# groupoid representation

Let $q:E\u27f6M$ be a vector bundle^{}, with $E$ the *total space*, and $M$ a smooth manifold. Then, consider the representation ${R}_{G}$ of a group $G$ as an action on a vector space $V$, that is, as a homomorphism^{} $h:G\u27f6End(V)$, 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 $E\u27f6M$.

###### Definition 0.1.

Let $\mathcal{G}$ be a groupoid, and given a vector bundle $q:E\u27f6M$ consider the *frame groupoid ^{}*

$$\mathrm{\Phi}(E)=s,t:\varphi (E)\u27f6M,$$ |

with $\varphi (E)$ being the set of all vector space isomorphisms^{}
$\eta :{E}_{x}\u27f6{E}_{y}$ over all pairs $(x,y)\in {M}^{2}$, also with the associated structure maps^{}. Then, a general *representation* ${R}_{d}$ of a groupoid $\mathcal{G}$ is defined as a homomorphism ${R}_{d}:\mathcal{G}\u27f6\mathrm{\Phi}(E)$

Example 0.1: Lie groupoid representations

###### Definition 0.2.

Let ${\mathcal{G}}_{L}=s,t:{G}_{1}\u27f6M$ be a Lie groupoid. A *representation of a Lie groupoid* ${\mathcal{G}}_{L}=s,t:{G}_{1}\u27f6M$ on a vector bundle $q:E\u27f6M$ is defined as a smooth homomorphism (or a functor^{}) $\rho :{\mathcal{G}}_{L}\u27f6\mathrm{\Phi}(E)$ of Lie groupoids over $M$.

Note:
A *Lie groupoid representation* $\rho $ thus yields a functor, $R:{\mathcal{G}}_{L}\u27f6\mathrm{\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}},$ with $\mathrm{\mathbf{V}\mathbf{e}\mathbf{c}\mathbf{t}}$ being the category of vector spaces and $R(x)={E}_{x}$ being the fiber at each $x\in M$, as well as an isomorphism $R(g)$ for each $g:x\to y$.

Example 0.2: Group representations If 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 |