Discussions On Lie Groupoids And Its Actions

Lie groupoids and Lie algebroids can be regarded as a natural generalization of the concept of Lie group and Lie algebras, but just as we know, they also have many characteristics of bundles. In this paper, we did some researches on its basic properties in terms of the two aspects, and hope to deepen our comprehensions and reorganizations of Lie groupoids. This paper was divided into the following three parts:1.Bundle structure of Lie groupoids;2.The forms field and morphisms of Lie groupoids;3. Actions on Lie groupoids.In the section one of the first part, the spaces which consisted of the α—fiber and β—fiber of a Lie groupoids were called α—bundle and β—bundle respectively. And by using the locally trivial conditions, we proved that the α—fiber and β—fiber of a transitive Lie groupoid are fiber bundles. In the section two, we defined the concept Associated groupoids of a fiber bundles, and studied the relationship between principal bundles and Lie groupoids from its actions.The algebra structures of a Lie groupoids were invested in the second part. In the section one, we obtained a conclusion that the results of the Maurer-Cartan forms of a Lie groupoids acts on its left-invariant vector...