Enveloping algebras

Abstract

Enveloping algebras define a functor $\mathfrak{g}\mapsto U(\mathfrak {g})$ from the category of Lie algebras to the category of associative unital algebras in such a way that representations of $\mathfrak{g}$ on vector spaces V are equivalent to algebra representations of $U(\mathfrak{g})$ on V. A fundamental result in the theory of enveloping algebras is the Poincaré–Birkhoff–Witt Theorem, which states that the natural “quantization map” $q_{U}:\ S(\mathfrak{g})\to U(\mathfrak{g})$ is an isomorphism of vector spaces. One of the goals of this chapter is to present a proof of this result, due to E. Petracci, which is similar to the proof that the quantization map for Clifford algebras is an isomorphism. The proof builds on a discussion of the Hopf algebra structure on the enveloping algebra, and the fact that the quantization map q _U preserves the comultiplication.