Prolongations of Convenient Lie Algebroids

Abstract

We first define the concept of Lie algebroid in the convenient setting. In reference to the finite dimensional context, we adapt the notion of prolongation of a Lie algebroid over a fibred manifold to a convenient Lie algebroid over a fibred manifold. Then we show that this construction is stable under projective and direct limits under adequate assumptions.

Appendices

Appendix A: Projective Limits

1.1 Projective Limits of Topological Spaces

Definition 57

A projective sequence of topological spacesprojective sequence!of topological spaces is a sequence

$\left( \left( X_{i},\delta _{i}^{j}\right) \right) _{(i,j) \in {\mathbb {N}}^2,\; j \ge i}$ where

(PSTS 1):

For all $i\in {\mathbb {N}},$ $X_{i}$ is a topological space;

(PSTS 2):

For all $\left( i,j \right) \in {\mathbb {N}}^2$ such that $j\ge i$, $\delta _{i}^{j}:X_{j}\rightarrow X_{i}$ is a continuous map;

(PSTS 3):

For all $i\in {\mathbb {N}}$, $\delta _{i}^{i}={Id}_{X_{i}}$;

(PSTS 4):

For all $\left( i,j,k \right) \in {\mathbb {N}}^3$ such that $k \ge j \ge i$, $\delta _{i}^{j}\circ \delta _{j}^{k}=\delta _{i}^{k}$.

Notation 58

For the sake of simplicity, the projective sequence $\left( \left( X_{i},\delta _{i}^{j}\right) \right) _{(i,j) \in {\mathbb {N}}^2,\; j \ge i}$ will be denoted $\left( X_{i},\delta _{i}^{j} \right) _{j\ge i}$.

An element $\left( x_{i}\right) _{i\in {\mathbb {N}}}$ of the product ${\displaystyle \prod \limits _{i\in {\mathbb {N}}}}X_{i}$ is called a threadthread if, for all $j\ge i$, $\delta _{i}^{j}\left( x_{j}\right) =x_{i}$.

Definition 59

The set $X=\underleftarrow{\lim }X_{i}$ $X=\underleftarrow{\lim }X_{i}$ of all threads, endowed with the finest topology for which all the projections $\delta _{i}:X\rightarrow X_{i}$ are continuous, is called the projective limit of the sequenceprojective limit!of a sequence $\left( X_{i},\delta _{i}^{j} \right) _{j\ge i}$.

A basisbasis!of a topology of the topology of X is constituted by the subsets $\left( \delta _{i} \right) ^{-1}\left( U_{i}\right)$ where $U_{i}$ is an open subset of $X_{i}$ (and so $\delta _i$ is open whenever $\delta _i$ is surjective).

Definition 60

Let $\left( X_{i},\delta _{i}^{j} \right) _{j\ge i}$ and $\left( Y_{i},\gamma _{i}^{j} \right) _{j\ge i}$ be two projective sequences whose respective projective limits are X and Y.

A sequence $\left( f_{i}\right) _{i\in {\mathbb {N}}}$ of continuous mappings $f_{i}:X_{i}\rightarrow Y_{i}$, satisfying, for all $(i,j) \in {\mathbb {N}}^2,$ $j \ge i,$ the coherence conditioncoherence condition

$$\begin{aligned} \gamma _{i}^{j}\circ f_{j}=f_{i}\circ \delta _{i}^{j} \end{aligned}$$

is called a projective sequence of mappingsprojective sequence!of mappings.

The projective limit of this sequence is the mapping

$$\begin{aligned} \begin{array}{cccc} f: &{} X &{} \rightarrow &{} Y\\ &{} \left( x_{i}\right) _{i\in {\mathbb {N}}} &{} \mapsto &{} \left( f_{i}\left( x_{i}\right) \right) _{i\in {\mathbb {N}}} \end{array} \end{aligned}$$

The mapping f is continuous if all the $f_{i}$ are continuous (cf. [1]).

1.2 Projective Limits of Banach Spaces

Consider a projective sequence $\left( {\mathbb {E}}_{i},\delta _{i}^{j} \right) _{j\ge i}$ of Banach spaces.

Remark 61

Since we have a countable sequence of Banach spaces, according to the properties of bonding maps, the sequence $\left( \delta _i^j\right) _{(i,j)\in {\mathbb {N}}^2, \;j\ge i}$ is well defined by the sequence of bonding maps $\left( \delta _i^{i+1}\right) _{i\in {\mathbb {N}}}$.

