then since we also have h ‚ Cg (t), we can ¬nd a BSA, b of Cg (t) containing h,

and conjugate b to b , since we are assuming that t = 0 and hence Cg (t) = g.

Since b © b ⊃ h has bigger dimension than b © b, we can further conjugate to

b by the induction hypothesis.

If

b ‚ Cg (t)

then there is a common non-zero eigenvector for ad t in b , call it x. So there

is a t ∈ t such that [t , x] = c x, c = 0. Setting

1

t := t

c

we have [t, x] = x. Let ¦t ‚ ¦ consist of those roots for which β(t) is a positive

rational number. Then

s := h • gβ

β∈¦t

is a solvable subalgebra and so lies in a BSA, call it b . Since t ‚ b , x ∈ b

we see that b © b has strictly larger dimension than b © b . Also b © b has

strictly larger dimension than b © b since h ‚ b © b . So we can conjugate b

to b and then b to b.

This leaves only the case b © b = 0 which we will show is impossible. Let

t be a maximal toral subalgebra of b . We can not have t = 0, for then b

would consist entirely of nilpotent elements, hence nilpotent by Engel, and also

self-normalizing as is every BSA. Hence it would be a CSA which is impossible

since every CSA in a semi-simple Lie algebra is toral. So choose a CSA, h

containing t, and then a standard BSA containing h . By the preceding, we

know that b is conjugate to b and, in particular has the same dimension as b .

But the dimension of each standard BSA (relative to any Cartan subalgebra)

is strictly greater than half the dimension of g, contradicting the hypothesis

g ⊃b•b. QED

92 CHAPTER 5. CONJUGACY OF CARTAN SUBALGEBRAS.

Chapter 6

The simple ¬nite

dimensional algebras.

In this chapter we classify all possible root systems of simple Lie algebras. A

consequence, as we shall see, is the classi¬cation of the simple Lie algebras

´

themselves. The amazing result - due to Killing with some repair work by Elie

Cartan - is that with only ¬ve exceptions, the root systems of the classical

algebras that we studied in Chapter III exhaust all possibilities.

The logical structure of this chapter is as follows: We ¬rst show that the

root system of a simple Lie algebra is irreducible (de¬nition below). We then

develop some properties of the of the root structure of an irreducible root system,

in particular we will introduce its extended Cartan matrix. We then use the

Perron-Frobenius theorem to classify all possible such matrices. (For the expert,

this means that we ¬rst classify the Dynkin diagrams of the a¬ne algebras of

the simple Lie algebras. Surprisingly, this is simpler and more e¬cient than

the classi¬cation of the diagrams of the ¬nite dimensional simple Lie algebras

themselves.) From the extended diagrams it is an easy matter to get all possible

bases of irreducible root systems. We then develop a few more facts about root

systems which allow us to conclude that an isomorphism of irreducible root

systems implies an isomorphism of the corresponding Lie algebras. We postpone

the the proof of the existence of the exceptional Lie algebras until Chapter VIII,

where we prove Serre™s theorem which gives a uni¬ed presentation of all the

simple Lie algebras in terms of generators and relations derived directly from

the Cartan integers of the simple root system.

Throughout this chapter we will be dealing with semi-simple Lie algebras

over the complex numbers.

93

94 CHAPTER 6. THE SIMPLE FINITE DIMENSIONAL ALGEBRAS.

6.1 Simple Lie algebras and irreducible root sys-

tems.

We choose a Cartan subalgebra h of a semi-simple Lie algebra g, so we have the

corresponding set ¦ of roots and the real (Euclidean) space E that they span.

We say that ¦ is irreducible if ¦ can not be partitioned into two disjoint

subsets

¦ = ¦1 ∪ ¦2

such that every element of ¦1 is orthogonal to every element of ¦2 .

Proposition 17 If g is simple then ¦ is irreducible.

Proof. Suppose that ¦ is not irreducible, so we have a decomposition as above.

