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.

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

References in corpus (6)

• Walls of Marginal Stability and Dyon Spectrum in N=4 Supersymmetric String Theories
• Comments on the Spectrum of CHL Dyons
• Wall Crossing, Discrete Attractor Flow and Borcherds Algebra
• An E9 multiplet of BPS states
• Symmetries,Singularities and the De-Emergence of Space
• Embeddings of hyperbolic Kac-Moody algebras into ${\mathbf{E}}_{10}$}

Cited by in corpus (11)

• Supersymmetric quantum cosmological billiards
• Irregular connections and Kac-Moody root systems
• Relativistic Wavepackets in Classically Chaotic Quantum Cosmological Billiards
• The arithmetic geometry of AdS${}_2$ and its continuum limit
• On fundamental domains and volumes of hyperbolic Coxeter-Weyl groups
• The continuum limit of the modular discretization of AdS${}_2$
• Automorphic correction of the hyperbolic Kac-Moody algebra $E_{10}$
• Modular realizations of hyperbolic Weyl groups
• Hidden Kac-Moody Structures in the Fermionic Sector of Five-Dimensional Supergravity
• Weyl groups of some hyperbolic Kac-Moody algebras
• Some Hidden Structure in BKL