle is isomorphic to the dual vector bundle $(A/\psi ({\mathfrak g}))^*$ of $A/\psi ({\mathfrak g})\rightarrow M.$

Therefore, using the equivalences (3.13), we deduce that the three vector bundles

$$\begin{equation*} \fl A_0=A/TG\rightarrow M/G, \quad(A/\psi ({\mathfrak g}))/G\rightarrow M/G,\quad \big(J_{A^*}^{-1}(0)/G\big)^*\cong \big(J_{A^*}^{-1}(0)\big)^*/G\rightarrow M/G \end{equation*}$$

are isomorphic. Thus, we may induce isomorphic Lie algebroid structures on these vector bundles.

The description of the Lie algebroid isomorphism between $({\mathcal T}^{A}A^*)_0$ and ${\mathcal T}^{A_0}{A_0^*}$. In what follows, we identify A₀ = A/TG with $(J_{A^*}^{-1}(0)/G)^*\cong (J_{A^*}^{-1}(0))^*/G.$ Under this identification, we denote by $\varphi :A\rightarrow (J_{A^*}^{-1}(0)/G)^*$ the epimorphism of vector bundles corresponding to the quotient projection A → A/TG.

Let us consider the following epimorphism of vector bundles over π₀: J⁻¹_A*(0) → A*₀ = J_A*⁻¹(0)/G

$$\begin{equation} \varphi ^T:{\mathcal T}^{A}J_{A^*}^{-1}(0)\rightarrow {\mathcal T}^{A_0}A_0^* \quad (a_x,v_{\alpha _x})\mapsto (\varphi (a_x), T_{\alpha _x}\pi _0(v_{\alpha _x})). \end{equation} \tag{ 5.2 }$$

Note that, using (5.1) and the fact that $\widetilde{\tau }_*^0\circ \pi _0=\pi \circ \tau _*^0,$ we have that φ^T is well defined. Here $\widetilde{\tau }_*^0:J_{A^*}^{-1}(0)/G\rightarrow M/G$ is the dual vector bundle of $(J_{A^*}^{-1}(0)/G)^*\rightarrow M/G$.

Now, since φ: A → A₀ is a Lie algebroid epimorphism, φ^T is also a Lie algebroid epimorphism. Furthermore, it is easy to prove, using that φ is TG-invariant, that φ^T is TG-invariant with respect to the action (Φ, TΦ*)^T of TG restricted to ${\mathcal T}^AJ_{A^*}^{-1}(0).$ In fact,

$$\begin{equation*} \fl\begin{array}{r@{\,}c@{\,}l}\varphi ^T\big((\Phi , T\Phi ^*)^T_{(g,\xi )}(a_x,v_{\alpha _x})\big)&=& (\varphi (\Phi _g(a_x)) + \varphi (\Phi _g(\psi (\xi ))), T_{\alpha _x}\pi _0(T_x\Phi _g(v_{\alpha _x}+\xi _{A^*})))\\\ms &=& (\varphi (a_x), T_{\alpha _x}\pi _0(v_{\alpha _x}))=\varphi ^T(a_x,v_{\alpha _x}) \end{array} \end{equation*}$$

for all $(g,\xi )\in G\times {\mathfrak g}\cong TG$ and $(a_x,v_{\alpha _x})\in {\mathcal T}^A_{\alpha _x}J_{A^*}^{-1}(0).$ Note that

$$\begin{equation*} \varphi (\psi (\xi ))=\varphi (\Phi _{(e,\xi )}(0))=0. \end{equation*}%$$

Thus, we have the following Lie algebroid epimorphism $\bar{\varphi }^T$ between $({\mathcal T}^AA^*)_0$ and ${\mathcal T}^{A_0}A_0^*$ over the identity of A*₀:

In fact,

$$\begin{equation} \bar{\varphi }^T[(a_x,v_{\alpha _x})]=(\varphi (a_x), (T_{\alpha _x}\pi _0)(v_{\alpha _x}))\quad\mbox{for } (a_x,v_{\alpha _x})\in {\mathcal T}^A_{\alpha _{x}} J_{A^*}^{-1}(0). \end{equation} \tag{ 5.4 }$$

Finally, we will prove that $\bar{\varphi }^T$ is an isomorphism, that is, $\bar{\varphi }^T$ is injective.

If $(e_x,v_{\alpha _x})\in \ker \varphi ^T$, then $e_x\in \ker \varphi =\psi _x({\mathfrak g})$ and $v_{\alpha _x}$ is a vertical vector with respect to $\pi _0:(J_{A^*})^{-1}(0)\rightarrow (J_{A^*})^{-1}(0)/G.$ Then, there are $\xi ,\xi ^{\prime }\in {\mathfrak g}$ such that

$$\begin{equation*} e_x=\psi _x(\xi )\quad\mbox{and } v_{\alpha _x}=\xi ^{\prime }_{A^*}(\alpha _x). \end{equation*}$$

Then,

$$\begin{equation*} \xi _M(x)=\rho (e_x)=T_{\alpha _x}\tau ^0_*(v_{\alpha _x})=\xi ^{\prime }_M(x) \end{equation*}$$

and, since ϕ: G × M → M is a free action, we conclude that ξ = ξ'. Therefore,

$$\begin{equation*} (e_x,v_{\alpha _x})=(\psi _x(\xi ), \xi _{A^*}(\alpha _x))=(\Phi ,T\Phi ^*)^T((e,\xi ), (0,0)) \end{equation*}$$

where e is the identity element of G. Thus, $\bar{\varphi }^T$ is injective.

