g’1 is irreducible under the (adjoint) action of g0 . (3.6)

Condition (3.4) means that if x ∈ gi , i ≥ 0 is such that [y, x] = 0 for all y ∈ g’1

then x = 0.

We wish to show that any non-zero g satisfying these six conditions is simple.

We know that g’1 , g0 and g1 are all non-zero, since 0 = d ∈ g0 by (3.5) and

45

46 CHAPTER 3. THE CLASSICAL SIMPLE ALGEBRAS.

[g’1 , g1 ] = g0 by (3.3). So g can not be the one dimensional commutative

algebra, and hence what we must show is that any non-zero ideal I of g must

be all of g.

We ¬rst show that any ideal I must be a graded ideal, i.e. that

I = I’1 • I0 • I1 • · · · , where Ij := I © gj .

Indeed, write any x ∈ g as x = x’1 + x0 + x1 + · · · + xk and successively bracket

by d to obtain

x’1 + x0 + x1 + · · · + xk

x =

’x’1 + 0 + x1 + · · · + kxk

[d, x] =

x’1 + 0 + x1 + · · · + k 2 xk

[d, [d, x]] =

. . .

. . .

. . .

(ad d)k x (’1)k x’1 + 0 + x1 + · · · + k k xk

=

(ad d)k+1 x (’1)k+1 x’1 + 0 + x1 + · · · + k k+1 xk .

=

The matrix «

1 1 ···

1 1

’1 0 1 ··· k

¬ ·

¬ ·

. .. .

. . . ··· .

¬ ·

. .. .

¬ ·

¬ ·

(’1)k kk

0 1 ···

(’1)k+1 k k+1

0 1 ···

is non singular. Indeed, it is a van der Monde matrix, that is a matrix of the

form «

1 ···

1 1 1

1 ···

¬ t1 t2 tk+2 ·

¬. . . .·

¬ ·

¬. . . ··· .·

. . . .·

¬k k k

1 ···

t1 t2 tk+2

k+1 k+1

tk+1

1 ···

t1 t2 k+2

whose determinant is

(ti ’ tj )

i<j

and hence non-zero if all the tj are distinct. Since t1 = ’1, t2 = 0, t3 = 1 etc. in

our case, our matrix is invertible, and so we can solve for each of the components

of x in terms of the (ad d)j x. In particular, if x ∈ I then all the (ad d)j x ∈ I

since I is an ideal, and hence all the component xj of x belong to I as claimed.

The subspace I’1 ‚ g’1 is invariant under the adjoint action of g0 on

g’1 , and since we are assuming that this action is irreducible, there are two

possibilities: I’1 = 0 or I’1 = g’1 . We will show that in the ¬rst case I = 0

and in the second case that I = g.

Indeed, if I’1 = 0 we will show inductively that Ij = 0 for all j ≥ 0. Suppose

0 = y ∈ g0 . Since every element of [I’1 , y] belongs to I and to g’1 we conclude

3.2. SL(N + 1) 47

that [g’1 , y] = 0 and hence that y = 0 by (3.4). Thus I0 = 0. Suppose that we

know that Ij’1 = 0. Then the same argument shows that any y ∈ Ij satis¬es

[g’1 , y] = 0 and hence y = 0. So Ij = 0 for all j, and since I is the sum of all

the Ij we conclude that I = 0.

Now suppose that I’1 = g’1 . Then g0 = [g’1 , g1 ] = [I’1 , g1 ] ‚ I. Fur-

thermore, since d ∈ g0 ‚ I we conclude that gk ‚ I for all k = 0 since every

1

element y of such a gk can be written as y = k [d, y] ∈ I. Hence I = g. QED

For example, the Lie algebra of all polynomial vector ¬elds, where

‚

Xi X i homogenous polynomials of degree k + 1}

gk = {

‚xi

is a simple Lie algebra. Here d is the Euler vector ¬eld

‚ ‚

+ · · · + xn

d = x1 .

‚x1 ‚xn

This algebra is in¬nite dimensional. We are primarily interested in the ¬nite

dimensional Lie algebras.

3.2 sl(n + 1)

Write the most general matrix in sl(n + 1) as

w—

’ tr A

v A

where A is an arbitrary n—n matrix, v is a column vector and w— = (w1 , . . . , wn )

is a row vector. Let g’1 consist of matrices with just the top row, i.e. with

v = A = 0. Let g1 consist of matrices with just the left column, i.e. with

A = w— = 0. Let g0 consist of matrices with just the central block, i.e. with

v = w— = 0. Let

1 ’n 0

d=

0I

n+1

where I is the n — n identity matrix. Thus g0 acts on g’1 as the algebra of all

endomorphisms, and so g’1 is irreducible. We have

0 w— ’ w— , v

0 0 0

[ , ]= ,

v — w—

v 0 00 0

where w— , v denotes the value of the linear function w— on the vector v, and

this is precisely the trace of the rank one linear transformation v — w— . Thus

all our axioms are satis¬ed. The algebra sl(n + 1) is simple.

48 CHAPTER 3. THE CLASSICAL SIMPLE ALGEBRAS.

3.3 The orthogonal algebras.

The algebra o(2) is one dimensional and (hence) commutative. In our (real)

Euclidean three dimensional space, the algebra o(3) has a basis X, Y, Z (in-

¬nitesimal rotations about each of the axes) with bracket relations

[X, Y ] = Z, [Y, Z] = X, [Z, X] = Y,