On Bitopological Lie Algebroids

1. Introduction

Lie algebroids provide a common geometric framework for Lie algebras, tangent bundles, foliations, and infinitesimal structures associated with Lie groupoids. Their theory has been developed systematically in the work of Mackenzie [1,2,3].

A Lie algebroid consists of a vector bundle equipped with a Lie bracket on its space of sections and an anchor map relating that bracket to the usual Lie bracket of vector fields. This combination of algebraic and differential geometric structures makes Lie algebroids useful in differential geometry and mathematical physics.

The purpose of this paper is to formulate a framework in which two topologies on a common smooth manifold are considered simultaneously with Lie algebroid structures.

Recall that a bitopological space is a triple

$$(M,{\tau }_1,{\tau }_2),$$

where ${\tau }_1$ and ${\tau }_2$ are topologies on the same set M. Such spaces were introduced as a natural extension of ordinary topological spaces and provide a setting for studying properties that depend on two distinct notions of openness.

In the present setting, we assume that M is a smooth manifold whose topology is denoted by $\tau$, and that ${\tau }_1,{\tau }_2$ are two topologies on M compatible with the fixed smooth structure. In particular, when the manifold is finite-dimensional and Hausdorff, we assume that both topologies coincide with the manifold topology unless otherwise stated.

This assumption separates the topological aspect of the problem from the differential structure needed to define smooth vector bundles, sections, and anchors.

The principal objectives are:

(i)

to define bitopological Lie algebroids;

(ii)

to describe their basic examples;

(iii)

to formulate morphisms and compatibility conditions;

(iv)

to compare the associated cohomology complexes;

(v)

to identify questions for further research.

The classical theory of Lie algebroids and their integrations provides the background for our constructions [3]. We emphasize that the pair of structures introduced here is additional data: two topologies alone do not determine two Lie algebroid structures.

2. Preliminaries

Throughout this paper, all manifolds and vector bundles are finite-dimensional, smooth, and real, unless otherwise specified.

Definition 1. 

A bitopological space is a triple

$$(M,{\tau }_1,{\tau }_2),$$

where M is a set and ${\tau }_1,{\tau }_2$ are topologies on M.

The two topologies need not be comparable. If ${\tau }_1\subseteq {\tau }_2$, then every ${\tau }_1$-open set is ${\tau }_2$-open, but the converse need not hold.

Definition 2. 

Let M be a smooth manifold with its fixed smooth structure. A bitopological smooth manifold is a quadruple

$$(M,\mathcal{S},{\tau }_1,{\tau }_2),$$

where $\mathcal{S}$ is the smooth structure and ${\tau }_1,{\tau }_2$ are topologies on M compatible with the underlying smooth manifold structure.

Remark 1. 

A smooth structure determines a topology on a manifold. Consequently, if both ${\tau }_1$ and ${\tau }_2$ are required to be the topology induced by the same smooth atlas, then they coincide. A genuinely distinct pair of topologies therefore requires an additional convention, such as considering auxiliary topologies while retaining a fixed smooth structure.

In this paper, differential operations are defined using the fixed smooth structure. The two topologies are recorded as additional topological data.

Definition 3. 

Let $E\to M$ be a smooth vector bundle. The space of smooth sections of E is denoted by

$$\mathrm{\Gamma }\left(E\right).$$

The space of smooth real-valued functions on M is denoted by $C^{\infty }\left(M\right)$.

Definition 4. 

A Lie algebroid over M is a vector bundle

$$\pi :E⟶M$$

together with a Lie bracket

$${[·,·]}_E:\mathrm{\Gamma }\left(E\right)\times \mathrm{\Gamma }\left(E\right)⟶\mathrm{\Gamma }\left(E\right)$$

and a vector bundle morphism

$${\rho }_E:E⟶TM,$$

called the anchor, such that

$${\rho }_E\left({[s,t]}_E\right)=[{\rho }_E\left(s\right),{\rho }_E\left(t\right)]$$

and

$${[s,ft]}_E=f{[s,t]}_E+\left({\rho }_E\left(s\right)f\right)t$$

for all $s,t\in \mathrm{\Gamma }\left(E\right)$ and $f\in C^{\infty }\left(M\right)$.

