Universal enveloping algebra

Summary

The universal enveloping algebra $U(\mathfrak g)$ of a Lie algebra $\mathfrak g$ is the most general associative algebra containing $\mathfrak g$. In it the Lie bracket becomes the commutator $xy - yx$. Representations of $\mathfrak g$ are the same as modules over $U(\mathfrak g)$, so Lie theory can use the tools of ring theory. By the Poincaré–Birkhoff–Witt theorem, $U(\mathfrak g)$ has a basis of ordered monomials in a basis of $\mathfrak g$. It is also a Hopf algebra, and its deformations give the quantum groups.