# The Octonions

by John Huerta (CAMGSD, Instituto Superior Técnico)

The octonions are the largest and most eccentric of the four normed division algebras. These are algebras with a norm that satisfies the following key identity, familiar from the study of real and complex numbers:

|*xy*| = |*x*| |*y*|.

The octonions are noncommutative and, worse, nonassociative. Yet they play an essential role in many parts of mathematics and physics, from giving string theory 10 dimensions to reigning over the theory of exceptional Lie groups. In this talk, we introduce the octonions, prove they are the largest normed division algebra, and sketch their deep relationship with the exceptional Lie groups, from the humble G2 to the enigmatic E8.

*Apoio:
Seminário financiado por Fundos Nacionais através da FCT Fundação para a
Ciência e a Tecnologia no âmbito do projeto PEst-OE/MAT/UI0208/2013*