Proof. We know that every irreducible ¬nite dimensional representation is a

cyclic module with integer highest weight, that those with even highest weight

contain 0 as an eigenvalue of h with multiplicity one and do not contain 1 as

an eigenvalue of h, and that those with odd highest weight contain 1 as an

eigenvalue of h with multiplicity one, and do not contain 0 as an eigenvalue. So

2), 3) and 4) follow from 1). We must prove 1).

We ¬rst prove

Proposition 1 Let 0 ’ V ’ W ’ k ’ 0 be an exact sequence of sl(2)

modules and such that the action of sl(2) on k is trivial (as it must be, since

sl(2) has no non-trivial one dimensional modules). Then this sequence splits,

i.e. there is a line in W supplementary to V on which sl(2) acts trivially.

This proposition is, of course, a special case of the theorem we want to prove.

But we shall see that it is su¬cient to prove the theorem.

Proof of proposition. It is enough to prove the proposition for the case

that V is an irreducible module. Indeed, if V1 is a submodule, then by induction

on dim V we may assume the theorem is known for 0 ’ V /V1 ’ W/V1 ’ k ’ 0

so that there is a one dimensional invariant subspace M in W/V1 supplementary

42 CHAPTER 2. SL(2) AND ITS REPRESENTATIONS.

to V /V1 on which the action is trivial. Let N be the inverse image of M in W .

By another application of the proposition, this time to the sequence

0 ’ V1 ’ N ’ M ’ 0

we ¬nd an invariant line, P , in N complementary to V1 . So N = V1 • P . Since

(W/V1 ) = (V /V1 ) • M we must have P © V = {0}. But since dim W = dim

V + 1, we must have W = V • P . In other words P is a one dimensional

subspace of W which is complementary to V .

Next we are reduced to proving the proposition for the case that sl(2) acts

faithfully on V . Indeed, let I = the kernel of the action on V . Since sl(2) is

simple, either I = sl(2) or I = 0. Suppose that I = sl(2). For all x ∈ sl(2) we

have, by hypothesis, xW ‚ V , and for x ∈ I = sl(2) we have xV = 0. Hence

[sl(2), sl(2)] = sl(2)

acts trivially on all of W and the proposition is obvious. So we are reduced

to the case that V is irreducible and the action, ρ, of sl(2) on V is injective.

We have our Casimir element C whose image in End W must map W ’ V

since every element of sl(2) does. On the other hand, C = 1 n(n + 2) Id = 0

2

since we are assuming that the action of sl(2) on the irreducible module V is

not trivial. In particular, the restriction of C to V is an isomorphism. Hence

ker Cρ : W ’ V is an invariant line supplementary to V . We have proved the

proposition.

Proof of theorem from proposition. Let 0 ’ E ’ E be an exact

sequence of sl(2) modules, and we may assume that E = 0. We want to ¬nd an

invariant complement to E in E. De¬ne W to be the subspace of Homk (E, E )

whose restriction to E is a scalar times the identity, and let V ‚ W be the

subspace consisting of those linear transformations whose restrictions to E is

zero. Each of these is a submodule of End(E). We get a sequence

0’V ’W ’k’0

and hence a complementary line of invariant elements in W . In particular, we

can ¬nd an element, T which is invariant, maps E ’ E , and whose restriction

to E is non-zero. Then ker T is an invariant complementary subspace. QED

2.6 The Weyl group.

We have

1 1 1 0

exp ’f =

exp e = and

’1

0 1 1

so

1 1 1 0 1 1 01

(exp e)(exp ’f )(exp e) = = .

’1 ’1 0

0 1 1 0 1

Since

exp ad x = Ad(exp x)

2.6. THE WEYL GROUP. 43

we see that

„ := (exp ad e)(exp ad(’f ))(exp ad e)

consists of conjugation by the matrix

0 1

.

’1 0

Thus

0 ’1 ’1

0 1 10 0

= ’h,

„ (h) = =

’1 0 ’1

0 10 0 1

0 ’1

0 1 0 1 0 0

= ’f

„ (e) = =

’1 ’1

0 0 0 10 0

and similarly „ (f ) = ’e. In short

„ : e ’ ’f, f ’ ’e, h ’ ’h.

In particular, „ induces the “re¬‚ection” h ’ ’h on Ch and hence the re¬‚ection

µ ’ ’µ (which we shall also denote by s) on the (one dimensional) dual space.

In any ¬nite dimensional module V of sl(2) the action of the element „ =

(exp e)(exp ’f )(exp e) is de¬ned, and

(„ )’1 h(„ ) = Ad „ ’1 (h) = s’1 h = sh

so if

hu = µu

then

h(„ u) = „ („ )’1 h(„ )u = „ s(h)u = ’µ„ u = (sµ)„ u.

So if

Vµ : {u ∈ V |hu = µu}

then

„ (Vµ ) = Vsµ . (2.8)

The two element group consisting of the identity and the element s (acting

as a re¬‚ection as above) is called the Weyl group of sl(2). Its generalization

to an arbitrary simple Lie algebra, together with the generalization of formula

(2.8) will play a key role in what follows.

44 CHAPTER 2. SL(2) AND ITS REPRESENTATIONS.

Chapter 3

The classical simple

algebras.

In this chapter we introduce the “classical” ¬nite dimensional simple Lie al-

gebras, which come in four families: the algebras sl(n + 1) consisting of all

traceless (n + 1) — (n + 1) matrices, the orthogonal algebras, on even and odd

dimensional spaces (the structure for the even and odd cases are di¬erent) and

the symplectic algebras (whose de¬nition we will give below). We will prove

that they are indeed simple by a uniform method - the method that we used

in the preceding chapter to prove that sl(2) is simple. So we axiomatize this

method.

3.1 Graded simplicity.

We introduce the following conditions on the Lie algebra g:

∞

g = gi (3.1)

i=’1

‚

[gi , gj ] gi+j (3.2)

[g1 , g’1 ] = g0 (3.3)

’ x = 0, ∀ x ∈ gi , ∀i ≥ 0

[g’1 , x] = 0 (3.4)

There exists a d ∈ g0 satisfying [d, x] = kx, x ∈ gk , ∀k, (3.5)