Mackenzie theory and Q-manifolds

Linear duals of graded bundles and higher analogues of (Lie) algebroids

Graded bundles are a class of graded manifolds which represent a natural generalisation of vector bundles and include the higher order tangent bundles as canonical examples. We present and study the concept of the linearisation of graded bundle which allows us to define the notion of the linear dual of a graded bundle. They are examples of double structures, graded-linear ($\mathcal{GL}$) bundles, including double vector bundles as a particular case. On $\mathcal{GL}$-bundles we define what we shall call weighted algebroids, which are to be understood as algebroids in the category of graded bu…

Distributions and quotients on degree 1 NQ-manifolds and Lie algebroids

It is well-known that a Lie algebroid A is equivalently described by a degree 1 Q-manifold M. We study distributions on M, giving a characterization in terms of A. We show that involutive Q-invariant distributions on M correspond bijectively to IM-foliations on A (the infinitesimal version of Mackenzie's ideal systems). We perform reduction by such distributions, and investigate how they arise from non-strict actions of strict Lie 2-algebras on M.

Jacobi algebroids and quasi Q-manifolds

We reformulate the notion of a Jacobi algebroid in terms of weighted odd Jacobi brackets. We then show how a Jacobi algebroid can be understood in terms of a kind of curved Q-manifold. In particular the homological condition on the odd vector field is deformed in a very specific way. This leads to the notion of a quasi Q-manifold.

Q-groupoids and their cohomology

We approach Mackenzie's LA-groupoids from a supergeometric point of view by introducing Q-groupoids, which are groupoid objects in the category of Q-manifolds. There is a faithful functor from the category of LA-groupoids to the category of Q-groupoids. We associate to every Q-groupoid a double complex that provides a model for the Q-cohomology of the classifying space. As examples, we obtain models for equivariant Q- and orbifold Q-cohomology, and for equivariant Lie algebroid and orbifold Lie algebroid cohomology. We obtain double complexes associated to Poisson groupoids and groupoid-algebr…

Odd Jacobi manifolds: general theory and applications to generalised Lie algebroids

In this paper we define a Grassmann odd analogue of Jacobi structure on a supermanifold. The basic properties are explored. The construction of odd Jacobi manifolds is then used to reexamine the notion of a Jacobi algebroid. It is shown that Jacobi algebroids can be understood in terms of a kind of curved Q-manifold, which we will refer to as a quasi Q-manifold.

（预）多重辛流形上李代数胚的相容 E-微分形式

Compatible $E$-Differential Forms on Lie Algebroids over (Pre-)Multisymplectic Manifolds

本文在（预）多重辛流形上的李代数胚中引入相容 E-n-形式，统一刻画扭曲泊松结构、动量映射及其同伦推广等结构。

N-manifolds of degree 2 and metric double vector bundles

This paper shows the equivalence of the categories of $N$-manifolds of degree $2$ with the category of double vector bundles endowed with a linear metric. Split Poisson $N$-manifolds of degree $2$ are shown to be equivalent to self-dual representations up to homotopy. As a consequence, the equivalence above induces an equivalence between so called metric VB-algebroids and Poisson $N$-manifolds of degree $2$. Then a new description of split Lie $2$-algebroids is given, as well as their "duals", the Dorfman $2$-representations. We show that Dorfman $2$-representations are equivalent in a simple …

李子代数胚的Serre谱序列

The Serre spectral sequence of a Lie subalgebroid

本文提出并研究李子代数胚的Serre谱序列，统一推广了纤维化、李代数及李代数胚扩张中的经典谱序列，并证明其在宽子代数胚和真子流形情形下分别收敛到相应上同调，同时应用于泊松几何。

VB-Courant algebroids, E-Courant algebroids and generalized geometry

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 …

$\mathbb{Z}$-graded supergeometry: Differential graded modules, higher algebroid representations, and linear structures

This thesis studies the representation theory and linear structures of $\mathcal{Q}$-manifolds and higher Lie algebroids. We introduce differential graded modules (or for short DG-modules) of $\mathcal{Q}$-manifolds and the equivalent notion of representations up to homotopy in the case of Lie $n$-algebroids ($n\in\mathbb{N}$), as generalisations of the homonymous structures that exist already in the case of ordinary Lie algebroids. The adjoint and coadjoint modules are described, and the corresponding split versions of the adjoint and coadjoint representations up to homotopy of Lie $n$-algebr…

