Izu Vaisman
We extend R. Fernandes' construction of secondary characteristic classes of a Lie algebroid to the case of a base-preserving morphism between two Lie algebroids. Like in the case of a Lie algebroid, the simplest characteristic class of our construction coincides with the modular class of the morphism.
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We propose a definition of Jacobi quasi-Nijenhuis algebroid and show that any such Jacobi algebroid has an associated quasi-Jacobi bialgebroid. Therefore, also an associated Courant-Jacobi algebroid is obtained. We introduce the notions of quasi-Jacobi bialgebroid morphism and Courant-Jacobi algebroid morphism providing also some examples of Courant-Jacobi algebroid morphisms.
We study the behavior of the modular class of a Lie algebroid under general Lie algebroid morphisms by introducing the relative modular class. We investigate the modular classes of pull-back morphisms and of base-preserving morphisms associated to Lie algebroid extensions. We also define generalized morphisms, including Morita equivalences, that act on the 1-cohomology, and observe that the relative modular class is a coboundary on the category of Lie algebroids and generalized morphisms with values in the 1-cohomology.
We propose a definition of Poisson quasi-Nijenhuis Lie algebroids as a natural generalization of Poisson quasi-Nijenhuis manifolds and show that any such Lie algebroid has an associated quasi-Lie bialgebroid. Therefore, also an associated Courant algebroid is obtained. We introduce the notion of a morphism of quasi-Lie bialgebroids and of the induced Courant algebroids morphism and provide some examples of Courant algebroid morphisms. Finally, we use paired operators to deform doubles of Lie and quasi-Lie bialgebroids and find an application to generalized complex geometry.
Comomentum sections and Poisson maps in Hamiltonian Lie algebroids
本文在哈密顿李代数胚中引入共动量截面概念，证明其诱导李代数胚态射，并推广了动量映射为泊松映射的经典结论，最终将动量截面重新解释为狄拉克态射。
We show in this paper that the correspondence between $2$-term representations up to homotopy and $\mathcal{VB}$-algebroids, established by Gracia-Saz and Mehta, holds also at the level of morphisms. This correspondence is hence an equivalence of categories. As an application, we study foliations and distributions on a Lie algebroid, that are compatible both with the linear structure and the Lie algebroid structure. In particular, we show how infinitesimal ideal systems in a Lie algebroid $A$ are related with subrepresentations of the adjoint representation of $A$.
$L_\infty$-morphisms between twisted Courant $r$-Lie algebras and untwisted Courant $(r{+}1)$-Lie algebroids
针对Marco Zambon提出的高阶闭形式扭化Courant代数胚是否存在典范$L_\infty$-态射的问题，本文给出肯定回答，提出通用构造框架并阐明其底层结构，验证框架可适配相关已有代数间的态射。
We study the relative modular classes of Lie algebroids, and we determine their relationship with the modular classes of Lie algebroids with a twisted Poisson structure.
For a Lie algebroid, divergences chosen in a classical way lead to a uniquely defined homology theory. They define also, in a natural way, modular classes of certain Lie algebroid morphisms. This approach, applied for the anchor map, recovers the concept of modular class due to S. Evans, J.-H. Lu, and A. Weinstein.
A category of locally convex Lie algebroids
研究一阶局部凸李代数胚，在巴斯蒂亚尼微积分背景下，定义相关层和态射证明其构成范畴，还研究表示等，表明局部凸李群胚有一阶李代数胚及态射，给出相关应用如证明流代数胚性质及李 II 定理。
We study Hamiltonian spaces associated with pairs (E,A), where E is a Courant algebroid and A\subset E is a Dirac structure. These spaces are defined in terms of morphisms of Courant algebroids with suitable compatibility conditions. Several of their properties are discussed, including a reduction procedure. This set-up encompasses familiar moment map theories, such as group-valued moment maps, and it provides an intrinsic approach from which different geometrical descriptions of moment maps can be naturally derived. As an application, we discuss the relationship between quasi-Poisson and pres…
Courant algebroids provide a useful mathematical tool (not only) in string theory. It is thus important to define and examine their morphisms. To some extent, this was done before using an analogue of canonical relations known from symplectic geometry. However, it turns out that applications in physics require a more general notion. We aim to provide a self-contained and detailed treatment of Courant algebroid relations and morphisms. A particular emphasis is placed on providing enough motivating examples. In particular, we show how Poisson-Lie T-duality and Kaluza-Klein reduction of supergrav…
We study VB-groupoids and VB-algebroids, which are vector bundles in the realm of Lie groupoids and Lie algebroids. Through a suitable reformulation of their definitions, we elucidate the Lie theory relating these objects, i.e., their relation via differentiation and integration. We also show how to extend our techniques to describe the more general Lie theory underlying double Lie algebroids and LA-groupoids.
We introduce Riemannian Lie algebroids as a generalization of Riemannian manifolds and we show that most of the classical tools and results known in Riemannian geometry can be stated in this setting. We give also some new results on the integrability of Riemannian Lie algebroids.
