Draft:Cartier-Kostant Theorem

The Cartier–Kostant theorem (also called the Cartier–Gabriel–Kostant theorem) is a structure theorem for cocommutative Hopf algebras over fields of characteristic zero.

Let $H$ be a cocommutative Hopf algebra over an algebraically closed field $k$ of characteristic zero. Let

$G(H)=\{g\in H:\Delta (g)=g\otimes g,\ \varepsilon (g)=1\}$

be its group of group-like elements, and let

$P(H)=\{x\in H:\Delta (x)=x\otimes 1+1\otimes x\}$

be its Lie algebra of primitive elements. Then

$H\cong U(P(H))\#k[G(H)],$

where $U(P(H))$ is the universal enveloping algebra of $P(H)$, $k[G(H)]$ is the group algebra of $G(H)$, and $\#$ is the smash product induced by the conjugation action

$g\cdot x=gxg^{-1}.$

Thus a cocommutative Hopf algebra in characteristic zero is described by its primitive elements, its group-like elements, and the action of the latter on the former.

When $H$ is connected, $G(H)$ is trivial and

$H\cong U(P(H)),$

which is the connected case associated with the Milnor–Moore theorem.

Morally, this says that cocommutative Hopf algebras over fields of characteristic zero are classified by only three pieces of datum alone: a Lie algebra, a group, and some twisting information (more specifically, a group action on the Lie algebra).

See also

• Hopf algebra
• Milnor–Moore theorem
• Universal enveloping algebra
• Group algebra