Presentations of affine Kac-Moody groups

TL;DR

This paper provides explicit finite presentations for affine Kac-Moody groups over finite fields, including specific cases like ${ m SL}_n(F_q[t,t^{-1}])$, with minimal generators and relations, advancing understanding of their algebraic structure.

Contribution

It offers explicit, field-independent finite presentations of affine Kac-Moody groups over finite fields, including Chevalley groups, with bounds on generators and relations, and describes these presentations in detail.

Findings

${ m SL}_n(F_q[t,t^{-1}])$ has a presentation with 9 generators and 44 relations.

Any simply connected affine Kac-Moody group over a finite field has a presentation with at most 11 generators and 70 relations.

Explicit presentations of Chevalley groups over Laurent polynomial rings and their profinite completions are derived.

Abstract

How many generators and relations does ${\mathrm SL}_n({\mathbb F}_q[t, t^{-1}])$ need? In this paper we exhibit its explicit presentation with $9$ generators and $44$ relations. We investigate presentations of affine Kac-Moody groups over finite fields. Our goal is to derive finite presentations, independent of the field and with as few generators and relations as we can achieve. It turns out that any simply connected affine Kac-Moody group over a finite field has a presentation with at most 11 generators and 70 relations. We describe these presentations explicitly type by type. As a consequence, we derive explicit presentations of Chevalley groups $G({\mathbb F}_q[t, t^{-1}])$ and explicit profinite presentations of profinite Chevalley groups $G({\mathbb F}_q[[t]])$.