1.3 Projective Limits of Differential Maps

The following proposition (cf. [9, Lemma 1.2] and [4, Chap. 4]) is essential

Proposition 62

Let $\left( {\mathbb {E}}_i,\delta _i^j \right) _{j\ge i}$ be a projective sequence of Banach spaces whose projective limit is the Fréchet space ${\mathbb {F}}=\underleftarrow{lim} {\mathbb {E}}_i$ and $\left( f_i : {\mathbb {E}}_i \rightarrow {\mathbb {E}}_i \right) _{i \in {\mathbb {N}}}$ a projective sequence of differential maps whose projective limit is $f=\underleftarrow{\lim } f_i$. Then the following conditions hold:

• (1) f is smooth in the convenient sense (cf. [13])

• (2) For all $x = \left( x_i \right) _{i \in {\mathbb {N}}}$, $df_x = \underleftarrow{\lim } { \left( df_i \right) }_{x_i}$.

• (3) $df = \underleftarrow{\lim }df_i$.

1.4 Projective Limits of Banach Manifolds

Definition 63

The projective sequence $\left( M_{i},\delta _{i}^{j} \right) _{j\ge i}$ is called projective sequence of Banach manifoldsprojective sequence!of Banach manifolds if

(PSBM 1):

$M_{i}$ is a manifold modelled on the Banach space ${\mathbb {M}}_{i}$;

(PSBM 2):

$\left( {\mathbb {M}}_{i},\overline{\delta _{i}^{j}}\right) _{j\ge i}$ is a projective sequence of Banach spaces;

(PSBM 3):

For all $x=\left( x_{i}\right) \in M=\underleftarrow{\lim }M_{i}$, there exists a projective sequence of local charts $\left( U_{i},\xi _{i}\right) _{i\in {\mathbb {N}}}$ such that $x_{i}\in U_{i}$ where one has the relation

$$\begin{aligned} \xi _{i}\circ \delta _{i}^{j}=\overline{\delta _{i}^{j}}\circ \varphi _{j}; \end{aligned}$$

(PSBM 4):

$U=\underleftarrow{\lim }U_{i}$ is a non empty open set in M.

Under the assumptions (PSBM 1) and (PSBM 2) in Definition 63, the assumptions (PSBM 3)] and (PSBM 4) around $x\in M$ is called the projective limit chart property around $x\in M$ and $(U=\underleftarrow{\lim }U_{i}, \phi =\underleftarrow{\lim }\phi _{i})$ is called a projective limit chart.

The projective limit $M=\underleftarrow{\lim }M_{i}$ has a structure of Fréchet manifold modelled on the Fréchet space ${\mathbb {M}} =\underleftarrow{\lim }{\mathbb {M}}_{i}$ and is called a $\mathsf {PLB}$-manifold $\mathsf {PLB}$-manifold. The differentiable structure is defined via the charts $\left( U,\varphi \right)$ where $\varphi =\underleftarrow{\lim }\xi _{i}:U\rightarrow \left( \xi _{i}\left( U_{i}\right) \right) _{i \in {\mathbb {N}}}.$

$\varphi$ is a homeomorphism (projective limit of homeomorphisms) and the charts changings $\left( \psi \circ \varphi ^{-1}\right) _{|\varphi \left( U\right) }=\underleftarrow{\lim }\left( \left( \psi _{i}\circ \left( \xi _{i}\right) ^{-1}\right) _{|\xi _{i}\left( U_{i}\right) }\right)$ between open sets of Fréchet spaces are smooth in the sense of convenient spaces.

1.5 Projective Limits of Banach Vector Bundles

Let $\left( M_{i},\delta _{i}^{j}\right) _{j\ge i}$ be a projective sequence of Banach manifolds where each manifold $M_{i}$ is modelled on the Banach space ${\mathbb {M}}_{i}$.

For any integer i, let $\left( E_{i},\pi _{i},M_{i} \right)$ be the Banach vector bundle whose type fibre is the Banach vector space ${\mathbb {E}}_{i}$ where $\left( {\mathbb {E}}_{i},\lambda _{i}^{j}\right) _{j\ge i}$ is a projective sequence of Banach spaces.

