Hyperbolic Weyl Groups and the Four Normed Division Algebras

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Weyl Groups; Hyperbolic Kac–Moody Algebras; Normed Division Algebras


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.

Publisher Attribution

Final version published in the Journal of Algebra, 322 (4) pp. 1295-1339, published by Elsevier. doi: 10.1016/j.jalgebra.2009.05.006