A complex Lie algebroid is a complex vector bundle over a smooth (real) manifold M with a bracket on sections and an anchor to the complexified tangent bundle of M which satisfy the usual Lie algebroid axioms. A proposal is made here to integrate analytic complex Lie algebroids by using analytic continuation to a complexification of M and integration to a holomorphic groupoid. A collection of diverse examples reveal that the holomorphic stacks presented by these groupoids tend to coincide with known objects associated to structures in complex geometry. This suggests that the object integrating a complex Lie algebroid should be a holomorphic stack.
We define a general notion of abstract double Lie algebroid. We show (1) that the double Lie algebroid of a double Lie groupoid is a double Lie algebroid in this sense; (2) that the double cotangent constructed from Lie algebroid structures on a vector bundle A and its dual A* is a double Lie algebroid if and only if (A, A*) is a Lie bialgebroid; (3) that a vacant double Lie algebroid structure is equivalent to a matched pair structure on the side Lie algebroids.
This paper provides an alternative, much simpler, definition for Li-Bland's LA-Courant algebroids, or Poisson Lie 2-algebroids, in terms of split Lie 2-algebroids and self-dual 2-representations. This definition generalises in a precise sense the characterisation of (decomposed) double Lie algebroids via matched pairs of 2-representations. We use the known geometric examples of LA-Courant algebroids in order to provide new examples of Poisson Lie 2-algebroids, and we explain in this general context Roytenberg's equivalence of Courant algebroids with symplectic Lie 2-algebroids. We study furthe…
A hom-Lie algebroid is a vector bundle together with a Lie algebroid like structure which is twisted by a homomorphism. In this paper we use the idea of representations up to homotopy of Lie algebroids to construct a same structure for hom-Lie algebroids and we will explain how representations up to homotopy of length 1 are related to extensions of hom-Lie algebroids.
We give a new proof of the equivalence between the existence of a complete action of a Lie algebroid on a surjective submersion and its integrability. The main tools in our approach are double Lie groupoids and multiplicative foliations, our proof relies on a simple characterization of those vacant double Lie groupoids which induce a Lie groupoid structure on their orbit spaces.
Almost Lie algebroids are generalizations of Lie algebroids, when the Jacobiator is not necessary null. A simple example is given, for which a Lie algebroid bracket or a Courant bundle is not possible for the given anchor, but a natural extension of the bundle and the new anchor allows a Lie algebroid bracket. A cohomology and related characteristic classes of an almost Lie algebroid are also constructed. We prove that these characteristic classes are all pull-backs of the characteristic classes of the base space, as in the case of a Lie algebroid.
We develop a theory of Lie algebroids over differentiable stacks that extends the standard theory of Lie algebroids over manifolds. In particular we show that Lie algebroids satisfy descent for submersions, define the category of Lie algebroids over a differentiable stack, construct a cohomology theory for these objects, and explain the relation to the theory of $\mathcal{LA}$-groupoids. We construct a number of examples.
We define Lie and Courant algebroids on Fréchet manifolds. Moreover, we construct a Dirac structure on the generalized tangent bundle of a Fréchet manifold and show that it inherits a Fréchet Lie algebroid structure. We show that the Lie algebroid cohomology of the $\bb$-cotangent bundle Lie algebroid of a weakly symplectic Fréchet manifold $M$ is the Lichnerowicz-Poisson cohomology of $M$.
We define covariant Lie derivatives acting on vector-valued forms on Lie algebroids and study their properties. This allows us to obtain a concise formula for the Frölicher-Nijenhuis bracket on Lie algebroids.
The core diagram of a double Lie algebroid consists in the core of the double Lie algebroid, together with the two core-anchor maps to the sides of the double Lie algebroid. If these two core anchors are surjective, then the double Lie algebroid and its core diagram are called transitive. This paper establishes an equivalence between transitive double Lie algebroids, and transitive core diagrams over a fixed base manifold. In other words, it proves that a transitive double Lie algebroid is completely determined by its core diagram. The comma double Lie algebroid associated to a morphism of Lie…
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.
First, we extend the notion of second order differential equations (SODE) on a smooth manifold to anchored Banach vector bundles. Then we define the Banach Lie algebroids as Lie algebroids structures modeled on anchored Banach vector bundles and prove that they form a category.
