Basic Lie Theory
This chapter is devoted to the subject proper of this book: Lie groups, defined as smooth manifolds with a group structure such that all structure maps (multiplication and inversion) are smooth. Here we use vector fields to build the key tools of Lie theory. The Lie functor which associates a Lie algebra with a Lie group and the exponential function from the Lie algebra to the Lie group. They provide the means to translate global problems into infinitesimal ones and to lift infinitesimal solutions to local ones. Passing from the local to the global level usually requires tools from covering theory, resp., topology. In the process, we introduce smooth group actions and the adjoint representation, and provide a number of topological facts about Lie groups.
KeywordsAdjoint Representation Smooth Action Identity Neighborhood Trotter Product Formula Complete Vector Field
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