Lie groups: Poincare-Birkhoff-Witt theorem
Channel: Richard E Borcherds
Date: 2021-02-19

This lecture is part of an online graduate course on Lie groups.

We state the Poincare-Birkhoff Witt theorem, which shows that the universal enveloping algebra (UEA) of a Lie algebra is the same size as a polynomial algebra. We prove it for Lie algebras of Lie groups and sketch a proof of the general case. 

As an application we show that in characteristic 0 the primitive elements of the UEA are just the original Lie algebra. The case when L is a free Lie algebra on 2 generators was used in the proof of the Baker-Campbell-Hausdorff formula in the previous lecture. 


For the other lectures in the course see https://www.youtube.com/playlist?list=PL8yHsr3EFj53RWBkiHKoOsTw-dGHAoJ-h