Let $G$ be an algebraic group. We can try to reconstruct $G$ from its lie algebra $\mathfrak{g}$, but the best we get in general is a formal group scheme $\operatorname{Spf}(U(\mathfrak{g})^\*)$, where $U(\mathfrak{g})$ is the universal enveloping algebra of $\mathfrak{g}$.


However, I have heard that certain conditions provide for a full reconstruction of $G$. Namely, if $G$ is connected and simply connected, then there is an isomorphism $\mathcal{O}(G) \rightarrow U(\mathfrak{g})^{\circ}$, where $U(\mathfrak{g})^{\circ}$ is the Hopf dual of $U(\mathfrak{g})$.


I can't seem to find a reference for this. Could someone please provide one?



Reference for showing that $\mathcal{O}(G) \cong U(\mathfrak{g})^{\circ}$ when $G$ is connected and simply connected
In general (if the characteristic is not zero) you should use the "hyperalgebra" (or the "algebra of distributions" in modern terms) instead of U(g). The reference is Mitsuhiro Takeuchi's paper "On coverings and hyperalgebras of affine algebraic groups", Trans. AMS 211 (1975), 179-196. Possibly you can also find the result in Jantzen's book "Representation of algebraic groups".


