I'm reading Mukai's book "An introduction to invariants and moduli", and I am having trouble understanding one of his examples. It is example 3.49 on page 101.


The setup is as follows. Let $G$ be a finite group, considered as an algebraic group over a field $k$. The coordinate ring of $G$ is then just the set of functions $G \rightarrow k$ with the usual pointwise addition and multiplication. This can be identified with the group ring $k[G]$ in the obvious way (an element $[g] \in k[G]$ corresponds to the function $G \rightarrow k$ that takes $g$ to $1$ and $h$ to $0$ for $h \neq g$). Under this identification, it seems to me that the coproduct is the function


$$\phi : k[G] \rightarrow k[G] \otimes k[G]$$


$$\phi([g]) = \sum\_{h \in G} [h] \otimes [h^{-1} g]$$


However, Mukai asserts that if $G$ is the finite cyclic group of order $n$, then the coordinate ring of $G$ is $k[t]/(t^n-1)$ with the coproduct $t \mapsto t \otimes t$. These do not seem like the same thing to me -- what am I doing wrong?



Coproduct on coordinate ring of finite algebraic group
I think the author accidentally described the dual of the Hopf algebra you're thinking of. 
Finite group rings are usually endowed with multiplication $(g,h)\mapsto gh$ and comultiplication $g \mapsto g\otimes g$ (see [here](http://en.wikipedia.org/wiki/Group_Hopf_algebra)). 


The coordinate ring $k[G]$ is obtained by dualizing. Then $g \mapsto g\otimes g$ becomes 
$e\_g^2 = e\_g$, where $e\_g$ is the function on $G$ that maps $g$ to $1$ and all other group elements to $0$. Comultiplication will look exactly the way you described it (i.e. $e\_g \mapsto \sum\_h e\_{gh^{-1}}\otimes e\_h$).


