Constantin M. Arcuş
In this paper we introduce the notion of generalized Lie algebroid and we develop a new formalism necessary to obtain a new solution for the Weistein's Problem. Many applications emphasize the importance and the utility of this new framework determined by the introduction of generalized Lie algebroids.
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This paper develops the notion of implicit Lagrangian systems on Lie algebroids and a Hamilton--Jacobi theory for this type of system. The Lie algebroid framework provides a natural generalization of classical tangent bundle geometry. We define the notion of an implicit Lagrangian system on a Lie algebroid $E$ using Dirac structures on the Lie algebroid prolongation $\T^EE^*$. This setting includes degenerate Lagrangian systems with nonholonomic constraints on Lie algebroids.
In this paper, some of formulations of Hamilton-Jacobi equations for Hamiltonian system on Lie algebroids are given. Here we use the general properties of Lie algebroids to express and prove two geometric version of the Hamilton-Jacobi theorem for Hamiltonian system on Lie algebroids. Then this results are generalized and two types of time-dependent Hamilton-Jacobi theorem of Hamiltonian system on Lie algebroids are obtained.
In this paper we discuss the relation between the unimodularity of a Lie algebroid $τ_{A}: A\to Q$ and the existence of invariant volume forms for the hamiltonian dynamics on the dual bundle $A^{*}$. The results obtained in this direction are applied to several hamiltonian systems on different examples of Lie algebroids.
In this paper we extend the almost complex Poisson structures from almost complex manifolds to almost complex Lie algebroids. Examples of such structures are also given and the almost complex Poisson morphisms of almost complex Lie algebroids are studied.
We develop a theory of gauge and dynamical equivalence for Lagrangian systems on Lie algebroids, also studying its relationship with Noether and non-Noether conserved quantities.
We will prove that the generalized Lie algebroid is a distinguished example by Lie algebroid. The generality of it with respect to the Lie algebroid is similar with the generality of the pull-back vector bundle with respect to the vector bundle. Next, we will prove that the proof of Theorem 3.1 from [15] is a misconception and the mentioned Theorem has no validity. Finally, we anatomize an optimal control problem solvable in the generalized Lie algebroid framework whereas Lie algebroid instrumentation can not solve it.
In this paper, the geometric approach to the virial theorem developed in \cite{CFR12} is written in terms of quasi-velocities (see \cite{CNCS07}). A generalization of the virial theorem for mechanical systems on Lie algebroids is also given, using the geometric tools of Lagrangian and Hamiltonian mechanics on the prolongation of the Lie algebroid.
A Survey of Lagrangian Mechanics and Control on Lie algebroids and groupoids
在这篇综述中，我们给出了李代数胚上拉格朗日力学和哈密顿力学的几何描述。李代数体形式主义的灵活性，使我们能够分析系统的非完整约束，机械控制系统，离散力学和扩展经典场论在一个单一的框架。讨论中的各种例子说明了这种方法的合理性。
In this paper, first we introduce the notion of a $\VB$-Lie $2$-algebroid, which can be viewed as the categorification of a $\VB$-Lie algebroid. The tangent prolongation of a Lie $2$-algebroid is a $\VB$-Lie $2$-algebroid naturally. We show that after choosing a splitting, there is a one-to-one correspondence between $\VB$-Lie $2$-algebroids and flat superconnections of a Lie 2-algebroid on a 3-term complex of vector bundles. Then we introduce the notion of a $\VB$-$\LWX$ 2-algebroid, which can be viewed as the categorification of a $\VB$-Courant algebroid. We show that there is a one-to-one c…
In this paper, we study representations of Hom-Lie algebroids, give some properties of Hom-Lie algebroids and discuss equivalent statements of Hom-Lie algebroids. Then, we prove that two known definitions of Hom-Lie algebroids can be transformed into each other under some conditions.
In this paper, first we modify the definition of a Hom-Lie algebroid introduced by Laurent-Gengoux and Teles and give its equivalent dual description. Many results that parallel to Lie algebroids are given. In particular, we give the notion of a Hom-Poisson manifold and show that there is a Hom-Lie algebroid structure on the pullback of the cotangent bundle of a Hom-Poisson manifold. Then we give the notion of a Hom-Lie bialgebroid, which is a natural generalization of a purely Hom-Lie bialgebra and a Lie bialgebroid. We show that the base manifold of a Hom-Lie bialgebroid is a Hom-Poisson man…
In this paper, first we introduce a new approach to the notion of $F$-algebroids, which is a generalization of $F$-manifold algebras and $F$-manifolds, and show that $F$-algebroids are the corresponding semi-classical limits of pre-Lie formal deformations of commutative associative algebroids. Then we use the deformation cohomology of pre-Lie algebroids to study pre-Lie infinitesimal deformations and extension of pre-Lie $n$-deformations to pre-Lie $(n+1)$-deformations of a commutative associative algebroid. Next we develop the theory of Dubrovin's dualities of $F$-algebroids with eventual ide…
We introduce and study a class of Lie algebroids associated to faithful modules which is motivated by the notion of cotangent Lie algebroids of Poisson manifolds. We also give a classification of transitive Lie algebroids and describe Poisson algebras by using the notions of algebroid and Lie connections.
