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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?

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    $\begingroup$ You can also try to reconstruct $G$ using Tannaka-Krein duality starting from the category of finite-dimensional representations of $\mathfrak{g}$. $\endgroup$ – Qiaochu Yuan Dec 29 '18 at 19:43
  • $\begingroup$ Thank you Qiaochu, glad to hear from one of my favorite expositors ... that seems like a better approach (but I only just read the wiki article). I would accept an answer elaborating along the lines of this comment. $\endgroup$ – Dean Young Dec 29 '18 at 20:05
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    $\begingroup$ Adding to Qiaochu's comment, one way of defining the Hopf dual of an Hopf algebra $H$, is as the coalgebra you get from Tanaka-Krein applied to the category of finite dimensional $H$-modules, ie f.d. $H^\circ$-comodules are (monoidally) the same as f.d. $H$-modules, and this characterizes $H^\circ$. $\endgroup$ – Adrien Dec 30 '18 at 17:40
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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".

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