frame groupoid
Definition 0.1.
Let 𝒢 be a groupoid, defined as usual by a category
in which all morphisms
are invertible
, with the structure maps
s,t:G1⟶G0, and u:G0⟶G1. Given a vector bundle
q:E⟶G0, the frame groupoid
is defined as
Φ(E)=s,t:ϕ(E)⟶G0 |
, with ϕ(E) being the set of all vector space isomorphisms η:Ex⟶Ey 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 representation is then defined as a homomorphism
h:G⟶End(V), |
with End(V) being the group of endomorphisms e:V⟶V of the vector space V.
Note:
With the notation used above, let us consider q:E⟶G0 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 action on the vector space V, or as the homomorphism h:G⟶End(V), with End(V) being the group of endomorphisms of the vector space V. The generalization
of group representations to the representations of groupoids then occurs naturally by considering the groupoid action on a vector bundle q:E⟶G0. 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 |