# Applications of a theorem of Cartier and Gabriel

In a representation theory course I took we stated and proved the following Theorem due to Cartier and Gabriel:

Theorem: Suppose $H$ is a cocommutative Hopf algebra over a field $k$ such that $\operatorname{char} k = 0$ and $k$ is algebraically closed. Then there is an isomorphism $H \cong U \left( P (H) \right) \rtimes G(H)$ where $P(H)$ is the set of primitive elements of $H$ and $G(H)$ is the set of group-like elements.

We spent a few lectures proving this statement and then moved on to other things and I never saw any applications of this theorem. So I would like to know some nice applications of this theorem!

• The case when $H$ is connected graded (thus $G\left(H\right)=\left\lbrace 1\right\rbrace$) has a lot of applications. (This case of the theorem is called the Cartier-Milnor-Moore theorem, and does not require $k$ to be algebraically closed; it is fundamental in studying combinatorial Hopf algebras.) I admit I haven't seen any applications of the general case, though I can imagine some to exist. Maybe the theory of affine algebraic groups has some uses for this. – darij grinberg Jul 16 '13 at 22:42
• BTW, I know this theorem under the name "Cartier-Kostant theorem"; is there some convoluted history going on here? – darij grinberg Jul 16 '13 at 22:48
• I think you are correct! I must have copied it down incorrectly, it probably also explains why when I searched on google I got nothing... – Anette Jul 16 '13 at 23:23