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

@article{Amiri2018LinkingLG, title={Linking Lie groupoid representations and representations of infinite-dimensional Lie groups}, author={H. Amiri and Alexander Schmeding}, journal={Annals of Global Analysis and Geometry}, year={2018}, volume={55}, pages={749-775} }

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 associated Lie groups. The groups envisaged here are the bisection group and a group of groupoid self-maps. Then, representations of the Lie groupoids give rise to representations of the infinite-dimensional Lie groups on spaces of (compactly supported) bundle sections. Endowing the spaces of bundle… Expand

#### 3 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

Lie groupoids of mappings taking values in a Lie groupoid

- Mathematics
- 2018

Endowing differentiable functions from a compact manifold to a Lie group with the pointwise group operations one obtains the so-called current groups and, as a special case, loop groups. These are… 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

#### References

SHOWING 1-10 OF 46 REFERENCES

VB-groupoids and representation theory of Lie groupoids

- Mathematics
- 2010

A VB-groupoid is a Lie groupoid equipped with a compatible linear structure. In this paper, we describe a correspondence, up to isomorphism, between VB-groupoids and 2-term representations up to… 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

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

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

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

The Lie group of bisections of a Lie groupoid

- Mathematics
- 2014

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… 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

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

Representations of the group of smooth mappings of a manifold X into a compact Lie group

- Mathematics
- 1977

Some important nonlocal representations of the group Gx consisting of C°°-mappings of a Riemannian manifold X to a compact semisimple Lie group G are constructed. The irreducibility, as well as… Expand

STRONG TOPOLOGIES FOR SPACES OF SMOOTH MAPS WITH INFINITE-DIMENSIONAL TARGET

- Mathematics
- 2016

In this article we study two "strong" topologies for spaces of smooth functions from a finite-dimensional manifold to a (possibly infinite-dimensional) manifold modeled on a locally convex space.… Expand