groupoid representation

Let q:EM be a vector bundleMathworldPlanetmath, with E the total space, and M a smooth manifold. Then, consider the representation RG of a group G as an action on a vector space V, that is, as a homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath h:GEnd(V), with End(V) being the group of endomorphisms of the vector space V. The generalizationPlanetmathPlanetmath of the group representationMathworldPlanetmath to general representations of groupoidsPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath then occurs somewhat naturally by considering the groupoid action ( on a vector bundle EM.

Definition 0.1.

Let 𝒢 be a groupoid, and given a vector bundle q:EM consider the frame groupoidPlanetmathPlanetmath


with ϕ(E) being the set of all vector space isomorphismsMathworldPlanetmathPlanetmath η:ExEy over all pairs (x,y)M2, also with the associated structure mapsPlanetmathPlanetmath. Then, a general representation Rd of a groupoid 𝒢 is defined as a homomorphism Rd:𝒢Φ(E)

Example 0.1: Lie groupoid representations

Definition 0.2.

Let 𝒢L=s,t:G1M be a Lie groupoid. A representation of a Lie groupoid 𝒢L=s,t:G1M on a vector bundle q:EM is defined as a smooth homomorphism (or a functorMathworldPlanetmath) ρ:𝒢LΦ(E) of Lie groupoids over M.

Note: A Lie groupoid representation ρ thus yields a functor, R:𝒢L𝐕𝐞𝐜𝐭, with 𝐕𝐞𝐜𝐭 being the category of vector spaces and R(x)=Ex being the fiber at each xM, as well as an isomorphism R(g) for each g:xy.

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 representationPlanetmathPlanetmathPlanetmath
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