Juan Carlos Marrero, Edith Padron, Miguel Rodriguez-Olmos
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.
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For any nonempty, compact and fiberwise convex set $K$ in $T^*\mathbb{R}^n$, we prove an isomorphism between symplectic homology of $K$ and a certain relative homology of loop spaces of $\mathbb{R}^n$. We also prove a formula which computes symplectic homology capacity (which is a symplectic capacity defined from symplectic homology) of $K$ using homology of loop spaces. As applications, we prove (i) symplectic homology capacity of any convex body is equal to its Ekeland-Hofer-Zehnder capacity, (ii) a certain subadditivity property of the Hofer-Zehnder capacity, which is a generalization of a …
Several types of generically-nondegenerate Poisson structures can be effectively studied as symplectic structures on naturally associated Lie algebroids. Relevant examples of this phenomenon include log-, elliptic, $b^k$-, scattering and elliptic-log Poisson structures. In this paper we discuss topological obstructions to the existence of such Poisson structures, obtained through the characteristic classes of their associated symplectic Lie algebroids. In particular we obtain the full obstructions for surfaces to carry such Poisson structures.
Symplectic reduction and a Darboux-Moser-Weinstein theorem for Lie algebroids
本文将 Marsden-Weinstein 约化定理与 Darboux-Moser-Weinstein 定理推广至辛李代数胚，并提出了相应的余迷向嵌入定理。
We construct an infinite-dimensional symplectic 2-groupoid as the integration of an exact Courant algebroid. We show that every integrable Dirac structure integrates to a "Lagrangian" sub-2-groupoid of this symplectic 2-groupoid. As a corollary, we recover a result of Bursztyn-Crainic-Weinstein-Zhu that every integrable Dirac structure integrates to a presymplectic groupoid.
Transitive Courant algebroids and double symplectic groupoids
该研究将双重辛群胚的Lu-Weinstein构造推广至传递Courant代数胚且Atiyah代数胚可积的李双代数胚，分类了李群胚上的正合扭曲Courant代数胚，证明了扭曲CA-群胚基上的叶状结构，推广了现有双重辛群胚例子。
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…
Systolic inequalities on the sphere from symplectic embeddings
本文利用 Hutchings 的辛容量估计 $T^*S^2$ 中纤维星形超曲面的 Reeb 轨道最小作用量，并引入纤维 $β$-平衡概念，证明相关收缩上界及对跖对称下 $δ$-挤扁度量的非精确估计。
Symplectic Hodge Theory on Lie Algebroids
本文研究了辛几何中李代数簇上的辛Hodge理论，探讨了Brylinksi条件、强Lefschetz条件和dδ引理的类比，并通过代数方法证明其等价性及几何应用。
Shifted symplectic Lie and $L_\infty$ algebroids model formal neighbourhoods of manifolds in shifted symplectic stacks, and serve as target spaces for twisted variants of classical AKSZ topological field theory. In this paper, we classify zero-, one- and two-shifted symplectic algebroids and their higher gauge symmetries, in terms of classical geometric "higher structures", such as Courant algebroids twisted by $Ω^2$-gerbes. As applications, we produce new examples of twisted Courant algebroids from codimension-two cycles, and we give symplectic interpretations for several well known features …
Symplectic algebroids, groupoid Toeplitz operators and deformation quantization
本文利用Toeplitz算子在辛Lie代数余结构诱导的泊松流形上定义星积，推广了Guillemin和Melrose在辛情形下的方法。
We show that the leaves of an LA-groupoid which pass through the unit manifold are, modulo a connectedness issue, Lie groupoids. We illustrate this phenomenon by considering the cotangent Lie algebroids of Poisson groupoids thus obtaining an interesting class of symplectic groupoids coming from their symplectic foliations. In particular, we show that for a (strict) Lie 2-group the coadjoint orbits of the units in the dual of its Lie 2-algebra are symplectic groupoids, meaning that the classical Kostant-Kirillov-Souriau symplectic forms on these special coadjoint orbits are multiplicative.