Transitive double Lie algebroids via core diagrams

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…

On Morphic Actions and Integrability of LA-Groupoids

Lie theory for the integration of Lie algebroids to Lie groupoids, on the one hand, and of Poisson manifolds to symplectic groupoids, on the other, has undergone tremendous developements in the last decade, thanks to the work of Mackenzie-Xu, Moerdijk-Mrcun, Cattaneo-Felder and Crainic-Fernandes, among others. In this thesis we study - part of - the categorified version of this story, namely the integrability of LA-groupoids (groupoid objects in the category of Lie algebroids), to double Lie groupoids (groupoid objects in the category of Lie groupoids) providing a first set of sufficient condi…

Hom-Lie algebroids, Hom-Lie bialgebroids and Hom-Courant algebroids

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…

Dirac结构上的Hamilton李代数和sigma模型

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。

$Q$-manifolds and Higher Analogs of Lie Algebroids

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.

Strong homotopy Lie algebras, homotopy Poisson manifolds and Courant algebroids

We study Maurer-Cartan elements on homotopy Poisson manifolds of degree $n$. They unify many twisted or homotopy structures in Poisson geometry and mathematical physics, such as twisted Poisson manifolds, quasi-Poisson $\g$-manifolds, and twisted Courant algebroids. Using the fact that the dual of an $n$-term $L_\infty$-algebra is a homotopy Poisson manifold of degree $n-1$, we obtain a Courant algebroid from a $2$-term $L_\infty$-algebra $\g$ via the degree $2$ symplectic NQ-manifold $T^*[2]\g^*[1]$. By integrating the Lie quasi-bialgebroid associated to the Courant algebroid, we obtain a Lie…

Deformations of Courant Algebroids and Dirac Structures via Blended Structures

Deformations of a Courant Algebroid E and its Dirac subbundle A have been widely considered under the assumption that the pseudo-Euclidean metric is fixed. In this paper, we attack the same problem in a setting that allows the pseudo-Euclidean metric to deform. Thanks to Roytenberg, a Courant algebroid is equivalent to a symplectic graded Q-manifold of degree 2. From this viewpoint, we extend the notions of graded Q-manifold, DGLA and L_\infty-algebra all to "blended" version so that Poisson manifold, Lie algebroid and Courant algebroid are unified as blended Q-manifolds; and define a submanio…

Jacobi quasi-Nijenhuis Algebroids

In this paper, for a Jacobi algebroid $A$, by introducing the notion of Jacobi quasi-Nijenhuis algebroids, which is a generalization of Poisson quasi-Nijenhuis manifolds introduced by Stiénon and Xu, we study generalized complex structures on the Courant-Jacobi algebroid $A\oplus A^*$, which unifies generalized complex (contact) structures on an even(odd)-dimensional manifold.

Nambu structures on Lie algebroids and their modular classes

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…

Graded contact manifolds and contact Courant algebroids

We develop a systematic approach to contact and Jacobi structures on graded supermanifolds. In this framework, contact structures are interpreted as symplectic principal GL(1,R)-bundles. Gradings compatible with the GL(1,R)-action lead to the concept of a graded contact manifold, in particular a linear (more generally, n-linear) contact structure. Linear contact structures are proven to be exactly the canonical contact structures on first jets of line bundles. They provide linear Kirillov (or Jacobi) brackets and give rise to the concept of a Kirillov algebroid, an analog of a Lie algebroid, f…

The standard cohomology of regular Courant algebroids

For any regular Courant algebroid $E$ over a smooth manifold $M$ with characteristic distribution $F$ and ample Lie algebroid $A_E$, we prove that there exists a canonical homological vector field on the graded manifold $A_E[1] \oplus (TM/F)^\ast[2]$ such that the resulting dg manifold $\mathcal{M}_E$, which we call the minimal model of the Courant algebroid $E$, encodes all cohomological information of $E$. Indeed, the standard cohomology of $E$ can be identified with the cohomology of the function space on $\mathcal{M}_E$, which can be computed by a Hodge-to-de Rham type spectral sequence. W…

Reduction of Poisson-Nijenhuis Lie algebroids to symplectic-Nijenhuis Lie algebroids with nondegenerate Nijenhuis tensor

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.

