frame groupoid

Definition 0.1.

Let 𝒢 be a groupoidPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, defined as usual by a categoryMathworldPlanetmath in which all morphismsMathworldPlanetmath are invertiblePlanetmathPlanetmath, with the structure mapsPlanetmathPlanetmathPlanetmath s,t:G1G0, and u:G0G1. Given a vector bundleMathworldPlanetmath q:EG0, the frame groupoidPlanetmathPlanetmath is defined as


, with ϕ(E) being the set of all vector space isomorphismsMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath η:ExEy over all pairs (x,y)G02, also with the usual conditions for the structure maps of the groupoid.

Definition 0.2.

Let G be a group and V a vector space. A group representationMathworldPlanetmath is then defined as a homomorphismMathworldPlanetmathPlanetmathPlanetmath


with End(V) being the group of endomorphisms e:VV of the vector space V.

Note: With the notation used above, let us consider q:EG0 to be a vector bundle. Then, consider a group representation– which was here defined as the representation RG of a group G via the group actionMathworldPlanetmath on the vector space V, or as the homomorphism h:GEnd(V), with End(V) being the group of endomorphisms of the vector space V. The generalizationPlanetmathPlanetmath of group representations to the representations of groupoids then occurs naturally by considering the groupoid action on a vector bundle q:EG0. Therefore, the frame groupoid enters into the definition of groupoid representationsPlanetmathPlanetmathPlanetmath (

Title frame groupoid
Canonical name FrameGroupoid
Date of creation 2013-03-22 19:19:14
Last modified on 2013-03-22 19:19:14
Owner bci1 (20947)
Last modified by bci1 (20947)
Numerical id 29
Author bci1 (20947)
Entry type Definition
Classification msc 55N33
Classification msc 55N20
Classification msc 55P10
Classification msc 22A22
Classification msc 20L05
Classification msc 18B40
Classification msc 55U40
Related topic GroupAction
Related topic VectorBundle
Related topic GroupoidRepresentation4
Related topic RepresentationsOfLocallyCompactGroupoids
Related topic FunctorMathworldPlanetmath
Related topic FunctionalBiology
Defines group representation
Defines End(V)
Defines group endomorphism
Defines Lie groupoid representation
Defines structure maps