Computations with the dual of a Hopf Group algebra

Question

If $G$ is a finite group, it is well-known that the set $\mathbb{K}^G$ of all functions $f: G\to \mathbb{K} $ is a Hopf algebra over $\mathbb{K}$, with

$m(f \otimes g)(x)=f(x)g(x)$,

$\Delta(f)(x,y)=f(xy)$,

$S(f)(x)=f(x^{-1})$,

for all $x,y\in G$, $f,g\in\mathbb{K}^G$. I have to do this computation

$W=(id_{\mathbb{K}^G} \otimes m)(\Delta \otimes id_{\mathbb{K}^G})$,

but I have some problems with the identification $\mathbb{K}^{G \times G} \cong \mathbb{K}^G \otimes \mathbb{K}^G$.

Can anybody give me a hint or write the computations?

Answer 1

Concerning the isomorphism $\mathbb{K}^{G\times G}\cong \mathbb{K}^G\otimes \mathbb{K}^G$, consider the following facts.

1. The assignment $$\mathbb{K}^G\times \mathbb{K}^G \to \mathbb{K}^{G\times G}, \quad (f,g)\mapsto \Big[(x,y)\mapsto f(x)g(y)\Big]$$ is a well-defined, $\mathbb{K}$-bilinear morphism and hence it factors through a $\mathbb{K}$-linear morphism $\mathbb{K}^G\otimes \mathbb{K}^G\to \mathbb{K}^{G\times G}$;
2. In $\mathbb{K}^G$ there is a family of "distinguished" morphisms: for every $x\in G$ we consider $x^*:G \to \mathbb{K}$ as the $\mathbb{K}$-linear map sending $x$ to $1$ and every other $y\in G$ to $0$;
3. The assignment $$\mathbb{K}^{G\times G} \to \mathbb{K}^G\otimes \mathbb{K}^G,\quad f\mapsto \sum_{x,y\in G}f(x,y)x^*\otimes y^*$$ is well-defined, because $G$ is finite, and it is an inverse for the above $\mathbb{K}$-linear morphism.

It could be interesting to notice that you always have a bijection $$ \mathsf{Fun}(G,\mathbb{K})\cong \mathsf{Hom}_{\mathbb{K}}\left(\mathbb{K}G, \mathbb{K}\right), $$ which is simply a restatement of the fact that the elements of $G$ form a basis of the group algebra $\mathbb{K}G$. Then $$ \mathbb{K}^{G\times G}\cong \mathsf{Hom}_{\mathbb{K}}\left(\mathbb{K}(G\times G),\mathbb{K}\right)\cong \mathsf{Hom}_{\mathbb{K}}\left(\mathbb{K}G\otimes \mathbb{K}G,\mathbb{K}\right)=\left(\mathbb{K}G\otimes \mathbb{K}G\right)^*. $$ Since $G$ is finite, $\mathbb{K}G$ is finite-dimensional over $\mathbb{K}$ and hence $$ \left(\mathbb{K}G\otimes \mathbb{K}G\right)^*\cong \left(\mathbb{K}G\right)^*\otimes\left( \mathbb{K}G\right)^* \cong \mathbb{K}^G \otimes \mathbb{K}^G. $$ If you follow the chain of isomorphisms, you will eventualy run into the one above.

With this at hand, you may convince yourself that $$ \Delta(f) = \sum_{x,y\in G}f(xy)x^*\otimes y^* $$ for every $f\in \mathbb{K}^G$.