The previous lecture in this series is here. The next lecture in this series is here.
We state the Poincaré-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 two generators was used in the proof of the Baker-Campbell-Hausdorff formula in the previous lecture.
This video is part of a lecture course by Richard Borcherds from 2021.