groupoid representation
Let be a vector bundle, with the total space, and a smooth manifold. Then, consider the representation of a group as an action on a vector space , that is, as a homomorphism , with being the group of endomorphisms of the vector space . 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 .
Definition 0.1.
Let be a groupoid, and given a vector bundle consider the frame groupoid
with being the set of all vector space isomorphisms over all pairs , also with the associated structure maps. Then, a general representation of a groupoid is defined as a homomorphism
Example 0.1: Lie groupoid representations
Definition 0.2.
Let be a Lie groupoid. A representation of a Lie groupoid on a vector bundle is defined as a smooth homomorphism (or a functor) of Lie groupoids over .
Note: A Lie groupoid representation thus yields a functor, with being the category of vector spaces and being the fiber at each , as well as an isomorphism for each .
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 |