Presentation of hyperbolic Kac-Moody groups over rings

TL;DR

This paper provides explicit presentations for hyperbolic Kac-Moody and Steinberg groups over rings, extending known structures and establishing finite presentability under certain conditions, with implications for superstring theory.

Contribution

It introduces simple explicit presentations for hyperbolic Kac-Moody and Steinberg groups over rings, generalizing Curtis-Tits presentations and proving finite presentability in key cases.

Findings

Explicit presentations for hyperbolic Kac-Moody groups over rings.

Finite presentability of Steinberg groups over finitely generated rings.

Simplified presentations for the case when the ground ring is Z.

Abstract

Tits has defined Kac-Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac-Moody groups whose Dynkin diagrams are hyperbolic and simply laced. Our presentations are analogues of the Curtis-Tits presentation of the finite groups of Lie type. When the ground ring is finitely generated, we derive the finite presentability of the Steinberg group, and similarly for the Kac-Moody group when the ground ring is a Dedekind domain of arithmetic type. These finite-presentation results need slightly stronger hypotheses when the rank is smallest possible, namely 4. The presentations simplify considerably when the ground ring is Z, a case of special interest because of the conjectured role of the Kac-Moody group E10(Z) in superstring theory.