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.