On Slim Double Lie Groupoids (original) (raw)

Coverings and Actions of Structured Lie Groupoids I

2009

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-groupoids on Lie groups and the category of smooth cov- erings of Lie group-groupoids. Further, we prove a theorem which denotes how is obtained a covering Lie group-groupoid and a smooth covering morphism of Lie group-groupoids from a Lie group-groupoid.

On the linearization theorem for proper Lie groupoids

2011

We revisit the linearization theorems for proper Lie groupoids around general orbits (statements and proofs). In the the fixed point case (known as Zung's theorem) we give a shorter and more geometric proof, based on a Moser deformation argument. The passing to general orbits (Weinstein) is given a more conceptual interpretation: as a manifestation of Morita invariance. We also clarify the precise conditions needed for the theorem to hold (which often have been misstated in the literature).

SOME PROPERTIES OF SLIM AG-GROUPOIDS

In this paper, we study slim AG-groupoids. We explore some properties of slim AG-groupoids by studying its super classes. At the end, we connect slim AG-groupoids with commutative semigroups and prove some results which gives necessay and sufficient conditions for commutative semi-groups.

Actions of Double Group-groupoids and Covering Morphism

GAZI UNIVERSITY JOURNAL OF SCIENCE, 2020

• Covering morphism of double groupoids derived by action double groupoid was considered. • Action double group-groupoid on a group-groupoid was characterized. • Covering morphism of double group-groupoids was obtained. • A categorical equivalence was proved.

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-Mrčun, 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 conditions for the integration to be possible. Mackenzie's double Lie structures arise naturally from lifting processes, such as the cotangent lift or the path prolongation, on ordinary Lie theoretic and Poisson geometric objects and we use them to study the integrability of quotient Poisson bivector fields, the relation between "local" and "global" duality of Poisson groupoids and Lie theory for Lie bialgebroids and Poisson groupoids. In the first Chapter we prove suitable versions of Lie's 1-st and 2-nd theorem for Lie bialgebroids, that is, the integrability of subobjects (coisotropic subalgebroids) and morphisms, extending earlier results by Cattaneo and Xu, obtained using different techniques. We develop our functorial approach to the integration of LA-groupoids [65] in the second Chapter, where we also obtain partial results, within the program, proposed by Weinstein, for the integration of Poisson groupoids to symplectic double groupoids. The task of integrating quotients of Poisson manifolds with respect to Poisson groupoid actions motivates the study we undertake in third Chapter of what we refer to as morphic actions, i.e. groupoid actions in the categories of Lie algebroids and Lie groupoids, where we obtain general reduction and integrability results. In fact, applying suitable proceduresà la Marsden-Weinstein zero level reduction to "moment morphisms", respectively of Lie bialgebroids or Poisson groupoids, canonically associated to a Poisson G-space, we derive two approches to the integration of the quotient Poisson bivector fields. The first, a kind of integration via symplectic double groupoids, is not always effective but reproduces the "symplectization functor" approch to Poisson actions of Lie groups, very recently developed by Fernandes-Ortega-Ratiu, from quite a different perspective. We earlier implemented this approach successfully in the special case of complete Poisson groups [64]. The second approach, relying both on a cotangent lift of the Poisson G-space and on a prolongation of the original action to an action on suitable spaces of Lie algebroid homotopies, produces necessary and sufficient integrability conditions for the integration and gives a positive answer to the integrability problem under the most natural assumptions. Könnte jeder brave Mann solche Glöcken finden, seine Feinde würden dann ohne Mühe schwinden 1 [ ? ] 2

On the Lie 2-algebra of sections of an LA-groupoid

Journal of Geometry and Physics, 2019

In this work we introduce the category of multiplicative sections of an LA-groupoid. We prove that these categories carry natural strict Lie 2-algebra structures, which are Morita invariant. As applications, we study the algebraic structure underlying multiplicative vector fields on a Lie groupoid and in particular vector fields on differentiable stacks. We also introduce the notion of geometric vector field on the quotient stack of a Lie groupoid, showing that the space of such vector fields is a Lie algebra. We describe the Lie algebra of geometric vector fields in several cases, including classifying stacks, quotient stacks of regular Lie groupoids and in particular orbifolds, and foliation groupoids.

On the existence of global bisections of Lie groupoids

Acta Mathematica Sinica-english Series, 2007

We show that every source connected Lie groupoid always has global bisections through any given point. This bisection can be chosen to be the multiplication of some exponentials as close as possible to a prescribed curve. The existence of bisections through more than one prescribed points is also discussed. We give some interesting applications of these results.

Proper actions of groupoids on C*-algebras

In 1990, Rieffel defined a notion of proper action of a group H on a C *-algebra A. He then defined a generalized fixed point algebra A α for this action and showed that A α is Morita equivalent to an ideal of the reduced crossed product. We generalize Rieffel's notion to define proper groupoid dynamical systems and show that the generalized fixed point algebra for proper groupoid actions is Morita equivalent to a subalgebra of the reduced crossed product. We give some nontrivial examples of proper groupoid dynamical systems and show that if (A , G, α) is a groupoid dynamical system such that G is principal and proper, then the action of G on A is saturated, that is the generalized fixed point algebra in Morita equivalent to the reduced crossed product.