In this work we deal with coverings and actions of Lie group- groupoids being a sort of the structured Lie groupoids. Firstly, we define an action of a Lie group-groupoid on some Lie group and the smooth coverings of Lie group-groupoids. Later, we show the equivalence of the category of smooth actions of Lie group-grou…
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48 results for “Lie groupoid actions”
Locally convex bialgebroids reconstruct Lie groupoids of orbits.
problem Reconstructing Lie groupoids from their orbits using bialgebroids.
method Constructing a locally convex bialgebroid over the space of orbits.
result Groupoids can be reconstructed from their spectral action Lie groupoids.
Stacky Lie groupoids are generalizations of Lie groupoids in which the "space of arrows" of the groupoid is a differentiable stack. In this paper, we consider actions of stacky Lie groupoids on differentiable stacks and their associated quotients. We provide a characterization of principal actions of stacky Lie groupoi…
Extends Cheeger's method to Lie groupoid actions on manifolds.
problem Smooth Lie group actions on manifolds with singularities.
method Extension of Cheeger's deformation techniques to Lie groupoid actions.
result Explicit sectional curvature description of the deformation.
New proof of Lie algebroid action equivalence and integrability.
problem Equivalence between Lie algebroid action and integrability.
method Double Lie groupoids and multiplicative foliations.
result Simple characterization of double Lie groupoids inducing Lie groupoid structures.
New findings on how foliations and Lie group actions interact.
problem Understanding interactions between foliations and Lie group actions.
method Functorial properties of holonomy groupoids and quotient manifolds.
result Holonomy groupoids of quotients by Lie group actions exhibit similar properties.
Equivalent bicategories constructed from action Lie groupoids.
problem Equivalence of bicategories constructed from action Lie groupoids.
method Localizing at equivariant weak equivalences, surjective submersive equivariant weak equivalences, and all weak equivalences.
result Weak equivalences between action Lie groupoids are isomorphic to compositions of nice forms of equivariant weak equivalences.
The paper constructs automorphisms of Lie groupoids and applies them to symplectic reductions on orbifolds.
problem Formulating Hamiltonian actions of Lie 2-groups on orbifolds.
method Constructing automorphisms of Lie groupoids, using 2-group actions and Kan fibrations.
result Symplectic reductions of Lie 2-group actions on orbifolds are Lie 2-groupoids under certain conditions.
Study Lie 2-group actions on Riemannian groupoids, proving existence and developing geometric Killing vector fields.
problem Understanding isometric actions of Lie 2-groups on Riemannian groupoids.
method Exhibit properties, prove existence, construct bi-invariant metrics, provide infinitesimal description.
result Existence of 2-equivariant Slice Theorem and Equivariant Tubular Neighborhood Theorem.
Symplectic leaves of Lie groupoids realized as orbits of coadjoint action.
problem Understanding symplectic leaves of Lie groupoids.
method Realization of symplectic leaves as orbits of the affine coadjoint action of a fiber bundle.
result Symplectic leaves are groupoid orbits of the symplectic groupoid .
In this thesis, we employ simplicial methods to study actions, principal bundles, and bibundles of higher groupoids. Roughly, we use Kan fibrations to model actions of higher groupoids, we use pairs of a Kan fibration and a special acyclic fibration to model principal bundles of higher groupoids, we use inner Kan fibra…
PB-groupoids vs VB-groupoidsmath.DG
Paper defines PB-groupoids and their relation to VB-groupoids.
problem Establishing PB-groupoids as a Lie 2-groupoid counterpart.
method Defining PB-groupoids as Lie 2-groupoid actions on diagrams of Lie groupoids and principal bundles.
result PB-groupoids generalize VB-groupoids and provide a new correspondence between them and principal bundles.
On slim double Lie groupoidsmath.DG
We prove that every slim double Lie groupoid with proper core action is completely determined by a factorization of a certain canonically defined "diagonal" Lie groupoid.
Extends distribution algebra concept to Lie groupoids.
problem Distribution algebra on Lie groupoids.
method Construction of convolution C_c(M)/R-bialgebra associated with Lie groupoid adjoint action.
result Extension of Cartier-Gabriel decomposition to Lie groupoids.
