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Hyperbolic Weyl groups and the four normed division algebras

par Feingold, Alex A.J.;Kleinschmidt, Axel ;Nicolai, Hermann
Référence Journal of algebra, 322, 4, page (1295-1339)
Publication Publié, 2009-08

Article révisé par les pairs

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Résumé :

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. © 2009 Elsevier Inc. All rights reserved.