Definition 64

$\left( (E_i,\pi _i,M_i),\left( \xi _i^j,\delta _i^j \right) \right) _{j \ge i}$, where $\xi _i^j:E_j \rightarrow E_i$ is a morphism of vector bundles, is called a projective sequence of Banach vector bundlesprojective sequence!of Banach vector bundles on the projective sequence of manifolds $\left( M_{i},\delta _{i}^{j}\right) _{j\ge i}$ if, for all $\left( x_{i} \right)$, there exists a projective sequence of trivializations $\left( U_{i},\tau _{i}\right)$ of $\left( E_{i},\pi _{i},M_{i}\right)$, where $\tau _{i}:\left( \pi _{i}\right) ^{-1}\left( U_{i}\right) \rightarrow U_{i}\times {\mathbb {E}}_{i}$ are local diffeomorphisms, such that $x_{i}\in U_{i}$ (open in $M_{i}$) and where $U=\underleftarrow{\lim }U_{i}$ is a non empty open set in M where, for all $(i,j) \in {\mathbb {N}}^2$ such that $j\ge i,$ we have the compatibility condition

• (PLBVB) $\left( \delta _{i}^{j}\times \lambda _{i}^{j}\right) \circ \tau _{j}=\tau _{i}\circ \xi _i^j$.

With the previous notations, $(U=\underleftarrow{\lim }U_{i}, \tau =\underleftarrow{\lim }\tau _i)$ is called a projective bundle chart limitprojective bundle chart limit. The triple of projective limit $(E=\underleftarrow{\lim }E_{i}, \pi =\underleftarrow{\lim }\pi _{i}, M=\underleftarrow{\lim }M_{i}))$ is called a projective limit of Banach bundles or $\mathsf {PLB}$-bundle$\mathsf {PLB}$-bundle for short.

The following proposition generalizes the result of [9] about the projective limit of tangent bundles to Banach manifolds (cf. [4, 8]).

Proposition 65

Let $\left( (E_i,\pi _i,M_i),\left( \xi _i^j,\delta _i^j \right) \right) _{j \ge i}$ be a projective sequence of Banach vector bundles.

Then $\left( \underleftarrow{\lim }E_i,\underleftarrow{\lim }\pi _i,\underleftarrow{\lim }M_i \right)$ is a Fréchet vector bundle.

Definition 66

$\left( \left( E_{i},\pi _{i},M_{i},\rho _{i}, [.,.]_{i} \right) ,\left( \xi _i^j, \delta _{i}^{j} \right) \right) _{(i,j)\in {\mathbb {N}}^2, j \ge i}$ is called a projective sequence of Lie algebroids projective sequence!of Lie algebroids if

(PSBLA 1):

$\left( E_{i},\xi _i^j\right) _{j\ge i}$ is a projective sequence of Banach vector bundles ($\pi _{i}:E_{i}\rightarrow M_{i})_{i\in {\mathbb {N}}}$ over the projective sequence of manifolds $\left( M_{i},\delta _{i}^{j}\right) _{j\ge i}$;

(PSBLA 2):

For all $\left( i,j \right) \in {\mathbb {N}}^2$ such that $j\ge i$, one has

$$\begin{aligned} \rho _{i}\circ \xi _i^j=T\delta _{i}^{j}\circ \rho _{j} \end{aligned}$$

(PSBLA 3):

$\xi _i^j:E_{j}\rightarrow E_{i}$ is a Lie morphism from $\left( E_{j},\pi _{j},M_{j},\rho _{j} \right)$ to $\left( E_{i},\pi _{i},M_{i},\rho _{i}\right)$

We then have the following result (cf. [4]):

Theorem 67

Let $\left( \left( E_{i},\pi _{i},M_{i},\rho _{i}, [.,.]_{i} \right) ,\left( \xi _i^j, \delta _{i}^{j} \right) \right) _{(i,j)\in {\mathbb {N}}^2, j \ge i}$ be a projective sequence of Banach–Lie algebroids. If $(M_i,\delta _i^j)_{(i,j)\in {\mathbb {N}}^2, j \ge i}$ is a submersive projective sequence, then $\left( \underleftarrow{\lim }E_{i},\underleftarrow{\lim }\pi _{i},\underleftarrow{\lim }M_{i},\underleftarrow{\lim }\rho _{i}\right)$ is a strong partial Fréchet Lie algebroid.

Appendix B: Direct Limits

1.1 Direct Limits of Topological Spaces

Let $\left\{ \left( Y_{i},\varepsilon _{i}^{j}\right) \right\} _{(i,j)\in I^2,\ i \preccurlyeq j}$ be a direct system of topological spaces and continuous maps. The direct limit $\left\{ \left( X,\varepsilon _{i}\right) \right\} _{i\in I}$ of the sets becomes the direct limit in the category $\mathbf{TOP}$ of topological spaces if X is endowed with the direct limit topology (DL-topology for short)DL-topologytopology!DL-, i.e. the final topology with respect to the inclusion maps $\varepsilon _{i}:X_{i}\rightarrow X$ which corresponds to the finest topology which makes the maps $\varepsilon _{i}$ continuous. So $O\subset X$ is open if and only if $\varepsilon _{i}^{-1}\left( O\right)$ is open in $X_{i}$ for each $i\in I$.