Holomorphic Poisson Manifolds and Holomorphic Lie Algebroids

We study holomorphic Poisson manifolds and holomorphic Lie algebroids from the viewpoint of real Poisson geometry. We give a characterization of holomorphic Poisson structures in terms of the Poisson Nijenhuis structures of Magri-Morosi and describe a double complex which computes the holomorphic Poisson cohomology. A holomorphic Lie algebroid structure on a vector bundle $A\to X$ is shown to be equivalent to a matched pair of complex Lie algebroids $(T^{0,1}X,A^{1,0})$, in the sense of Lu. The holomorphic Lie algebroid cohomology of $A$ is isomorphic to the cohomology of the elliptic Lie alge…

Deformation of algebroid bracket of differential forms and Poisson manifold

We construct the family of algebroid brackets $[\cdot,\cdot]_{c,v}$ on the tangent bundle $T^*M$ to a Poisson manifold $(M,π)$ starting from an algebroid bracket of differential forms. We use these brackets to generate Poisson structures on the tangent bundle $TM$. Next, in the case when $M$ is equipped with a bi-Hamiltonian structure $(M,π_1, π_2)$ we show how to construct another family of Poisson structures. Moreover we present how to find Casimir functions for those structures and we discuss some particular examples.

具有哈密顿李代数胚的Poisson流形的约化

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。

Equivariant cohomology and localization for Lie algebroids

Let M be a manifold carrying the action of a Lie group G, and A a Lie algebroid on M equipped with a compatible infinitesimal G-action. Out of these data we construct an equivariant Lie algebroid cohomology and prove for compact G a related localization formula. As an application we prove a Bott-type formula.

通过栈积分李代数胚及其在雅可比流形上的应用

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。

Stability of higher order singular points of Poisson manifolds and Lie algebroids

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 …

Jacobi-左对称代数胚上的 Jacobi-Koszul-Vinberg 结构

Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids

本文在 Jacobi-左对称代数胚上引入 Jacobi-Koszul-Vinberg 结构，作为 Jacobi 结构的对称类比，并研究其性质，进而定义 Jacobi-Koszul-Vinberg 流形。

Graded manifolds of type $Δ$ and $n$-fold vector bundles

Vector bundles and double vector bundles, or $2$-fold vector bundles, arise naturally for instance as base spaces for algebraic structures such as Lie algebroids, Courant algebroids and double Lie algebroids. It is known that all these structures possess a unified description using the language of super\-geometry and graded manifolds of degree $\leq 2$. Indeed, a link has been established between the super and classical pictures by the geometrization process, leading to an equivalence of the category of graded manifolds of degree $\leq 2$ and the category of (double) vector bundles with additi…

On Lie Algebroids and Poisson Algebras

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.

Algebroid Structures on Para-Hermitian Manifolds

We present a global construction of a so-called D-bracket appearing in the physics literature of Double Field Theory (DFT) and show that if certain integrability criteria are satisfied, it can be seen as a sum of two Courant algebroid brackets. In particular, we show that the local picture of the extended space-time used in DFT fits naturally in the geometrical framework of para-Hermitian manifolds and that the data of an (almost) para-Hermitian manifold is sufficient to construct the D-bracket. Moreover, the twists of the bracket appearing in DFT can be interpreted in this framework geometric…

扭Courant r-Lie代数与非扭Courant (r+1)-Lie代数胚之间的$L_\infty$-态射

$L_\infty$-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。

On the connection theory of transitive Lie groupoids

Due to a result by Mackenzie, extensions of transitive Lie groupoids are equivalent to certain Lie groupoids which admit an action of a Lie group. This paper is a treatment of the equivariant connection theory and holonomy of such groupoids, and shows that such connections give rise to the transition data necessary for the classification of their respective Lie algebroids.

On higher-dimensional Courant algebroids

We define the transgression functor which associates to a (higher-dimensional) Courant algebroid on a manifold a Lie algebroid on the shifted tangent bundle of the manifold.

哈密顿李代数胚中的共动量截面与泊松映射

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。

Lie algebroid Fibrations

A degree 1 non-negative graded super manifold equipped with a degree 1 vector field Q satisfying [Q, Q]=1, namely a so-called NQ-1 manifold is, in plain differential geometry language, a Lie algebroid. We introduce a notion of fibration for such super manifols, that essentially involves a complete Ehresmann connection. As it is the case for Lie algebras, such fibrations turn out not to be just locally trivial products. We also define homotopy groups and prove the expected long exact sequence associated to a fibration. In particular, Crainic and Fernandes's obstruction to the integrability of L…

Algebroids - general differential calculi on vector bundles

A notion of an algebroid - a generalization of a Lie algebroid structure is introduced. We show that many objects of the differential calculus on a manifold M associated with the canonical Lie algebroid structure on T^M can be obtained in the framework of a general algebroid. Also a compatibility condition which leads, in general, to a concept of a bialgebroid.

Momentum section on Courant algebroid and constrained Hamiltonian mechanics

We propose a generalization of the momentum map on a symplectic manifold with a Lie algebra action to a Courant algebroid structure. The theory of a momentum section on a Lie algebroid is generalized to the theory compatible with a Courant algebroid. As an example, we identify the momentum section in a constrained Hamiltonian mechanics with Courant algebroid symmetry. Moreover, we construct cohomological formulations by considering the BFV and BV formalism of this Hamiltonian system. The Weil algebra for this structure is constructed.

