I am working through some of my first exercises regarding Hopf algebras and I am kind of stuck with this one:
Given an algebraically closed field $k$ and a cocommutative $k$-Hopf algebra $A$ finite as $k$-vector space, the group functor the Hopf algebra represents is constant. As you can notice I am looking at an entirely algebraic approach, not using scheme theory language. Here's what I know:
I proved that the tangent space $T_0A \cong Hom_{k-HopfAlg}(k[X],A^\vee)$, where $k[X]$ represents $\mathbb{G}_{a,k}$, and $A^\vee$ is the dual Hopf algebra i.e. the one representing the Cartier dual of the functor $h^A(-)$ . Also proved: $A$ represents a constant group functor iff $A^0 \cong k$, where $A^0$ is the "connected part" in the canonical decomposition $A \cong A^0 \otimes A^{const}$. I hope that it is clear what I mean by this notation. I realise that it is a bit strange to talk about this without using group scheme theory, but my course is structured in that manner.
I hope it's not too impolite to ask that any hints and/or answers be detailed about what they are trying to get across, since, as I mentioned, I don't have a lot of intuition on this. My main problem is not having enough time to go through a Group Scheme Theory textbook before I have to turn in my homework.