Lie Groupoid - Examples

Examples

  • Any Lie group gives a Lie groupoid with one object, and conversely. So, the theory of Lie groupoids includes the theory of Lie groups.
  • Given any manifold, there is a Lie groupoid called the pair groupoid, with as the manifold of objects, and precisely one morphism from any object to any other. In this Lie groupoid the manifold of morphisms is thus .
  • Given a Lie group acting on a manifold, there is a Lie groupoid called the translation groupoid with one morphism for each triple with .
  • Any foliation gives a Lie groupoid.
  • Any principal bundle with structure group G gives a groupoid, namely over M, where G acts on the pairs componentwise. Composition is defined via compatible representatives as in the pair groupoid.

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