In this dissertation we study Courant algebroids, objects that first appeared in the work of T. Courant on Dirac structures; they were later studied by Liu, Weinstein and Xu who used Courant algebroids to generalize the notion of the Drinfeld double to Lie bialgebroids. As a first step towards understanding the complicated properties of Courant algebroids, we interpret them by associating to each Courant algebroid a strongly homotopy Lie algebra in a natural way. Next, we propose an alternative construction of the double of a Lie bialgebroid as a homological hamiltonian vector field on an even…
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…
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.
Algebroid Desingularizable Poisson Structures
本文引入了可微化泊松流形，这是一种由具有近可积锚映射的辛李余结构诱导的泊松流形，扩展了包括对数辛、$b^m$-辛、$E$-辛流形和超曲面余结构在内的结构。作者证明了实数、有限维、非阿贝尔、可约李代数的对偶体 never 允许此类余结构，并给出了两个无限族的2-step nilpotent李代数，其中一个是可微化的，另一个不是。
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 propose a definition of symplectic 2-groupoid which includes integrations of Courant algebroids that have been recently constructed. We study in detail the simple but illustrative case of constant symplectic 2-groupoids. We show that the constant symplectic 2-groupoids are, up to equivalence, in one-to-one correspondence with a simple class of Courant algebroids that we call constant Courant algebroids. Furthermore, we find a correspondence between certain Dirac structures and Lagrangian sub-2-groupoids.
Emphasizing the role of Gerstenhaber algebras and of higher derived brackets in the theory of Lie algebroids, we show that the several Lie algebroid brackets which have been introduced in the recent literature can all be defined in terms of Poisson and pre-symplectic functions in the sense of Roytenberg and Terashima. We prove that in this very general framework there exists a one-to-one correspondence between non-degenerate Poisson functions and symplectic functions. We determine the differential associated to a Lie algebroid structure obtained by twisting a structure with background by both …
Deformation quantization algebroids over a complex symplectic manifold X are locally given by rings of WKB operators, that is, microdifferential operators with an extra central parameter τ. In this paper, we will show that such algebroids are classified by H^2(X;k^*), where k^* is a subgroup of the group of invertible formal Laurent series in τ^-1.
In previous work with M.C. Fernandes, we found a Lie algebroid symmetry for the Einstein evolution equations of general relativity. The present work was motivated by the effort to explain the coisotropic structure of the constraint subset for the initial value problem by extending the notion of hamiltonian structure from Lie algebra actions to general Lie algebroids over presymplectic manifolds. After comparing possible compatibility conditions between the anchor $A\to TM$ and the presymplectic structure on the base $M$, we choose the most natural of them, given by a suitably chosen connection…
The paper starts with an interpretation of the complete lift of a Poisson structure from a manifold M to its tangent bundle TM by means of the Schouten- Nijenhuis bracket of covariant symmetric tensor fields defined by the co- tangent Lie algebroid of M. Then, we discuss Poisson structures of TM which have a graded restriction to the fiberwise polynomial algebra; they must be p-related (p is the projection of TM onto M) with a Poisson structure on M. Furthermore, we define transversal Poisson structures of a foliation, and discuss bivector fields of TM which produce graded brackets on the fibe…
On the classification of symplectic DQ-algebroids
本文通过适配 P. Deligne 的方法，证明了辛流形上局部定义的 DQ-algebroids 构成的 2-gerbe 存在典范整体截面，并借助非阿贝尔上同调与 Weyl 代数局部计算，实现了辛 DQ-algebroids 的分类。
The canonical involution of a double (=iterated) tangent bundle may be dualized in different ways to yield relations between the Tulczyjew diffeomorphism, the Poisson anchor associated with the standard symplectic structure on the cotangent space,and the reversal diffeomorphism. We show that the constructions which yield these maps extend very generally to the double Lie algebroids of double Lie groupoids, where they play a crucial role in the relations between double Lie algebroids and Lie bialgebroids.