The Lie algebroid isomorphism between $({\mathcal T}^{A}{A^*})_0$ and ${\mathcal T}^{A_0}{A_0^*}$ is canonical. We will see that $\bar{\varphi }^T$ is canonical, i.e.

$$\begin{equation} (\bar{\varphi }^T)^*\Omega _{A_0}=\Omega _0, \end{equation} \tag{ 5.5 }$$

where Ω₀ is the reduced symplectic-like structure on $({\mathcal T}^{A}{A^*})_0$ given in theorem 3.11 and $\Omega _{A_0}$ is the canonical symplectic-like structure on ${\mathcal T}^{A_0}A_0^*$.

From (4.3), (5.2) and the definition of the morphism φ: A → A₀, we obtain that

$$\begin{equation*} (\varphi ^T)^*\lambda _{A_0}=\widetilde{\iota }_0^*\lambda _A, \end{equation*}$$

where $\lambda _{A_0}$ (respectively, λ_A) is the Liouville section of A₀ → M/G (respectively, of A → M) and $\widetilde{\iota }_0:{\mathcal T}^AJ_{A^*}^{-1}(0)\rightarrow {\mathcal T}^AA^*$ is the inclusion.

Thus, since φ^T and $\widetilde{\iota }_0$ are Lie algebroid morphisms, we obtain that

$$\begin{equation*} (\varphi ^T)^*\Omega _{A_0}=\widetilde{\iota }_0^*\Omega _A. \end{equation*}$$

On the other hand, if $\widetilde{\pi }_0: {\mathcal T}^AJ_{A^*}^{-1}(0)\rightarrow ({\mathcal T}^AA^*)_0$ is the canonical projection, it is clear that $\bar{\varphi }^T\circ \widetilde{\pi }_0=\varphi ^T$ which implies that

$$\begin{equation*} \widetilde{\pi }_0^*((\bar{\varphi }^T)^*\Omega _{A_0})=\widetilde{\pi }_0^*\Omega _0 \end{equation*}$$

and, therefore,

$$\begin{equation} (\bar{\varphi }^T)^*\Omega _{A_0}=\Omega _0. \end{equation} \tag{ 5.6 }$$

In the following theorem, we summarize the results obtained in the case μ = 0.

Theorem 5.1. Let (A, [[ ·, ·]], ρ) be a Lie algebroid on the manifold M and Φ: G × A → A a free and proper action of a connected Lie group by complete lifts. Then, the reduced symplectic-like Lie algebroid

$$\begin{equation*} ({\mathcal T}^AA^*)_0=\big({\mathcal T}^AJ_{A^*}^{-1}(0)\big)\big/TG\rightarrow J_{A^*}^{-1}(0)/G \end{equation*}$$

is canonically isomorphic to the Lie algebroid ${\mathcal T}^{A_0}A_0^*$, equipped with the standard symplectic-like structure, where the Lie algebroid A₀ is the vector bundle

$$\begin{equation*} A_0=A/TG\rightarrow M/G \end{equation*}$$

endowed with the quotient Lie algebroid structure characterized by (5.1).

Suppose that the assumptions of theorem 5.1 hold. Additionally, we consider a principal G-connection ${\mathcal A}: TM\rightarrow {\mathfrak g}$ for the corresponding principal bundle π: M → M/G. In such a case, we have a vector bundle morphism ${\mathcal A}^A:A\rightarrow {\mathfrak g}$ given by

$$\begin{equation*} {\mathcal A}^A(a_x)={\mathcal A}(\rho _x(a_x)),\quad\forall a_x\in A_x, \end{equation*}$$

which satisfies the following properties:

• (i)  
  ${\mathcal A}^A$ is equivariant with respect to Φ: G × A → A and the adjoint action, that is,

  $$\begin{equation*} {\mathcal A}^A(\Phi _g(a_x))=Ad_g^G({\mathcal A}^A(a_x)),\quad \forall a_x\in A_x, \end{equation*}$$
• (ii)  
  ${\mathcal A}^A(\psi (\xi )(x))=\xi ,$ for all $\xi \in {\mathfrak g}$ and x ∈ M.

Note that if π: M → M/G is the quotient projection, then we have

$$\begin{equation*} TQ=V\pi \oplus H\mbox{ and } A=\psi ({\mathfrak g})\oplus H^A, \end{equation*}$$

where Vπ is the vertical bundle of π and H^A (respectively, H) is the vector bundle on M whose fiber at x ∈ M is the vector space

$$\begin{equation*} H^A_x=\lbrace a_x\in A_x/{\mathcal A}^A(a_x)=0\rbrace \ \mbox{ ({\rm respectively}, $H_x=\lbrace v_x\in T_xM/{\mathcal A}(v_x)=0\rbrace $}). \end{equation*}$$

Moreover, H and H^A are G-invariant vector bundles, that is,

$$\begin{equation*} H^A_{\phi _g(x)}=\Phi _g\big(H_x^A\big)\quad\mbox{and } H_{\phi _g(x)}=T_x\phi _g(H_x), \quad \forall g\in G. \end{equation*}$$

Now, if $\mu \in {\mathfrak g}^*$, we consider the section α_μ of A* given by

$$\begin{equation*} \alpha _\mu (a_x)=\mu ({\mathcal A}^A(a_x)), \end{equation*}$$

with x ∈ M and a_x ∈ A_x. This section has the following properties:

