254A, Notes 1: Lie groups, Lie algebras, and the Baker-Campbell-Hausdorff formula

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In this set of notes, we describe the basic analytic structure theory of Lie groups, by relating them to the simpler concept of a Lie algebra. Roughly speaking, the Lie algebra encodes the “infinitesimal” structure of a Lie group, but is a simpler object, being a vector space rather than a nonlinear manifold. Nevertheless, thanks to the fundamental theorems of Lie, the Lie algebra can be used to reconstruct the Lie group (at a local level, at least), by means of the exponential map and the Baker-Campbell-Hausdorff formula. As such, the local theory of Lie groups is completely described (in principle, at least) by the theory of Lie algebras, which leads to a number of useful consequences, such as the following:

  • (Local Lie implies Lie) A topological group $latex {G}&fg=000000$ is Lie (i.e. it is isomorphic to a Lie group) if and only if it is locally…

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