# Reference for showing that $\mathcal{O}(G) \cong U(\mathfrak{g})^{\circ}$ when $G$ is connected and simply connected

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?

• You can also try to reconstruct $G$ using Tannaka-Krein duality starting from the category of finite-dimensional representations of $\mathfrak{g}$. – Qiaochu Yuan Dec 29 '18 at 19:43
• 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. – Dean Young Dec 29 '18 at 20:05
• 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 $H^\circ$. – Adrien Dec 30 '18 at 17:40