TITLE: Abelian category from the category of Hopf algebras
QUESTION [7 upvotes]: The kernel of a Hopf algebra map $\phi:H_1 \to H_2$ is in general not a Hopf 
sub-algebra of $H_1$. Is there some replacement or alteration of the notion 
of a kernel in the Hopf algebra setting. Same question for cokernels. So can we construct an abelian category from the category of Hopf algebras (over a fixed field $k$).

REPLY [2 votes]: This is a delicate matter, and there are in principle many possible answers depending on exactly what you wish to apply things to.
Frequently, one of two variations of the Hopf subalgebra mentioned by Konstantinos is used: Given $\pi\colon H\to K$ a morphism of Hopf algebras, we have the left coinvariants and right coinvariants (of $\pi$) defined respectively by:
$$ {}^{\text{co}\,\pi}H = \{ h\in H  \ | \ (\pi\otimes\operatorname{id})\Delta(h) = 1\otimes h\},\\
H^{\text{co}\,\pi} = \{ h \in H \ | \ (\operatorname{id}\otimes\pi)(\Delta(h))= h\otimes 1\}.$$
These need not be the same subobjects of $H$, however.
One could then define a short exact sequence of Hopf algebras (over $k$)
$$ k\longrightarrow K \overset{i}{\longrightarrow} H \overset{\pi}{\longrightarrow} L\longrightarrow k$$
to be a sequence of morphisms of Hopf algebras such that

$i$ is injective and $\pi$ is surjective;
$\ker(\pi) = H\, i(K)^+$;
$i(K) = {}^{\text{co}\,\pi}H$.

Note that the more classical notion of kernel is still involved: it will generally not be enough to conduct any of the usual arguments you'd like to do with a "short exact sequence" to assume only the first and third (or first and second) conditions.  The first and third are enough when $H$ is faithfully coflat, and the first and second are enough when $H$ is faithfully flat.
This is a good definition for short exact sequences and results that normally rely on them, such as looking for Jordan-Holder analogues.  As such, coinvariants-as-kernels will be prominent when working in a category of Hopf algebras.  But as I said it's not the only answer, and in other categorical contexts (generalizing/lifting to functors between representation categories, namely) it is the wrong answer.  The categorical kernels and cokernels of $\pi\colon H\to K$ are
$$ \text{Hker}(\pi) = \{ h\in H \ | \ h_{(1)}\otimes \pi(h_{(2)})\otimes h_{(3)} = h_{(1)}\otimes 1 \otimes h_{(2)}\},\\
\text{Hcoker}(\pi) = K/(K \pi(H^+) K).$$
And this kernel (and cokernel) is not necessarily the same thing as the left or right coinvariants (and associated cokernel). By applying counits to the left/right of the defining relation, we see that $$\text{Hker}(\pi)\subseteq {}^{\text{co}\,\pi}H\cap H^{\text{co}\,\pi},$$ at least. Though equality (of all three subobjects) can happen, and assuring it does is usually the key to making sure attempts at defining exact sequences of tensor categories works out well.
Sonia Natale recently posted a review of these notions, and the related issues with tensor categories, that should be helpful.