# The Lie group of bisections of a Lie groupoid

@article{Schmeding2014TheLG, title={The Lie group of bisections of a Lie groupoid}, author={Alexander Schmeding and Christoph Wockel}, journal={Annals of Global Analysis and Geometry}, year={2014}, volume={48}, pages={87-123} }

In this article, we endow the group of bisections of a Lie groupoid with compact base with a natural locally convex Lie group structure. Moreover, we develop thoroughly the connection to the algebra of sections of the associated Lie algebroid and show for a large class of Lie groupoids that their groups of bisections are regular in the sense of Milnor.

#### 23 Citations

The Lie group of vertical bisections of a regular Lie groupoid

- Mathematics
- 2019

Abstract In this note we construct an infinite-dimensional Lie group structure on the group of vertical bisections of a regular Lie groupoid. We then identify the Lie algebra of this group and… Expand

FUNCTORIAL ASPECTS OF THE RECONSTRUCTION OF LIE GROUPOIDS FROM THEIR BISECTIONS

- Mathematics
- Journal of the Australian Mathematical Society
- 2016

To a Lie groupoid over a compact base $M$ , the associated group of bisection is an (infinite-dimensional) Lie group. Moreover, under certain circumstances one can reconstruct the Lie groupoid from… Expand

Re)constructing Lie groupoids from their bisections and applications to prequantisation

- Mathematics
- 2015

This paper is about the relation of the geometry of Lie groupoids over a fixed compact manifold and the geometry of their (infinite-dimensional) bisection Lie groups. In the first part of the paper… Expand

Banach-Lie groupoids and generalized inversion

- Mathematics
- 2018

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… Expand

n-transitivity of bisection groups of a Lie groupoid

- Mathematics
- 2016

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… Expand

A differentiable monoid of smooth maps on Lie groupoids

- Mathematics
- 2017

In this article we investigate a monoid of smooth mappings on the space of arrows of a Lie groupoid and its group of units. The group of units turns out to be an infinite-dimensional Lie group which… Expand

Linking Lie groupoid representations and representations of infinite-dimensional Lie groups

- Mathematics
- 2018

The present paper links the representation theory of Lie groupoids and infinite-dimensional Lie groups. We show that smooth representations of Lie groupoids give rise to smooth representations of… Expand

A cohomology theory for Lie 2-algebras and Lie 2-groups

- Mathematics
- 2018

In this thesis, we introduce a new cohomology theory associated to a Lie 2-algebras and a new cohomology theory associated to a Lie 2-group. These cohomology theories are shown to extend the… Expand

Lie groups of Poisson diffeomorphisms

- Mathematics
- 2021

While the symplectomorphismgroup of a symplecticmanifold is well-understood as an infinite-dimensional Lie group, little is known about the Poisson diffeomorphism group of other classes of Poisson… Expand

Lie rackoids

- Mathematics
- 2015

We define a new differential geometric structure, called Lie rackoid. A Lie rackoid differentiates to Leibniz algebroids in a similar way as Lie groupoids differentiate to Lie algebroids. Its main… Expand

#### References

SHOWING 1-10 OF 40 REFERENCES

On the group of lagrangian bisections of a symplectic groupoid

- Mathematics
- 2001

The group of lagrangian bisections of a symplectic groupoid extends the concept of the symplectomorphism group. The flux homomorphism is a basic invariant of this group. It is shown that this group… Expand

The Lie Algebra of the Group of Bisections -A Chapter in Synthetic Differential Geometry of Groupoids-

- Mathematics
- 2006

Groupoids provide a more appropriate framework for differential geometry than principal bundles. Synthetic differential geometry is the avantgarde branch of differential geometry, in which nilpotent… Expand

Towards a Lie theory of locally convex groups

- Mathematics
- 2006

Abstract.In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras,… Expand

Lie group structures on symmetry groups of principal bundles

- Mathematics, Physics
- 2007

Abstract In this paper we describe how one can obtain Lie group structures on the group of (vertical) bundle automorphisms for a locally convex principal bundle P over the compact manifold M . This… Expand

General theory of lie groupoids and lie algebroids

- Mathematics
- 2005

Part I. The General Theory: 1. Lie groupoids: fundamental theory 2. Lie groupoids: algebraic constructions 3. Lie algebroids: fundamental theory 4. Lie algebroids: algebraic constructions Part II.… Expand

A Smooth Model for the String Group

- Mathematics
- 2013

We construct a model for the string group as an infinite-dimensional Lie group. In a second step we extend this model by a contractible Lie group to a Lie 2-group model. To this end we need to… Expand

Notes on regularity properties of infinite-dimensional Lie groups

- Mathematics
- 2012

The evolution of a C-regular Lie group G is shown to be smooth as a map to Ck+1([0, 1], G). We also give examples of non-regular Lie groups (modelled on non-Mackey complete spaces) for which some… Expand

Differential calculus, manifolds and lie groups over arbitrary infinite fields

- Mathematics
- 2003

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of… Expand

Infinite-dimensional Lie groups without completeness restrictions

- Mathematics
- 2002

We describe a setting of infinite-dimensional smooth (resp., analytic) Lie groups modelled on arbitrary, not necessarily sequentially complete, locally convex spaces, generalizing the framework of… Expand

Differential Calculus over General Base Fields and Rings

- Mathematics
- 2004

Abstract We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic… Expand