On Complex Lie Algebroids with Constant Real Rank
该研究为复李代数胚定义实元素分布与实秩不变量，推导常实秩下的实李代数胚、极小子复李代数胚及局部分解，还引入斜代数胚复匹配对刻画实向量丛复化上的所有复李代数胚结构。
In this paper, we develop the theory of Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids introduced by Cai, Liu and Sheng. Specifically, we introduce the notions of Hom-Poisson, Hom-Nijenhuis and Hom-Poisson-Nijenhuis structures on a Hom-Lie algebroid and the notion of Hom-Dirac structures on a Hom-Courant algebroid. We show that these structures satisfy similar properties to structures non "Hom-"version. For example, there exists the hierarchy of a Hom-Poisson-Nijenhuis structure and we have a relation between Hom-Dirac structures and Maurer-Cartan type equation. Moreover w…
We show that the path construction integration of Lie algebroids by Lie groupoids is an actual equivalence from the category of integrable Lie algebroids and complete Lie algebroid comorphisms to the category of source 1-connected Lie groupoids and Lie groupoid comorphisms. This allows us to construct an actual symplectization functor in Poisson geometry. We include examples to show that the integrability of comorphisms and Poisson maps may not hold in the absence of a completeness assumption.
We prove the existence of a local analytic Levi decomposition for analytic Poisson structures and Lie algebroids.
It is a classical fact in Poisson geometry that the cotangent bundle of a Poisson manifold has the structure of a Lie algebroid. Manifestations of this structure are the Lichnerowicz differential on multivector fields (calculating Poisson cohomology) and the Koszul bracket of differential forms. "Raising indices" by the Poisson tensor maps the de Rham differential to the Lichnerowicz differential and the Koszul bracket to the Schouten bracket. In this paper, we present a homotopy analog of the above results. When an ordinary Poisson structure is replaced by a homotopy one, instead of a single …
We show that well known structures on Lie algebroids can be viewed as Nijenhuis tensors or pairs of compatible tensors on Courant algebroids. We study compatibility and construct hierarchies of these structures.
We define an abstract notion of double Lie algebroid, which includes as particular cases: (1) the double Lie algebroid of a double Lie groupoid in the sense of the author, such as the iterated tangent bundle of an ordinary manifold, and various iterated tangent/cotangent constructions in symplectic and Poisson geometry; (2) the double of a Lie bialgebra; (3) the double of a matched pair, or twilled extension, of Lie algebras, and a corresponding construction for matched pairs of general Lie algebroids in the sense of Mokri and of J.-H. Lu. We prove that, given any abstract Lie bialgebroid, the…
Finite-Rank Lie Algebroids for Singular Foliations of Prescribed Vanishing Order
该研究构造了一族有限秩李代数胚，其锚映射可实现原点处指定消失阶向量场的奇异叶状结构，该构造源自一类左对称代数胚的次相邻李代数胚。
Chern-Simons gauge theories in 3 dimensions and the Poisson Sigma Model (PSM) in 2 dimensions are examples of the same theory, if their field equations are interpreted as morphisms of Lie algebroids and their symmetries (on-shell) as homotopies of such morphisms. We point out that the (off-shell) gauge symmetries of the PSM in the literature are not globally well-defined for non-parallelizable Poisson manifolds and propose a covariant definition of them as left action of some finite-dimensional Lie algebroid. Our approach allows to avoid complications arising in the infinite dimensional super-…
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…
We extend the notion of connection in order to be able to study singular geometric structures, namely, we consider a notion of connection on a Lie algebroid which is a natural extension of the usual concept of connection. Using connections, we are able to define holonomy of the orbit foliation of a Lie algebroid and prove a stability theorem. We also introduce secondary or exotic characteristic classes that generalize the modular class of a Lie algebroid.
We study the stability of singular points for smooth Poisson structures as well as general Lie algebroids. We give sufficient conditions for stability lying on the first (not necessarily linear) approximation of the given Poisson structure or Lie algebroid at a singular point. The main tools used here are the classical Lichnerowicz-Poisson cohomology and the deformation cohomology for Lie algebroids recently introduced by Crainic and Moerdijk. We also provide several examples of stable singular points of order $k \geq 1$ for Poisson structures and Lie algebroids. Finally, we apply our results …
We define hom-Lie algebroids, a definition that may seem cumbersome at first, but which is justified, first, by a one-to-one corespondence with hom-Gerstenhaber algebras, a notion that we also introduce, and several examples, including hom-Poisson structures.
