When is a Hopf Algebra isomorphic to a group ring k[G]?

Question

Let $H$ be a Hopf algebra over a field $k$. What are some nice conditions for when $H$ is isomorphic to $k[G]$ for a finite group $G$?

The co-multiplication structure on the group algebra $k[G]$ is given the by map sending $g$ to $g \otimes g$.

Answer 1

Two straightforward necessary conditions are that $H$ be finite-dimensional and that it be cocommutative. At this point it will be cleaner to work with the dual Hopf algebra $H^{\ast}$, which is finite-dimensional and commutative. Because it is commutative, we can use the language of algebraic geometry: $\text{Spec } H^{\ast}$ makes sense and is a finite group scheme over $k$. Now the question is: when is a finite group scheme the constant group scheme $G$ for $G$ a finite group? This corresponds to $H^{\ast}$ being the Hopf algebra of functions $G \to k$, dual to the group algebra $H \cong k[G]$.

The answer is almost never. If $k$ isn't separably closed, then you can construct interesting nonconstant group schemes from pairs consisting of a finite group $G$ and an action of the absolute Galois group $\text{Gal}(k_s/k)$ on $G$, which are constant iff the action is trivial. These are precisely the finite étale group schemes. If $k$ is separably closed, then every finite étale group scheme over $k$ is constant, but in general there are finite group schemes that are not étale. For example, if $k$ has characteristic $p$, then $k[x]/x^p$ can be equipped with a comultiplication $\Delta(x) = x \otimes 1 + 1 \otimes x$ making $\text{Spec } k[x]/x^p$ a finite nonconstant group scheme called $\alpha_p$. It arises naturally as the kernel of the Frobenius map $\mathbb{G}_a \to \mathbb{G}_a$.

So now let's assume that $k$ is separably closed and has characteristic $0$ (so $k$ is algebraically closed). Finally some good news: in this case every finite group scheme is étale and hence constant. In other words,

If $k$ is an algebraically closed field of characteristic zero, then every finite-dimensional cocommutative Hopf algebra over $k$ is a group algebra.

This is a special case of the Cartier-Kostant-Milnor-Moore theorem, although presumably it has a more elementary proof.