Uploaded February 2021 | Updated September 2026, 2 weeks ago
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 youtube.com/playlist?list=PL8yHsr3EFj53RWBkiHKoOsTw-dGHAoJ-h
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 youtube.com/playlist?list=PL8yHsr3EFj53RWBkiHKoOsTw-dGHAoJ-h
