Universal Enveloping Algebra - Motivation

Motivation

An important topic in Lie algebras studies and probably the main source of their appearance in applications is representation of the Lie algebra. A representation ρ assigns to any element x of a Lie algebra a linear operator ρ(x). The space of linear operators is not only a Lie algebra, but also an associative algebra and so one can consider products ρ(x)ρ(y). The main point to introduce the universal enveloping algebra is to study such products in various representations of a Lie algebra. One obstacle can be immediately seen in a naive attempt to do this: properties of products drastically depend on the representation, not only on the Lie algebra itself. For example for one representation we might have ρ(x)ρ(y) = 0, while in another representation this product may not be zero. Nevertheless it appears to be true that certain properties are universal for all representations, i.e. they hold true for all representations simultaneously. The universal enveloping algebra is a way to grasp all such properties and only them.