TITLE: Meaning of the antipode in Hopf algebras?
QUESTION [10 upvotes]: What I understand so far is that Hopf algebra is a vector space which is both algebra and coalgebra. In addition to this, there is a linear operation $S$, which for each element gives a so-called 'anitpode'.
Can anyone give an intuitive explanation of what is the 'antiopde' element? Why is it essential for the representation theory? What is the meaning of the axiom related to it (#3 on wiki)? (looks like without that axiom the product and co-product would be unrelated to each other)
Thanks.

REPLY [2 votes]: A Hopf algebra is an example of an object called a Hopf monoid, which you can define in any symmetric monoidal category. If you have a look at the definition, it looks almost exactly like the definition of a group object, but with tensor products instead of the usual products, and with compatible comultiplication. 
However, in any category with products, every object automatically comes with a comultiplication given by the diagonal map, and every monoid object is canonically a bimonoid. Thus, if you consider the category of sets with the Cartesian product as the monoidal product, then a Hopf monoid in the category of sets is precisely a group, and the antipode map is just the inverse map $g \mapsto g^{-1}$ of the group! Similarly Hopf monoids in the category of smooth manifolds are precisely Lie groups.
In the category of $R$-modules ($R$ commutative) with the usual tensor product, a Hopf monoid is just a Hopf algebra. So Hopf algebras are really like "groups" in the category of $R$-modules, and the antipode should be thought of as being like the group inverse. 
From the point of view of representation theory, the antipode is essential because it allows you to form the dual representation (just as the antipode=inversion for group representations). This is why, for instance, Lie algebras have dual reps: the universal enveloping algebra is a Hopf algebra!
As an aside: thinking about groups as Hopf monoids in the category of sets is very convenient, because it immediately follows that the tensor algebra $T(V)$ of an $R$-module $V$ is a Hopf algebra, and so is the group ring $R[G]$ of a group $G$.