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arXiv:0805.3018 [math.RT]AbstractReferencesReviewsResources

Hyperbolic Weyl groups and the four normed division algebras

Alex J. Feingold, Axel Kleinschmidt, Hermann Nicolai

Published 2008-05-20, updated 2017-07-14Version 2

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.

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
Journal: J.Algebra 322:1295-1339 (2009)
Categories: math.RT, hep-th, math.GR
Subjects: 17A35, 17B67, 20F55, 11F06
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