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

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 $g \otimes g$.

• – Watson Jun 28 '18 at 20:05
• @Watson: it's not that closely related. The functor from groups to Hopf algebras is fully faithful; you can recover the underlying group from $k[G]$ as the grouplike elements. So the situation is pretty different from if you only have access to the ring structure. – Qiaochu Yuan Jun 28 '18 at 22:04
• Your description of the comultiplication is incorrect; what you've described is the comultiplication on the dual Hopf algebra of functions $G \to k$. The comultiplication on $k[G]$ is $\Delta(g) = g \otimes g$. – Qiaochu Yuan Jun 28 '18 at 22:08
• My apologies, it's fixed. – Dean Young Jun 28 '18 at 23:59

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.