• (i)  
  α_μ(M)⊆J⁻¹_A*(μ). In fact,

  $$\begin{equation*} J_{A^*}(\alpha _\mu (x))(\xi )=\alpha _\mu (\psi (\xi )(x))=\mu ({\mathcal A}^A(\psi (\xi )(x)))=\mu (\xi ) \end{equation*}$$

  for all x ∈ M and $\xi \in {\mathfrak g}.$
• (ii)  
  $\Phi _g^*\alpha _\mu =\alpha _{Coad^G_g\mu },$ for all g ∈ G, which is a consequence from the equivariance properties of ${\mathcal A}^A.$

Thus, since G_μ = G, we deduce that α_μ is G-invariant, i.e.

$$\begin{equation} \Phi _g^*(\alpha _\mu )=\alpha _\mu . \end{equation} \tag{ 5.7 }$$

So, in what follows, we assume that there is a G-invariant 1-section α_μ ∈ Γ(A*) of A* with values in $J_{A^*}^{-1}(\mu ).$ Using (5.7), proposition 2.2 and the fact that the flow of ψ(ξ) is $\lbrace \Phi _{\exp (t\xi )}\rbrace _{t\in {\mathbb R}}$, we obtain that

$$\begin{equation} {\mathcal L}^A_{\psi (\xi )}\alpha _\mu =0. \end{equation} \tag{ 5.8 }$$

On the other hand,

$$\begin{equation*} i_{\psi (\xi )}\alpha _\mu =\mu ({\mathcal A}^A(\psi (\xi ))=\mu (\xi ). \end{equation*}$$

Then,

$$\begin{equation} 0={\mathcal L}_{\psi (\xi )}^A\alpha _\mu =i_{\psi (\xi )}d^A\alpha _\mu . \end{equation} \tag{ 5.9 }$$

Denote β_μ = d^Aα_μ. From (5.7) and since Φ_g: A → A is a Lie algebroid morphism we deduce that the 2-section β_μ of A* is G-invariant. Moreover, it satisfies i_ψ(ξ)β_μ  =  0 which implies that

$$\begin{equation*} \big(\Phi ^T_{(g,\xi )}\big)^*\beta _\mu =\Phi _g^*\beta _\mu =\beta _\mu \end{equation*}$$

for all $(g,\xi )\in G\times {\mathfrak g}\cong TG$.

Therefore, there exists a unique B_μ ∈ Γ(∧²A*₀) with the following property:

$$\begin{equation} \widetilde{\pi }^*B_\mu =\beta _\mu =d^A\alpha _\mu , \end{equation} \tag{ 5.10 }$$

where $\widetilde{\pi }:A\rightarrow A_0$ is the corresponding projection. It is clear that $d^{A_0}B_\mu =0$.

The 2-section B_μ of A*₀ is said to be the magnetic term associated with α_μ.

Now, we will prove that there is a Lie algebroid isomorphism, $\Upsilon _{\alpha _\mu }:({\mathcal T}^AA^*)_\mu \rightarrow {\mathcal T}^{A_0}A_0^*$, between the reduced Lie algebroid $({\mathcal T}^AA^*)_\mu$ and ${\mathcal T}^{A_0}A_0^*$ such that the symplectic-like section $\Omega _{A_0}$ on ${\mathcal T}^{A_0}A_0^*$ and the reduced symplectic-like section Ω_μ on $({\mathcal T}^AA^*)_\mu$ are related by the following formula:

$$\begin{equation*} \Upsilon _{\alpha _\mu }^*(\Omega _{A_0} - pr_1^*B_\mu )=\Omega _\mu , \end{equation*}$$

where $pr_1:{\mathcal T}^{A_0}A_0^*\rightarrow A_0$ is the Lie algebroid morphism induced by the first projection.

The description of the Lie algebroid isomorphism $\Upsilon _{\alpha _\mu }:({\mathcal T}^AA^*)_\mu \rightarrow {\mathcal T}^{A_0}A_0^*.$ Firstly, we will describe a Lie algebroid morphism between the reduced spaces $({\mathcal T}^AA^*)_\mu$ and $({\mathcal T}^AA^*)_0$. Then, we may use theorem 5.1 in order to construct the isomorphism $\Upsilon _{\alpha _\mu }$.

Using the fact that α_μ(M)⊆J⁻¹_A*(μ), we deduce that $J_{A^*}^{-1}(\mu )\rightarrow M$ is an affine bundle on M such that

$$\begin{equation*} J_{A^*}^{-1}(\mu )\cap A_x^*=\big\lbrace \beta _x\in A^*_x/\beta _x-\alpha _\mu (x)\in J_{A^*}^{-1}(0)\big\rbrace \end{equation*}$$

for all x ∈ M.

