Modular realizations of hyperbolic Weyl groups

Abstract

We study the recently discovered isomorphisms between hyperbolic Weyl groups and unfamiliar modular groups. These modular groups are defined over integer domains in normed division algebras, and we focus on the cases involving quaternions and octonions. We outline how to construct and analyse automorphic forms for these groups; their structure depends on the underlying arithmetic properties of the integer domains. We also give a new realization of the Weyl group W(E8) in terms of unit octavians and their automorphism group.

35+1 pages, 1 figure. v2: Extended results. In particular, the octonionic realization of W(E10) is completed. Sections and appendices rearranged. References added

References in corpus (8)

• Duality and higher derivative terms in M theory
• Automorphic properties of low energy string amplitudes in various dimensions
• $R^4$ couplings and automorphic unipotent representations
• Supersymmetric quantum cosmological billiards
• Hyperbolic Weyl groups and the four normed division algebras
• Perturbation Theory From Automorphic Forms
• Two Conjectures on Gauge Theories, Gravity, and Infinite Dimensional Kac-Moody Groups
• Arithmetical Chaos and Quantum Cosmology