Cartan-Lie algebroids, i.e. Lie algebroids equipped with a compatible connection, permit the definition of an adjoint representation, on the fiber as well as on the tangent of the base. We call (positive) quadratic Lie algebroids, Cartan-Lie algebroids with ad-invariant (Riemannian) metrics on their fibers and base $κ$ and $g$, respectively. We determine the necessary and sufficient conditions for a positive quadratic Lie algebroid to integrate to a Riemmanian Cartan-Lie groupoid. Here we mean a Cartan-Lie groupoid $\mathcal{G}$ equipped with a bi-invariant and inversion invariant metric $η$ o…
Given a G-structure with connection satisfying a regularity assumption we associate to it a classifying Lie algebroid. This algebroid contains all the information about the equivalence problem and is an example of a G-structure Lie algebroid. We discuss the properties of this algebroid, the G-structure groupoids integrating it and the relationship with Cartan's realization problem.
The Chern-Simons forms for R-linear connections on Lie algebroids are considered. A generalized Chern-Simons formula for such R-linear connections is obtained. We it apply to define Chern character and secondary characteristic classes for R-linear connections of Lie algebroids.
We show that double Lie algebroids, together with a chosen linear splitting, are equivalent to pairs of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the tangent of a Lie algebroid.
A class of metrizable vector bundles in the general framework of generalized Lie algebroids have been presented in the eight reference. Using a generalized Lie algebroid we obtain the Lie algebroid generalized tangent bundle of a vector bundle. This Lie algebroid is a new example of metrizable vector bundle. A new class of Lagrange spaces, called by use, generalized Lagrange (ρ?;η?)-space, Lagrange (ρ?;η?)-space and Finsler (ρ?;η?)-space are presented. In the particular case of Lie algebroids, new and important results are presented. In particular, if all morphisms are identities morphisms, th…
Considering prolongation of a Lie algebroid equipped with a spray, defining some classical tensors, we show that a Lie symmetry of a spray is a curvature collineation for these tensors.
The author presents the generalized Stokes theorem for R-linear forms on Lie algebroids (which can be non-local). We apply the Stokes formula on forms to prove that two homotopic homomorphisms of Lie algebroids implies the existence of a chain operator joining their pullback operators.
Horizontal endomorphisms, almost complex structures, vertical, horizontal and complete lifts on prolongation of a Lie algebroid are considered. Then using exact sequences, semisprays are constructed. Moreover, important geometrical objects such as classical distinguished connections, torsions and partial curvatures are studied on prolongation of Lie algebroids. Considering pullback bundle, covariant derivatives are scrutinized based on anchor map. Several Finsler geometry models on Lie algebroid structures, will organized via recent arguments. Finally, it will be overhaulled some special Finsl…
Li-Bland's correspondence between linear Courant algebroids and Lie $2$-algebroids is explained and shown to be an equivalence of categories. Decomposed VB-Courant algebroids are shown to be equivalent to split Lie 2-algebroids in the same manner as decomposed VB-algebroids are equivalent to 2-term representations up to homotopy (Gracia-Saz and Mehta). Several classes of examples are discussed, yielding new examples of split Lie 2-algebroids. We prove that the bicrossproduct of a matched pair of $2$-representations is a split Lie $2$-algebroid and we explain this result geometrically, as a con…
We construct local coordinates for the Weinstein groupoid of a non-integrable Lie algebroid. To this end, we reformulate the notion of bi-submersion in a completely algebraic way and prove the existence of bi-submersions as such for the Weinstein groupoid. This implies that a $C^*$-algebra can be attached to every Lie algebroid.
The almost complex Lie algebroids over smooth manifolds are introduced in the paper. In the first part we give some examples and we obtain a Newlander-Nirenberg type theorem on almost complex Lie algebroids. Next the almost Hermitian Lie algebroids and some related structures on the associated complex Lie algebroid are studied. For instance, we obtain that the $E$-Chern form of $E^{1,0}$ associated to an almost complex connection $\nabla$ on $E$ can be expressed in terms of the matrix $J_ER$, where $J_E$ is the almost complex structure of $E$ and $R$ is the curvature of $\nabla$. Also, we cons…
In this note we motivate the definition and use of Lie algebroids by revisiting the problem of reconstructing a hypersurface in Euclidean space from infinitesimal data.
In this paper we generalize the main notions from the geometry of (almost) contact manifolds in the category of Lie algebroids. Also, using the framework of generalized geometry, we obtain an (almost) contact Riemannian Lie algebroid structure on a vertical Liouville distribution over the big-tangent manifold of a Riemannain manifold.
In this short note we define a new cohomology for a Lie algebroid $\mathcal{A}$, that we call the \emph{twisted cohomology} of $\mathcal{A}$ by an odd cocycle $θ$ in the Lie algebroid cohomology of $\mathcal{A}$. We proof that this cohomology only depends on the Lie algebroid cohomology class $[θ]$ of the odd cocycle $θ$. We give a few examples showing that this new cohomology encompasses various well-known cohomology theories.
