Universal Enveloping Algebra

The universal enveloping algebra provides an algebraic framework for studying Lie algebra representations through associative algebra theory.

DefinitionUniversal Enveloping Algebra

Let $\mathfrak{g}$ be a Lie algebra. The universal enveloping algebra $U(\mathfrak{g})$ is the associative algebra generated by $\mathfrak{g}$ with relations: $XY - YX = [X,Y] \text{ for all } X, Y \in \mathfrak{g}$

Formally, $U(\mathfrak{g}) = T(\mathfrak{g}) / I$ where $T(\mathfrak{g})$ is the tensor algebra and $I$ is the ideal generated by $X \otimes Y - Y \otimes X - [X,Y]$ .

TheoremPoincare-Birkhoff-Witt (PBW) Theorem

Let $\{X_1, \ldots, X_n\}$ be an ordered basis of $\mathfrak{g}$ . Then the monomials: $X_1^{k_1} X_2^{k_2} \cdots X_n^{k_n} \quad (k_i \geq 0)$ form a basis for $U(\mathfrak{g})$ as a vector space.

In particular, the canonical map $\mathfrak{g} \to U(\mathfrak{g})$ is injective, and $\dim U(\mathfrak{g}) = \infty$ unless $\dim \mathfrak{g} = 0$ .

Example$U(\mathfrak{sl}_2)$

For $\mathfrak{sl}_2$ with basis $\{H, E, F\}$ and relations $[H,E] = 2E$ , $[H,F] = -2F$ , $[E,F] = H$ :

A PBW basis is $\{H^a E^b F^c : a, b, c \geq 0\}$ .

The element $EF \in U(\mathfrak{sl}_2)$ satisfies $EF = FE + H$ from the relation $[E,F] = H$ .

TheoremUniversal Property

For any associative algebra $A$ and Lie algebra homomorphism $\phi: \mathfrak{g} \to A$ (where $A$ has Lie bracket $[a,b] = ab - ba$ ), there exists a unique algebra homomorphism $\Phi: U(\mathfrak{g}) \to A$ extending $\phi$ .

This universal property characterizes $U(\mathfrak{g})$ up to isomorphism.

DefinitionRepresentations via Modules

A representation of $\mathfrak{g}$ on $V$ is equivalent to a $U(\mathfrak{g})$ -module structure on $V$ . The category of $\mathfrak{g}$ -representations is isomorphic to the category of $U(\mathfrak{g})$ -modules.

This allows us to apply powerful techniques from ring theory and homological algebra to study Lie algebra representations.

Remark

The universal enveloping algebra is crucial for:

Verma modules: Highest weight theory via $U(\mathfrak{g})$ -modules
Homological algebra: Ext and Tor functors for representations
Quantum groups: Deformations $U_q(\mathfrak{g})$ with quantum parameters
D-modules: Differential operators as enveloping algebras

The PBW theorem shows that $U(\mathfrak{g})$ is a filtered deformation of the symmetric algebra $S(\mathfrak{g})$ , connecting Lie theory to commutative algebra and Poisson geometry.