Universal enveloping algebra of a Lie algebra

Definition

Let $\mathfrak g$ be a Lie algebra over a field $k$ . Its universal enveloping algebra is a unital associative $k$ -algebra $U(\mathfrak g)$ with a Lie homomorphism $i:\mathfrak g\to U(\mathfrak g)^{-}$ such that every Lie homomorphism $f:\mathfrak g\to A^{-}$ , for a unital associative $k$ -algebra $A$ , factors uniquely as $f=\widetilde f\circ i$ through a unital algebra homomorphism $\widetilde f:U(\mathfrak g)\to A$ . Here $A^{-}$ carries the commutator bracket. Concretely,

U(\mathfrak g)=T(\mathfrak g)/\langle x\otimes y-y\otimes x-[x,y]\rangle .

Universal property and representations

The universal property determines $U(\mathfrak g)$ uniquely up to a unique compatible isomorphism. Giving a representation $\mathfrak g\to\operatorname{End}_k(V)^{-}$ is therefore equivalent to giving a unital left $U(\mathfrak g)$ -module structure on $V$ . This converts iterated actions of Lie-algebra elements into multiplication in one associative algebra.

Filtration and the PBW theorem

Tensor degree induces an increasing filtration on $U(\mathfrak g)$ . The Poincaré–Birkhoff–Witt theorem identifies its associated graded algebra with the symmetric algebra $S(\mathfrak g)$ ; in particular, the canonical map $i:\mathfrak g\to U(\mathfrak g)$ is injective. Over an ordered basis of $\mathfrak g$ , ordered monomials form a vector-space basis of $U(\mathfrak g)$ Dixmier, §2.1.

Examples and conventions

If $\mathfrak g$ is abelian, then $U(\mathfrak g)\cong S(\mathfrak g)$ . For a one-dimensional abelian Lie algebra this is the polynomial algebra $k[t]$ . The adjective “universal” refers to the displayed factorization property; it does not mean a universal $C^*$ -completion. Unless stated otherwise, homomorphisms and algebras here are unital and $k$ -linear.

References

Jacques Dixmier, Enveloping Algebras, Graduate Studies in Mathematics 11, American Mathematical Society, 1996. DOI record. Relevant: Chapter 2, especially the construction and PBW theorem.
Anthony W. Knapp, Lie Groups Beyond an Introduction, 2nd ed., Birkhäuser, 2002. DOI record. Relevant: Chapter V on enveloping algebras.