Trialities and Exceptional Lie Algebras: DECONSTRUCTING the Magic Square

Abstract

A construction of the magic square, and hence of exceptional Lie algebras, is carried out using trialities rather than division algebras. By way of preparation, a comprehensive discussion of trialities is given, incorporating a number of novel results and proofs. Many of the techniques are closely related to, or derived from, ideas which are commonplace in theoretical physics. The importance of symmetric spaces in the magic square construction is clarified, allowing the Jacobi property to be verified for each algebra in a uniform and transparent way, with no detailed calculations required. A variation on the construction, corresponding to other symmetric spaces, is also given.