Now, we consider the affine bundle isomorphism

where sh_μ(β_x) = β_x − α_μ(x) for all $\beta _x\in J^{-1}_{A^*}(\mu )\cap A_x^*.$

From the G-invariance of α_μ, we deduce that sh_μ is equivariant with respect to the action Φ*: G × A* → A*, i.e.

$$\begin{equation*} (sh_\mu \circ \Phi _g^*)({\beta _x})=(\Phi _g^*\circ {sh_\mu )(\beta _x}) \end{equation*}$$

for all β_x ∈ A*_x∩J⁻¹_A*(μ) and g ∈ G. Moreover, one may induce a morphism of vector bundles

where ${\mathcal T}^Ash_\mu (a_x,X_{\beta _x})=(a_x,T_{\beta _x}sh_\mu (X_{\beta _x})),$ with $(a_x,X_{\beta _x})\in {{\mathcal T}^AJ_{A^*}^{-1}(\mu )}.$ Note that, since

$$\begin{equation*} \tau _{*|J_{A^*}^{-1}(0)}\circ sh_\mu =\tau _{*|J_{A^*}^{-1}(\mu )}, \end{equation*}$$

${\mathcal T}^Ash_\mu$ is well defined. Furthermore, a direct proof shows that ${\mathcal T}^Ash_\mu$ is an isomorphism of vector bundles. In fact, one can easily see that ${\mathcal T}^Ash_\mu$ is a Lie algebroid isomorphism, taking into account that

$$\begin{equation*} \fl\begin{array}{@{}l@{}}\mathcal T^Ash_\mu ([\! [X,Y]\! ],[U,V])=([\! [X,Y]\! ],[Tsh_\mu \circ U\circ sh_\mu ^{-1},Tsh_\mu \circ {V}\circ sh_\mu ^{-1}])\circ sh_\mu ,\\ [8pt] \rho _{{\mathcal T}^AJ_{A^*}^{-1}(0)}(({\mathcal T}^Ash_\mu )(X,U))=(T sh_\mu \circ \rho _{{\mathcal T}^AJ_{A^*}^{-1}(\mu )})(X,U) \end{array} \end{equation*}$$

for all X, Y ∈ Γ(A) and $U,V\in {\mathfrak X}(J_{A^*}^{-1}(\mu ))$ which are $(\tau _*)_{|J_{A^*}^{-1}(\mu )}$-projectable on ρ(X) and ρ(Y), respectively.

Moreover, since sh_μ is equivariant, we deduce that ${\mathcal T}^Ash_\mu$ is equivariant with respect to the action (Φ, TΦ*)^T of G restricted to ${\mathcal T}^AJ_{A^*}^{-1}(\mu )$ and ${\mathcal T}^AJ_{A^*}^{-1}(0)$, respectively.

Thus, one induces a Lie algebroid isomorphism

Finally, the isomorphism $\Upsilon _{\alpha _\mu }:({\mathcal T}^AA^*)_\mu \rightarrow {\mathcal T}^{A_0}A_0^*$ is defined as follows:

$$\begin{equation*} \Upsilon _{\alpha _\mu }=\bar{\varphi }^T\circ \widetilde{{\mathcal T}^Ash_\mu ,} \end{equation*}$$

where $\bar{\varphi }^T:({\mathcal T}^AA^*)_0\rightarrow {\mathcal T}^{A_0}A_0^*$ is the Lie algebroid isomorphism defined by (5.4).

Relation between the symplectic-like structures on $({\mathcal T}^AA^*)_\mu$ and ${\mathcal T}^{A_0}A_0^*$. Let λ_A be the Liouville section on ${\mathcal T}^AA^*$ and $\iota _0:{\mathcal T}^A(J_{A^*})^{-1}(0)\rightarrow {\mathcal T}^AA^*$ (respectively, $\iota _\mu :{\mathcal T}^A(J_{A^*})^{-1}(\mu )\rightarrow {\mathcal T}^AA^*$) be the corresponding inclusion. Then,

$$\begin{equation*} (({\mathcal T}^Ash_\mu )^*(\iota _0^*\lambda _A))(a_x,X_{\beta _x}) =(\iota _\mu ^*\lambda _A)(a_x,X_{\beta _x})-\alpha _\mu (a_x) \end{equation*}$$

for all β_x ∈ J⁻¹_A*(μ) and $(a_x,X_{\beta _x})\in {\mathcal T}^A_{\beta _x}J_{A^*}^{-1}(\mu ).$

On the other hand, if $pr^0_1:{\mathcal T}^AJ_{A^*}^{-1}(0)\rightarrow A$ is the Lie algebroid morphism induced by the first projection, we have that

$$\begin{equation*} \big(\big(pr_1^0\circ {\mathcal T}^Ash_\mu \big)^*\alpha _\mu \big)(a_x,X_{\beta _x})=\alpha _\mu (a_x). \end{equation*}$$

This implies that

$$\begin{equation*} ({\mathcal T}^Ash_\mu )^*\big(\iota _0^*\lambda _A + \big(pr_1^0\big)^*\alpha _\mu \big)=\iota _\mu ^*\lambda _A \end{equation*}$$

and thus, from theorem 3.11, we deduce that

$$\begin{equation} ({\mathcal T}^Ash_\mu )^*(\widetilde{\pi }_0^*\Omega _0-\big(pr_1^0\big)^*\beta _\mu )=\widetilde{\pi }_\mu ^*\Omega _\mu , \end{equation} \tag{ 5.11 }$$

where Ω₀ (respectively, Ω_μ) is the symplectic-like structure on $({\mathcal T}^AJ_{A^*}^{-1}(0))/TG$ (respectively, $({\mathcal T}^AJ_{A^*}^{-1}(\mu ))/TG$) and $\widetilde{\pi }_0:{\mathcal T}^AJ_{A^*}^{-1}(0)\rightarrow ({\mathcal T}^AJ_{A^*}^{-1}(0))/TG$ (respectively, $\widetilde{\pi }_\mu :{\mathcal T}^AJ_{A^*}^{-1}(\mu )\rightarrow ({\mathcal T}^AJ_{A^*}^{-1}(\mu ))/TG$) is the canonical projection.

