Modular groups over real normed division algebras and over-extended hyperbolic Weyl groups

Abstract

Feingold and Frenkel came up with a very insightful way to study the structure of the rank 3 hyperbolic root system AE3 by realizing the AE3 root system as the set of 2 × 2 symmetric integral matrices X with determinant greater than -1. Kac, Moody and Wakimoto generalized the structural results to the hyperbolic root system E10, which is the over-extension of E8. There has then been very little new insight into the structure of the Weyl groups of hyperbolic root systems until Feingold, Kleinschmidt, and Nicolai presented a coherent picture for many higher rank hyperbolic Kac-Moody root systems which was based on the relation to modular groups associated with lattices and subrings of the four normed division algebras over the real numbers. However, it still remained an outstanding problem to find a more manageable realization of the group W(E10) directly in terms of 2 × 2 matrices with Octavian entries.In this thesis, I will illustrate the relationships between modular groups over real normed division algebras and hyperbolic Weyl groups arising from overextending finite root systems inside the four real normed division algebras. I will give some modular group presentations for those over-extended hyperbolic Weyl groups.