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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?


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These are two totally different questions. I would suggest to seperate them. And for the first one, the spectrum of an algebraic group is a hopf algebra. – Marc Palm Mar 30 '13 at 18:58
oh right, you mean the coordinate ring? that makes sense... Also I have removed the second question. – Daniel Barter Mar 30 '13 at 19:07
Instance 2 is only true when $G$ is simply-connected; the smallest counterexample is $G = S^1$. – Qiaochu Yuan Mar 30 '13 at 20:07
@Qiaochu: ahh i didn't know this. At the very least a representation of G gives you a representation of the universal cover – Daniel Barter Mar 30 '13 at 20:40
@Qiaochu: the smallest counterexample is $G = \mathbf{Z}/(2)$ since there has been some sloppiness about connectedness hypotheses (a genuine issue in view of Instance 1), though $S^1$ is the smallest "interesting" counterexample. Actually, the compact form $PSU(2)$ of $PGL_2$ is the smallest genuinely interesting counterexample. – user30379 Mar 31 '13 at 4:34
up vote 9 down vote accepted

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 $G$.

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For semisimple $\mathfrak{g}$ in char. 0, Hochschild's 1959 paper "Algebraic Lie algebras and representative functions" addresses the algebra structure of $U(\mathfrak{g})^{\ast}$, and studies the $k$-subspace $A(\mathfrak{g}) = \varinjlim (U(\mathfrak{g})/J)^{\ast}$ for the 2-sided ideals $J$ of finite codimension. He shows $A(\mathfrak{g})$ consists of the "matrix coefficients" of finite-dimensional representations of $\mathfrak{g}$ and so is a $k$-subalgebra, and that Spec($A(\mathfrak{g})$) is simply connected semisimple with Lie algebra $\mathfrak{g}$. – user30379 Mar 31 '13 at 4:23
Thanks! I figured something along those lines must be true. – Sam Gunningham Mar 31 '13 at 5:13
By the way, of course Hochschild has to first prove that $A(\mathfrak{g})$ is actually finitely generated over $k$, which he gets from considerations with fundamental weights (if I remember correctly). – user30379 Mar 31 '13 at 6:08
@Sam: thanks for the great answer! – Daniel Barter Mar 31 '13 at 14:15

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.

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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).

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