is a reductive Lie algebra with an invariant symmetric bilinear form, and the

action is the adjoint action, i.e. x · y = [x, y]. Let h be a Cartan subalgebra of g

154CHAPTER 9. CLIFFORD ALGEBRAS AND SPIN REPRESENTATIONS.

and let ¦ denote the set of roots and suppose that we have chosen root vectors

eφ , e’φ , φ ∈ ¦ so that

(eφ , e’φ ) = 1.

Let h1 , . . . , hs be a basis of h and k1 , . . . ks the dual basis. Let

ψ : g ’ §2 g

be the map ν when applied to this adjoint action. Then (9.14) becomes

«

s

1

hi § [ki , x] + e’φ § [eφ , x] .

ψ(x) = (9.15)

4 i=1

φ∈¦

In case x = h ∈ h this formula simpli¬es. The [ki , h] = 0, and in the second

sum we have

e’φ § [eφ , h] = ’φ(h)e’φ § eφ

which is invariant under the interchange of φ and ’φ. So let us make a choice

¦+ of positive roots. Then we can write (9.15) as

1

ψ(h) = ’ φ(h)e’φ § eφ , h ∈ h. (9.16)

2

φ∈¦+

Now

e’φ § eφ = ’1 + e’φ eφ .

So if

1

ρ := φ (9.17)

2

φ∈¦+

is one half the sum of the positive roots we have

1

ψ(h) = ρ(h) ’ h ∈ h.

φ(h)e’φ eφ , (9.18)

2

φ∈¦+

In this equation, the multiplication on the right is in the Cli¬ord algebra.

9.3 The spin representations.

If

p = p1 • p2

is a direct sum decomposition of a vector space p with a symmetric bilinear

form into two orthogonal subspaces then it follows from the de¬nition of the

Cli¬ord algebra that

C(p) = C(p1 ) — C(p2 )

9.3. THE SPIN REPRESENTATIONS. 155

where the multiplication on the tensor product is taken in the sense of superal-

gebras, that is

(a1 — a2 )(b1 — b2 ) := a1 b1 — a2 b2

if either a2 or b1 are even, but

(a1 — a2 )(b1 — b2 ) := ’a1 b1 — a2 b2

if both a2 and b1 are odd. It costs a sign to move one odd symbol past another.

9.3.1 The even dimensional case.

Suppose that p is even dimensional. If the metric is split (which is always the

case if the metric is non-degenerate and we are over the complex numbers) then

p is a direct sum of two dimensional mutually orthogonal split spaces, Wi , so

let us examine ¬rst the case of a two dimensional split space p, spanned by ι,

with (ι, ι) = ( , ) = 0, (ι, ) = 1 . Let T be a one dimensional space with basis

2

t and consider the linear map of p ’ End (§T ) determined by

ι ’ ι(t— )

’ (t),

where (t) denotes exterior multiplication by t and ι(t— ) denotes interior multi-

plication by t— , the dual element to t in T — . This is a Cli¬ord map since

(t)2 = 0 = ι(t— )2 , (t)ι(t— ) + ι(t— ) (t) = id.

This therefore extends to a map of C(p) ’ End(§T ). Explicitly, if we use

1 ∈ §0 T, t ∈ §1 T as a basis of §T this map is given by

1 0

1’

0 1

0 1

ι’

0 0

0 0

’

1 0

1 0

’

ι .

0 0

This shows that the map is an isomorphism. If now

p = W1 • · · · • Wm

is a direct sum of two dimensional split spaces, and we write

T = T1 • · · · • Tm

156CHAPTER 9. CLIFFORD ALGEBRAS AND SPIN REPRESENTATIONS.

where the C(Wi ) ∼ End(§Ti ) as above, then since

=

§T = §T1 — · · · — §Tm

we see that

C(p) ∼ End (§T ).

=

In particular, C(p) is isomorphic to the full 2m — 2m matrix algebra and hence

has a unique (up to isomorphism) irreducible module. One model of this is

S = §T.

We can write

S = S+ • S’

as a supervector space, where we choose the standard Z2 grading on §T to

determine the grading on S if m is even, but use the opposite grading (for

reasons which will become apparent in a moment) if m is odd.

The even part, C0 (p) of C(p) acts irreducibly on each of S± . Since §2 p

together with the constants generates C0 (p) we see that the action of §2 p on

each of S± is irreducible. Since §2 p under Cli¬ord commutation is isomorphic

to o(p) the two modules S± give irreducible modules for the even orthogonal

algebra o(p). These are the half spin representations of the even orthogonal

algebras.

We can identify S = S+ • S’ as a left ideal in C(p) as follows: Suppose

that we write

p = p+ • p’

where p± are complementary isotropic subspaces. Choose a basis e+ , . . . , e+ of

m

1

p+ and let

e+ := e+ · · · e+ = e+ § · · · § e+ ∈ §m p+ .