The paper defines connections on Lie groupoid bundles and their properties.
problem Defining connections on Lie groupoid bundles and their properties.
method Introducing a suitable definition for connections on Lie groupoid bundles and proving their existence.
result Existence of connections on Lie groupoid bundles and their properties.
Paper develops equivariant basic cohomology for Lie groupoids.
problem Equivariant cohomology for Lie groupoids with weak actions.
method Using Kan fibrations and fiber structures, constructing models and comparing with existing theories.
result Equivariant basic cohomology theory for orbifolds and Lie groupoids.
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 alm…
Lie Groupoidsmath.DG
Lie groupoids generalize Lie groups with multiplication defined for certain pairs.
problem Generalizing Lie groups to include more complex structures.
method Definition and examples, including actions and representations.
result Lie groupoids have significant connections to Poisson geometry and mathematical physics.
The paper defines a stratification for Lie groupoids in a tame topology context.
problem Presenting a tame topology counterpart to canonical stratification of Lie groupoids.
method Using Shiota's isotopy lemma and approximation theorem, the paper defines a canonical Whitney stratification of definable Lie groupoids into invariant strata.
result A canonical Whitney stratification of the Lie groupoid into definable strata invariant under the groupoid action.
New Morita equivalence for diffeological groupoids defined.
problem Defining Morita equivalence for diffeological groupoids.
method Developed diffeological groupoid actions, -bundles, and -bibundles; introduced principality; defined Hilsum-Skandalis tensor product.
result Biprincipal bibundles are weakly invertible in the bicategory DiffBiBund.
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…
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 …
On n-tuple principal bundlesmath.DG
Develops n-tuple principal bundles from Lie group actions.
problem No specific problem stated; focuses on concept development.
method Compatibility condition for Lie group action on groupoid.
result Concept of n-tuple principal bundles established.
New cohomology theory shows compact Lie group actions are Morita invariant.
problem Establishing Morita invariance for cohomology of compact Lie group actions.
method Using bibundles to transfer coefficient systems between Morita equivalent groupoids.
result Twisted Bredon-Illman cohomology is Morita invariant for compact Lie group actions.
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 …
Given a proper, cocompact action of a Lie groupoid, we define a higher index pairing between invariant elliptic differential operators and smooth groupoid cohomology classes. We prove a cohomological index formula for this pairing by applying the van Est map and algebraic index theory. Finally we discuss in examples th…
We establish a 1:1 correspondence between Poisson-Lie group actions on integrable Poisson manifolds and twisted multiplicative hamiltonian actions on source 1-connected symplectic groupoids. For an action of a Poisson-Lie group on a Poisson manifold , we find an explicit description of the lifted hamiltonian act…
We prove the following result, conjectured by Alan Weinstein: every smooth proper Lie groupoid near a fixed point is locally linearizable, i.e. it is locally isomorphic to the associated groupoid of a linear action of a compact Lie group. In combination with a slice theorem of Weinstein, our result implies the smooth l…
Characterizes algebraic integrability and minimality of Lie equations for non-commutative pseudogroups.
problem Understanding algebraic integrability and minimality of Lie equations for non-commutative pseudogroups.
method Algebraic characterization and differential Galois theory of rational connections.
result Equivalence of algebraic integrability to the triviality of the differential Galois group and demonstration of minimality under certain conditions.
To a closed wide Lie subgroupoid of a Lie groupoid , i.e. a Lie groupoid pair, we associate an Atiyah class which we interpret as the obstruction to the existence of -invariant fibrewise affine connections on the homogeneous space . For Lie groupoid pairs with…
This paper is concerned with symmetries of closed multiplicative 2-forms on Lie groupoids and their infinitesimal counterparts. We use them to study Lie group actions on Dirac manifolds by Dirac diffeomorphisms and their lifts to presymplectic groupoids, building on recent work of Fernandes-Ortega-Ratiu \cite{FOR} on P…
Study smooth structures and stratifications on orbit spaces of Lie groupoids.
problem Understanding the smooth structure and stratification of orbit spaces of Lie groupoids.
method Utilizing Spallek's framework and extending classical theories of proper Lie group actions.
result Established Morita invariance of smooth structures and stratifications on orbispaces.
