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

What's new

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…

View original post 10,833 more words