Definition 68

ascending sequence!of topological spaces Let ${\mathcal {S}}=\left( \left( X_{n},\varepsilon _{n}^{m}\right) \right) _{(m,n) \in {\mathbb {N}}^2,\ n\le m}$ be a direct sequence of topological spaces such that each $\varepsilon _{n}^{m}$ is injective. Without loss of generality, we may assume that we have

$$\begin{aligned} X_{1}\subset X_{2}\subset \cdots \subset X_{n}\subset X_{n+1}\subset \cdots \end{aligned}$$

and $\varepsilon _{n}^{n+1}$ becomes the natural inclusion.

(ASTS):

${\mathcal {S}}$ will be called an ascending sequence of topological spaces.

(SASTS):

Moreover, if each $\varepsilon _{n}^{m}$ is a topological embedding, then we will say that ${\mathcal {S}}$ is a strict ascending sequence of topological spaces (expanding sequence

Notation 69

The direct sequence $\left( \left( X_{n},\varepsilon _{n}^{m}\right) \right) _{(n,m) \in {\mathbb {N}}^2,\; n \le m}$ will be denoted $\left( X_{n},\varepsilon _{n}^{m} \right) _{n \le m}$.

If $\left( X_{n},\varepsilon _{n}^{m}\right) _{n\le m}$ is a strict ascending sequence of topological spacesascending sequence!of topological spaces, each $\varepsilon _{n}$ is a topological embedding from $X_{n}$ into $X=\underrightarrow{\lim }X_{n}$.

1.2 Direct Limits of Ascending Sequences of Banach Manifolds

direct limit!of Banach manifolds

Definition 70

${\mathcal {M}}=(M_{n},\varepsilon _{n}^{n+1}) _{n\in {\mathbb {N}}}$ is called an ascending sequence of Banach manifolds if, for any $n\in {\mathbb {N}}$, $\left( M_{n},\varepsilon _{n}^{n+1}\right)$ is a closed submanifold of $M_{n+1}$.

Proposition 71

(cf. [4, 6]) Let ${\mathcal {M}}=\left( M_{n},\varepsilon _{n}^{n+1} \right) _{n\in {\mathbb {N}}}$ be an ascending sequence of Banach manifolds.

Assume that for $x\in M=\underrightarrow{\lim }M_{n}$, there exists a sequence of charts $\left( (U_{n},\phi _{n})\right) _{n \in {\mathbb {N}}}$ of $\left( M_{n}\right) _{n \in {\mathbb {N}}}$, such that:

(ASC 1):

$(U_{n})_{n\in {\mathbb {N}}}$ is an ascending sequence of chart domains;

(ASC 2):

$\forall n\in {\mathbb {N}},\ \phi _{n+1}\circ \varepsilon _{n}^{n+1}=\iota _{n}^{n+1}\circ \phi _{n}$.

Then $U=\underrightarrow{\lim }U_{n}$ is an open set of M endowed with the $\mathrm {DL}$-topology and $\phi =\underrightarrow{\lim }\phi _{n}$ is a well defined map from U to ${\mathbb {M}}=\underrightarrow{\lim }{\mathbb {M}}_{n}$.

Moreover, $\phi$ is a continuous homeomorphism from U onto the open set $\phi (U)$ of ${\mathbb {M}}$.

Definition 72

We say that an ascending sequence ${\mathcal {M}}= (M_{n},\varepsilon _{n}^{n+1}) _{n\in {\mathbb {N}}}$ of Banach manifolds has the direct limit chart property (DLCP)direct!limit chart at x if it satisfies both (ASC 1) and (ASC 2).

We then have the fundamental result (cf. [6]).

Theorem 73

Let $\left( M_{n} \right) _{n\in {\mathbb {N}}}$ be an ascending sequence of Banach $C^\infty$-manifolds, modelled on the Banach spaces ${\mathbb {M}}_{n}$. Assume that

(ASBM 1):

$\left( M_{n} \right) _{n\in {\mathbb {N}}}$ has the direct limit chart property (DLCP) at each point $x\in M=\underrightarrow{\lim }M_{n}$

(ASBM 2):

${\mathbb {M}}=\underrightarrow{\lim }{\mathbb {M}}_{n}$ is a convenient space.

Then there is a unique n.n.H. convenient manifold structure on $M=\underrightarrow{\lim }M_{n}$ modelled on the convenient space ${\mathbb {M}}$ such that the topology associated to this structure is the DL-topology on M.

In particular, for each $n\in {\mathbb {N}}$, the canonical injection $\varepsilon _{n}:M_{n}\longrightarrow M$ is an injective conveniently smooth map and $(M_{n},\varepsilon _{n})$ is a closed submanifold of M.