Now, using the relations

$$\begin{equation*} \widetilde{\pi }_0\circ {\mathcal T}^Ash_\mu =\widetilde{{\mathcal T}^Ash_\mu }\circ \widetilde{\pi }_\mu \quad\mbox{and }\widetilde{\pi }\circ pr_1^0\circ {\mathcal T}^Ash_\mu =pr_1\circ \Upsilon _{\alpha _\mu }\circ \widetilde{\pi }_\mu , \end{equation*}$$

and the facts

$$\begin{equation*} (\bar{\varphi }^T)^*\Omega _{A_0}=\Omega _0\quad\mbox{and } \widetilde{\pi }^*B_\mu =\beta _\mu , \end{equation*}$$

we conclude that (5.11) is equivalent to

$$\begin{equation*} \widetilde{\pi }_\mu ^*(\Upsilon _{\alpha _\mu }^*\Omega _{A_0}-\Upsilon ^*_{\alpha _\mu }(pr_1^*(B_\mu )))=\widetilde{\pi }_\mu ^*\Omega _\mu . \end{equation*}$$

Therefore,

$$\begin{equation*} \Upsilon _{\alpha _\mu }^*(\Omega _{A_0}-pr_1^*(B_\mu ))=\Omega _\mu . \end{equation*}$$

The results obtained in this case may be summarized in the following theorem.

Theorem 5.2. Let (A, [[ ·, ·]], ρ) be a Lie algebroid on the manifold M and Φ: G × A → A a free and proper action of a connected Lie group G by complete lifts. Suppose that we consider $\mu \in {\mathfrak g}^*$ such that G = G_μ. Then, choosing any G-invariant section α_μ of A* with values in $J_{A^*}^{-1}(\mu )$, there is a canonical Lie algebroid isomorphism

$$\begin{equation*} \Upsilon _{\alpha _\mu }:(({\mathcal T}^AA^*)_\mu ,\Omega _\mu )\rightarrow ({\mathcal T}^{A_0}A_0^*, \Omega _{A_0} - (pr_1)^*B_\mu ) \end{equation*}$$

where A₀ is the vector bundle

$$\begin{equation*} A_0=A/TG\rightarrow M/G \end{equation*}$$

endowed with the Lie algebroid structure characterized by (5.1), $\Omega _{A_0}$ is the canonical symplectic-like structure on ${\mathcal T}^{A_0}A_0^*$, $pr_1:{\mathcal T}^{A_0}A_0^*\rightarrow A_0$ is the projection on the first factor and B_μ ∈ Γ(∧²A*₀) is the corresponding magnetic term associated with α_μ which is characterized by (5.10).

Let (A, [[ ·, ·]], ρ) be a Lie algebroid on the manifold M and Φ: G × A → A a free and proper action of a connected Lie group G on A by complete lifts with respect to the Lie algebra anti-morphism $\psi :{\mathfrak g}\rightarrow \Gamma (A).$

Let $\mu \in {\mathfrak g}^*$ and denote by ${\mathfrak g}_\mu$ the isotropy algebra of μ. Then, the induced action Φ: G_μ × A → A is a free and proper action by complete lifts with respect to the restriction $\psi : {\mathfrak g}_\mu \rightarrow \Gamma (A)$ of ψ to ${\mathfrak g}_\mu .$

Now, denote by $\bar{\mu }\in {\mathfrak g}_\mu ^*$ the restriction of μ to ${\mathfrak g}_\mu$ and by $J_{A^*}^\mu :A^*\rightarrow {\mathfrak g}_\mu ^*$ the map given by

$$\begin{equation*} J_{A^*}^\mu =i^*\circ J_{A^*}, \end{equation*}$$

where $i^*:{\mathfrak g}^*\rightarrow {\mathfrak g}_\mu ^*$ is the dual of the inclusion $i:{\mathfrak g}_\mu \rightarrow {\mathfrak g}.$ Then, $J_{A^*}^\mu$ is the momentum map associated with the action of G_μ on A.

A direct computation proves that the isotropy group of $\bar{\mu }\in {\mathfrak g}_\mu$ with respect to the coadjoint action of G_μ, $(G_\mu )_{\bar{\mu }},$ is just G_μ. Therefore, we are in the conditions of section 5.2 if we choose as the starting Lie group G_μ. Next, we choose a G_μ-invariant section α_μ ∈ Γ(A*) such that

$$\begin{equation*} \alpha _{\mu }(M)\subset (J_{A^*}^\mu )^{-1}(\bar{\mu }). \end{equation*}$$

This is always possible as we have shown in section 5.2. If A_{0, μ} is the vector bundle A/TG_μ → M/G_μ associated with the action Φ^T: TG_μ × A → A, we denote by B_μ ∈ Γ(∧²A*_{0, μ}) the corresponding magnetic term associated with α_μ. Then, from theorem 5.2, we conclude that the reduced symplectic-like Lie algebroid

$$\begin{equation*} ({\mathcal T}^AA^*)_{\bar{\mu }}=\big({\mathcal T}^A\big(J_{A^*}^{\mu }\big)^{-1}(\bar{\mu })\big)\big/TG_\mu \rightarrow J^{-1}_{A^*}(\mu )/G_\mu \end{equation*}$$

is isomorphic to the symplectic-like Lie algebroid $({\mathcal T}^{A_{0,\mu }}A_{0,\mu }^*, \Omega _{A_{0,\mu }}-pr_1^*(B_{{\mu }})),$ where $\Omega _{A_{0,\mu }}$ is the canonical symplectic-like structure on ${\mathcal T}^{A_{0,\mu }}A_{0,\mu }^*$ and $pr_1:{\mathcal T}^{A_{0,\mu }}A_{0,\mu }^*\rightarrow A_{0,\mu }$ is the projection on the first factor.

