groupoid representation (original) (raw)
Let q:E⟶M be a vector bundle
, with E the total space, and M a smooth manifold. Then, consider the representation RG of a group G as an action on a vector space V, that is, as a homomorphism
h:G⟶End(V), with End(V) being the group of endomorphisms of the vector space V. 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 E⟶M.
Definition 0.1.
Let 𝒢 be a groupoid, and given a vector bundle q:E⟶M consider the frame groupoid
| Φ(E)=s,t:ϕ(E)⟶M, |
|---|
with ϕ(E) being the set of all vector space isomorphisms η:Ex⟶Ey over all pairs (x,y)∈M2, also with the associated structure maps
. Then, a general representation Rd of a groupoid 𝒢 is defined as a homomorphism Rd:𝒢⟶Φ(E)
Example 0.1: Lie groupoid representations
Definition 0.2.
Let 𝒢L=s,t:G1⟶M be a Lie groupoid. A representation of a Lie groupoid 𝒢L=s,t:G1⟶M on a vector bundle q:E⟶M is defined as a smooth homomorphism (or a functor) ρ:𝒢L⟶Φ(E) of Lie groupoids over M.
Note:A Lie groupoid representation ρ thus yields a functor, R:𝒢L⟶𝐕𝐞𝐜𝐭, with 𝐕𝐞𝐜𝐭 being the category of vector spaces and R(x)=Ex being the fiber at each x∈M, as well as an isomorphism R(g) for each g:x→y.
Example 0.2: Group representationsIf 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 |