Hyperbolic Weyl groups and the four normed division algebras

TL;DR

This paper explores the connection between hyperbolic Kac-Moody algebra Weyl groups and the four normed division algebras, revealing their structure as generalized modular groups and suggesting new avenues in modular form theory.

Contribution

It establishes a link between Weyl groups of hyperbolic Kac-Moody algebras and generalized modular groups over the four division algebras, extending the understanding of their algebraic and geometric properties.

Findings

Even subgroups of Weyl groups are isomorphic to generalized modular groups over division algebras.

The structure differs between simply laced and non-simply laced algebras, involving finite extensions.

The results suggest a broader theory of modular forms and functions related to these algebraic structures.

Abstract

We study the Weyl groups of hyperbolic Kac-Moody algebras of `over-extended' type and ranks 3, 4, 6 and 10, which are intimately linked with the four normed division algebras K=R,C,H,O, respectively. A crucial role is played by integral lattices of the division algebras and associated discrete matrix groups. Our findings can be summarized by saying that the even subgroups, W^+, of the Kac-Moody Weyl groups, W, are isomorphic to generalized modular groups over K for the simply laced algebras, and to certain finite extensions thereof for the non-simply laced algebras. This hints at an extended theory of modular forms and functions.