Naive question about the representation theory of algebraic groups and hopf algebras

Question

I have been learning some representation theory and have some questions about the following pattern:

Instance 1: If we have a finite group $G$ and a field $k$, a representation of $G$ over $k$ consists of a finite dimensional $k$-vector space $V$ and a homomorphism $G \to GL(V)$. We have a Hopf algebra $kG$ and an equivalence between representations of $G$ over $k$ and finite dimensional $kG$-modules.

Instance 2: If we have a Lie group $G$, a representation of $G$ consists of a finite dimensional $\mathbb{R}$-vector space $V$ and a smooth homomorphism $ G \to GL(V) $. Let $\mathfrak{g}$ be the lie algebra of $G$. Then we have a Hopf algebra $U(\mathfrak{g})$ and an equivalence between representations of $G$ and finite dimensional $U(\mathfrak{g})$-modules.

Question: If $G$ is an algebraic group over some field $k$, is there a Hopf algebra floating around?

Thanks!

Answer 1

If $G$ is an affine algebraic group, its coordinate ring $\mathcal O(G)$ is a Hopf algebra. The multiplication is the usual (commutative) pointwise multiplication of functions. The comultiplication is pullback under the map $m:G\times G \to G$ given by the group structure (this is only cocommutative if $G$ is abelian).

The algebraic group $G$ is determined by the Hopf algebra $\mathcal O(G)$.

Representations of $G$ are the same as comodules for $\mathcal O(G)$. So this Hopf algebra is dual to the notion of group algebra for finite groups.

Similarly, the universal enveloping algebra $U(\mathfrak g)$ is (kind of) dual to the coordinate ring as Hopf algebras. More precisely (I think!), the dual of $U(\mathfrak g)$ is the coordinate ring of the formal neighbourhood of the identity in $G$.

Answer 2

To supplement Sam's answer, I'd call attention to the different behavior of algebraic groups (or group schemes) over fields of prime characteristic $p$. Here the Lie algebra of the group again has a universal enveloping algebra, with Hopf algebra structure, but it poorly reflects the "rational" representations of the group. Instead it's essential to modify the construction to get a hyperalgebra. The general theory is well covered in Part I of Jantzen's book Representations of Algebraic Groups (2nd edition, AMS, 2003). The larger Part II focuses on semisimple (or more generally reductive) groups, where there is a nice description of the hyperalgebra showing how it comes by a sort of reduction mod $p$ process using Kostant's $\mathbb{Z}$-form of the usual enveloping algebra in characteristic 0.

There are also good analogues for quantum groups (quantized enveloping algebras) at a root of unity.

Answer 3

There is a direct way to define group algebras for different classes of groups (locally compact groups, Lie groups, algebraic groups, etc.).

For an (affine) algebraic group $G$ one should take the algebra ${\mathcal R}(G)$ of polynomials (i.e. regular functions) on $G$ and then consider the dual algebra ${\mathcal R}^\star(G)$ of linear functionals on ${\mathcal R}(G)$ with the natural topology (in this case the topology of poitwise convergence on ${\mathcal R}(G)$). Then ${\mathcal R}^\star(G)$ becomes a (topological) group algebra for $G$: each "polynomial representation" of $G$ corresponds bijectively to a continuous representation of ${\mathcal R}^\star(G)$. At the same time both ${\mathcal R}(G)$ and ${\mathcal R}^\star(G)$ are Hopf algebras in the monoidal category of stereotype spaces, see details in: S.S.Akbarov, Pontryagin duality in the theory of topological vector spaces and in topological algebra, Journal of Mathematical Sciences, 113(2): 179-349, 2003 (Theorem 10.12).