a non-zero multiply of p2 , and then get any monomial from p2 by bracketing

1 1

1

with pi q1 appropriately. The element d is given by 2 (p1 q1 + · · · + pn qn ).

We have shown that the symplectic algebra is simple, but we haven™t really

explained what it is. Consider the space of V of homogenous linear polynomials,

i.e all polynomials of the form

= a1 q1 + · · · + an qn + b1 pq + · · · + bn pn .

De¬ne an anti-symmetric bilinear form ω on V by setting

ω( , ) := { , ).

From the formula (3.8) it follows that the Poisson bracket of two linear functions

is a constant, so ω does indeed de¬ne an antisymmetric bilinear form on V ,

and we know that this bilinear form is non-degenerate. Furthermore, if f is a

homogenous quadratic polynomial, and is linear, then {f, } is again linear,

and if we denote the map

’ {f, }

by A = Af , then Jacobi™s identity translates into

ω(A , ) + ω( A ) = 0 (3.9)

since { , } is a constant. Condition (3.9) can be interpreted as saying that A

belongs to the Lie algebra of the group of all linear transformations R on V

which preserve ω, i.e. which satisfy

ω(R , R ) = ω( , ).

This group is known as the symplectic group. The form ω induces an isomor-

phism of V with V — and hence of Hom(V, V ) = V — V — with V — V , and this

time the image of the set of A satisfying (3.9) consists of all symmetric ten-

sors of degree two, i.e. of S 2 (V ). (Just as in the orthogonal case we got the

anti-symmetric tensors). But the space S 2 (V ) is the same as the space of ho-

mogenous polynomials of degree two. In other words, the symplectic algebra as

de¬ned above is the same as the Lie algebra of the symplectic group.

It is an easy theorem in linear algebra, that if V is a vector space which

carries a non-degenerate anti-symmetric bilinear form, then V must be even

dimensional, and if dim V = 2n then it is isomorphic to the space constructed

above. We will not pause to prove this theorem.

52 CHAPTER 3. THE CLASSICAL SIMPLE ALGEBRAS.

3.5 The root structures.

We are going to choose a basis for each of the classical simple algebras which

generalizes the basis e, f, h that we chose for sl(2). Indeed, for each classical

simple algebra g we will ¬rst choose a maximal commutative subalgebra h all

of whose elements are semi-simple = diagonizable in the adjoint representation.

Since the adjoint action of all the elements of h commute, this means that they

can be simultaneously diagonalized. Thus we can decompose g into a direct

sum of simultaneous eigenspaces

g =h• g± (3.10)

±

where 0 = ± ∈ h— and

g± := {x ∈ g|[h, x] = ±(h)x ∀ h ∈ h}.

The linear functions ± are called roots (originally because the ±(h) are roots

of the characteristic polynomial of ad(h)). The simultaneous eigenspace g± is

called the root space corresponding to ±. The collection of all roots will usually

be denoted by ¦.

Let us see how this works for each of the classical simple algebras.

3.5.1 An = sl(n + 1).

We choose h to consist of the diagonal matrices in the algebra sl(n + 1) of all

(n + 1) — (n + 1) matrices with trace zero. As a basis of h we take

«

1 0 ··· 00

¬0 ’1 0 · · · 0·

h1 := ¬ .

¬ ·

. . . .·

. . . . .

. . . . .

···

00 0 0

«

···

00 0 0

···

¬0 1 0 0·

¬ ·

h2 := ¬0 0 ’1 ··· 0·

¬

·

¬. . . . .·

. . . . .

.. . . .

···

00 0 0

.

. := .

.

. .

«

···

00 0 0

¬. . . . .·

¬. . . . .·

.. . . . ·.

hn := ¬

0 0 · · · 1 0

0 0 ··· ’1

0

Let Li denote the linear function which assigns to each diagonal matrix its

i-th (diagonal) entry,

3.5. THE ROOT STRUCTURES. 53

Let Eij denote the matrix with one in the i, j position and zero™s elsewhere.

Then

[h, Eij ] = (Li (h) ’ Lj (h))Eij ∀ h∈h

so the linear functions of the form

Li ’ Lj , i = j

are the roots.

We may subdivide the set of roots into two classes: the positive roots

¦+ := {Li ’ Lj ; i < j}

and the negative roots

¦’ := ’¦+ = {Lj ’ Li , i < j}.

Every root is either positive or negative. If we de¬ne

±i := Li ’ Li+1

then every positive root can be written as a sum of the ±i :

Li ’ Lj = ±i + · · · + ±j’1 .

We have

±i (hi ) = 2,

and for i = j

±i (hi±1 ) = ’1, ±i (hj ) = 0, j = i ± 1. (3.11)

The elements

Ei,i+1 , hi , Ei+1,i

form a subalgebra of sl(n + 1) isomorphic to sl(2). We may call it sl(2)i .

Cn = sp(2n), n ≥ 2.

3.5.2

Let h consist of all linear combinations of p1 q1 , . . . , pn qn and let Li be de¬ned

by