On the other hand, the inclusion $i_{\mu ,\bar{\mu }}:J_{A^*}^{-1}(\mu )\rightarrow (J^\mu _{A^*})^{-1}(\bar{\mu })$ is G_μ-invariant and induces a Lie algebroid TG_μ-invariant monomorphism $I:{\mathcal T}^AJ_{A^*}^{-1}(\mu )\rightarrow {\mathcal T}^A(J^\mu _{A^*})^{-1}(\bar{\mu })$ over $i_{\mu ,\bar{\mu }}$. Therefore, we have a Lie algebroid monomorphism $(\widetilde{I},\widetilde{i}_{\mu ,\bar{\mu }})$:

which is canonical with respect to Ω_μ and $\Omega _{\bar{\mu }}$ on the reduced spaces $({\mathcal T}^A(J_{A^*})^{-1}(\bar{\mu }))/TG_\mu$ and $({\mathcal T}^A(J^\mu _{A^*})^{-1}(\bar{\mu }))/TG_\mu ,$ respectively.

Denote by $\tilde{\iota }_{\bar{\mu }}:{\mathcal T}^A(J_{A^*}^\mu )^{-1}(\bar{\mu })\rightarrow {\mathcal T}^AA^*$ and by $\tilde{\iota }_\mu :{\mathcal T}^A(J_{A^*})^{-1}(\mu )\rightarrow {\mathcal T}^AA^*$ the corresponding inclusions which are related by

$$\begin{equation*} \tilde{\iota }_\mu =\tilde{\iota }_{\bar{\mu }}\circ {I}. \end{equation*}$$

Now, if

$$\begin{eqnarray*} &&\pi _{\bar{\mu }}: {\mathcal T}^A\big(J_{A^*}^\mu \big)^{-1}(\bar{\mu })\rightarrow \big({\mathcal T}^A\big(J_{A^*}^\mu \big)^{-1}(\bar{\mu })\big)/TG_\mu \\&&\ms \pi _\mu : {\mathcal T}^A(J_{A^*})^{-1}(\mu )\rightarrow ({\mathcal T}^A(J_{A^*})^{-1}(\mu ))/TG_\mu \end{eqnarray*}$$

are the corresponding projections, we have that

$$\begin{equation*} \pi _\mu ^*\Omega _\mu =\tilde{\iota }_\mu ^*\Omega _A ={I}^*(\tilde{\iota }_{\bar{\mu }}^*\Omega _A)={I}^*(\pi _{\bar{\mu }}^*\Omega _{\bar{\mu }}). \end{equation*}$$

Then, using that $\pi _{\bar{\mu }}\circ {I}=\widetilde{I}\circ \pi _\mu$ we conclude that

$$\begin{equation*} \widetilde{I}^*\Omega _{\bar{\mu }}=\Omega _\mu . \end{equation*}$$

Therefore, $\widetilde{I}$ is a canonical Lie algebroid monomorphism. Thus, we have proved the main result of this section.

Theorem 5.3. Let (A, [[ ·, ·]], ρ) be a Lie algebroid over the manifold M and Φ: G × A → A a free and proper action of a connected Lie group by complete lifts. If $\mu \in {\mathfrak g}^*$ and $\bar{\mu }$ is the restriction of μ to ${\mathfrak g}_\mu$, then, choosing a G_μ-invariant section α_μ of A* with values in $(J^\mu _{A^*})^{-1}(\bar{\mu })$, there exists a canonical embedding

$$\begin{equation*} ({\mathcal T}^AA^*)_\mu \rightarrow {\mathcal T}^{A_{0,\mu }}A_{0,\mu }^* \end{equation*}$$

from the reduced algebroid $({\mathcal T}^AA^*)_\mu$ equipped with the canonical reduced symplectic-like structure Ω_μ to the Lie algebroid ${\mathcal T}^{A_{0,\mu }}A_{0,\mu }^*$ endowed with the symplectic-like structure

$$\begin{equation*} \bar{\Omega }_{\mu }=\Omega _{A_{0,\mu }} - (pr_1)^*B_\mu . \end{equation*}$$

Moreover, this embedding is an isomorphism if and only if ${\mathfrak g}={\mathfrak g}_\mu .$

Here A_{0, μ} is the vector bundle

$$\begin{equation*} A_{0,\mu }=A/TG_\mu \rightarrow M/G_\mu , \end{equation*}$$

$\Omega _{A_{0,\mu }}$ is the canonical symplectic-like structure on ${\mathcal T}^{A_{0,\mu }}A_{0,\mu }^*$, $pr_1:{\mathcal T}^{A_{0,\mu }}A_{0,\mu }^*\rightarrow A_{0,\mu }$ is the projection on the first factor and B_μ ∈ Γ(∧²A*₀) is the corresponding magnetic term associated with α_μ which is characterized by (5.10).

Examples 5.4. 

