Presentation of affine Kac-Moody groups over rings
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.
Keywords
Cite
@article{arxiv.1409.0176,
title = {Presentation of affine Kac-Moody groups over rings},
author = {Daniel Allcock},
journal= {arXiv preprint arXiv:1409.0176},
year = {2016}
}
Comments
Major revision: section 2 is new. Theorem and equation numbering changed. Case 4 in section 5 rewritten. Many other minor changes, and additional references