Embeddings of algebraic groups in Kac-Moody groups

Abstract

Let $k_1,k_2$ be two fields of characteristic 0. Let $G_1$ be a split semisimple algebraic group over $k_1$, $G_2$ a split Kac--Moody group over $k_2$ and $\phi\colon G_1(k_1)\to G_2(k_2)$ an abstract embedding. We show that $\im \phi$ is a bounded subgroup whenever $k_1$ is an algebraic extension of the rational numbers, while there are embeddings with unbounded image if $k_1$ has infinite transcendence degree over the rational numbers.