• (i)  
  If we apply the previous theorem to the particular case when A is the standard Lie algebroid TM → M then we recover a classical result in cotangent bundle reduction theory (see [1, 12]).
• (ii)  
  For the case $A={\mathfrak g}\times TM$ from (ii) in examples 3.2, we have seen that the vector bundle ${\mathcal T}^AA^*\rightarrow A^*$ can be identified with ${\mathfrak g}\times T({\mathfrak g}^*\times T^*M)\rightarrow {\mathfrak g}^*\times T^*M$ (see example 4.1). Moreover, the Lie algebroid $({\mathfrak g}\times TM)/TG\rightarrow M/G$ is isomorphic to the Atiyah algebroid associated with the principal bundle π_M: M → M/G (see (3.15)).

Then, the reduced symplectic-like Lie algebroid $({\mathcal T}^AA^*)_0,$ for the value $\mu =0\in {\mathfrak g}^*$, is simplectically isomorphic to the canonical cover of the fiberwise linear Poisson structure of (T*M)/G induced by the Atiyah Lie algebroid (TM)/G → M/G, i.e. ${\mathcal T}^{(TM)/G}{((T^*M)/G)}\rightarrow (T^*M)/G.$ In fact, this last Lie algebroid is just the Atiyah algebroid associated with the principal bundle $\pi _{T^*M}: T^*M\rightarrow (T^*M)/G$ (see [13]) and its symplectic-like structure $\Omega _{(T^*M)/G}\in \Gamma (\wedge ^2(T^*(T^*M)/G))$ is the one induced by the G-invariant symplectic structure on T*M.

Now, we choose $\mu \in {\mathfrak g}^*$ such that G = G_μ and a G-invariant 1-form α_μ ∈ Ω¹(M) on M such that α_μ(M)⊂J⁻¹(μ), where $J:T^*M\rightarrow {\mathfrak g}^*$ is the momentum map given as in (1.1). Then, the reduced symplectic-like Lie algebroid $({\mathcal T}^AA^*)_\mu$ is simplectically isomorphic to the Atiyah algebroid associated with the principal bundle $\pi _{T^*M}: T^*M\rightarrow (T^*M)/G$ endowed with the symplectic-like structure

$$\begin{equation*} \Omega _{(T^*M)/G} - \gamma _\mu , \end{equation*}$$

where γ_μ ∈ Γ(∧²(T*M/G)) is the 2-section obtained from a magnetic term defined as follows. We consider the epimorphism

$$\begin{equation*} \widetilde{\pi }:{\mathfrak g}\times TM\rightarrow TM/G,\quad \tilde{\pi }(\xi ,v_x)=[v_x+\xi _M(x)]. \end{equation*}$$

Then, we have that there exists B_μ ∈ Γ(∧²((T*M)/G)) such that

$$\begin{equation*} \widetilde{\pi }^*(B_\mu )=(0,d\alpha _\mu ). \end{equation*}$$

Finally, γ_μ is just

$$\begin{equation*} (T\tau _{T^*Q}/G)^*B_\mu , \end{equation*}$$

where $T\tau _{T^*Q}/G:(T(T^*M))/G\rightarrow (TM)/G$ is the vector bundle induced by the equivariant tangent lift $T\tau _{T^*M}:T(T^*M)\rightarrow TM$ of $\tau _{T^*M}:T^*M\rightarrow M.$

We finish this paper with an application which is related with the reduction of non-autonomous Hamiltonian systems.

Example 5.5. Let $p:M\rightarrow {\mathbb {R}}$ be a fibration. We denote by τ_Vp: Vp → M the vertical bundle associated with p. Note that the sections of this vector bundle may be identified with the vector fields X on M such that η(X) = 0, where η is the exact 1-form p*(dt) on M, t being the standard coordinate on ${\mathbb {R}}.$

This vector bundle admits, in a natural way, a Lie algebroid structure where the Lie bracket is the standard Lie bracket of vector fields and the anchor map is the inclusion of vertical vectors with respect to p into TM.

Now, suppose that we additionally have a free and proper action ϕ: G × M → M of a Lie group G on M which is fibered, i.e.

$$\begin{equation*} p\circ \phi _g=p \quad \mbox{ for all } g\in G. \end{equation*}$$

Then,

• (i)  
  the infinitesimal generators of this last action are vertical vector fields;
• (ii)  
  the tangent lifted action Tϕ: G × TM → TM induces a free and proper action

  $$\begin{equation*} \Phi :G\times Vp\rightarrow Vp \end{equation*}$$

  of G on the vertical vector bundle Vp of p;
• (iii)  
  p induces a new fibration $\widetilde{p}:M/G\rightarrow {\mathbb {R}}$ on the quotient manifold M/G.

Then, Φ: G × Vp → Vp is an action by complete lifts with respect to the Lie algebra anti-morphism

$$\begin{equation*} \psi : {\mathfrak g}\rightarrow Vp, \quad \psi (\xi )=\xi _{M}. \end{equation*}$$