These identities express the compatibility between the Lie bracket on sections and the differential structure of the base. See [2,3] for the classical theory.

3. Bitopological Lie Algebroids

We now introduce the principal object of this paper.

Definition 5. 

Let

$$(M,\mathcal{S},{\tau }_1,{\tau }_2)$$

be a bitopological smooth manifold. A bitopological Lie algebroid over M is a pair

$$\mathcal{A}=(A_1,A_2)$$

of Lie algebroids over the same smooth manifold M, where

$$A_i=(E_i,{[·,·]}_i,{\rho }_i),i=1,2.$$

The structure $A_i$ is regarded as the Lie algebroid structure associated with the i-th topological viewpoint.

The definition does not require $E_1$ and $E_2$ to be isomorphic. When they are isomorphic, additional compatibility conditions can be imposed.

Definition 6. 

A bitopological Lie algebroid $\mathcal{A}=(A_1,A_2)$ is called strongly compatible if there exists a vector bundle isomorphism

$$\mathrm{\Phi }:E_1⟶E_2$$

over ${id}_M$ such that

$$\mathrm{\Phi }\left({[s,t]}_1\right)={[\mathrm{\Phi }\left(s\right),\mathrm{\Phi }\left(t\right)]}_2$$

for all $s,t\in \mathrm{\Gamma }\left(E_1\right)$, and

$${\rho }_2\circ \mathrm{\Phi }={\rho }_1.$$

Remark 2. 

Strong compatibility identifies the two Lie algebroid structures as Lie algebroids over the same base. It does not require the two topologies to be equal. Conversely, equality of the two topologies does not imply strong compatibility.

Proposition 1. 

If a bitopological Lie algebroid $\mathcal{A}=(A_1,A_2)$ is strongly compatible, then the Lie algebras of sections $\mathrm{\Gamma }\left(E_1\right)$ and $\mathrm{\Gamma }\left(E_2\right)$ are isomorphic.

Proof. 

The bundle isomorphism $\mathrm{\Phi }$ induces a linear isomorphism

$${\mathrm{\Phi }}_{*}:\mathrm{\Gamma }\left(E_1\right)⟶\mathrm{\Gamma }\left(E_2\right),{\mathrm{\Phi }}_{*}\left(s\right)=\mathrm{\Phi }\circ s.$$

By the bracket compatibility condition,

$${\mathrm{\Phi }}_{*}\left({[s,t]}_1\right)={[{\mathrm{\Phi }}_{*}\left(s\right),{\mathrm{\Phi }}_{*}\left(t\right)]}_2.$$

Thus ${\mathrm{\Phi }}_{*}$ is a Lie algebra isomorphism. □

4. The Anchor Maps

The anchor is the principal link between a Lie algebroid and the tangent bundle of its base.

For a bitopological Lie algebroid $\mathcal{A}=(A_1,A_2)$, we have two anchor maps

$${\rho }_1:E_1⟶TM,{\rho }_2:E_2⟶TM.$$

Their images determine generalized distributions

$${\mathcal{D}}_i={\rho }_i\left(E_i\right)\subseteq TM,i=1,2.$$

These distributions need not have constant rank.

Proposition 2. 

Suppose that $\mathrm{\Phi }:E_1\to E_2$ is an isomorphism of strongly compatible Lie algebroids. Then

$${\rho }_1\left(E_1\right)={\rho }_2\left(E_2\right).$$

Proof. 

