How to think of the group ring as a Hopf algebra?

Given a finite group $G$ and a field $K$, one can form the group ring $K[G]$ as the free vector space on $G$ with the obvious multiplication. This is very useful when studying the representation theory of $G$ over $K$, as for instance if $K=\mathbb{C}$, by Maschke’s Theorem and Wedderburn’s Theorem we can write $\mathbb{C}[G] = \bigoplus_i \mathrm{M}_{n_i}(\mathbb{C})$, and each factor corresponds to an $n_i$-dimensional irreducible representation of $G$.

However, this decomposition of the group ring doesn’t remember as much as one would initially hope, for instance one has that $\mathbb{C}[D_4] \cong \mathbb{C}[Q_8]$, where $D_4$ is the dihedral group of order $8$ and $Q_8$ is the quaternion group. So one can’t recover the group from the group ring in general.

One way to remedy this is by imposing more structure on the group ring $K[G]$. For instance, it is a cocommutative Hopf algebra, and one can recover the group as the set of group-like elements in $K[G]$.

Given that we have more information here to keep track of, I’m not sure what the Hopf algebra “looks like”. Is there some structure theorem that tells us what the group ring looks like as a Hopf algebra, especially in terms of the representation theory of $G$?

(Any answers providing general intuition about how to think of Hopf algebras in general are more than welcome.)

Answer

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:

  1. An exterior algebra generated by homogeneous elements of odd degree if $H^\bullet(X;k)$ is finite dimensional
  2. 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

  1. $A_0=k$ (we say $A$ is connected in this case [HINT: think about motivating example above!])
  2. 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.

Attribution
Source : Link , Question Author : Will , Answer Author : L Gavroche

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