Hopf algebra of function on G

Question

I am interested about Hopf algebra and I need this definition, K(G) is Hopf algebra of function on G ,but I do not understand K(G),where G is finite group.

Answer 1

A Hopf algebra is a structure that is both an algebra and a coalgebra. So for functions on our finite group, we need to know how to multiply them, and how to comultiply.

The algebra structure is pointwise. The set of functions $K(S)=K^S$ is an algebra over the field $K$ for functions from any set $S$ into the field. So multiplication is given as $f\cdot g=s\mapsto f(s)g(s).$ This operation has a unit, is associative; it is a commutative algebra structure.

The coalgebra structure exists because the domain of these functions has its own algebraic structure. Comultiplication is a map $\Delta\colon K(G)\to K(G)\otimes K(G)$. Since $G$ is finite, we have an isomorphism $K(G\times G)\cong K(G)\otimes K(G)$, which means we just need to product a function of two variables in $G$.

So if $f\colon G\to K$, then $\Delta(f)$ needs to take two group elements. The natural thing to do is $\Delta(f)(gh)=f(gh),$ using the group multiplication.

Comultiplication must have a counit, and satisfy a coassociativity axiom. It will be cocommutative if $G$ is abelian.

Additionally a Hopf algebra is equipped with an antipode, which relates the algebra and coalgebra structures. For $K(G)$ we may take $f(g^{-1})$ as the antipode of $f$.