If ± ∈ ¦1 and β ∈ ¦2 then

(± + β, ±) = (±, ±) > 0 and (± + β, β) = (β, β) > 0

which means that ± + β can not belong to either ¦1 or ¦2 and so is not a root.

This means that

[g± , gβ ] = 0.

In other words, the subalgebra g1 of g generated by all the g± , ± ∈ ¦1 is

centralized by all the gβ , so g1 is a proper subalgebra of g, since if g1 = g this

would say that g has a non-zero center, which is not true for any semi-simple

Lie algebra. The above equation also implies that the normalizer of g1 contains

all the gγ where γ ranges over all the roots. But these gγ generate g. So g1 is

a proper ideal in g, contradicting the assumption that g is simple. QED

Let us choose a base ∆ for the root system ¦ of a semi-simple Lie algebra.

We say that ∆ is irreducible if we can not partition ∆ into two non-empty

mutually orthogonal sets as in the de¬nition of irreducibility of ¦ as above.

Proposition 18 ¦ is irreducible if and only if ∆ is irreducible.

Proof. Suppose that ¦ is not irreducible, so has a decomposition as above.

This induces a partition of ∆ which is non-trivial unless ∆ is wholly contained

in ¦1 or ¦2 . If ∆ ‚ ¦1 say, then since E is spanned by ∆, this means that all

the elements of ¦2 are orthogonal to E which is impossible. So if ∆ is irreducible

so is ¦. Conversely, suppose that

∆ = ∆1 ∪ ∆2

is a partition of ∆ into two non-empty mutually orthogonal subsets. We have

proved that every root is conjugate to a simple root by an element of the Weyl

group W which is generated by the simple re¬‚ections. Let ¦1 consist of those

roots which are conjugate to an element of ∆1 and ¦2 consist of those roots

which are conjugate to an element of ∆2 . The re¬‚ections sβ , β ∈ ∆2 commute

with the re¬‚ections s± , ± ∈ ∆1 , and furthermore

sβ (±) = ±

6.2. THE MAXIMAL ROOT AND THE MINIMAL ROOT. 95

since (±, β) = 0. So any element of ¦1 is conjugate to an element of ∆1 by

an element of the subgroup W1 generated by the s± , ± ∈ ∆1 . But each such

re¬‚ection adds or subtracts ±. So ¦1 is in the subspace E1 of E spanned by ∆1

and so is orthogonal to all the elements of ¦2 . So if ¦1 is irreducible so is ∆.

QED

We are now into the business of classifying irreducible bases.

6.2 The maximal root and the minimal root.

Suppose that ¦ is an irreducible root system and ∆ a base, so irreducible.

Recall that once we have chosen ∆, every root β is an integer combination of

the elements of ∆ with all coe¬cients non-negative, or with all coe¬cients non-

positive. We write β 0 in the ¬rst case, and β 0 in the second case. This

de¬nes a partial order on the elements of E by

» if and only if » ’ µ =

µ k± ±, (6.1)

±∈∆

where the k± are non-negative integers. This partial order will prove very im-

portant to us in representation theory.

Also, for any β = k± ± ∈ ¦+ we de¬ne its height by

ht β = k± . (6.2)

±

Proposition 19 Suppose that ¦ is an irreducible root system and ∆ a base.

Then

• There exists a unique β ∈ ¦+ which is maximal relative to the ordering

.

• This β = k± ± where all the k± are positive.

• (β, ±) ≥ 0 for all ± ∈ ∆ and (β, ±) > 0 for at least one ± ∈ ∆.

Proof. Choose a β = k± ± which is maximal relative to the ordering.

At least one of the k± > 0. We claim that all the k± > 0. Indeed, suppose

not. This partitions ∆ into ∆1 , the set of ± for which k± > 0 and ∆2 , the

set of ± for which k± = 0. Now the scalar product of any two distinct simple

roots is ¤ 0. (Recall that this followed from the fact that if (±1 , ±2 ) > 0,