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

TL;DR

This paper investigates the relationship between hyperbolic Kac--Moody groups over integers and their representation-theoretic counterparts, establishing conditions under which a natural map extends to a homomorphism and analyzing its kernel.

Contribution

It proves the extension of a finite presentation map to a group homomorphism over integers and characterizes the kernel in relation to the group's structure.

Findings

The map extends to a homomorphism over  when R=.

The kernel of this map is contained in a specific subgroup H().

If a certain injectivity condition holds, the kernel is isomorphic to a product of  groups.

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 $R=\Z$, we prove that this map extends to a group homomorphism $\rho_{\lambda,\Z}: G(\Z) \to G^{\lambda}(\Z).$ We prove that the kernel $K^{\lambda}$ of the map $\rho_{\lam,\Z}: G(\Z)\to G^{\lam}(\Z)$ lies in $H(\C)$ and if the group homomorphism $\varphi:G(\Z)\to G(\C)$ is injective, then $K^{\lambda}\leq H(\Z)\cong(\Z/2\Z)^{rank(G)}$.