Cartan subalgebra h. This determines the corresponding Euclidean space E

and root system ¦. Suppose we have a second such pair (g , h ). We would like

to show that an isomorphism of (E, ¦) with (E , ¦ ) determines a Lie algebra

isomorphism of g with g . This would then imply that the Dynkin diagrams

classify all possible simple Lie algebras. We would still be left with the problem

of showing that the exceptional Lie algebras exist. We will defer this until

Chapter VIII where we prove Serre™s theorem with gives a direct construction

of all the simple Lie algebras in terms of generators and relations determined

by the Cartan matrix.

We need a few preliminaries.

Proposition 24 Every positive root can be written as a sum of simple roots

±i1 + · · · ±ik

in such a way that every partial sum is again a root.

110 CHAPTER 6. THE SIMPLE FINITE DIMENSIONAL ALGEBRAS.

Proof. By induction (on say the height) it is enough to prove that for every

positive root β which is not simple, there is a simple root ± such that β ’ ± is

a root. We can not have (β, ±) ¤ 0 for all ± ∈ ∆ for this would imply that the

set {β} ∪ ∆ is independent (by the same method that we used to prove that ∆

was independent). So (β, ±) > 0 for some ± ∈ ∆ and so β ’ ± is a root. Since

β is not simple, its height is at least two, and so subtracting ± will not be zero

or a negative root, hence positive. QED

Proposition 25 Let g, h be a semi-simple Lie algebra with a choice of Cartan

subalgebra, Let ¦ be the corresponding root system, and let ∆ be a base. Then

g is generated as a Lie algebra by the subspaces g± , g’± , ± ∈ ∆.

From the representation theory of sl(2)± we know that [g± , gβ ] = g±+β if ± + β

is a root. Thus from the preceding proposition, we can successively obtain all

the gβ for β positive by bracketing the g± , ± ∈ ∆. Similarly we can get all

the gβ for β negative from the g’± . So we can get all the root spaces. But

[g± , g’± ] = Ch± so we can get all of h. The decomposition

g =h• gγ

γ∈¦

then shows that we have generated all of g.

Here is the big theorem:

Theorem 15 Let g, h and g , h be simple Lie algebras with choices of Cartan

subalgebras, and let ¦, ¦ be the corresponding root systems. Suppose there is

an isomorphism

f : (E, ¦) ’ (E , ¦ )

which is an isometry of Euclidean spaces. Extend f to an isomorphism of

h— ’ h —

via complexi¬cation. Let f : h ’ h denote the corresponding isomorphism on

the Cartan subalgebras obtained by identifying h and h with their duals using

the Killing form.

Fix a base ∆ of ¦ and ∆ of ¦ . Choose 0 = x± ∈ g± , ± ∈ ∆ and 0 = x± ∈

g± . Extend f to a linear map

f :h• g± ’ h • g±

± ∈∆

±∈∆

by

f (x± ) = x± .

Then f extends to a unique isomorphism of g ’ g .

Proof. The uniqueness is easy. Given x± there is a unique y± ∈ g’± for which

[x± , y± ] = h± so f , if it exists, is determined on the y± and hence on all of g

since the x± and y± generate g be the preceding proposition.

6.7. THE CLASSIFICATION OF THE POSSIBLE SIMPLE LIE ALGEBRAS.111

To prove the existence, we will construct the graph of this isomorphism.

That is, we will construct a subalgebra k of g • g whose projections onto the

¬rst and onto the second factor are isomorphisms:

Use the x± and y± as above, with the corresponding elements x± and y± in

g . Let

x± := x± • x± ∈ g • g

and similarly de¬ne

y ± := y± • y± ,

and

h± := h± • h± .

Let β be the (unique) maximal root of g, and choose x ∈ gβ . Make a similar

choice of x ∈ gβ where β is the maximal root of g . Set

x := x • x .

Let m ‚ g • g be the subspace spanned by all the

ad y ±i1 · · · ad y ±im x.

The element ad y ±i1 · · · ad y ±im x belongs to gβ’P ±ij • gβ so

P

’ ±i

j

m © (gβ • gβ ) is one dimensional .

In particular m is a proper subspace of g • g .

Let k denote the subalgebra of g • g generated by the x± the y ± and the

h± . We claim that

[k, m] ‚ m.

Indeed, it is enough to prove that m is invariant under the adjoint action of the

generators of k. For the ad y ± this follows from the de¬nition. For the ad h± we

use the fact that

[h, y± ] = ’±(h)y±

to move the ad h± past all the ad y γ at the cost of introducing some scalar

multiple, while

ad h± x = β, ± xβ + β , ± xβ = β, ± x

because f is an isomorphism of root systems.

Finally [x±1 , y±2 ] = 0 if ±1 = ±2 ∈ ∆ since ±1 ’ ±2 is not a root. On the

other hand [x± , y± ] = h± . So we can move the ad x± past the ad y γ at the

expense of introducing an ad h± every time γ = ±. Now ± + β is not a root,

since β is the maximal root. So [x± , xβ ] = 0. Thus ad x± x = 0, and we have

proved that [k, m] ‚ m. But since m is a proper subspace of g • g , this implies

that k is a proper subalgebra, since otherwise m would be a proper ideal, and

the only proper ideals in g • g are g and g .

112 CHAPTER 6. THE SIMPLE FINITE DIMENSIONAL ALGEBRAS.

Now the subalgebra k can not contain any element of the form z • 0, z = 0,

for it if did, it would have to contain all of the elements of the form u • 0 since

we could repeatedly apply ad x± ™s until we reached the maximal root space and

then get all of g • 0, which would mean that k would also contain all of 0 • g

and hence all of g • g which we know not to be the case. Similarly k can not

contain any element of the form 0•z . So the projections of k onto g and onto g

are linear isomorphisms. By construction they are Lie algebra homomorphisms.

Hence the inverse of the projection of k onto g followed by the projection of k

onto g is a Lie algebra isomorphism of g onto g . By construction it sends x±

to x± and h± to h± and so is an extension of f . QED

Chapter 7

Cyclic highest weight

modules.

In this chapter, g will denote a semi-simple Lie algebra for which we have chosen

a Cartan subalgebra, h and a base ∆ for the roots ¦ = ¦+ ∪ ¦’ of g.

We will be interested in describing its ¬nite dimensional irreducible repre-

sentations. If W is a ¬nite dimensional module for g, then h has at least one

simultaneous eigenvector; that is there is a µ ∈ h— and a w = 0 ∈ W such that

∀ h ∈ h.

hw = µ(h)w (7.1)

The linear function µ is called a weight and the vector v is called a weight

vector. If x ∈ g± ,

hxw = [h, x]w + xhw = (µ + ±)(h)xw.

This shows that the space of all vectors w satisfying an equation of the type

(7.1) (for varying µ) spans an invariant subspace. If W is irreducible, then the

weight vectors (those satisfying an equation of the type (7.1)) must span all of