Lattice, Trees and Group Actions

Rutgers University New Brunswick, New Brunswick NJ

Investigators

Abstract

Abstract for award DMS-0401107 of Carbone The objective of this program is to advance understanding in algebra, group theory and representation theory with particular emphasis on the mathematics underlying physics and geometry. Kac-Moody Lie algebras are infinite dimensional algebras that were first discovered by physicists. A mathematical characterization of these algebras obtained in the 1970's allowed for the existence of a wider class of Kac-Moody algebras, namely hyperbolic algebras. To this day, no classical interpretation of hyperbolic algebras is known in mathematics, nor has any physical interpretation of them been discovered. Yet these algebras display remarkable symmetry properties which encode deep and intricate relationships between numbers and geometry. These hyperbolic algebras and their groups are the objects of our study. We have discovered the beginnings of a theory of automorphic forms for lattices in Kac-Moody groups over finite fields. Although Kac-Moody groups have no obvious algebraic or arithmetic structure, our work demonstrates substantial analogies with Lie groups over fields of positive characteristic. Our approach, in part, has been to adapt discrete and combinatorial methods in order to solve problems in this infinite dimensional setting. The objective of this program is to advance understanding in algebra and symmetries of certain geometric objects with a particular emphasis on the mathematics underlying physics and geometry. "Kac-Moody Lie algebras" are infinite dimensional spaces that are studied both by mathematicians and physicists, having first been discovered in physics as algebras of "loops", that is, maps from the circle into finite dimensional Lie algebras. A mathematical characterization of these spaces, obtained in the 1970's, allowed for the existence of a wider class of Kac-Moody algebras, namely hyperbolic algebras. To this day, no classical interpretation of hyperbolic algebras is known in mathematics, nor has any physical interpretation of them been discovered. Yet these algebras display remarkable symmetry properties which encode deep and intricate relationships between numbers and geometry. These hyperbolic algebras and their symmetries are the objects of our study. A strong theme in this research program is to adapt discrete and combinatorial methods to solve problems in algebra. The proposer is developing a team of researchers in algebra, geometry, physics and combinatorics from a wide variety of countries, and including several women, in order to bring a wealth of diverse viewpoints to this research team.