Lie groupoids and their natural transformations

We discuss natural transformations in the context of Lie groupoids, and their infinitesimal counterpart. Our main result is an integration procedure that provides smooth natural transformations between Lie groupoid morphisms.     

关于群胚作用和李双代数胚态射

在这篇文章中，我们讨论了李双代数胚之间的态射，得到了一些李双代数胚之间态射的性质．研究了泊松群胚在泊松流形上的泊松作用，以及这个泊松作用与被作用流形的切李双代数胚到作用泊松群胚的切李双代数胚之间的态射的关系，得到了一些有用的结论。     

Towards a 2-dimensional notion of holonomy

Previous work (Pradines, C. R. Acad. Sci. Paris 263 (1966) 907; Aof and Brown, Topology Appl. 47 (1992) 97) has given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here we develop analogous 2-dimensional notions starting from a locally Lie crossed module of groupoids. This involves replacing the Ehresmann notion of a local smooth coadmissible section of a groupoid by a local smooth coadmissible homotopy (or free derivation) for the crossed module case. The development also has to use corresponding notions for certain types of double groupoids. This leads to a holonomy Lie groupoid rather than double groupoid, but one which involves the 2-dimensional information.     

Dirac submanifolds and Poisson involutions

Dirac submanifolds are a natural generalization in the Poisson category of the symplectic submanifolds of a symplectic manifold. They correspond to symplectic subgroupoids of the symplectic groupoid of the given Poisson manifold. In particular, Dirac submanifolds arise as the stable loci of Poisson involutions. In this paper, we make a general study of these submanifolds including both local and global aspects.In the second part of the paper, we study Poisson involutions and the induced Poisson structures on their stable loci. In particular, we discuss the Poisson involutions on a special class of Poisson groups, and more generally Poisson groupoids, called symmetric Poisson groups, and symmetric Poisson groupoids. Many well-known examples, including the standard Poisson group structures on semi-simple Lie groups, Bruhat Poisson structures on compact semi-simple Lie groups, and Poisson groupoid structures arising from dynamical r-matrices of semi-simple Lie algebras are symmetric, so they admit a Poisson involution. For symmetric Poisson groups, the relation between the stable locus Poisson structure and Poisson symmetric spaces is discussed. As a consequence, we prove that the Dubrovin Poisson structure on the space of Stokes matrices U₊ (appearing in Dubrovin's theory of Frobenius manifolds) is a Poisson symmetric space.     

Banach–Lie groupoids and generalized inversion

We study a few basic properties of Banach–Lie groupoids and algebroids, adapting some classical results on finite dimensional Lie groupoids. As an illustration of the general theory, we show that the notion of locally transitive Banach–Lie groupoid sheds fresh light on earlier research on some infinite-dimensional manifolds associated with Banach algebras.     

Geometric combinatorics of Weyl groupoids

We extend properties of the weak order on finite Coxeter groups to Weyl groupoids admitting a finite root system. In particular, we determine the topological structure of intervals with respect to weak order, and show that the set of morphisms with fixed target object forms an ortho-complemented meet semilattice. We define the Coxeter complex of a Weyl groupoid with finite root system and show that it coincides with the triangulation of a sphere cut out by a simplicial hyperplane arrangement. As a consequence, one obtains an algebraic interpretation of many hyperplane arrangements that are not reflection arrangements.     

A tale of two categories: Inductive groupoids and cross-connections

A groupoid is a small category in which all morphisms are isomorphisms. An inductive groupoid is a specialized groupoid whose object set is a regular biordered set and the morphisms admit a partial order. A normal category is a specialized small category whose object set is a strict preorder and the morphisms admit a factorization property. A pair of ‘related’ normal categories constitutes a cross-connection. Both inductive groupoids and cross-connections were identified by Nambooripad as categorical models of regular semigroups. We explore the inter-relationship between these seemingly different categorical structures and prove a direct category equivalence between the category of inductive groupoids and the category of cross-connections.     

Proper groupoids and momentum maps: Linearization, affinity, and convexity

We show that proper Lie groupoids are locally linearizable. As a consequence, the orbit space of a proper Lie groupoid is a smooth orbispace (a Hausdorff space which locally looks like the quotient of a vector space by a linear compact Lie group action). In the case of proper (quasi-)symplectic groupoids, the orbit space admits a natural integral affine structure, which makes it into an affine orbifold with locally convex polyhedral boundary, and the local structure near each boundary point is isomorphic to that of a Weyl chamber of a compact Lie group. We then apply these results to the study of momentum maps of Hamiltonian actions of proper (quasi-)symplectic groupoids, and show that these momentum maps preserve natural transverse affine structures with local convexity properties. Many convexity theorems in the literature can be recovered from this last statement and some elementary results about affine maps.     

Groupoïde d'holonomie et géométrie globale

The purpose of this Note is to show that global geometry requires Lie groupoid theory. In this way we show that holonomy groupoid of a regular foliation leads to a satisfactory solution of many global existence problems. Three examples are given: Palais'theorem on local group actions, a characteristics method and links between Lie groupoid and algebroid morphisms. Complete proofs will be given elsewere.     

Non-abelian differentiable gerbes

We study non-abelian differentiable gerbes over stacks using the theory of Lie groupoids. More precisely, we develop the theory of connections on Lie groupoid G-extensions, which we call “connections on gerbes”, and study the induced connections on various associated bundles. We also prove analogues of the Bianchi identities. In particular, we develop a cohomology theory which measures the existence of connections and curvings for G-gerbes over stacks. We also introduce G-central extensions of groupoids, generalizing the standard groupoid S¹-central extensions. As an example, we apply our theory to study the differential geometry of G-gerbes over a manifold.     