Let μ be an element of ${\mathfrak g}^*$ and we denote by $J_{V^*p}:V^*p\rightarrow {\mathfrak g}^*$ the momentum map defined as in (4.8). Then, we have that the vector bundle ${\mathcal T}^{Vp}\big(J_{V^*p}^{-1}(\mu )\big)\rightarrow J_{V^*p}^{-1}(\mu )$ may be identified in a natural way with the vertical bundle Vp_μ → J⁻¹_V*p(μ), where $p_\mu : {J_{V^*p}^{-1}(\mu )}\rightarrow \mathbb {R}$ is the fibration given by

$$\begin{equation*} p_\mu =p\circ \tau _{J_{V^*p}^{-1}(\mu )}, \end{equation*}$$

$\tau _{J_{V^*p}^{-1}(\mu )}: {J_{V^*p}^{-1}(\mu )}\rightarrow M$ being the corresponding projection. Under this identification the action (Φ, TΦ*): G_μ × Vp_μ → Vp_μ given by (4.4) is described as follows. Consider the tangent lift $T\Phi ^*:G_\mu \times T(J_{V^*p}^{-1}(\mu ))\rightarrow T(J_{V^*p}^{-1}(\mu ))$ of the restricted dual action $\Phi ^*:G_\mu \times J^{-1}_{V^*p}(\mu )\rightarrow J^{-1}_{V^*p}(\mu )$. Since

$$\begin{equation*} p_\mu \circ \Phi ^*=p_\mu , \end{equation*}$$

we may induce an action of G_μ on Vp_μ which is just (Φ, TΦ*). Therefore, the action (Φ, TΦ*)^T: TG_μ × Vp_μ → Vp_μ is given by

$$\begin{equation*} (\Phi ,T\Phi ^*)^T((g,\xi ), v_{\alpha _x})=T_{\alpha _x}\Phi ^*_g(v_{\alpha _x}+ \xi _M^{*c}(\alpha _x)) \end{equation*}$$

for all $(g,\xi )\in G_\mu \times {\mathfrak g}_\mu \cong TG_\mu \mbox{ and } \alpha _x\in V^*_xp.$ Here, $\xi _M^{*c}\in {\mathfrak X}(V^*p)$ is the complete lift to V*p of the infinitesimal generator of ξ with respect to the action ϕ.

Finally, from theorem 3.11, we conclude that the reduced vector bundle

$$\begin{equation*} (Vp_\mu )/TG_\mu \rightarrow J_{V^*p}^{-1}(\mu )/G_\mu \end{equation*}$$

is a symplectic-like Lie algebroid.

If μ = 0, then using theorem 5.1, we have that this symplectic-like Lie algebroid is isomorphic to ${\mathcal T}^{A_0}A_0^*$ where A₀ is the quotient vector bundle over M/G with total space Vp/TG. We remark that the action of $TG\cong G\times {\mathfrak g}$ on Vp is given by

$$\begin{equation*} \Phi ^T((g,\xi ), v_x)=T_x\phi _g(v_x+\xi _{M}(x)) \end{equation*}$$

for $(g,\xi )\in G\times {\mathfrak g}$ and v_x ∈ V_xp. In fact, the vector bundle A₀ is isomorphic to the vertical bundle $V\widetilde{p}$ with respect to the fibration $\widetilde{p}:M/G\rightarrow {\mathbb {R}}$. The isomorphism is just

$$\begin{equation*} Vp/TG\rightarrow V\widetilde{p},\quad [v_x]\mapsto T_x\pi (v_x) \end{equation*}$$

where π: M → M/G is the canonical projection. Therefore, ${\mathcal T}^{A_0}A_0^*$ may be identified with the vertical bundle $V\bar{p}\rightarrow V^*\widetilde{p}$ with

$$\begin{equation*} {\bar{p}}=\widetilde{p}\circ \tau _{V^*\widetilde{p}}: V^*\widetilde{p}\rightarrow \mathbb {R}, \end{equation*}$$

where $\tau _{V^*\widetilde{p}}: V^*\widetilde{p}\rightarrow M/G$ is the corresponding vector bundle projection. In conclusion, the reduced Lie algebroid (Vp₀)/TG → J⁻¹_V*p(0)/G is canonically isomorphic to the Lie algebroid $V\bar{p}$ on $V^*\widetilde{p}$ with its standard symplectic-like structure.

Now, we consider $\mu \in {\mathfrak g}^*$ such that G_μ = G. Let α_μ be a G-invariant 1-form on M such that

$$\begin{equation*} \alpha _\mu (\xi _M)=\mu (\xi ) \quad \mbox{ for all } \xi \in {\mathfrak g}. \end{equation*}$$

Then, the restriction α_μ|Vp of α_μ to the vertical bundle Vp of the fibration $p:M\rightarrow \mathbb {R}$ determines a G-invariant section of V*p with values in $J_{V^*p}^{-1}(\mu ).$

Let β_μ = d^Vp(α_μ|Vp). Equivalently, β_μ is the restriction of dα_μ ∈ Ω²(M) to Vp × Vp. The magnetic term B_μ associated with α_μ is the restriction to $V\widetilde{p}\times V\widetilde{p}$ of the unique 2-form $\bar{B}_\mu$ of M/G such that

$$\begin{equation*} {\pi }^*\bar{B}_\mu =d\alpha _\mu , \end{equation*}$$

where π: M → M/G is the quotient projection. Moreover, using theorem 5.2, we have that (Vp_μ)/TG is a symplectic-like Lie algebroid on $J_{V^*p}^{-1}(\mu )/G$ isomorphic to the Lie algebroid $V\bar{p}$ endowed with the symplectic-like section

$$\begin{equation*} \Omega _{V\widetilde{p}}-pr_1^*B_\mu \in \Gamma (\wedge ^2 V^*\bar{p}) \end{equation*}$$

with $pr_1:V\bar{p}\rightarrow V\widetilde{p}$ the vector bundle morphism on $\tau _{V^*\widetilde{p}}: V^*\widetilde{p}\rightarrow M/G$ given by the restriction to $V\bar{p}$ of the tangent lift $T\tau _{V^*\widetilde{p}}:T(V^*\widetilde{p})\rightarrow T(M/G)$.