groupoid representation

Let $q:E\\longrightarrow M$ be a vector bundle, with $E$ the total space, and $M$ a smooth manifold. Then, consider the representation $R_{G}$ of a group $G$ as an action on a vector space $V$ , that is, as a homomorphism $h:G\\longrightarrow End(V)$ , with $End(V)$ being the group of endomorphisms of the vector space $V$ . The generalization of the group representation to general representations of groupoids then occurs somewhat naturally by considering the groupoid action (http://planetmath.org/GroupoidAction) on a vector bundle $E\\longrightarrow M$ .

Definition 0.1.

Let $\\mathcal{G}$ be a groupoid, and given a vector bundle $q:E\\longrightarrow M$ consider the frame groupoid

\\Phi(E)=s,t:\\phi(E)\\longrightarrow M,

with $\\phi(E)$ being the set of all vector space isomorphisms $\\eta:E_{x}\\longrightarrow E_{y}$ over all pairs $(x,y)\\in M^{2}$ , also with the associated structure maps. Then, a general representation $R_{d}$ of a groupoid $\\mathcal{G}$ is defined as a homomorphism $R_{d}:\\mathcal{G}\\longrightarrow\\Phi(E)$

Example 0.1: Lie groupoid representations

Definition 0.2.

Let $\\mathcal{G}_{L}=s,t:G_{1}\\longrightarrow M$ be a Lie groupoid. A representation of a Lie groupoid $\\mathcal{G}_{L}=s,t:G_{1}\\longrightarrow M$ on a vector bundle $q:E\\longrightarrow M$ is defined as a smooth homomorphism (or a functor) $\\rho:\\mathcal{G}_{L}\\longrightarrow\\Phi(E)$ of Lie groupoids over $M$ .

Note:A Lie groupoid representation $\\rho$ thus yields a functor, $R:\\mathcal{G}_{L}\\longrightarrow{\\bf Vect},$ with ${\\bf Vect}$ being the category of vector spaces and $R(x)=E_{x}$ being the fiber at each $x\\in M$ , as well as an isomorphism $R(g)$ for each $g:x\\to y$ .

Example 0.2: Group representationsIf one restricts the vector bundle to a single vector space in Definition 0.1 then one obtains a group representation, in the same manner as a groupoid that reduces to a group when its object space is reduced to a single object.

Title	groupoid representation
Canonical name	GroupoidRepresentation
Date of creation	2013-03-22 19:19:17
Last modified on	2013-03-22 19:19:17
Owner	bci1 (20947)
Last modified by	bci1 (20947)
Numerical id	40
Author	bci1 (20947)
Entry type	Definition
Classification	msc 55N33
Classification	msc 55N20
Classification	msc 55P10
Classification	msc 22A22
Classification	msc 20L05
Classification	msc 55U40
Related topic	GroupoidAction
Related topic	FrameGroupoid
Related topic	RepresentationsOfLocallyCompactGroupoids
Related topic	Functor
Related topic	FrameGroupoid
Related topic	LieGroupoid
Related topic	CategoryOfRepresentations
Related topic	FunctionalBiology
Defines	frame groupoid
Defines	Vect
Defines	End(V)
Defines	group endomorphism
Defines	Lie groupoid representation