Cartier–Milnor–Moore theorem

Remarks

The map $U(P(H)) \to H$ exists by the universal property of $U$, and it respects the coproducts because both are generated by primitive elements. Injectivity uses the Poincaré–Birkhoff–Witt theorem and the fact that in characteristic $0$ the primitive elements of $U(\mathfrak{g})$ are exactly $\mathfrak{g}$. Surjectivity is proved by induction along the coradical filtration of $H$. Example: $H = K[x]$ with $x$ primitive; then $P(H) = Kx$, an abelian Lie algebra of dimension $1$, and $U(Kx) = K[x]$. Characteristic $0$ is needed: if $K$ has characteristic $p > 0$, then $x^p$ is also primitive in $K[x]$, since $\Delta(x)^p = x^p \otimes 1 + 1 \otimes x^p$. Then $U(P(K[x]))$ is a polynomial ring in the independent variables $x, x^p, x^{p^2}, \dots$, and the map to $K[x]$ is not injective. Connectedness is needed: for a group $G \ne 1$, $KG$ is cocommutative but $P(KG) = 0$, so $U(P(KG)) = K \ne KG$. Milnor and Moore proved the graded version. A connected graded cocommutative Hopf algebra over a field of characteristic $0$ is the universal enveloping algebra of its graded Lie algebra of primitives.

Going deeper

For a student

The theorem is one half of the classification of cocommutative Hopf algebras in characteristic $0$. Over an algebraically closed field, the Cartier–Gabriel–Kostant theorem adds group algebras: $H$ is the smash product $U(P(H)) \# KG(H)$, where $G(H)$ is the group of group-like elements. In algebraic topology, let $X$ be a simply connected space with loop space $\Omega X$. The graded version identifies the rational homology of $\Omega X$ with the universal enveloping algebra of the Lie algebra $\pi_*(\Omega X) \otimes \Q$. It is also the reason why the Hopf algebras of renormalization in physics are dual to universal enveloping algebras.