Since $\mathrm{\Phi }$ is surjective, every element of $E_2$ has the form $\mathrm{\Phi }\left(e\right)$ for some $e\in E_1$. Hence

$${\rho }_2\left(E_2\right)={\rho }_2\left(\mathrm{\Phi }\left(E_1\right)\right)={\rho }_1\left(E_1\right),$$

where the last equality follows from ${\rho }_2\circ \mathrm{\Phi }={\rho }_1$. □

Definition 7. 

A bitopological Lie algebroid is called anchor-compatible if

$${\rho }_1\left(E_1\right)={\rho }_2\left(E_2\right)$$

as subsets of $TM$.

Anchor compatibility is weaker than strong compatibility: equality of the images of the anchors does not generally produce a vector bundle isomorphism preserving the brackets.

Example 1 

(The tangent Lie algebroid). Let M be a smooth manifold equipped with two auxiliary topologies ${\tau }_1$ and ${\tau }_2$. The tangent bundle has its standard Lie algebroid structure

$$A_i=TM,{[·,·]}_i=[·,·],{\rho }_i={id}_{TM},$$

for $i=1,2$.

The identity map

$$\mathrm{\Phi }={id}_{TM}$$

gives a strongly compatible bitopological Lie algebroid.

This example shows that the same differential structure can be considered together with two topological viewpoints.

5. Examples

5.1. Lie Algebroids Associated with Lie Algebra Actions

Let $\mathfrak{g}$ be a finite-dimensional Lie algebra acting infinitesimally on a smooth manifold M. The action determines a bundle map

$$\rho :M\times \mathfrak{g}⟶TM.$$

Under the standard convention for infinitesimal actions, the trivial bundle

$$A=M\times \mathfrak{g}$$

carries an action Lie algebroid structure. Its bracket is determined by the Lie bracket on $\mathfrak{g}$ and the action on smooth functions.

More explicitly, for constant sections associated with $u,v\in \mathfrak{g}$, one has

$${[u,v]}_A={[u,v]}_{\mathfrak{g}},$$

and the Leibniz rule determines the bracket on general sections.

This construction is a standard source of Lie algebroids; see [1,3].

If the same action is considered in two topological viewpoints, it gives a strongly compatible pair provided the underlying smooth action and the resulting Lie algebroid structures are unchanged.

5.2. The Cotangent Lie Algebroid of a Poisson Manifold

Let $(M,\pi )$ be a Poisson manifold, where

$$\pi \in \mathrm{\Gamma }\left({\wedge }^{2}TM\right)$$

satisfies

$${[\pi ,\pi ]}_{\mathrm{Sch}}=0.$$

The cotangent bundle $T^{*}M$ carries a Lie algebroid structure with anchor

$${\pi }^{♯}:T^{*}M⟶TM,$$

defined by

$$\beta \left({\pi }^{♯}\left(\alpha \right)\right)=\pi (\alpha ,\beta ).$$

The bracket on one-forms is given by

$${[\alpha ,\beta ]}_{\pi }={\mathcal{L}}_{{\pi }^{♯}\left(\alpha \right)}\beta -{\mathcal{L}}_{{\pi }^{♯}\left(\beta \right)}\alpha -d\left(\pi (\alpha ,\beta )\right).$$

Here $\mathcal{L}$ denotes the Lie derivative and d is the de Rham differential.

This Lie algebroid is fundamental in Poisson geometry and in the theory of Poisson groupoids [3,4].

For two Poisson structures ${\pi }_1$ and ${\pi }_2$ on the same smooth manifold, we obtain two cotangent Lie algebroids

$$A_i=(T^{*}M,{[·,·]}_{{\pi }_i},{\pi }_i^{♯}),i=1,2.$$

They form a bitopological Lie algebroid over $(M,{\tau }_1,{\tau }_2)$. In general, the identity on $T^{*}M$ does not define an isomorphism of these Lie algebroids. Such an isomorphism requires compatibility between the two Poisson structures.

Remark 3. 

