Triality, Exceptional Lie Groups and Dirac Operators

Summary

Abstract

In dimension eight there are three basic representations for the spin group. These representations lead to a concept of triality and hence lead to the construction of two exceptional commutative algebras: The Chevalley algebra A of dimension twenty-four and the Albert algebra I of dimension twenty-seven. All of the exceptional Lie groups can be described using triality, the octonians, and these two algebras.

On a complex four-manifold, with triality a parallel field, the Dirac and associated twistor operators can be constructed on either bundle of exceptional algebras. The geometry of triality leads to refinement of duality common to four dimensions.

In nine dimensions there is a weaker notion of triality which is related to several additional multiplicative structures on J.

Introduction

Cartan's classification of simple Lie groups yields all Lie groups with some exceptions. In his thesis Cartan described these exceptional groups but, save the one of smallest rank G₂, he was unable to describe the geometry of the groups. In 1950 Chevalley and Schafer successfully identified F₄ and E₆ as structure groups for the exceptional Jordan algebra and the Freudenthal cross product on J, respectively. Later Freudenthal successfully identified the geometry associated with the remaining exceptional groups, E₇ and E₈.