Infinite dimensional Chevalley groups and Kac--Moody groups over $\mathbb{Z}$

Let $A$ be a symmetrizable generalized Cartan matrix. Let $\mathfrak{g}$ be the corresponding Kac--Moody algebra over a commutative ring $R$ with 1. We construct an infinite dimensional analog $G_V$ of a finite dimensional Chevalley group over $R$, constructed using integrable highest weight modules $V$, a $\Z$--form of the universal enveloping algebra of $\mathfrak{g}$ and a $\Z$--form $V_{\Z}$. Our construction naturally gives rise to `arithmetic' subgroups $G_V(\Z)$, which we also construct using representation theory. We prove a Bruhat decomposition $G_V({\mathbb{Q}})=G_V({\mathbb{Z}})B({\mathbb{Q}})$ over $\Q$. We also consider a universal representation-theoretic Kac--Moody group $G$ and prove that its arithmetic subgroup $\Gamma(\Z)$ coincides with the subgroup of integral points $G(\Z)$.