Hopf algebras arising as Group Algebras

As Ben says, a commutative Hopf algebra $R$ is, by definition, the coordinate ring of an affine group scheme. There are also group schemes that are not affine, such as abelian varieties, so these are excluded from the question. Since the question is about algebras, let's say that the group scheme $G$ is defined over a field $k$. Then $G$ is morally, but not actually, the same as its group of $k$-rational points $G(k)$. Unless $G$ is a finite group with $R = k[G(k)]^*$, the dual of the usual group algebra, there are three possible differences between $R$ and a (dual) group algebra.

First, the group elements of $G(k)$ are ideals of $R$ with residue field $k$, and their group law is given by the coproduct on $R$. If $p$ is such an ideal, viewed as a point on $G$, then in general the function that is 1 on $p$ and $0$ on the rest of $G$ is not regular; it is not an element of $R$. This is one way to tell that $G$ must be 0-dimensional in order for $R$ to be a literal group algebra.

Second, if $k$ is not algebraically closed, there may be other closed points in $G$ whose residue field is a field extension of $k$. If the field extension is separable, then the group law on these points is multivalued. For instance if $G = text{GL}(n,mathbb{R})$, then it has complex points, which correspond to complex conjugate pairs of complex matrices. The way to multiply two of these points is to multiply conjugates in all possible ways. (This is an example of making a tensor product $E otimes_k F$ of two fields over a field, an operation that also came up in another MO question.)

Third, in characteristic $p$, $G$ may not be reduced. The simplest example is actually finite-dimensional: Take the universal enveloping algebra $U(L)$ of an abelian Lie algebra, which is to say a polynomial algebra, and divide by the ideal of $p$th powers of elements of $L$ to obtain a finite-dimensional Hopf algebra $u(L)$ which is a local ring. (This is similar and related to a common construction with quantum group Hopf algebras.) However, Cartier and Oort showed that algebraic group schemes in characteristic zero are reduced. You can always reduce $G$ as a scheme, but, as in the example $u(L)$, you may be throwing away everything interesting.

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