Nambu-Dirac Structures on Lie Algebroids
流形上的Nambu-Poisson结构理论被推广到李代数胚的上下文中，以一种自然的方式基于与李代数胚上同调相关的Vinogradov括号。证明了在一定的假设下，李代数胚上的Nambu-Poisson结构是可分解的，并引入了李代数胚上的高阶Dirac结构的概念。这允许在相同的设置中描述李代数胚上的Nambu-Poisson结构和流形上的Dirac结构。
Weighted Lie algebroids were recently introduced as Lie algebroids equipped with an additional compatible non-negative grading, and represent a wide generalisation of the notion of a VB -algebroid. There is a close relation between two term representations up to homotopy of Lie algebroids and VB - algebroids. In this paper we show how this relation generalises to weighted Lie algebroids and in doing so we uncover new and natural examples of higher term representations up to homotopy of Lie algebroids. Moreover, we show how the van Est theorem generalises to weighted objects.
A theorem of Maurer-Cartan type for Lie algebroids is presented. Suppose that any vector subbundle of a Lie algebroid is called interior differential system (IDS) for that Lie algebroid. A theorem of Cartan type is obtained. Extending the classical notion of exterior differential system (EDS) to Lie algebroids, a theorem of Cartan type is obtained.
We examine Lie (super)algebroids equipped with a homological section, i.e., an odd section that `self-commutes', we refer to such Lie algebroids as inner Q-algebroids: these provide natural examples of suitably "superised" Q-algebroids in the sense of Mehta. Such Lie algebroids are a natural generalisation of Q-manifolds and Lie superalgebras equipped with a homological element. Amongst other results, we show that, via the derived bracket formalism, the space of sections of an inner Q-algebroid comes equipped with an odd Loday-Leibniz bracket.
Lagrangian submanifolds and dynamics on Lie algebroids
在以前的一些论文中，拉格朗日力学的李代数胚的几何描述已经发展。本文给出了李代数胚上力学的一个Hamilton描述。此外，我们引入了辛李代数胚的拉格朗日子流形的概念，并证明了李代数胚上的拉格朗日（哈密顿）动力学可以用辛李代数胚的拉格朗日子流形来描述。李代数胚上的拉格朗日（哈密顿）形式主义允许处理不一定定义在切（余切）丛上的拉格朗日（哈密顿）函数。因此，我们可以将我们的结果应用于在对称李群作用下不变的拉格朗日（哈密顿）函数的投影。由此得到Lagrange-Poincare（Hamilton-Poincare）方程是与相应的Atiyah代数体相关联的Euler-Lagrange（Hamilton）方程。证明了Lagrange-Poincare（Hamilton-Poincare）方程是定义辛Atiyah代数胚的某些拉格朗日子流形的局部方程.
We consider homotopy actions of a Lie algebroid on a graded manifold, defined as suitable $L_{\infty}$-algebra morphisms. On the "semi-direct product" we construct a homological vector field that projects to the Lie algebroid. Our main theorem states that this construction is a bijection. Since several classical geometric structures can be described by homological vector fields as above, we can display many explicit examples, involving Lie algebroids (including extensions, representations up to homotopy and their cocycles) as well as transitive Courant algebroids.
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.
Omni-Lie algebroids
在直和丛D（E）+J（E）上定义了一个广义Courant代数胚结构，其中D（E）和J（E）分别是向量丛E的规范李代数胚和喷丛.这样的结构被称为全李代数胚，因为如果基流形是一个点，它被约化为A.Weinstein引入的全李代数。证明了E上的任何李代数体结构都被Dirac结构刻画为从J（E）到D（E）的丛映射的图。
We prove that "generalized Lie algebroid", a geometric object which appeared recently in the literature, is a misconception.