In this paper, we introduce the notion of modular class of a Lie algebroid $A$ equipped with a Nambu structure satisfying some suitable hypothesis. We also introduce cohomology and homology theories for such Lie algebroids and prove that these theories are connected by a duality isomorphism when the modular class is null.
We give numerous examples of almost Lie algebroids arising as Dirac structures in pre-Courant algebroids, e.g. from twisted Poisson structures, as well as from twisted actions of a Lie algebra. We moreover define a cohomology for them, motivated by a Q-structure, that is trivial for (bundles of) Lie algebras but characterizes the underlying integrable distribution the image of the anchor map.
The paper introduces Laplace-type operators for functions defined on the tangent space of a Finsler Lie algebroid, using a volume form on the prolongation of the algebroid. It also presents the construction of a horizontal Laplace operator for forms defined on the prolongation of the algebroid. All of the Laplace operators considered in the paper are also locally expressed using the Chern-Finsler connection of the algebroid.
In this paper we investigate the relations between semispray, nonlinear connection, dynamical covariant derivative and Jacobi endomorphism on Lie algebroids. Using these geometric structures, we study the symmetries of second order differential equations in the general framework of Lie algebroids.
For every Lie pair $(L,A)$ of algebroids we construct a dg-manifold structure on the $\mathbb{Z}$-graded manifold $\mathcal M=L[1]\oplus L/A$ such that the inclusion $ι: A[1] \to \mathcal M$ and the projection $p:\mathcal M\to L[1]$ are morphisms of dg-manifolds. The vertical tangent bundle $T^p\mathcal M$ then inherits a structure of dg-Lie algebroid over $\mathcal M$. When the Lie pair comes from a matched pair of Lie algebroids, we show that the inclusion $ι$ induces a quasi-isomorphism that sends the Atiyah class of this dg-Lie algebroid to the Atiyah class of the Lie pair. We also show ho…
We introduce the notion of hypersymplectic structure on a Courant algebroid and we prove the existence of a one-to-one correspondence between hypersymplectic and hyperkähler structures. This correspondence provides a simpler way to define a hyperkähler structure on a Courant algebroid. We show that hypersymplectic structures on Courant algebroids encompass hyperkähler and hyperkähler structures with torsion on Lie algebroids. In the latter, the torsion existing at the Lie algebroid level is incorporated in the Courant structure. Cases of hypersymplectic structures on Courant algebroids which a…
Hypersymplectic structures with torsion on Lie algebroids are investigated. We show that each hypersymplectic structure with torsion on a Lie algebroid determines three Nijenhuis morphisms. From a contravariant point of view, these structures are twisted Poisson structures. We prove the existence of a one-to-one correspondence between hypersymplectic structures with torsion and hyperkähler structures with torsion. We show that given a Lie algebroid with a hypersymplectic structure with torsion, the deformation of the Lie algebroid structure by any of the transition morphisms does not affect th…
We propose a definition of a "higher" version of the omni-Lie algebroid and study its isotropic and involutive subbundles. Our higher omni-Lie algebroid is to (multi)contact and related geometries what the higher generalized tangent bundle of Zambon and Bi/Sheng is to (multi)symplectic and related geometries.
This paper is a study of the relationship between two constructions associated with Cartan geometries, both of which involve Lie algebroids: the Cartan algebroid, due to [Blaom A.D., Trans. Amer. Math. Soc. 358 (2006), 3651-3671], and tractor calculus [Cap A., Gover A.R., Trans. Amer. Math. Soc. 354 (2001), 1511-1548].
A generalized notion of a Lie algebroid is presented. Using this, the Lie algebroid generalized tangent bundle is obtained. A new point of view over (linear) connections theory on a fiber bundle is presented. These connections are characterized by o horizontal distribution of the Lie algebroid generalized tangent bundle. Some basic properties of these generalized connections are investigated. Special attention to the class of linear connections is paid. The recently studied Lie algebroids connections can be recovered as special cases within this more general framework. In particular, all resul…
The authors define some secondary characteristic homomorphism for the triple (A,B,\bigtriangledown), in which B\subset A is a pair of regular Lie algebroids over the same foliated manifold and \bigtriangledown:L\rightarrow A is a homomorphism of Lie algebroids (i.e. a flat L-connection in A) where L is an arbitrary (not necessarily regular) Lie algebroid and show that characteristic classes from its image generalize known exotic characteristic classes for flat regular Lie algebroids (Kubarski) and flat principal fibre bundles with a reduction (Kamber, Tondeur). The generalization includes also…
We establish a relationship between two different generalizations of Lie algebroid representations: representation up to homotopy and Vaintrob's Lie algebroid modules. Specifically, we show that there is a noncanonical way to obtain a representation up to homotopy from a given Lie algebroid module, and that any two representations up to homotopy obtained in this way are equivalent in a natural sense. We therefore obtain a one-to-one correspondence, up to equivalence.
