Calculus on Lie algebroids, Lie groupoids and Poisson manifolds
Charles-Michel Marle
备注：
To appear in Dissertationes Mathematicae. 57 pages, 2 figures. Subsection 3.2.6 about integration of Lie algebroids rewritten, references added. More references added and minor typos corrected
发表信息：
Dissertationes Mathematicae 457, Institute of mathematics, Polish Academy of Sciences, Warszawa 2008
We begin with a short presentation of the basic concepts related to Lie groupoids and Lie algebroids, but the main part of this paper deals with Lie algebroids. A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its external powers, one can define an operator similar to the exterior derivative. We present the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for their most part, are already known) are given with detailed proofs. In the final sections, the results are applied to Poisson manifolds, whose links with Lie algebroids are very close.
In the paper we study the algebroid A of the groupoid of partially invertible elements over the lattice of orthogonal projections of a $W^*$-algebra. In particular the complex analytic manifold structure of these objects is investigated. The expressions on the Lie brackets for A and related algebroids are given in noncommutative operator coordinates in the explicit way. We also prove statements describing structure of the groupoid of partial isometries and the frame groupoid of A as well as the structure of their algebroids.
A Lie groupoid, called \textit{material Lie groupoid}, is associated in a natural way to any elastic material. The corresponding Lie algebroid, called \textit{material algebroid}, is used to characterize the uniformity and the homogeneity properties of the material. The relation to previous results in terms of $G-$structures is discussed in detail. An illustrative example is presented as an application of the theory.
Lie algebroids are by no means natural as an infinitesimal counterpart of groupoids. In this paper we propose a functorial construction called Nishimura algebroids for an infinitesimal counterpart of groupoids. Nishimura algebroids, intended for differential geometry, are of the same vein as Lawvere's functorial notion of algebraic theory and Ehresmann's functorial notion of theory called sketches. We study totally intransitive Nishimura algebroids in detail. Finally we show that Nishimura algebroids naturally give rise to Lie algebroids.
Lie groupoids and their associated algebroids arise naturally in the study of the constitutive properties of continuous media. Thus, Continuum Mechanics and Differential Geometry illuminate each other in a mutual entanglement of theory and applications. Given any material property, such as the elastic energy or an index of refraction, affected by the state of deformation of the material body, one can automatically associate to it a groupoid. Under conditions of differentiability, this material groupoid is a Lie groupoid. Its associated Lie algebroid plays an important role in the determination…
Groupoids are mathematical structures able to describe symmetry properties more general than those described by groups. They were introduced (and named) by H. Brandt in 1926. Around 1950, Charles Ehresmann used groupoids with additional structures (topological and differentiable) as essential tools in topology and differential geometry. In recent years, Mickael Karasev, Alan Weinstein and Stanisław Zakrzewski independently discovered that symplectic groupoids can be used for the construction of noncommutative deformations of the algebra of smooth functions on a manifold, with potential applica…
A Lie groupoid, called \textit{second-order non-holonomic material Lie groupoid}, is associated in a natural way to any Cosserat media. This groupoid is used to give a new definition of homogeneity which does not depend on a reference crystal. The corresponding Lie algebroid, called \textit{second-order non-holonomic material Lie algebroid}, is used to characterize the homogeneity property of the material. We also relate these results with the previously ones in terms of non-holonomic second-order $\overline{G}$-structures.
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…
VB-groupoids can be thought of as vector bundle objects in the category of Lie groupoids. Just as Lie algebroids are the infinitesimal counterparts of Lie groupoids, VB-algebroids correspond to the infinitesimal version of VB-groupoids. In this work we address the problem of the existence of a VB-groupoid admitting a given VB-algebroid as its infinitesimal data. Our main result is an explicit characterization of the obstructions appearing in this integrability problem as the vanishing of the spherical periods of certain cohomology classes. Along the way, we illustrate our result in concrete ex…
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.
VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids respectively, and they are related via the Lie functor. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. Additionally, they can be seen as models for vector bundles over singular spaces. In this paper we study their infinitesimal automorphisms, i.e. vector fields on them generating a flow by diffeomorphisms preserving both the linear and the groupoid/algebroid structures. For a special class of VB-groupoids/algebroids coming from representations of Lie…
We construct local coordinates for the Weinstein groupoid of a non-integrable Lie algebroid. To this end, we reformulate the notion of bi-submersion in a completely algebraic way and prove the existence of bi-submersions as such for the Weinstein groupoid. This implies that a $C^*$-algebra can be attached to every Lie algebroid.
We complete the construction of the double Lie algebroid of a double Lie groupoid begun in the first paper of this title. We show that the Lie algebroid structure of an LA--groupoid may be prolonged to the Lie algebroid of its Lie groupoid structure; in the case of a double groupoid this prolonged structure for either LA--groupoid is canonically isomorphic to the Lie algebroid structure associated with the other; this extends many canonical isomorphisms associated with iterated tangent and cotangent structures. We also show that the cotangent of a double Lie groupoid is a symplectic double gro…
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.