We introduce the notion of the modular class of a Lie algebroid equipped with a Nambu structure. In particular, we recover the modular class of a Nambu-Poisson manifold $M$ with its Nambu tensor $Λ$ as the modular class of the tangent Lie algebroid $TM$ with Nambu structure $Λ.$ We show that many known properties of the modular class of a Nambu-Poisson manifold that of a Lie algebropid extend to the setting of a Lie algebroid with Nambu structure. Finally, we prove that for a large class of Nambu-Poisson manifolds considered as tangent Lie algebroids with Nambu structures, the associated modul…
In this paper we study the local description of spaces of forms on transitive Lie algebroids. We use this local description to introduce global structures like metrics, $\ast$-Hodge operation and integration along the algebraic part of the transitive Lie algebroid (its kernel). We construct a Čech-de Rham bicomplex with a Leray-Serre spectral sequence. We apply the general theory to Atiyah Lie algebroids and to derivations on a vector bundle.
We study the extension of a Lie algebroid by a representation up to homotopy, including semidirect products of a Lie algebroid with such representations. The extension results in a higher Lie algebroid. We give exact Courant algebroids and string Lie 2-algebras as examples of such extensions. We then apply this to obtain a Lie 2-groupoid integration of an exact Courant algebroid.
For any Lie algebroid A, its 1-jet bundle JA is a Lie algebroid naturally and there is a representation π: JA ->DA. Denote by dJ the corresponding coboundary operator. In this paper, we realize the deformation cohomology of a Lie algebroid A introduced by M. Crainic and I. Moerdijk as the cohomology of a subcomplex (Γ(Hom(^\bulletJA,A)DA), dJ) of the cochain complex (Γ(Hom(^\bulletJA,A)), dJ).
We introduce the concept of a higher algebroid, generalizing the notions of an algebroid and a higher tangent bundle. Our ideas are based on a description of (Lie) algebroids as vector bundle comorphisms - differential relations of a special kind. In our approach higher algebroids are vector bundle comorphism between graded-linear bundles satisfying natural axioms. We provide natural examples and discuss applications in geometric mechanics.
The study of mechanical systems on Lie algebroids permits an understanding of the dynamics described by a Lagrangian or Hamiltonian function for a wide range of mechanical systems in a unified framework. Systems defined in tangent bundles, Lie algebras, principal bundles, reduced systems and constrained are included in such description. In this paper, we investigate how to derive the dynamics associated with a Lagrangian system defined on the set of admissible elements of a given Lie algebroid using Tulczyjew's triple on Lie algebroids and constructing a Lagrangian Lie subalgebroid of a symple…
Lie algebroid structures and Lagrangian systems on affine bundles
作为前几篇文章的继续，我们利用仿射丛的标准浸入到它的双对偶中的方法，研究了仿射丛上的李代数体结构的概念。我们特别注意的延长和各种提升程序，并在仿射李代数体的拉格朗日型动力学的几何结构。
Lie n-algebroids and Lie infinity algebroids are usually thought of exclusively in supergeometric or algebraic terms. In this work, we apply the higher derived brackets construction to obtain a geometric description of Lie n-algebroids by means of brackets and anchors. Moreover, we provide a geometric description of morphisms of Lie n-algebroids over different bases, give an explicit formula for the Chevalley-Eilenberg differential of a Lie n-algebroid, compare the categories of Lie n-algebroids and NQ-manifolds, and prove some conjectures of Sheng and Zhu [SZ11].
Poisson-Lie T-duality is explained using the language of Courant algebroids.
BRST Cohomology is Lie Algebroid Cohomology
本文证明Atiyah李代数胚的外代数推广了BRST复形，通过建立李代数胚同构与规范等价联络的对应关系，将主丛导出的代数胚划分为等价类，并利用平凡化代数胚的外代数重构BRST复形，从而为一致与协变量子反常提供统一的几何化理解。
This paper studies differential graded modules and representations up to homotopy of Lie $n$-algebroids, for general $n\in\mathbb{N}$. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy are explained. In particular, the case of Lie 2-algebroids is analysed in detail. The compatibility of a Poisson bracket with the homological vector field of a Lie $n$-algebroid is shown to be equivalent to a morphism from the coadjoint module to the adjoint module, leading to an alternative characterisation of non-de…
The language of Lagrangian submanifolds is used to extend a geometric characterization of the inverse problem of the calculus of variations on tangent bundles to regular Lie algebroids. Since not all closed sections are locally exact on Lie algebroids, the Helmholtz conditions on Lie algebroids are necessary but not sufficient, so they give a weaker definition of the inverse problem. As an application the Helmholtz conditions on Atiyah algebroids are obtained so that the relationship between the inverse problem and the reduced inverse problem by symmetries can be described. Some examples and c…
Modular classes of regular twisted Poisson structures on Lie algebroids
利用Poisson映射的象的典范李代数体表示，导出了具有正则扭Poisson结构的李代数体的模类的一个公式。我们利用这个公式计算了具有扭曲三角r-矩阵的李代数的模类。研究了与Frobenius李代数相关的r-矩阵的特殊情况。
Omni-Lie algebroids are generalizations of Alan Weinstein's omni-Lie algebras. A Dirac structure in an omni-Lie algebroid $\dev E\oplus \jet E$ is necessarily a Lie algebroid together with a representation on $E$. We study the geometry underlying these Dirac structures in the light of reduction theory. In particular, we prove that there is a one-to-one correspondence between reducible Dirac structures and projective Lie algebroids in $\huaT=TM\oplus E$; we establish the relation between the normalizer $N_{L}$ of a reducible Dirac structure $L$ and the derivation algebra $\Der(\pomnib (L))$ of …
In this paper, we introduce the notion of an omni $n$-Lie algebra and show that they are linearization of higher analogues of standard Courant algebroids. We also introduce the notion of a nonabelian omni $n$-Lie algebra and show that they are linearization of higher analogues of Courant algebroids associated to Nambu-Poisson manifolds.
