Lie groups: Poincare-Birkhoff-Witt theorem

The lecture introduces the Poincaré-Birkhoff-Witt (PBW) theorem, which describes the structure of the universal enveloping algebra (UEA) of a Lie algebra. The speaker begins by defining the UEA as the associative algebra generated by the Lie algebra with relations encoding the bracket. He illustrates with examples: abelian Lie algebras yield polynomial algebras, and the UEA of a Lie group’s Lie algebra corresponds to left-invariant differential operators. The main goal is to show that the UEA has the same size as a polynomial algebra in n variables, where n is the dimension of the Lie algebra. The proof strategy involves filtering the UEA and constructing a surjective map from the symmetric algebra (polynomial algebra) onto the associated graded algebra. The PBW theorem asserts this map is an isomorphism, providing a lower bound. The speaker proves the theorem for Lie algebras of Lie groups using local coordinates and left-invariant vector fields, showing linear independence of ordered monomials. For the general case, he sketches a proof using a representation on a space of formal expressions. Applications include recovering the Lie algebra from the UEA via primitive elements (elements with coproduct a⊗1+1⊗a), with a caveat about characteristic p where the theorem fails. The lecture concludes by applying this to free Lie algebras, which was used in the proof of the Baker-Campbell-Hausdorff formula.