Steinberg groups as amalgams

Allcock, Daniel

doi:10.2140/ant.2016.10.1791

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Steinberg groups as amalgams

Daniel Allcock

Vol. 10 (2016), No. 8, 1791–1843

DOI: 10.2140/ant.2016.10.1791

Abstract

For any root system and any commutative ring, we give a relatively simple presentation of a group related to its Steinberg group $S t$ . 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 $S t$ , giving a presentation with many advantages over the usual presentation of $S t$ . 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 nonfields 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.

Keywords

Kac–Moody group, Steinberg group, pre-Steinberg group, Curtis–Tits presentation

Mathematical Subject Classification 2010

Primary: 19C99

Secondary: 20G44, 14L15

Milestones

Received: 29 March 2016

Accepted: 11 June 2016

Published: 7 October 2016

Authors

Daniel Allcock
Department of Mathematics University of Texas at Austin RLM 8.100 2515 Speedway Stop C1200 Austin, TX 78712 United States

Vol. 10, No. 8, 2016