How to think of the group ring as a Hopf algebra?
So first of all the group algebra $K[G]$ has, in addition to its algebra structure, a coproduct,
$$ \Delta: k[G] \rightarrow k[G] \otimes k[G] \\ g \mapsto g \otimes g $$ an antipode, $$ S: k[G] \rightarrow k[G]\\ g \mapsto g^{-1} $$ and a counit $\epsilon:k[G] \rightarrow k$ sending $g \mapsto 1$ for all $g \in G$. It is these maps that give $k[G]$ the structure of a Hopf algebgra.
As far as structure theorems go, there are several. These are particularly striking in the case of the cohomology of a H-space $X$ over a field $k$ of characteristic 0, when Hopf proved that the Hopf algebra $H^\bullet(X; k)$ is:
- An exterior algebra generated by homogeneous elements of odd degree if $H^\bullet(X;k)$ is finite dimensional
- A free graded-commutative algebra if each $H^n(x;k)$ is finite dimensional.
I include these results as they provide both a motivation for what a structure theorem for Hopf algebras looks like, and some historical context, as these objects were the motivation for the definition of a Hopf algebra. As far as more general structure theorems for Hopf algebras go there are some nice results of Cartier, Gabriel and Milnor-Moore. Here are two theorems:
For the first theorem note that $U(\mathfrak{g})$ denotes the universal enveloping algebra. I remark here that $k[G]$ is an example of a cocommutative Hopf algebra.
Theorem (Cartier-Gabriel) Assume that $k$ is algebraically closed, that $A$ is a cocommutative Hopf algebra. Let $\mathfrak{g}$ be the space of primitive elements, and $\Gamma$ the group of group like elements in $A$. Then there is an isomorphism of $\Gamma \ltimes U(\mathfrak{g})$ onto $A$ as Hopf algebras, inducing the identity on $\Gamma$ and on $\mathfrak{g}$.
Theorem (Milnor-Moore) Let $A = \bigoplus_{n \geq0}A_n$ be a graded Hopf algebra over $k$. Assume
- $A_0=k$ (we say $A$ is connected in this case [HINT: think about motivating example above!])
- The product in $A$ is commutative
Then $A$ is a free commutative algebra (a polynomial algebra) generated by homogeneous elements.
Reference: Pierre Cartier has a survey paper called "A primer of Hopf algebras" which has all of this stuff and more. There are further structure theorems for Hopf algebras but they are too hard to state without the addition of more definitions (e.g. conilpotent), which would just take too long here.
If you have a Hopf algebra $H$, then the set $G(H):=\{0\neq g\in H : \Delta(g)=g\otimes g\}$ is always a group (the operation is the product in $H$, the inverse $g^{-1}$ is $S(g)$) (*). So, if you have a Hopf algebra admiting a basis with elements $x_i$ such that $\Delta(x_i)=x_i\otimes x_i$ (that is, it "looks like a group algebra"), then $H=k[G(H)]$ IS a group algebra. The elements in $G(H)$ are called the "group-like" elements.
If in $H$ there is an element $x$ such that $\Delta x=x\otimes 1+1\otimes x$, then $x$ is called PRIMITIVE. The set of primitive elements, denoted $P(H)$ is always a Lie subalgebra of $H$ (where the bracket is the commutator, $[x,y]=xy-yx$). Thus, you have a Hopf algebra map $U(P(H))\to H$ (where $U(P(H)$ = the universal envelopping algebra of the Lie algebra $P(H)$).
An "intuitive" way of looking at Cartier-Gabriel theorem is that cocommutativity of $H$ implies that $H$ is generated by group-likes and primitives.
A generalization of primitive elements is the following: if $g_1$ and $g_2$ are group-likes, a SKEW $g_1$-$g_2$-primitive is an element $x$ such that $\Delta x=g_1\otimes x+x\otimes g_2$.
A very interesting question is when a Hopf algebra is generated by group-likes and skew primitives. This can be though as a generalization of Cartier-Gabriel theorem for the non cocomutative case. In the finite dimensional case, it is an open conjecture that all Hopf algebras are of this form (say $k=\overline k$).
(*) if $H=O(G)$, the reglar functions on an affine group $G$, then the group-like elements in $H^*$ are the algebra maps $H\to k$. (In general $H^*$ is not a Hopf algebra -only when $H$ is finite dimensional- but the notion of group-likes in $H^*$ makes sense anyway, that is, elements in $H^*$ satisfying $m^*(g)=g\otimes g\in H^*\otimes H^*\subset (H\otimes H)^*$.