Steinberg groups as amalgams

TL;DR

This paper provides a new, simplified presentation of Steinberg groups related to root systems, including infinite cases, revealing structural automorphisms and enabling proofs of finite presentability over rings.

Contribution

It introduces a Dynkin diagram-based presentation for Steinberg groups applicable to various root systems, including infinite and Kac-Moody types, with advantages over traditional presentations.

Findings

The new presentation applies to all spherical and certain affine root systems.

It reveals automorphisms in characteristics 2 and 3 and their generalizations.

Many Steinberg and Kac-Moody groups over rings are shown to be finitely presented.

Abstract

For any root system and any commutative ring we give a relatively simple presentation of a group related to its Steinberg group St. This includes the case of infinite root systems used in Kac-Moody theory, for which the Steinberg group was defined by Tits and Morita-Rehmann. In most cases our group equals St, giving a presentation with many advantages over the usual presentation of St. This equality holds for all spherical root systems, all irreducible affine root systems of rank>2, and all 3-spherical root systems. When the coefficient ring satisfies a minor condition, the last condition can be relaxed to 2-sphericity. Our presentation is defined in terms of the Dynkin diagram rather than the full root system. It is concrete, with no implicit coefficients or signs. It makes manifest the exceptional diagram automorphisms in characteristics 2 and 3, and their generalizations to Kac-Moody groups. And it is a Curtis-Tits style presentation: it is the direct limit of the groups coming from 1- and 2-node subdiagrams of the Dynkin diagram. Over non-fields this description as a direct limit is new and surprising. Our main application is that many Steinberg and Kac-Moody groups over finitely-generated rings are finitely presented.