On Curvature and Torsion in Courant Algebroids

We study the graded geometric point of view of curvature and torsion of Q-manifolds (differential graded manifolds). In particular, we get a natural graded geometric definition of Courant algebroid curvature and torsion, which correctly restrict to Dirac structures. Depending on an auxiliary affine connection K, we introduce the K-curvature and K-torsion of a Courant algebroid connection. These are conventional tensors on the body. Finally, we compute their Ricci and scalar curvature.

Complex Lie algebroids and ACH manifolds

We propose the definition of a Manifold with a complex Lie structure at infinity. The important class of ACH manifolds enters into this class.

Tulczyjew triples and higher Poisson/Schouten structures on Lie algebroids

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.

Courant algebroids: Cohomology and Matched Pairs

We introduce Courant algebroids, providing definitions, some historical notes, and some elementary properties. Next, we summarize basic properties of graded manifolds. Then, drawing on the work of Roytenberg and others, we introduce the graded or supergraded language demonstrating a cochain complex / cohomology for (general) Courant algebroids. We review spectral sequences and show how this tool is used to compute cohomology of regular Courant algebroids with a split base. Finally we analyze matched pairs of Courant algebroids including the complexified standard Courant algebroid of a complex …

Foliated Lie and Courant Algebroids

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…

A construction of Courant algebroids on foliated manifolds

For any transversal-Courant algebroid $E$ on a foliated manifold $(M,\mathcal{F})$, and for any choice of a decomposition $TM=T\mathcal{F}\oplus Q$, we construct a Courant algebroid structure on $T\mathcal{F}\oplus T^*\mathcal{F}\oplus E$.

$L_{\infty}$-actions of Lie algebroids

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.

From $L_{\infty}$-algebroids to higher Schouten/Poisson structures

We show that $L_{\infty}$-algebroids, understood in terms of Q-manifolds can be described in terms of certain higher Schouten and Poisson structures on graded (super)manifolds. This generalises known constructions for Lie (super)algebras and Lie algebroids.

Modular classes of Poisson-Nijenhuis Lie algebroids

The modular vector field of a Poisson-Nijenhuis Lie algebroid $A$ is defined and we prove that, in case of non-degeneracy, this vector field defines a hierarchy of bi-Hamiltonian $A$-vector fields. This hierarchy covers an integrable hierarchy on the base manifold, which may not have a Poisson-Nijenhuis structure.

Homotopy momentum sections on multisymplectic manifolds

We introduce a notion of a homotopy momentum section on a Lie algebroid over a pre-multisymplectic manifold. A homotopy momentum section is a generalization of the momentum map with a Lie group action and the momentum section on a pre-symplectic manifold, and is also regarded as a generalization of the homotopy momentum map on a multisymplectic manifold. We show that a gauged nonlinear sigma model with Wess-Zumino term with Lie algebroid gauging has the homotopy momentum section structure.

Homological sections of Lie algebroids

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.

Dirac generating operators of split Courant algebroids

Given a vector bundle $A$ over a smooth manifold $M$ such that the square root $\mathcal{L}$ of the line bundle $\wedge^{\mathrm{top}}A^\ast \otimes \wedge^{\mathrm{top}}T^\ast M$ exists, the Clifford bundle associated to the split pseudo-Euclidean vector bundle $(E = A \oplus A^\ast, \langle \cdot, \cdot \rangle)$, admits a spinor bundle $\wedge^\bullet A \otimes \mathcal{L}$, whose section space can be thought of as that of Berezinian half-densities of the graded manifold $A^\ast[1]$. We give an explicit construction of Dirac generating operators of split Courant algebroid (or proto-bialgebr…

Compatible structures on Lie algebroids and Monge-Ampère operators

We study pairs of structures, such as the Poisson-Nijenhuis structures, on the tangent bundle of a manifold or, more generally, on a Lie algebroid or a Courant algebroid. These composite structures are defined by two of the following, a closed 2-form, a Poisson bivector or a Nijenhuis tensor, with suitable compatibility assumptions. We establish the relationships between such composite structures. We then show that the non-degenerate Monge-Ampère structures on 2-dimensional manifolds satisfying an integrability condition provide numerous examples of such structures, while in the case of 3-dime…

Higher dimensional Lie algebroid sigma model with WZ term

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…

Fréchet Lie algebroids and their cohomology

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 $M$.