# 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 representations If 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