A Lie groupoid, called \textit{material Lie groupoid}, is associated in a natural way to any elastic material. The corresponding Lie algebroid, called \textit{material algebroid}, is used to characterize the uniformity and the homogeneity properties of the material. The relation to previous results in terms of $G-$structures is discussed in detail. An illustrative example is presented as an application of the theory.
A linear connection $D$ in a Lie algebroid is said to be metrizable if there exists a Riemannian metric $h$ in the Lie algebroid such that $Dh=0$. Conditions for the linear connection $D$ to be metrizable are investigated.
A pre-Lie algebroid is an anchored bundle provided with an almost Lie bracket such that the anchor is compatible with the Lie bracket of vector fields. We firstly show how most geometrical structures intensively studied in the framework of Lie algebroid can easily be extended in the pre-Lie algebroid context. The principal purpose of this paper is to show that how all these results only depend of the foliated structure and do not depend of the pre-Lie algebroid structure that we can put on a foliated anchored bundle. As application, we obtain a Finsler connection on a foliated anchored bundle …
We examine functorial and homotopy properties of the exotic characteristic homomorphism in the category of Lie algebroids which was lastly obtained by the authors in [4]. This homomorphism depends on a triple (A,B,$\nabla$) where B $\subset$ A are regular Lie algebroids, both over the same regular foliated manifold (M,F), and $\nabla$ is a flat L-connection in A, where L is an arbitrary Lie algebroid over M. The Rigidity Theorem (i.e. the independence from the choice of homotopic Lie subalgebroids of B) is obtained. The exotic characteristic homomorphism is factorized by one (called universal)…
A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie algebroids theory is presented using the notion of generalized Lie algebra and examples of objects of the category of generalized Lie algebroids are presented. An exterior differential calculus on generalized Lie algebras is pre- sented and a theorem of Maurer-Cartan type is obtained. Supposing that any submodule(vector subbundle) of a generalized Lie algebra(algebroid) is an interior alg…
In this paper, we will study compatible triples on Lie algebroids. Using a suitable decomposition for a Lie algebroid, we construct an integrable generalized distribution on the base manifold. As a result, the symplectic form on the Lie algebroid induces a symplectic form on each integral submanifold of the distribution. The induced Poisson structure on the base manifold can be represented by means of the induced Poisson structures on the integral submanifolds. Moreover, for any compatible triple with invariant metric and admissible almost complex structure, we show that the bracket annihilate…
An exterior differential calculus in the general framework of generalized Lie algebroids is presented. A theorem of Maurer-Cartan type is obtained. All results with details proofs are presented and a new point of view over exterior differential calculus for Lie algebroids is obtained. Using the theory of linear connections of Ehresmann type presented in the firstt reference, the identities of Cartan and Bianchi type are presented. Supposing that any vector subbundle of the pull-back Lie algebroid of a generalized Lie algebroid is interior differential system (IDS) for that generalized Lie alge…
We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its external powers, one can define an operator similar to the exterior derivative. We present the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the genera…
Complex Finsler vector bundles have been studied mainly by T. Aikou, who defined complex Finsler structures on holomorphic vector bundles. In this paper, we consider the more general case of a holomorphic Lie algebroid E and we introduce Finsler structures, partial and Chern-Finsler connections on it. First, we recall some basic notions on holomorphic Lie algebroids. Then, using an idea from E. Martinez, we introduce the concept of complexified prolongation of such an algebroid. Also, we study nonlinear and linear connections on the tangent bundle of E and on the prolongation of E and we inves…
A VB-algebroid is essentially defined as a Lie algebroid object in the category of vector bundles. There is a one-to-one correspondence between VB-algebroids and certain flat Lie algebroid superconnections, up to a natural notion of equivalence. In this setting, we are able to construct characteristic classes, which in special cases reproduce characteristic classes constructed by Crainic and Fernandes. We give a complete classification of regular VB-algebroids, and in the process we obtain another characteristic class of Lie algebroids that does not appear in the ordinary representation theory…
We define involution algebroids which generalise Lie algebroids to the abstract setting of tangent categories. As a part of this generalisation the Jacobi identity which appears in classical Lie theory is replaced by an identity similar to the Yang-Baxter equation. Every classical Lie algebroid has the structure of an involution algebroid and every involution algebroid in a tangent category admits a Lie bracket on the sections of its underlying bundle. As an illustrative application we take the first steps in developing the homotopy theory of involution algebroids.
