# 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 ^{}* $s,t:{G}_{1}\u27f6{G}_{0}$, and $u:{G}_{0}\u27f6{G}_{1}$. Given a vector bundle

^{}$q:E\u27f6{G}_{0}$, the

*frame groupoid*is defined as

^{}$$\mathrm{\Phi}(E)=s,t:\varphi (E)\u27f6{G}_{0}$$ |

, with $\varphi (E)$ being the set of all vector space isomorphisms^{} $\eta :{E}_{x}\u27f6{E}_{y}$ 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

^{}

$$h:G\u27f6End(V),$$ |

with $End(V)$ being the group of endomorphisms $e:V\u27f6V$ of the vector space $V$.

Note:
With the notation used above, let us consider $q:E\u27f6{G}_{0}$ 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 $h:G\u27f6End(V)$, 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 $q:E\u27f6{G}_{0}$. 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 |