Hyperbolic Weyl groups and the four normed division algebras
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.
Cite
@article{arxiv.0805.3018,
title = {Hyperbolic Weyl groups and the four normed division algebras},
author = {Alex J. Feingold and Axel Kleinschmidt and Hermann Nicolai},
journal= {arXiv preprint arXiv:0805.3018},
year = {2017}
}
Comments
56 pages, 21 figures. Revised to: (1) Correct typo in formula (4.33), (2) Correct error in Prop. 15, (3) Add references [44] and [45], (4) Match other corrections in the published version