Presentation of hyperbolic Kac-Moody groups over rings

JOURNAL OF ALGEBRA

卷:445

Article

Allcock, Daniel¹  Carbone, Lisa² 

[1] Univ Texas Austin, Dept Math, Austin, TX 78712 USA

[2] Rutgers State Univ, Dept Math, Piscataway, NJ 08855 USA

关键词: Finite presentation;    Hyperbolic Kac-Moody group;   

DOI  :  10.1016/j.jalgebra.2015.08.012

来源: Elsevier

【 摘 要 】

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 E-10 (Z) in superstring theory. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free