Vol. 10, No. 3, 2016

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Presentation of affine Kac–Moody groups over rings

Daniel Allcock

Vol. 10 (2016), No. 3, 533–556
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 irreducible affine root system with rank 3, 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. In fact, we give a completely explicit presentation. Using this we show that St is finitely presented if the rank is 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