# frame groupoid

###### Definition 0.1.

Let $\mathcal{G}$ be a groupoid, defined as usual by a category in which all morphisms are invertible, with the structure maps $s,t:G_{1}\longrightarrow G_{0}$, and $u:G_{0}\longrightarrow G_{1}$. Given a vector bundle $q:E\longrightarrow G_{0}$, the frame groupoid is defined as

 $\Phi(E)=s,t:\phi(E)\longrightarrow G_{0}$

, with $\phi(E)$ being the set of all vector space isomorphisms $\eta:E_{x}\longrightarrow E_{y}$ over all pairs $(x,y)\in{G_{0}}^{2}$, 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

 $h:G\longrightarrow End(V),$

with $End(V)$ being the group of endomorphisms $e:V\longrightarrow V$ of the vector space $V$.

Note: With the notation used above, let us consider $q:E\longrightarrow G_{0}$ to be a vector bundle. Then, consider a group representation– which was here defined as the representation $R_{G}$ of a group $G$ via the group action on the vector space $V$, or as the homomorphism $h:G\longrightarrow End(V)$, 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 $q:E\longrightarrow G_{0}$. 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