---
tag: DHCHJNHD02
kind: statement
role: theorem
name: Cartier–Milnor–Moore theorem
msc: [16T05, 17B35, 57T05]
uses: [7XFYP0N84H, SCZ8PRVY00, 0F6ES3BRYP]
about: [7XFYP0N84H, SCZ8PRVY00]
sources:
  - url: https://en.wikipedia.org/wiki/Milnor–Moore_theorem
---

## Statement

Let $K$ be a field of [[characteristic|0F6ES3BRYP]] $0$.
Let $H$ be a [[Hopf algebra|7XFYP0N84H]] over $K$ that is cocommutative and connected, which means that $K \cdot 1$ is the only simple subcoalgebra of $H$.
Let $P(H) = \set{x \in H : \Delta(x) = x \otimes 1 + 1 \otimes x}$ be the space of primitive elements, a Lie algebra under the commutator $[x, y] = xy - yx$.
Then the inclusion $P(H) \to H$ extends to an isomorphism of Hopf algebras $U(P(H)) \to H$, where $U(P(H))$ is the [[universal enveloping algebra|SCZ8PRVY00]] of $P(H)$.

## 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.

## Explanations

### L1 human

Lie algebras and their universal enveloping algebras give many Hopf algebras.
The theorem says that, over a field of characteristic $0$, such as $\Q$ or $\C$, every cocommutative Hopf algebra that is connected arises in this way.
So such a Hopf algebra carries no information beyond the Lie algebra of its primitive elements.

### L2 human

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.