Moreover, if each $M_n$ is locally compact or is open in $M_{n+1}$ or is a paracompact Banach manifold closed in $M_{n+1}$, then $M=\underrightarrow{\lim }M_{n}$ is provided with a Hausdorff convenient manifold structure.

1.3 Direct Limits of Banach Vector Bundles

direct limit!of Banach vector bundles

Definition 74

$\left( (E_n,\pi _n,M_n),\left( \lambda _n^{n+1},\varepsilon _n^{n+1} \right) \right) _{n \in {\mathbb {N}}}$ is called a strong ascending sequence of Banach vector bundles if the following assumptions are satisfied:

(ASBVB 1):

${\mathcal {M}}=(M_{n})_{n\in {\mathbb {N}}}$ is an ascending sequence of Banach $C^\infty$-manifolds, where $M_{n}$ is modelled on the Banach space ${\mathbb {M}}_{n}$ such that ${\mathbb {M}}_{n}$ is a supplemented Banach subspace of ${\mathbb {M}}_{n+1}$ and the inclusion $\varepsilon _{n}^{n+1}:M_{n}\rightarrow M_{n+1}$ is a $C^\infty$ injective map such that $(M_{n},\varepsilon _{n}^{n+1})$ is a closed submanifold of $M_{n+1}$;

(ASBVB 2):

The sequence $(E_{n})_{n\in {\mathbb {N}}}$ is an ascending sequence such that the sequence of typical fibres $\left( {\mathbb {E}}_{n}\right) _{n\in {\mathbb {N}}}$ of $(E_{n})_{n\in {\mathbb {N}}}$ is an ascending sequence of Banach spaces and ${\mathbb {E}}_{n}$ is a supplemented Banach subspace of ${\mathbb {E}}_{n+1}$;

(ASBVB 3):

For each $n\in {\mathbb {N}}$, $\pi _{n+1}\circ \lambda _{n}^{n+1}=\varepsilon _{n}^{n+1}\circ \pi _{n}$ where $\lambda _{n}^{n+1}:E_{n}\rightarrow E_{n+1}$ is the natural inclusion;

(ASBVB 4):

Any $x\in M=\underrightarrow{\lim }M_{n}$ has the direct limit chart property (DLCP) for $(U=\underrightarrow{\lim }U_{n},\phi =\underrightarrow{\lim }\phi _{n})$;

(ASBVB 5):

For each $n\in {\mathbb {N}}$, there exists a trivialization $\Psi _{n}:\left( \pi _{n}\right) ^{-1}\left( U_{n}\right) \rightarrow U_{n}\times {\mathbb {E}}_{n}$ such that, for any $i\le j$, the following diagram is commutative:

For example, the sequence $\left( \left( TM_n,\pi _n,M_n \right) , \left( T\varepsilon _n^{n+1},\varepsilon _n^{n+1} \right) \right) _{n \in {\mathbb {N}}}$ is a strong ascending sequence of Banach vector bundles whenever $\left( M_{n} \right) _{n\in {\mathbb {N}}}$ is an ascending sequence which has the direct limit chart property at each point of $x \in M=\underrightarrow{\lim }M_{n}$ whose model ${\mathbb {M}}_{n}$ is supplemented in ${\mathbb {M}}_{n+1}$.

Notation 75

From now on and for the sake of simplicity, the strong ascending sequence of vector bundles $\left( (E_n,\pi _n,M_n),\left( \lambda _n^{n+1},\varepsilon _n^{n+1} \right) \right) _{n \in {\mathbb {N}}}$ will be denoted $\left( E_{n},\pi _{n},M_{n}\right) _{\underrightarrow{n}}$.

We then have the following result given in [6].

Proposition 76

Let $\left( E_{n},\pi _{n},M_{n}\right) _{\underrightarrow{n}}$ be a strong ascending sequence of Banach vector bundles. We have:

• (1) $\underrightarrow{\lim }E_{n}$ has a structure of not necessarily Hausdorff convenient manifold modelled on the LB-space $\underrightarrow{\lim }{\mathbb {M}}_{n}\times \underrightarrow{\lim }{\mathbb {E}}_{n}$ which has a Hausdorff convenient structure if and only if M is Hausdorff.

• (2) $\left( \underrightarrow{\lim }E_{n},\underrightarrow{\lim }\pi _{n},\underrightarrow{\lim }M_{n}\right)$ can be endowed with a structure of convenient vector bundle whose typical fibre is $\underrightarrow{\lim }{{\mathbb {E}}}_{n}$.