Uniqueness of representation--theoretic hyperbolic Kac--Moody groups over $\mathbb{Z}$

Abstract

For a simply laced and hyperbolic Kac--Moody group $G=G(R)$ over a commutative ring $R$ with 1, we consider a map from a finite presentation of $G(R)$ obtained by Allcock and Carbone to a representation--theoretic construction $G^{\lambda }(R)$ corresponding to an integrable representation $V^{\lambda }$ with dominant integral weight $\lambda$. When $R=\mathbb{Z}$, we prove that this map extends to a group homomorphism ${\rho }_{\lambda ,\mathbb{Z}}:G(\mathbb{Z})\to G^{\lambda }(\mathbb{Z}).$ We prove that the kernel $K^{\lambda }$ of the map $ρ_{\lam,\Z}: G(\Z)\to G^{\lam}(\Z)$ lies in $H(\C)$ and if the group homomorphism $φ:G(\Z)\to G(\C)$ is injective, then $K^{\lambda }\le H(\mathbb{Z})\cong (\mathbb{Z}/2\mathbb{Z}{)}^{rank(G)}$.