Lie algebra Lie group dictionary

The Lie algebra captures the infinitesimal structure of a Lie group and so can be used to study Lie groups themselves. We present the key "dictionary" that shows how this happens. In essence, we see how Lie algebra homomorphisms almost correspond to Lie group homomorphisms via the differential. In particular, we see how the study of finite dimensional complex representations of a Lie group reduce to the algebraic study of finite dimensional modules over the corresponding universal enveloping algebra.

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