Presentation of affine Kac–Moody groups over rings

Allcock, Daniel

doi:10.2140/ant.2016.10.533

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Presentation of affine Kac–Moody groups over rings

Daniel Allcock

Vol. 10 (2016), No. 3, 533–556

DOI: 10.2140/ant.2016.10.533

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 $S t$ of any irreducible affine root system with rank $\geq 3$ , for any $R$ . Namely, $S t$ is the direct limit of the Steinberg groups coming from the $1$ - and $2$ -node subdiagrams of the Dynkin diagram. In fact, we give a completely explicit presentation. Using this we show that $S t$ is finitely presented if the rank is $\geq 4$ 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.

Keywords

affine Kac–Moody group, Steinberg group, Curtis–Tits presentation

Mathematical Subject Classification 2010

Primary: 20G44

Secondary: 14L15, 22E67, 19C99

Milestones

Received: 23 September 2014

Revised: 21 June 2015

Accepted: 15 October 2015

Published: 12 June 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. 3, 2016