We show how the relation between $Q$-manifolds and Lie algebroids extends to ``higher'' or ``non-linear'' analogs of Lie algebroids. We study the identities satisfied by a new algebraic structure that arises as a replacement of operations on sections of a Lie algebroid. When the base is a point, we obtain a generalization of Lie superalgebras.
For any regular Lie algebroid $A$, the kernel $K$ and the image $F$ of its anchor map $ρ_A$, together with $A$ itself fit into a short exact sequence, called Atiyah sequence, of Lie algebroids. We prove that Atiyah and Todd classes of dg manifolds arising from regular Lie algebroids respect the Atiyah sequence. That is, the Atiyah and Todd classes of $A$ restrict to the Atiyah and Todd classes of the bundle $K$ of Lie algebras on the one hand, and project onto the Atiyah and Todd classes of the integrable distribution $F \subseteq T_M$ on the other hand.
A Lie algebroid framework for non-holonomic systems
为了得到一个既能描述非完整力学系统又能描述具有对称性的非完整力学系统的框架，本文引入了李代数体的子丛上的拉格朗日系统的概念.
The $L_\infty$-algebra is an algebraic structure suitable for describing deformation problems. In this paper we construct one $L_\infty$-algebra, which turns out to be a differential graded Lie algebra, to control the deformations of Lie algebroids and a second one to control the deformations of Lie subalgebroids. We also combine these two $L_\infty$-algebras into one to control the simultaneous deformations of a Lie algebroid and its Lie subalgebroids. The results generalize the deformation theory of Lie algebra and Lie subalgebras. Applications of our results include deformations of foliatio…
In this paper, first we give a detailed study on the structure of a transitive Lie 2-algebroid and describe a transitive Lie 2-algebroid using a morphism from the tangent Lie algebroid TM to a strict Lie 3-algebroid constructed from derivations. Then we introduce the notion of a quadratic Lie 2-algebroid and define its first Pontryagin class, which is a cohomology class in H^5(M). Associated to a CLWX 2-algebroid, there is a quadratic Lie 2-algebroid naturally. Conversely, we show that the first Pontryagin class of a quadratic Lie 2-algebroid is the obstruction class of the existence of a CLWX…
Given a Lie algebroid with a representation, we construct a graded Lie algebra whose Maurer-Cartan elements characterize relative Rota-Baxter operators on Lie algebroids. We give the cohomology of relative Rota-Baxter operators and study infinitesimal deformations and extendability of order $n$ deformations to order $n+1$ deformations of relative Rota-Baxter operators in terms of this cohomology theory. We also construct a graded Lie algebra on the space of multi-derivations of a vector bundle whose Maurer-Cartan elements characterize left-symmetric algebroids. We show that there is a homomorp…
In this paper, we construct a homotopy Poisson algebra of degree 3 associated to a split Lie 2-algebroid, by which we give a new approach to characterize a split Lie 2-bialgebroid. We develop the differential calculus associated to a split Lie 2-algebroid and establish the Manin triple theory for split Lie 2-algebroids. More precisely, we give the notion of a strict Dirac structure and define a Manin triple for split Lie 2-algebroids to be a CLWX 2-algebroid with two transversal strict Dirac structures. We show that there is a one-to-one correspondence between Manin triples for split Lie 2-alg…
We introduce the notion of G-algebroid, generalising both Lie and Courant algebroids, as well as the algebroids used in $E_{n(n)}\times\mathbb{R}^+$ exceptional generalised geometry for $n\in\{3,\dots,6\}$. Focusing on the exceptional case, we prove a classification of "exact" algebroids and translate the related classification of Leibniz parallelisable spaces into a tractable algebraic problem. After discussing the general notion of Poisson-Lie duality, we show that the Poisson-Lie U-duality is compatible with the equations of motion of supergravity.