Shifted symplectic structure on Poisson Lie algebroid and generalized complex geometry
本文在同伦意义下将广义复流形重新表述为全纯symplectic formal stack，发展shifted symplectic机制证明coisotropic intersection继承shifted Poisson structure，并研究了广义复brane。
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。
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 …
Homological reduction of Courant algebroids
该研究构建了与辛几何BFV公式体系平行的库朗代数簇哈密顿约化同伦模型，通过关联微分分次库朗-多夫曼代数，在正则假设下用其同调得到约化库朗代数簇的对应代数。
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。
In this paper, we develop a differential-graded symplectic (Batalin-Vilkovisky) version of the framework of Crawley-Boevey, Etingof and Ginzburg on noncommutative differential geometry based on double derivations to construct non-commutative analogues of the Courant algebroids introduced by Liu, Weinstein and Xu. Adapting geometric constructions of Ševera and Roytenberg for (commutative) graded symplectic supermanifolds, we express the BRST charge, given in our framework by a `homological double derivation', in terms of Van den Bergh's double Poisson algebras for graded bi-symplectic non-commu…
Poisson structures of divisor-type are those whose degeneracy can be captured by a divisor ideal, which is a locally principal ideal sheaf with nowhere-dense quotient support. This is a large class of Poisson structures which includes all generically-nondegenerate Poisson structures, such as log-, $b^k$-, elliptic, elliptic-log, and scattering Poisson structures. Divisor ideals are used to define almost-injective Lie algebroids of derivations preserving them, to which these Poisson structures can be lifted, often nondegenerately so. The resulting symplectic Lie algebroids can be studied using …
We present a theory of reduction for Courant algebroids as well as Dirac structures, generalized complex, and generalized Kähler structures which interpolates between holomorphic reduction of complex manifolds and symplectic reduction. The enhanced symmetry group of a Courant algebroid leads us to define \emph{extended} actions and a generalized notion of moment map. Key examples of generalized Kähler reduced spaces include new explicit bi-Hermitian metrics on $\CC P^2$.
We present Hausdorff versions for Lie Integration Theorems 1 and 2 and apply them to study Hausdorff symplectic groupoids arising from Poisson manifolds. To prepare for these results we include a discussion on Lie equivalences and propose an algebraic approach to holonomy. We also include subsidiary results, such as a generalization of the integration of subalgebroids to the non-wide case, and explore in detail the case of foliation groupoids.
A log symplectic manifold is a Poisson manifold which is generically nondegenerate. We develop two methods for constructing the symplectic groupoids of log symplectic manifolds. The first is a blow-up construction, corresponding to the notion of an elementary modification of a Lie algebroid along a subalgebroid. The second is a gluing construction, whereby groupoids defined on the open sets of an appropriate cover may be combined to obtain global integrations. This allows us to classify all Hausdorff symplectic groupoids of log symplectic manifolds in a combinatorial fashion, in terms of a cer…
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…
In this paper, we describe an integration of exact Courant algebroids to symplectic 2-groupoids, and we show that the differentiation procedure from [26] inverts our integration.
Using a basic idea of Sullivan's rational homotopy theory, one can see a Lie groupoid as the fundamental groupoid of its Lie algebroid. This paper studies analogues of Lie algebroids with non-trivial higher homotopy. Using various homotopy classes one can obtain e.g. central extensions of loop groups, or one can integrate a Lie biagebroid to a double symplectic groupoid. When combined with symplectic geometry, this idea leads to an infinite sequence of notions, starting with sympectic manifolds, Poisson manifolds and Courant algebroids. They are interrelated with higher-dimensional variational…
We find computable criteria for stability of symplectic leaves of Poisson manifolds. Using Poisson geometry as an inspiration, we also give a general criterion for stability of leaves of Lie algebroids, including singular ones. This not only extends but also provides a new approach (and proofs) to the classical stability results for foliations and group actions.