For a smooth (locally trivial) principal bundle in Ehresmann's sense, the relation between the commuting vertical and horizontal actions of the structural Lie group and the structural Lie groupoid (isomorphisms between vertical fibers) is regarded as a special case of a symmetrical concept of conjugation between "princ…
This is a concise introduction to the theory of Lie groupoids, with emphasis in their role as models for stacks. After some preliminaries, we review the foundations on Lie groupoids, and we carefully study equivalences and proper groupoids. Differentiable stacks are geometric objects which have manifolds and orbifolds …
In this paper we study the Lie groupoids which appear in foliation theory. A foliation groupoid is a Lie groupoid which integrates a foliation, or, equivalently, whose anchor map is injective. The first theorem shows that, for a Lie groupoid G, the following are equivalent: - G is a foliation groupoid, - G has discrete…
We observe that any connected proper Lie groupoid whose orbits have codimension at most two admits a globally effective representation on a smooth vector bundle, i.e., one whose kernel consists only of ineffective arrows. As an application, we deduce that any such groupoid can up to Morita equivalence be presented as a…
Solves integration problem for generalized complex manifolds.
problem Integration problem for generalized complex manifolds.
method Weakly holomorphic symplectic groupoid approach.
result Generalized complex manifolds are integrable if and only if their underlying real Poisson structure is integrable.
Formula derived for Dirac operators on Lie groupoids.
problem Equivariant Dirac operators on Lie groupoids.
method Getzler rescaling method to derive a fixed-point formula.
result Reduces to standard fixed-point formula for closed manifolds.
Paper unifies curvature concepts for Lie groupoids and algebroids.
problem Separate theories for Lie groups, bundles, and submersions.
method Using Lie groupoids with source-fibre metrics and Riemannian Lie algebroids.
result Derives sectional curvature formula for Lie groupoids and Lie algebroid curvature formulas.
A Lie 2-group's left-invariant vector fields are isomorphic to its Lie 2-algebra.
problem Understanding the relationship between Lie 2-groups and their associated Lie 2-algebras.
method Analyzing the Lie 2-group structure to derive left-invariant vector fields and comparing them to the Lie 2-algebra.
result The Lie 2-algebra of a Lie 2-group is isomorphic to the Lie 2-algebra of its left-invariant vector fields.
Paper defines weak representations and their equivalence to VB-groupoids.
problem Understanding representations of Lie groupoids and VB-groupoids.
method Introduces weak representations and shows equivalences between categories.
result Equivalence between 2-term representations up to homotopy and weak representations of G.
We demonstrate that the notions of derivative representation of a Lie algebra on a vector bundle, of semi-linear representations of a Lie group on a vector bundle, and related concepts, may be understood in terms of representations of Lie algebroids and Lie groupoids, and we indicate how these notions extend to derivat…
Post-groupoids help solve Yang-Baxter equation using quivers.
problem Solving the Yang-Baxter equation using algebraic structures.
method Introducing post-groupoids and showing their connection to quivers.
result Post-groupoids provide solutions to the Yang-Baxter equation.
A proper etale Lie groupoid is modelled as a (noncommutative) spectral geometric space. The spectral triple is built on the algebra of smooth functions on the groupoid base which are invariant under the groupoid action. Stiefel-Whitney classes in Lie groupoid cohomology are introduced to measure the orientability of th…
In these lectures notes I discuss the Linearization Theorem for Lie groupoids, and its relation to the various classical linearization theorems for submersions, foliations and group actions. In particular, I explain in some detail the recent metric approach to this problem.
We address the question of duality for the dynamical Poisson groupoids of Etingof and Varchenko over a contractible base. We also give an explicit description for the coboundary case associated with the solutions of the classical dynamical Yang-Baxter equation on simple Lie algebras as classified by the same authors. O…
Solves the invariant linearization problem for Lie groupoids.
problem Understanding invariant linearization for Lie groupoids.
method Introduced a counter-example and a sufficient criterion using compatible complete metrics and covers of proper group actions.
result Proved a sufficient criterion for invariant linearization of Lie groupoids.
We describe principal 3-bundles with adjusted connections using Lie algebras and groupoids.
problem Describing principal 3-bundles with adjusted connections.
method Derived explicit forms of adjustment data for 3-term -algebras, integrated action Lie 3-algebroids to Lie 3-groupoids, and used differential cohomology.
result Explicit description of principal 3-bundles with adjusted connections in terms of differential cohomology.