**Background:**


Let $G$ be a linear algebraic group over an algebraically closed field $k$ and let $I \subseteq k[G]$ be the ideal of the identity element. The hyperalgebra $U(G)$ of $G$ is defined to be the subspace of the linear dual of $k[G]$ consisting of all $f$ such that $f(I^n) = 0$ for some $n > 0$; then there is a natural Hopf algebra structure on $U(G)$.


Given any Hopf algebra $A$ over a field $k$, one can also define the Hopf dual $A^\*$ of as follows: Let $A^\*$ be the subspace of the full linear dual of $A$ consisting of elements that vanish on some two-sided ideal of $A$ of finite codimension. Then $A^\*$ has a natural Hopf algebra structure.


EDIT: As pointed out in comments, I am using what might be nonstandard terminology -- perhaps I should write $A^\circ$ instead of $A^\*$.


**Questions**:


I now have two related questions.


1) I assume that in general the hyperalgebra of $G$ is not the same as the Hopf algebra dual of $k[G]$. What *is* the Hopf dual of $k[G]$?


2) Although one obtains $U(G)$ from $k[G]$ by a dual construction, you can't in general obtain $k[G]$ from $U(G)$ -- for example, if $G$ is reductive, you can't get $k[G]$ from $U(G)$ by some sort of duality since the hyperalgebra doesn't see isogeny. Is there something one can say in the reductive case about the structure of the Hopf dual of $U(G)$? More generally, for which algebraic groups $G$ can we obtain $k[G]$ from $U(G)$ by some sort of duality? And when we can do so, is it as simple as just taking the Hopf dual?



Hopf algebra duality and algebraic groups
In prime characteristic (or for algebraic groups rather than Lie algebras in general), the comments already posted indicate a need for caution. Jantzen's
Part I covers a lot of the ground, but he refers back at a few delicate points to Demazure-Gabriel. 


Duality for general Hopf algebras is discussed in section 3.5 of Cartier's 2006 notes 


[<http://inc.web.ihes.fr/prepub/PREPRINTS/2006/M/M-06-40.pdf>](http://inc.web.ihes.fr/prepub/PREPRINTS/2006/M/M-06-40.pdf) 


Starting with a Hopf algebra $A$, its *reduced dual* Hopf algebra (denoted by him $R(A)$) lives in the linear dual (a coalgebra in its own right) but is often smaller. So it's complicated to go back and forth. On the other hand, dealing with algebraic groups rather than just Lie algebras makes life a little more complicated, as suggested in his earlier discussion of the algebra of representative functions on a group; see also:


"Remark 3.7.3. Let $k$ be an algebraically closed field of arbitrary characteristic.
As in subsection 3.2, we can define an algebraic group over $k$ as a pair
$(G,O(G))$ where $O(G)$ is an algebra of representative functions on $G$ with
values in $k$ satisfying the conditions stated in Lemma 3.2.1. Let 
$H(G)$ be the
reduced dual Hopf algebra of $O(G)$. It can be shown that $H(G)$ is a twisted
tensor product $G \times U(G)$ where $U(G)$ consists of the linear forms on $O(G)$ vanishing on some power $\mathfrak{m}^N$ of the maximal ideal $\mathfrak{m}$ corresponding to the unit element of $G$; here $\mathfrak{m}$ is the kernel of the counit 
 $\epsilon: O(G) \rightarrow k$. If $k$ is of characteristic 0, $U(G)$ is again the enveloping algebra of the Lie algebra $\mathfrak{g}$ of
$G$. For the case of characteristic $p >0$, we refer the reader to Cartier [18] or
Demazure-Gabriel [32]." 


[Note that the \times symbol should be the LaTeX symbol \ltimes, which apparently won't print here.]


ADDED. Given $G$, the hyperalgebra of $G$ is what Cartier denotes by $U(G)$; so it is not quite the reduced dual Hopf algebra of the algebra of regular functions on $G$ in general. In case $G$ is a connected reductive algebraic group over an algebraically closed field of characteristic $p>0$, the hyperalgebra has an explicit "divided power" construction starting with Kostant's integral form of the universal enveloping algebra of the complex analogue of the Lie algebra of $G$ (in the crucial case $G$ semisimple and simply connected), then reducing mod $p$. This is used heavily by Jantzen to investigate the rational representations of $G$ and relevant closed subgroups such as Borel subgroups, etc. (A similar construction is used by Lusztig for the quantum enveloping algebra at a root of unity.)