We prove that the cotangent of a double Lie groupoid S has itself a double groupoid structure with sides the duals of associated Lie algebroids, and double base the dual of the Lie algebroid of the core of S. Using this, we prove a result outlined by Weinstein in 1988, that the side groupoids of a general symplectic double groupoid are Poisson groupoids in duality. Further, we prove that any double Lie groupoid gives rise to a pair of Poisson groupoids (and thus of Lie bialgebroids) in duality. To handle the structures involved effectively we extend to this context the dualities and canonical …
This paper is devoted to a study of geometric structures expressible in terms of graded symplectic supermanifolds. We extend the classical BRST formalism to arbitrary pseudo-Euclidean vector bundles (E\to M_{0}) by canonically associating to such a bundle a graded symplectic supermanifold ((M,Ω)), with (\textrm{deg}(Ω)=2). Conversely, every such manifold arises in this way. We describe the algebra of functions on (M) in terms of (E) and show that ``BRST charges'' on (M) correspond to Courant algebroid structures on (E), thereby constructing the standard complex for the latter as a generalizati…
We construct Hermitian representations of Lie algebroids and associated unitary representations of Lie groupoids by a geometric quantization procedure. For this purpose we introduce a new notion of Hamiltonian Lie algebroid actions. The first step of our procedure consists of the construction of a prequantization line bundle. Next, we discuss a version of Kähler quantization suitable for this setting. We proceed by defining a Marsden-Weinstein quotient for our setting and prove a ``quantization commutes with reduction'' theorem. We explain how our geometric quantization procedure relates to a …
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。
The notions of holomorphic symplectic structures and hypercomplex structures on Courant algebroids are introduced and then proved to be equivalent. These generalize hypercomplex triples and holomorphic symplectic 2-forms on manifolds respectively. Basic properties of such structures are established.
Shifted lagrangian structures in Poisson geometry
本文通过建立移位辛几何与经典泊松几何之间的新联系，利用2-移位辛群中的拉格朗日态射，发展了狄拉克结构与拟泊松流形积分的理论。
A 2-plectic manifold is a manifold equipped with a closed nondegenerate 3-form, just as a symplectic manifold is equipped with a closed nondegenerate 2-form. In 2-plectic geometry we meet higher analogues of many structures familiar from symplectic geometry. For example, any 2-plectic manifold has a Lie 2-algebra consisting of smooth functions and Hamiltonian 1-forms. This is equipped with a Poisson-like bracket which only satisfies the Jacobi identity up to `coherent chain homotopy'. Over any 2-plectic manifold is a vector bundle equipped with extra structure called an exact Courant algebroid…
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.
We use persistence modules and their corresponding barcodes to quantitatively distinguish between different fiberwise star-shaped domains in the cotangent bundle of a fixed manifold. The distance between two fiberwise star-shaped domains is measured by a non-linear version of the classical Banach-Mazur distance, called symplectic Banach-Mazur distance and denoted by $d_{SBM}.$ The relevant persistence modules come from filtered symplectic homology and are stable with respect to $d_{SBM}.$ Our main focus is on the space of unit codisc bundles of orientable surfaces of positive genus, equipped w…
We identify the cotangent bundle Lie algebroid of a Poisson homogeneous space G/H of a Poisson Lie group G as a quotient of a transformation Lie algebroid over G. As applications, we describe the modular vector fields of G/H, and we identify the Poisson cohomology of G/H with coefficients in powers of its canonical line bundle with relative Lie algebra cohomology of the Drinfeld Lie algebra associated to G/H. We also construct a Poisson groupoid over G/H which is symplectic near the identity section. This note serves as preparation for forthcoming papers, in which we will compute explicitly th…
We show how the classical Moser Lemma from symplectic geometry extends to generalized complex structures (GCS) on arbitrary Courant algebroids. For this, we extend the notion of Lie derivative to sections of the tensor bundle $(\otimes^i E)\otimes(\otimes^j E^*)$ with respect to sections of the Courant algebroid $E$ using the Dorfman bracket. We then give a cohomological interpretation of the existence of one-parameter families of GCS on $E$ and of flows of automorphims of $E$ identifying all GCS of such a family. In the particular cases of symplectic, we recover the results of Moser. Finally,…
Fibrations and log-symplectic structures
本文通过将泊松结构视为 $b$-切丛中的辛形式，引入 $b$-超纤维化概念，证明其全空间上存在对数辛结构，并建立其与非手性 Lefschetz 纤维化和折叠辛结构的联系。
In this note we compute the cohomology of the elliptic tangent bundle, a Lie algebroid used to describe singular symplectic forms arising from generalized complex geometry.
