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Universal Enveloping Algebras , and the Poincaré-Birkhoff-Witt Theorem

We investigate the fundamental ideas behind Lie groups, Lie algebras, and universal enveloping algebras. In particular, we emphasize the useful properties of the exponential mapping, which allows us to transition between Lie groups and Lie algebras. From there, we discuss universal enveloping algebras, prove their existence and uniqueness, and after introducing the necessary machinery, we prove the Poincaré-Birkhoff-Witt Theorem.