5Cartan classification

III Symmetries, Fields and Particles

5 Cartan classification

We now move on to the grand scheme of classifying all complex simple Lie

algebras. The starting point of everything is that we define a natural inner

product on our Lie algebra

g

. We will find a subalgebra

h

of

g

that plays

the role of the

H

we had when we studied

su

(2). The remainder of

g

will be

controlled by things known as roots, which live in

h

∗

. We will see that the Killing

form induces a inner product on

h

∗

, which allows us to think of these roots as

“geometric” objects that live in

R

n

. We can then find some strong conditions

that restrict what these roots and their inner products can be, and it turns out

these completely characterize our possible Lie algebras.