Presentation of affine Kac-Moody groups over rings

TL;DR

This paper provides an explicit, finite presentation for affine Kac-Moody groups over rings, generalizing Steinberg groups, and establishes conditions for finite presentability based on rank and ring properties.

Contribution

It introduces a Curtis-Tits-style presentation for Steinberg groups of affine root systems over rings, enabling explicit descriptions and finite presentation results.

Findings

Steinberg groups are the direct limit of subgroups from Dynkin diagram subdiagrams.

The groups are finitely presented under certain rank and ring conditions.

Results extend to Kac-Moody groups over Dedekind domains of arithmetic type.

Abstract

Tits has defined Steinberg groups and Kac-Moody groups for any root system and any commutative ring R. We establish a Curtis-Tits-style presentation for the Steinberg group St of any rank > 2 irreducible affine root system, for any R. Namely, St is the direct limit of the Steinberg groups coming from the 1- and 2-node subdiagrams of the Dynkin diagram. This leads to a completely explicit presentation. Using this we show that St is finitely presented if the rank is > 3 and R is finitely generated as a ring, or if the rank is 3 and R is finitely generated as a module over a subring generated by finitely many units. Similar results hold for the corresponding Kac-Moody groups when R is a Dedekind domain of arithmetic type.