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arXiv · 0805.3018

Hyperbolic Weyl groups and the four normed division algebras

Alex J. Feingold·Axel Kleinschmidt·Hermann Nicolai
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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.

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Alex J. Feingold, Axel Kleinschmidt, Hermann Nicolai. 2008-05-20. Hyperbolic Weyl groups and the four normed division algebras. https://doi.org/10.1016/j.jalgebra.2009.05.006

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