Integrating Lie algebroids via stacks and applications to Jacobi manifolds
针对李代数胚无法总能积分成李群胚的问题，本文引入“温斯坦群胚”（类群胚公理的可微栈）解决其积分问题，并将结果应用于雅可比流形构造接触群胚，还拓展到预量子化等领域。
We introduce the category of holomorphic string algebroids, whose objects are Courant extensions of Atiyah Lie algebroids of holomorphic principal bundles, as considered by Bressler, and whose morphisms correspond to inner morphisms of the underlying holomorphic Courant algebroids in the sense of Severa. This category provides natural candidates for Atiyah Lie algebroids of holomorphic principal bundles for the (complexified) string group and their morphisms. Our main results are a classification of string algebroids in terms of Cech cohomology, and the construction of a locally complete famil…
If $A$ is a Lie algebroid over a foliated manifold $(M,\mathcal{F})$, a foliation of $A$ is a Lie subalgebroid $B$ with anchor image $T\mathcal{F}$ and such that $A/B$ is locally equivalent with Lie algebroids over the slice manifolds of $\mathcal{F}$. We give several examples and, for foliated Lie algebroids, we discuss the following subjects: the dual Poisson structure and Vaintrob's super-vector field, cohomology and deformations of the foliation, integration to a Lie groupoid. In the last section, we define a corresponding notion of a foliation of a Courant algebroid $A$ as a bracket-close…
In this thesis we study geometric structures from Poisson and generalized complex geometry with mild singular behavior using Lie algebroids. The process of lifting such structures to their Lie algebroid version makes them less singular, as their singular behavior is incorporated in the anchor of the Lie algebroid. We develop a framework for this using the concept of a divisor, which encodes the singularities, and show when structures exhibiting such singularities can be lifted to a Lie algebroid built out of the divisor. Once one has successfully lifted the structure, it becomes possible to st…
Reduction of Poisson manifolds with Hamiltonian Lie algebroids
本文提出了具有哈密顿李代数胚的Poisson流形的约化定理，通过引入相容动量截面概念并证明其为李代数同态，得出其零水平集商空间为Poisson流形。
This article addresses the problem of developing an extension of the Marsden- Weinstein reduction process to symplectic Lie algebroids, and in particular to the case of the symplectic cover of a fiberwise linear Poisson structure, whose reduction process is the analogue to cotangent bundle reduction in the context of Lie algebroids.
Following Sullivan's spacial realization of a differential algebra, we construct a universal integrating Lie 2-groupoid for every Lie algebroid. Then We show that unlike Lie algebras which one-to-one correspond to simply connected Lie groups, Lie algebroids (integrable or not) one-to-one correspond to a sort of etale Lie 2-groupoids with 2-connected source fibres. Finally, we discuss how to lift Lie algebroid morphisms to Lie 2-groupoid morphisms (Lie II Theorem).
Inspired by the Poisson Sigma Model and its relation to 2d gravity, we consider models governing morphisms from TSigma to any Lie algebroid E, where Sigma is regarded as d-dimensional spacetime manifold. We address the question of minimal conditions to be placed on a bilinear expression in the 1-form fields, S^ij(X) A_i A_j, so as to permit an interpretation as a metric on Sigma. This becomes a simple compatibility condition of the E-tensor S with the chosen Lie algebroid structure on E. For the standard Lie algebroid E=TM the additional structure is identified with a Riemannian foliation of M…
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…
The word `double' was used by Ehresmann to mean `an object X in the category of all X'. Double categories, double groupoids and double vector bundles are instances, but the notion of Lie algebroid cannot readily be doubled in the Ehresmann sense, since a Lie algebroid bracket cannot be defined diagrammatically. In this paper we use the duality of double vector bundles to define a notion of double Lie algebroid, and we show that this abstracts the infinitesimal structure (at second order) of a double Lie groupoid. We further show that the cotangent of either Lie algebroid in a Lie bialgebroid h…
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 has been withdrawn by UCHINO.
Morphisms of double Lie groupoids: a simplicial approach
本文从单纯观点研究双李群胚的态射，扩展余对角函子至主双纤维丛和方形态射，并应用于非阿贝尔gerbes模型等价及传递Courant algebroid中Manin三元组的积分细化，证明积分在辛Morita等价意义下良定义。
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 show how to reduce, under certain regularities conditions, a Poisson-Nijenhuis Lie algebroid to a symplectic-Nijenhuis Lie algebroid with nondegenerate Nijenhuis tensor. We generalize the work done by Magri and Morosi for the reduction of Poisson-Nijenhuis manifolds. The choice of the more general framework of Lie algebroids is motivated by the geometrical study of some reduced bi-Hamiltonian systems. An explicit example of reduction of a Poisson-Nijenhuis Lie algebroid is also provided.
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…
In this paper, we investigate families of singular holomorphic Lie algebroids on complex analytic spaces. We introduce and study a special type of deformation called unfoldings of Lie algebroids, which generalizes the theory of singular holomorphic foliations developed by T. Suwa. We show that a one to one correspondence between transversal unfoldings and holomorphic flat connections on a natural Lie algebroid on the bases exists.
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 $M$.
Constructions of $3$-Lie algebroids
本文通过由有限族微分算子和对偶截面生成的联络，研究李代数胚和$3$-李代数胚的构造，给出判定条件并应用于泊松李代数胚。
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.
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.
Contact Tulczyjew Geometry for Continuous and Discrete Dissipative Dynamics on Skew Algebroids
本文发展了斜代数胚上耗散动力学的接触Tulczyjew形式体系，通过引入Euler向量场贡献得到局部接触Tulczyjew态射，并推广到离散情形，建立了隐式耗散动力学和离散Herglotz极值条件。
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…
We prove that direct limits of finite dimensional Lie algebroids and their prolongations can be endowed with structures of convenient spaces.
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 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…
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.
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