In this paper, we introduce the notion of a pre-symplectic algebroid, and show that there is a one-to-one correspondence between pre-symplectic algebroids and symplectic Lie algebroids. This result is the geometric generalization of the relation between left-symmetric algebras and symplectic (Frobenius) Lie algebras. Although pre-symplectic algebroids are not left-symmetric algebroids, they still can be viewed as the underlying structures of symplectic Lie algebroids. %We study three classes of pre-symplectic algebroids in detail. Then we study exact pre-symplectic algebroids and show that the…
Equivariant cohomology and localization for Lie algebroids
设M是一个带有李群G作用的流形，A是M上具有相容无穷小G作用的李代数胚.从这些数据中，我们构造了一个等变李代数体上同调，并证明了紧G的相关局部化公式。作为应用，我们证明了一个Bott型公式。
In this paper, we first discuss the relation between VB-Courant algebroids and E-Courant algebroids and construct some examples of E-Courant algebroids. Then we introduce the notion of a generalized complex structure on an E-Courant algebroid, unifying the usual generalized complex structures on even-dimensional manifolds and generalized contact structures on odd-dimensional manifolds. Moreover, we study generalized complex structures on an omni-Lie algebroid in detail. In particular, we show that generalized complex structures on an omni-Lie algebra $\gl(V)\oplus V$ correspond to complex Lie …
The equations for the critical points of the action functional defined by a Lagrangian depending on higher-order derivatives of admissible curves on a Lie algebroid are found. The relation with Euler-Poincaré and Lagrange Poincaré type equations is studied. Reduction and reconstruction results for such systems are established.
We define and examine the notion of a Killing section of a Riemannian Lie algebroid as a natural generalisation of a Killing vector field. We show that the various expression for a vector field to be Killing naturally generalise to the setting of Lie algebroids. As an application we examine the internal symmetries of a class of sigma models for which the target space is a Riemannian Lie algebroid. Critical points of these sigma models are interpreted as generalised harmonic maps.
Variational calculus on Lie algebroids
证明了李代数体上的拉格朗日系统的欧拉-拉格朗日方程可以作为定义在Banach曲线流形上的作用泛函的临界点的方程。研究了降阶理论及其与拉格朗日乘子法的关系。
Poisson structure on predual of Banach Lie algebroid
本文在Banach李代数胚的预对偶丛上构造线性Poisson括号，作为对偶丛线性次Poisson结构研究的补充，还讨论了奇异Banach李代数胚的存在性并给出相关例子与情形分析。
Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids
本文从实Poisson几何的角度研究了全纯Poisson流形和全纯李代数胚。利用Magri-Morosi的Poisson Nijenhuis结构给出了全纯Poisson结构的一个特征，并描述了一个计算全纯Poisson上同调的二重复形.证明了向量丛$A\to X$上的全纯李代数胚结构等价于一对匹配的复李代数胚$(T^{0,1}X,A^{1,0})$，在Lu意义下. $A$的全纯李代数胚上同调同构于椭圆李代数胚$T^{0,1}X\bowtie A^{1,0}$的上同调。当$(X,π)$是一个全纯Poisson流形且$A=(T^*X)_π$时，这样的椭圆李代数体与对应于全纯Poisson流形的广义复结构的Dirac结构一致。
Hamilton Lie algebroids over Dirac structures and sigma models
本文提出Hamilton李代数和动量截面作为Dirac结构上的推广，应用于构造sigma模型，扩展了Poisson流形上的Hamilton结构。
We show how to extend the construction of Tulczyjew triples to Lie algebroids via graded manifolds. We also provide a generalisation of triangular Lie bialgebroids as higher Poisson and Schouten structures on Lie algebroids.
We generalize the $(n+1)$-dimensional twisted $R$-Poisson topological sigma model with flux on a target Poisson manifold to a Lie algebroid. Analyzing consistency of constraints in the Hamiltonian formalism and the gauge symmetry in the Lagrangian formalism, geometric conditions of the target space to make the topological sigma model consistent are identified. The geometric condition is an universal compatibility condition of a Lie algebroid with the multi-symplectic structure. This condition is a generalization of the momentum map theory of a Lie group and is regarded as a generalization of t…
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