Embeddings of algebraic groups in Kac-Moody groups

TL;DR

This paper investigates embeddings of algebraic groups into Kac-Moody groups over fields of characteristic zero, showing boundedness of images under certain field conditions and unboundedness in others.

Contribution

It establishes conditions under which embeddings of algebraic groups into Kac-Moody groups have bounded or unbounded images based on the field's transcendence degree.

Findings

Bounded subgroup images when $k_1$ is an algebraic extension of $\,\mathbb{Q}$.

Existence of unbounded embeddings when $k_1$ has infinite transcendence degree.

Differentiation of embedding behavior based on field properties.

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.