This is a commutative Hopf algebra. Define $$N := \mathrm{max\,Spec}(U(\n)_{\mathrm{gr}}^*) = {\operatorname{Hom}}_{\mathrm{alg}}(U(\n)_{\mathrm{gr}}^*,\C).$$ The comultiplication of $U(\n)_{\mathrm{gr}}^*$ gives $N$ a group structure. As a group, $N$ can be identified with the pro-unipotent pro-group with Lie algebra $$\widehat{\n} = \prod_{{\a}\in\De^+} \g_{\a}.$$ By construction, we can identify $U(\n)_{\mathrm{gr}}^*$ with the coordinate ring $\C[N]$ of $N$.

For $w\in W$, let $N(w)$ be the subgroup of $N$ with Lie algebra $\n(w)$. Define also $N'(w)$ to be the subgroup of $N$ with Lie algebra $$\n'(w) := \prod_{{\a}\not\in\De_{w}} {\g}_{\a} 
\subset \widehat{{\n}}.$$ Multiplication yields a bijection $N({w})\times N'({w}) \stackrel{\sim}{\to} N$.

\[Prop3.2\] The coordinate ring $\C[{N}({w})]$ is isomorphic to the invariant subring $$\C[{N}]^{{N}'({w})} = 
\left\{f\in\C[{N}] \mid f(nn') = f(n),\ n\in {N},\ n'\in N'(w)\right\}.$$

Let $G$ be the group attached to $\g$ by Kac and Peterson [@KP]. This is an affine ind-variety. It has a refined Tits system $$({G}, \mathrm{Norm}_{{G}}({H}),{N_+},{N_-},{H}),$$ where $\mathrm{Lie}({H})={\h}$, $\mathrm{Lie}({N_+})={\n_+}$, and $\mathrm{Lie}({N_-})={\n_-}$ (see [@Ku]). Note that in general $N \not \subset G$. Both $N$ and $G$ can be regarded as subgroups of a bigger group ${G}^{\mathrm{max}}$ constructed by Tits. Then ${N_+} = {N} \cap {G}$.

\[Example2\] (a) If $\g$ is finite-dimensional of Dynkin type $X_n$, then $G = G_{X_n}(\C)$ is a connected simply-connected algebraic group of type 