We construct a topological sigma model and a current algebra based on a Courant algebroid structure on a Poisson manifold. In order to construct models, we reformulate the Poisson Courant algebroid by supergeometric construction on a QP-manifold. A new duality of Courant algebroids which transforms H-flux and R-flux is proposed, where the transformation is interpreted as a canonical transformation of a graded symplectic manifold.
In this paper, we study deformations of coisotropic submanifolds in a symplectic manifold. First we derive the equation that governs $C^\infty$ deformations of coisotropic submanifolds and define the corresponding $C^\infty$-moduli space of coisotropic submanifolds modulo the Hamiltonian isotopies. This is a non-commutative and non-linear generalization of the well-known description of the local deformation space of Lagrangian submanifolds as the set of graphs of {\it closed} one forms in the Darboux-Weinstein chart of a given Lagrangian submanifold. We then introduce the notion of {\it strong…
Pre-Courant algebroids are `Courant algebroids' without the Jacobi identity for the Courant-Dorfman bracket. In this paper we examine the corresponding supermanifold description of pre-Courant algebroids and some direct consequences thereof - such as the definition of (sub-)Dirac structures and the notion of the naive quasi-cochain complex. In particular we define symplectic almost Lie 2-algebroids and show how they correspond to pre-Courant algebroids. Moreover, the framework of supermanifolds allows us to economically define and work with pre-Courant algebroids equipped with a compatible non…
In the framework of the connection theory, a contravariant analog of the Sternberg coupling procedure is developed for studying a natural class of Poisson structures on fiber bundles, called coupling tensors. We show that every Poisson structure near a closed symplectic leaf can be realized as a coupling tensor. Our main result is a geometric criterion for the neighborhood equivalence between Poisson structures over the same leaf. This criterion gives a Poisson analog of the relative Darboux theorem due to Weinstein. Within the category of the algebroids, coupling tensors are introduced on the…
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…
Motivated by the study of symplectic Lie algebroids, we study a describe a type of algebroid (called an $E$-tangent bundle) which is particularly well-suited to study of singular differential forms and their cohomology. This setting generalizes the study of $b$-symplectic manifolds, foliated manifolds, and a wide class of Poisson manifolds. We generalize Moser's theorem to this setting, and use it to construct symplectomorphisms between singular symplectic forms. We give applications of this machinery (including the study of Poisson cohomology), and study specific examples of a few of them in …
Jets of foliations and $b^k$-algebroids
本文引入并分类了 $b^k$-型奇异叶状结构，揭示其与形式邻域中射流及和乐表示的关系，并研究了阶延拓的示性类阻碍、超曲面情形下的代数胚上同调以及代数胚辛形式与诱导辛变分的联系。
In this note, we prove that intrinsic characteristic classes of Lie algebroids - which in degree one recover the modular class - behave functorially with respect to arbitrary transverse maps, and in particular are weak Morita invariants. In the modular case, this result appeared in [Kosmann-Schwarzbach Y., Laurent-Gengoux C., Weinstein A., Transform. Groups 13 (2008), 727-755], and with a connectivity assumption which we here show to be unnecessary, it appeared in [Crainic M., Comment. Math. Helv. 78 (2003), 681-721] and [Ginzburg V.L., J. Symplectic Geom. 1 (2001), 121-169].
Scattering symplectic manifolds are (closed) manifolds with a mildly degenerate Poisson structure. In particular they can be viewed as symplectic structures on a Lie algebroid which is almost everywhere isomorphic to the tangent bundle. In this paper we prove that all scattering symplectic manifolds arise as glueings of weak symplectic fillings of contact manifolds, and all pairs of weak symplectic fillings with matching boundaries can be glued to a scattering symplectic manifold.
We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, have a symplectic structure. Particularly, if compatibility with a generalized Riemannian metric is required, the symplectic structure is of the Kaehler type.
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