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 .
Example 0.1: Lie groupoid representations
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.
|Date of creation||2013-03-22 19:19:17|
|Last modified on||2013-03-22 19:19:17|
|Last modified by||bci1 (20947)|
|Defines||Lie groupoid representation|