2014/09/20 by Daniel Allcock, Allcock, Daniel, Lisa Carbone +1
Mathematics · #19C99 #22E67 #FOS: Mathematics #Group Theory (math.GR) #Primary 20G44 #Representation Theory (math.RT) #Secondary 14L15 #math.GR #math.RT #msc:14L15 #msc:19C99 #msc:20G44 #msc:22E67
paper · pdf · doi:10.48550/arxiv.1409.5918
Minor revisions
arxiv created 2015/07/31 · arxiv updated 2015/08/04
Tits has defined Kac-Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac-Moody groups whose Dynkin diagrams are hyperbolic and simply laced. Our presentations are analogues of the Curtis-Tits presentation of the finite groups of Lie type. When the ground ring is finitely generated, we derive the finite presentability of the Steinberg group, and similarly for the Kac-Moody group when the ground ring is a Dedekind domain of arithmetic type. These finite-presentation results need slightly stronger hypotheses when the rank is smallest possible, namely 4. The presentations simplify considerably when the ground ring is Z, a case of special interest because of the conjectured role of the Kac-Moody group E10(Z) in superstring theory.