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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.)

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    I don't understand what "looks like" means here. If I remember correctly, Hopf algebraists are often happy to say that something "looks like" a group Hopf algebra, so group Hopf algebras are considered as something rather basic already.2011-10-20
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    What I can say is that *as a coalgebra*, a group Hopf algebra looks extremely simply: It is a direct sum of $1$-dimensional coalgebras, each of which is just given by $\Delta g=g\otimes g$ for some generator $g$ (this is, of course, the only possible form of a $1$-dimensional coalgebra). This direct sum decomposition is unique (up to the order of the addends) and gives you your group $G$ back, at least as a set (you then get the group operation from the algebra structure).2011-10-20
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    As Darij says, Hopf theorists consider group algebras as *building blocks*, so it is not quite clear what you want here.2011-10-20
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    Yes, well that's fair enough. It's just that, as someone who might be scared of general group rings might be reassured that its structure as an algebra can be easily understood with Wedderburn's Theorem, someone who's a bit scared of Hopf algebras could be reassured by some similar simple classification result. But it's become rather clear that the information will flow the other way: one can't get a good classification as finite groups are hard to classify, so in the end you instead just get to understand these particularly simple Hopf algebras in terms of the group ring, and not conversely.2011-10-21
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    Finite dimensional Hopf algebras are (much!) harder to classify than finite groups2011-10-21
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    In response to your last paragraph, I learned that one way to think about Hopf algebras in general is to think of them as those algebras whose representations behave like the representations of a finite group in certain ways. Namely, having a coproduct means that you can define the tensor product of representations and having an antipode means that you can define the dual of a representation. So Hopf algebras can be thought of loosely as "those algebras whose categories of representations have tensor products and duals".2011-10-25
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    One way to be able to think of any finite dimensional cocommutative Hopf algebra as something that "looks like" a group algebra is by taking the dual instead. This is then a commutative Hopf algebra, which means that it is the algebra of regular functions of an affine group scheme.2012-09-26

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