A pair of Poisson structures is not automatically a Poisson–Nijenhuis structure or a Lie bialgebroid. Those notions impose additional compatibility conditions. The same distinction applies to bitopological Lie algebroids.

6. Morphisms

We next define morphisms between bitopological Lie algebroids.

Definition 8. 

Let

$$\mathcal{A}=(A_1,A_2)\mathrm{and}\mathcal{B}=(B_1,B_2)$$

be bitopological Lie algebroids over smooth manifolds M and N, respectively.

A morphism of bitopological Lie algebroids is a pair of Lie algebroid morphisms

$$F_i:A_i⟶B_i,i=1,2,$$

covering the same smooth map

$$f:M⟶N.$$

Recall that a Lie algebroid morphism is understood in the standard sense: it is a vector bundle map compatible with the Lie algebroid structures. One equivalent formulation uses the induced morphism of the corresponding differential graded algebras of differential forms.

Definition 9. 

Suppose that ${\mathrm{\Phi }}_A:E_1\to E_2$ and ${\mathrm{\Phi }}_B:F_1\to F_2$ are strong compatibility isomorphisms for $\mathcal{A}$ and $\mathcal{B}$. A morphism $(F_1,F_2)$ is called compatibility-preserving if

$$F_2\circ {\mathrm{\Phi }}_A={\mathrm{\Phi }}_B\circ F_1.$$

Proposition 3. 

The composition of compatibility-preserving morphisms is compatibility-preserving.

Proof. 

Let

$$(F_1,F_2):\mathcal{A}⟶\mathcal{B}$$

and

$$(G_1,G_2):\mathcal{B}⟶\mathcal{C}$$

be compatibility-preserving morphisms.

Write ${\mathrm{\Phi }}_A,{\mathrm{\Phi }}_B,{\mathrm{\Phi }}_C$ for the respective compatibility isomorphisms. Then

$$F_2{\mathrm{\Phi }}_A={\mathrm{\Phi }}_B{F}_1$$

and

$$G_2{\mathrm{\Phi }}_B={\mathrm{\Phi }}_C{G}_1.$$

Consequently,

$$\begin{array}{cc}\hfill \left(G_2{F}_2\right){\mathrm{\Phi }}_A& =G_2\left(F_2{\mathrm{\Phi }}_A\right)\hfill \\ & =G_2{\mathrm{\Phi }}_B{F}_1\hfill \\ & ={\mathrm{\Phi }}_C{G}_1{F}_1.\hfill \end{array}$$

Thus the composite is compatibility-preserving. □

7. Lie Algebroid Cohomology

Let $A\to M$ be a Lie algebroid. Its space of differential forms is

$${\mathrm{\Omega }}^k\left(A\right)=\mathrm{\Gamma }\left({\wedge }^k{A}^{*}\right).$$

The Lie algebroid differential

$$d_A:{\mathrm{\Omega }}^k\left(A\right)⟶{\mathrm{\Omega }}^{k+1}\left(A\right)$$

is defined by

$$\begin{array}{cc}\hfill \left(d_A\omega \right)(s_0,\dots ,s_k)=& {\displaystyle \sum _{i=0}^k}{(-1)}^i\rho \left(s_i\right)\left(\omega (s_0,\dots ,\widehat{s_i},\dots ,s_k)\right)\hfill \\ & +\sum _{0\le i<j\le k}{(-1)}^{i+j}\omega \left([s_i,s_j],s_0,\dots ,\widehat{s_i},\dots ,\widehat{s_j},\dots ,s_k\right),\hfill \end{array}$$

where the arguments in the second sum are arranged in the usual order.

The Lie algebroid axioms imply

$$d_A^2=0.$$

The resulting cohomology is denoted by

$$H^k\left(A\right).$$

For a bitopological Lie algebroid $\mathcal{A}=(A_1,A_2)$, we obtain two cochain complexes

$$\left({\mathrm{\Omega }}^{•}\left(A_1\right),d_{A_1}\right),\left({\mathrm{\Omega }}^{•}\left(A_2\right),d_{A_2}\right).$$

Their cohomology groups are

$$H^k\left(A_1\right)\mathrm{and}{H}^k\left(A_2\right).$$

Proposition 4. 