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).
Lie algebroids can not always be integrated into Lie groupoids. We introduce a new object--``Weinstein groupoid'', which is a differentiable stack with groupoid-like axioms. With it, we have solved the integration problem of Lie algebroids. It turns out that every Weinstein groupoid has a Lie algebroid and every Lie algebroid can be integrated into a Weinstein groupoid.
This thesis deals with deformations of VB-algebroids and VB-groupoids. They can be considered as vector bundles in the categories of Lie algebroids and groupoids and encompass several classical objects, including Lie algebra and Lie group representations, 2-vector spaces and the tangent and the cotangent algebroid (groupoid) to a Lie algebroid (groupoid). Moreover, they are geometric models for some kind of representations of Lie algebroids (groupoids), namely 2-term representations up to homotopy. Finally, it is well known that Lie groupoids are "concrete" incarnations of differentiable stack…
Let $(G\rr P, \mathsf D_G)$ be a Dirac groupoid. We show that there are natural Lie algebroid structures on the units $\lie A(\mathsf D_G)$ and on the core $I^\tg(\mathsf D_G)$ of the multiplicative Dirac structure. In the Poisson case, the Lie algebroid $A^*G$ is isomorphic to $\lie A(\mathsf D_G)$ and in the case of a closed 2-form, the $IM$-2-form is equivalent to the core algebroid that we find. We construct a vector bundle $\lie B(\mathsf D_G)\to P$ associated to any (almost) Dirac structure. In the Dirac case, $\lie B(\mathsf D_G)$ has the structure of a Courant algebroid that generalize…
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…
A complex Lie algebroid is a complex vector bundle over a smooth (real) manifold M with a bracket on sections and an anchor to the complexified tangent bundle of M which satisfy the usual Lie algebroid axioms. A proposal is made here to integrate analytic complex Lie algebroids by using analytic continuation to a complexification of M and integration to a holomorphic groupoid. A collection of diverse examples reveal that the holomorphic stacks presented by these groupoids tend to coincide with known objects associated to structures in complex geometry. This suggests that the object integrating…
In this survey, we discuss a series of linearization problems--for Poisson structures, Lie algebroids, and Lie groupoids. The last problem involves a conjecture on the structure of proper groupoids. Attempting to prove this by the method of averaging leads to problems concerning almost actions of compact groups and almost invariant submanifolds for compact group actions. The paper ends with a discussion of possible extensions of the convexity theorems for momentum maps of hamiltonian actions of compact groups.
We define multiplicative Poisson-Nijenhuis structures on a Lie groupoid which extends the notion of symplectic-Nijenhuis groupoid introduced by Stié23non and Xu. We also introduce a special class of Lie bialgebroid structure on a Lie algebroid $A$, called P-N Lie bialgebroid, which defines a hierarchy of compatible Lie bialgebroid structures on $A$. We show that under some topological assumption on the groupoid, there is a one-to-one correspondence between multiplicative Poisson-Nijenhuis structures on a Lie groupoid and P-N Lie bialgebroid structures on the corresponding Lie algebroid.
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.
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 …
Any etale Lie groupoid G is completely determined by its associated convolution algebra C_c(G) equipped with the natural Hopf algebroid structure. We extend this result to the generalized morphisms between etale Lie groupoids: we show that any principal H-bundle P over G is uniquely determined by the associated C_c(G)-C_c(H)-bimodule C_c(P) equipped with the natural coalgebra structure. Furthermore, we prove that the functor C_c gives an equivalence between the Morita category of etale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.
Transitive Lie algebroids and \textbf{Q}-manifolds
Institute of Mathematics “Simion Stoilow” of the Romanian Academy（罗马尼亚科学院西米翁·斯托伊洛数学研究所）；Unité Mixte de Recherche 5127 CNRS, Université Savoie Mont Blanc, Laboratoire de Mathématiques (LAMA)（CNRS联合研究单位5127，萨瓦-蒙布朗大学，数学实验室）
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.
Inspired by the work of Molino, we show that the integrability obstruction for transitive Lie algebroids can be made to vanish by adding extra dimensions. In particular, we prove that the Weinstein groupoid of a non-integrable transitive and abelian Lie algebroid, is the quotient of a finite dimensional Lie groupoid. Two constructions as such are given: First, explaining the counterexample to integrability given by Almeida and Molino, we see that it can be generalized to the construction of an "Almeida-Molino" integrable lift when the base manifold is simply connected. On the other hand, we no…
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.
