Enveloping algebras

Abstract

AbstractEnveloping 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.KeywordsHopf AlgebraClifford AlgebraAlgebra HomomorphismSymmetric AlgebraAlgebra MorphismThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.