If $\mathrm{\Phi }:A_1\to A_2$ is an isomorphism of Lie algebroids, then it induces an isomorphism

$$H^k\left(A_2\right)\cong H^k\left(A_1\right)$$

for every $k\ge 0$.

Proof. 

The vector bundle isomorphism $\mathrm{\Phi }$ induces pullback maps on exterior powers of the dual bundles and hence a cochain map

$${\mathrm{\Phi }}^{*}:{\mathrm{\Omega }}^{•}\left(A_2\right)⟶{\mathrm{\Omega }}^{•}\left(A_1\right).$$

Compatibility with the anchors and brackets gives

$$d_{A_1}\circ {\mathrm{\Phi }}^{*}={\mathrm{\Phi }}^{*}\circ d_{A_2}.$$

Thus ${\mathrm{\Phi }}^{*}$ induces a map on cohomology. Since $\mathrm{\Phi }$ is an isomorphism, its inverse induces the inverse cohomology map. □

Remark 4. 

The two cohomology groups need not be isomorphic when the two Lie algebroid structures are unrelated. The existence of two topologies alone does not imply any relation between their cohomologies.

8. Compatibility and Further Structures

The preceding constructions lead naturally to stronger compatibility conditions.

Definition 10. 

A bitopological Lie algebroid $\mathcal{A}=(A_1,A_2)$ is called bracket-compatible if there is a specified identification of the underlying vector bundles under which

$${[s,t]}_1={[s,t]}_2$$

for all smooth sections $s,t$.

If the anchors also agree under this identification, the two Lie algebroid structures coincide.

More generally, one may investigate pairs of brackets related by a deformation. For example, if the underlying vector bundle is a fixed bundle E, one can consider

$${[·,·]}_{\lambda }={[·,·]}_1+\lambda {[·,·]}_2,\lambda \in \mathbb{R}.$$

However, this expression defines a Lie algebroid bracket only when suitable Jacobi, Leibniz, and anchor compatibility conditions hold. Such conditions are not automatic.

The study of compatible algebraic structures is closely related to Lie bialgebroids and their integrations. Mackenzie and Xu established a fundamental relationship between Lie bialgebroids and Poisson groupoids [4]. This suggests investigating whether additional compatibility data on a bitopological Lie algebroid can produce corresponding groupoid structures.

Problem 1. 

Find necessary and sufficient conditions under which a pair of Lie algebroid structures on a common vector bundle admits a common integration by a Lie groupoid.

Problem 2. 

Determine conditions under which two Lie algebroid differentials can be combined into a single differential on a graded algebra.

Problem 3. 

Investigate whether distinct auxiliary topologies on a common smooth base can yield useful topological invariants of Lie algebroid representations and their cohomology.

9. Concluding Remarks

We have introduced a framework for studying pairs of Lie algebroid structures over a common smooth manifold equipped with two topologies.

The basic construction consists of a pair

$$\mathcal{A}=(A_1,A_2),$$

where each component is an ordinary Lie algebroid. We have defined strong compatibility, anchor compatibility, and compatibility-preserving morphisms. We have also described the associated cohomology complexes and considered examples arising from tangent bundles, Lie algebra actions, and Poisson structures.

The framework makes explicit an important distinction: the existence of two topologies does not by itself determine two Lie algebroid structures. The differential geometric data must be specified, and any relationship between the two structures must be imposed separately.

Further work may focus on a genuinely topology-sensitive theory, in which the two topologies determine distinct classes of admissible sections, continuity conditions, or cohomological invariants. Another direction is the study of common integrations and compatibility with Lie bialgebroid structures.