In this master's thesis, we will go into the (projective) blow-up construction for Lie groupoids and Lie algebroids. In the literature, there are different methods to be found on how to do this, especially for Lie groupoids. The main goal of the thesis is to explain, in detail, the Lie groupoid and the Lie algebroid blow-up constructions, but also to examine and compare different points of view. More explicitly, we will rigorously explain the blow-up construction for Lie groupoids by Claire Debord and Georges Skandalis. Moreover, we will show that the blow-up construction for Lie groupoids by …
We provide a complete solution to the problem of extending a local Lie groupoid to a global Lie groupoid. First, we show that the classical Mal'cev's theorem, which characterizes local Lie groups that can be extended to global Lie groups, also holds in the groupoid setting. Next, we describe a construction that can be used to obtain any local Lie groupoid with integrable algebroid. Last, our main result establishes a precise relationship between the integrability of a Lie algebroid and the failure in associativity of a local integration. We give a simplicial interpretation of this result showi…
This thesis treats two main topics: calibrated symplectic foliations, and local Lie groupoids. Calibrated symplectic foliations are one possible generalization of taut foliations of 3-manifolds to higher dimensions. Their study has been popular in recent years, and we collect several interesting results. We then show how de Rham's theory of currents, and Sullivan's theory of structure currents, can be applied in trying to understand the calibratability of symplectic foliations. Our study of local Lie groupoids begins with their definition and an exploration of some of their basic properties. N…
We show that the integrability obstruction of a transitive Lie algebroid coincides with the lifting obstruction of a crossed module of groupoids associated naturally with the given algebroid. Then we extend this result to general extensions of integrable transitive Lie algebroids by Lie algebra bundles. Such a lifting obstruction is directly related with the classification of extensions of transitive Lie groupoids. We also give a classification of such extensions which differentiates to the classification of transitive Lie algebroids discussed in \cite{KCHM:new}.
Given a representation up to homotopy of a Lie algebroid on a 2-term complex of vector bundles, we define the corresponding holonomy as a strict 2-functor from a Weinstein path 2-groupoid to the gauge 2-groupoid of the underlying 2-term complex. We construct a corresponding transformation 2-groupoid and we prove that the 1-truncation of this 2-groupoid is isomorphic to the Weinstein groupoid of the VB-algebroid associated to a representation up to homotopy. As applications, we describe alternative integration schemes for semi-direct products of Lie 2-algebras and string algebras.
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…
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.
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.
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.
A symplectic Lie group is a Lie group with a left-invariant symplectic form. Its Lie algebra structure is that of a quasi-Frobenius Lie algebra. In this note, we identify the groupoid analogue of a symplectic Lie group. We call the aforementioned structure a \textit{$t$-symplectic Lie groupoid}; the "$t$" is motivated by the fact that each target fiber of a $t$-symplectic Lie groupoid is a symplectic manifold. For a Lie groupoid $\mathcal{G}\rightrightarrows M$, we show that there is a one-to-one correspondence between quasi-Frobenius Lie algebroid structures on $A\mathcal{G}$ (the associated …
A new category of Lie algebras, called generalized Lie algebras, is presented such that classical Lie algebras and Lie-Rinehart algebras are objects of this new category. A new philosophy over generalized Lie algebroids theory is presented using the notion of generalized Lie algebra and examples of objects of the category of generalized Lie algebroids are presented. An exterior differential calculus on generalized Lie algebras is pre- sented and a theorem of Maurer-Cartan type is obtained. Supposing that any submodule(vector subbundle) of a generalized Lie algebra(algebroid) is an interior alg…
The van Est map is a map from Lie groupoid cohomology (with respect to a sheaf taking values in a representation) to Lie algebroid cohomology. We generalize the van Est map to allow for more general sheaves, namely to sheaves of sections taking values in a (smooth or holomorphic) $G$-module, where $G$-modules are structures which differentiate to representations. Many geometric structures involving Lie groupoids and stacks are classified by the cohomology of sheaves taking values in $G$-modules and not in representations, including $S^1$-groupoid extensions and equivariant gerbes. Examples of …
A Lie algebroid over a manifold is a vector bundle over that manifold whose properties are very similar to those of a tangent bundle. Its dual bundle has properties very similar to those of a cotangent bundle: in the graded algebra of sections of its external powers, one can define an operator similar to the exterior derivative. We present in this paper the theory of Lie derivatives, Schouten-Nijenhuis brackets and exterior derivatives in the general setting of a Lie algebroid, its dual bundle and their exterior powers. All the results (which, for their most part, are already known) are given …
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.
In this paper we consider deformations of an algebroid stack on an etale groupoid. We construct a differential graded Lie algebra (DGLA) which controls this deformation theory. In the case when the algebroid is a twisted form of functions we show that this DGLA is quasiisomorphic to the twist of the DGLA of Hochschild cochains on the algebra of functions on the groupoid by the characteristic class of the corresponding gerbe.
We define the "localized index" of longitudinal elliptic operators on Lie groupoids associated to Lie algebroid cohomology classes. We derive a topological expression for these numbers using the algebraic index theorem for Poisson manifolds on the dual of the Lie algebroid. Underlying the definition and computation of the localized index, is an action of the Hopf algebroid of jets around the unit space, and the characteristic map it induces on Lie algebroid cohomology. This map can be globalized to differentiable groupoid cohomology, giving a definition as well as a computation of the "global …
