frame groupoid
Definition 0.1.
Let be a groupoid, defined as usual by a category in which all morphisms are invertible, with the structure maps , and . Given a vector bundle , the frame groupoid is defined as
, with being the set of all vector space isomorphisms over all pairs , also with the usual conditions for the structure maps of the groupoid.
Definition 0.2.
Let be a group and a vector space. A group representation is then defined as a homomorphism
with being the group of endomorphisms of the vector space .
Note: With the notation used above, let us consider to be a vector bundle. Then, consider a group representation– which was here defined as the representation of a group via the group action on the vector space , or as the homomorphism , with being the group of endomorphisms of the vector space . The generalization of group representations to the representations of groupoids then occurs naturally by considering the groupoid action on a vector bundle . Therefore, the frame groupoid enters into the definition of groupoid representations (http://planetmath.org/GroupoidRepresentation4).
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 | Functor |
Related topic | FunctionalBiology |
Defines | group representation |
Defines | End(V) |
Defines | group endomorphism |
Defines | Lie groupoid representation |
Defines | structure maps |