Presentation of affine KacMoody groups over rings
Abstract
Tits has defined Steinberg groups and KacMoody groups for any root system and any commutative ring R. We establish a CurtisTitsstyle 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 2node 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 KacMoody groups when R is a Dedekind domain of arithmetic type.
 Publication:

arXiv eprints
 Pub Date:
 August 2014
 arXiv:
 arXiv:1409.0176
 Bibcode:
 2014arXiv1409.0176A
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Representation Theory;
 20G44
 EPrint:
 Major revision: section 2 is new. Theorem and equation numbering changed. Case 4 in section 5 rewritten. Many other minor changes, and additional references