<Emphasis Type="Italic">n</Emphasis>-transitivity of bisection groups of a Lie groupoid

The notion of n-transitivity can be carried over from groups of diffeomorphisms on a manifold M to groups of bisections of a Lie groupoid over M. The main theorem states that the n-transitivity is fulfilled for all n ∈ N by an arbitrary group of C^r-bisections of a Lie groupoid Γ of class C^r, where 1 ≤ r ≤ ω, under mild conditions. For instance, the group of all bisections of any Lie groupoid and the group of all Lagrangian bisections of any symplectic groupoid are n-transitive in the sense of this theorem. In particular, if Γ is source connected for any arrow γ ∈ Γ, there is a bisection passing through γ.     

On the de Rham cohomology of differential and algebraic stacks

We introduce the notion of cofoliation on a stack. A cofoliation is a change of the differentiable structure which amounts to giving a full representable smooth epimorphism. Cofoliations are uniquely determined by their associated Lie algebroids.Cofoliations on stacks arise from flat connections on groupoids. Connections on groupoids generalize connections on gerbes and bundles in a natural way. A flat connection on a groupoid is an integrable distribution of the morphism space compatible with the groupoid structure and complementary to both source and target fibres. A cofoliation of a stack determines the flat groupoid up to étale equivalence.We show how a cofoliation on a stack gives rise to a refinement of the Hodge to De Rham spectral sequence, where the E₁-term consists entirely of vector bundle valued cohomology groups.Our theory works for differentiable, holomorphic and algebraic stacks.     

Commutator theory, action groupoids, and an intrinsic Schreier-Mac Lane extension theorem

Beyond groups of automorphisms in the category Gp of groups and Lie-algebras of derivations in the category K-Lie of Lie algebras, there are structures of internal groupoids (called action groupoids) in both categories. They allow a synthesis of the notion of obstruction to extensions. This leads, in any pointed protomodular category C with split extension classifiers, to a general treatment of non-abelian extensions which can be understood as morphisms in a certain groupoid TorsC.     

Inverse semigroup actions as groupoid actions

To an inverse semigroup, we associate an étale groupoid such that its actions on topological spaces are equivalent to actions of the inverse semigroup. Both the object and the arrow space of this groupoid are non-Hausdorff. We show that this construction provides an adjoint functor to the functor that maps a groupoid to its inverse semigroup of bisections, where we turn étale groupoids into a category using algebraic morphisms. We also discuss how to recover a groupoid from this inverse semigroup.     

Representations of étale Lie groupoids and modules over Hopf algebroids

The classical Serre-Swan’s theorem defines an equivalence between the category of vector bundles and the category of finitely generated projective modules over the algebra of continuous functions on some compact Hausdorff topological space. We extend these results to obtain a correspondence between the category of representations of an étale Lie groupoid and the category of modules over its Hopf algebroid that are of finite type and of constant rank. Both of these constructions are functorially defined on the Morita category of étale Lie groupoids and we show that the given correspondence represents a natural equivalence between them.     

Equivalence between the Morita categories of étale Lie groupoids and of locally grouplike Hopf algebroids

Any étale Lie groupoid G is completely determined by its associated convolution algebra C_c^∞(G) equipped with the natural Hopfalgebroid structure. We extend this result to the generalized morphisms between étale 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 étale Lie groupoids and the Morita category of locally grouplike Hopf algebroids.     

Linearization of Poisson groupoids

Motivated by a search for Lie group structures on groups of Poisson diffeomorphisms, we investigate linearizability of Poisson structures of Poisson groupoids around the unit section. After extending the Lagrangian neighbourhood theorem to the setting of cosymplectic Lie algebroids, we establish that dual integrations of triangular bialgebroids are always linearizable. Additionally, we show that the (non-dual) integration of a triangular Lie bialgebroid is linearizable whenever the $r$-matrix is of so-called cosymplectic type. The proof relies on the integration of a triangular Lie bialgebroid to a symplectic LA-groupoid, and in the process we define interesting new examples of double Lie algebroids and LA-groupoids. We also show that the product Poisson groupoid can only be linearizable when the Poisson structure on the unit space is regular.     

关于李群胚的几点讨论

文章讨论了李群胚作为丛的一些性质,得出李群胚的内子群胚是主丛的结论;研究了李群胚在其内子群胚上的作用,并证明了李群胚上的Maurer-cartan形式在其任意左不变向量场上作用的结果为常数.文末推广了关于李代数胚态射的一个结论.     

On Local Integrability Conditions of Jet Groupoids

A jet groupoid ℛ _q over a manifold X is a special Lie groupoid consisting of q-jets of local diffeomorphisms X→X. As a subbundle of J ^q (X,X), a jet groupoid can be considered as a system of nonlinear partial differential equations (PDE). This leads to the question if ℛ _q is formally integrable. On the other hand, each jet groupoid is the symmetry groupoid of a geometric object, which is a section ω of a natural bundle ℱ. Using the jet groupoids, we give a local characterisation of formal integrability for transitive jet groupoids in terms of their corresponding geometric objects. Thanks to M. Barakat and W. Plesken for discussions. The author was supported by DFG Grant Graduiertenkolleg 775.     

On the 1-Homotopy Type of Lie Groupoids

We propose a notion of 1-homotopy for generalized maps. This notion generalizes those of natural transformation and ordinary homotopy for functors. The 1-homotopy type of a Lie groupoid is shown to be invariant under Morita equivalence. As an application we consider orbifolds as groupoids and study the notion of orbifold 1-homotopy type induced by a 1-homotopy between presentations of the orbifold maps.