I've been working through Federico Ardila's online Hopf algebra lectures and hoped to check my understanding so far by constructing the Hopf algebra of a very small group ring from scratch. But I've failed, which has left me in a state of perplexity.
I chose $G=\mathbb{Z}/2\mathbb{Z}$ and $\mathbb{k}=\mathbb{Z}/3\mathbb{Z}$. So my group ring $\mathbb{k}G$ is the set of all formal $\mathbb{k}$-linear combinations of the two elements of $G$ (I'll call them $e$ and $a$ to avoid confusion with the coefficients, which I'll write as 0, 1, 2).
On top of this structure we have:
- Multiplication as defined here.
- Comultiplication as $r\mapsto r\otimes r$, for each $r\in \mathbb{k}G$
- The unit as $g\mapsto 1_\mathbb{k}$ for $g\in G$, extended linearly (so, for example, it maps $2e + a\mapsto 2 + 1 = 3 = 0 \mod 3$)
- The counit as $\lambda \mapsto \lambda e$ for all $\lambda\in \mathbb{k}$
- The antipode as $g\mapsto g^{-1}$, extended linearly. But in this case, that's just the identity.
With these definitions, I hoped this diagram would commute:
but it blinking doesn't. Example:
- Along the top path: $(e + a)\to (e + a)\otimes (e + a)\to (e + a)\otimes (e + a)\to (2e + 2a)$
- Through the middle: $(e + a)\to 2\to (2e)$.
And it's obvious why: products in the group ring don't always annihilate the $a$ element, but the counit-unit sequence does. So this could never work.
Hopefully I've explained what I've done in enough detail that whatever fundamental thing I've misunderstood isn't elided in the process. I'm hoping someone will be kind and patient enough to unpick it and point to the thing I've stupidly misunderstood!
The definition of comultiplication you have isn't quite right. $\Delta(g)=g\otimes g$ is true only for group elements, not every element of the algebra (that wouldn't even be a linear map!). Thus $\Delta (e+a) = \Delta(e) + \Delta(a) = e\otimes e + a \otimes a$, this goes to $e\otimes e +a\otimes a$ again, then to $e + a^2 = 2e$ agreeing with the path across the middle.