C(q) ∼ C(p) — C(C) ∼ End(S(q))

= =

where

S(q) := S(p) — S(C)

all tensor products being taken in the sense of superalgebra. We have a decom-

position

S(q) = S+ (q) • S’ (q)

as a super vector space where

S+ (q) = S+ (p) • xS’ (p), S’ (q) = S’ (p) • xS+ (p).

These two spaces are equivalent and irreducible as C0 (q) modules. Since the

even part of the Cli¬ord algebra is generated by §2 q together with the scalars,

we see that either of these spaces is a model for the irreducible spin representa-

tion of o(q) in this odd dimensional case.

Consider the decomposition p = p+ • p’ that we used to construct a model

for S(p) as being the left ideal in C(p) generated by §m p+ where m = dim p+ .

We have

§(C • p’ ) = §(C) — §p’ ,

and

Proposition 28 The left ideal in the Cli¬ord algebra generated by §m p+ is a

model for the spin representation.

Notice that this description is valid for both the even and the odd dimensional

case.

9.3.3 Spin ad and Vρ .

We want to consider the following situation: g is a simple Lie algebra and we

take ( , ) to be the Killing form. We have

¦ : g ’ §2 g ‚ C(g)

160CHAPTER 9. CLIFFORD ALGEBRAS AND SPIN REPRESENTATIONS.

which is the map ν associated to the adjoint representation of g. Let h be

a Cartan subalgebra and ¦ the collection of roots. We choose root vectors

eφ , φ ∈ ¦ so that

(eφ , e’φ ) = 1.

Then it follows from (9.14) that

«

1

hi § [ki , x]g + e’φ § [eφ , x]g

¦(x) = (9.24)

4

φ∈¦

where the brackets are the Lie brackets of g, where the hi range over a basis

of h and the ki over a dual basis. This equation simpli¬es in the special cases

where x = h ∈ h and in the case where x = eψ , ψ ∈ ¦+ relative to a choice,

¦+ of positive roots. In the case that x = h ∈ h we have seen that [ki , h] = 0

and the equation simpli¬es to

1

¦(h) = ρ(h)1 ’ φ(h)e’φ eφ (9.25)

2

φ∈¦+

where

1

ρ= φ

2

φ∈¦+

is one half the sum of then positive roots.

We claim that for ψ ∈ ¦+ we have

¦(eψ ) = xγ eψ (9.26)

where the sum is over pairs (γ , ψ ) such that either

1. γ = 0, ψ = ψ and xγ ∈ h or

2. γ ∈ ¦, ψ ∈ ¦+ and γ + ψ = ψ, and xγ ∈ gγ .

To see this, ¬rst observe that this ¬rst sum on the right of (9.24) gives

ψ(ki )hi § eψ

and so all these summands are of the form 1). For each summand

e’φ § [eφ , eψ ]

of the second sum, we may assume that either φ + ψ = 0 or that φ + ψ ∈ ¦ for

otherwise [eφ , eψ ] = 0. If φ + ψ = 0, so ψ = ’φ = 0, we have [eφ , eψ ] ∈ h which

is orthogonal to e’φ since φ = 0. So

e’φ § [eφ , eψ ] = ’[eφ , eψ ]eψ

again has the form 1).

9.3. THE SPIN REPRESENTATIONS. 161

If φ + ψ = „ = 0 is a root, then (e’φ , e„ ) = 0 since φ = „ . If „ ∈ ¦+ then

e’φ § [eφ , eψ ] = e’φ y„ ,

where y„ is a multiple of e„ so this summand is of the form 2). If „ is a negative

root, the φ must be a negative root so ’φ is a positive root, and we can switch

the order of the factors in the preceding expression at the expense of introducing

a sign. So again this is of the form 2), completing the proof of (9.26).

Let n+ be the subalgebra of g generated by the positive root vectors and

similarly n’ the subalgebra generated by the negative root vectors so

g = n+ • b’ , b’ := n’ • h

is an h stable decomposition of g into a direct sum of the nilradical and its

opposite Borel subalgebra.

Let N be the number of positive roots and let

0 = n ∈ §N n+ .

Clearly

∀y ∈ n+ .

yn = 0

Hence by (9.26) we have

¦(n+ )n = 0

while by (9.25)

¦(h)n = ρ(h)n ∀ h ∈ h.

This implies that the cyclic module

¦(U (g))n

is a model for the irreducible representation Vρ of g with highest weight ρ. Left

multiplication by ¦(x), x ∈ g gives the action of g on this module.

Furthermore, if nc = 0 for some c ∈ C(g) then nc has the same property:

¦(h)nc = ρ(h)nc, ∀ h ∈ h.

¦(n+ )nc = 0,

Thus every nc = 0 also generates a g module isomorphic to Vρ .

Now the map

§n+ — §b’ ’ C(g), x — b ’ xb

is a linear isomorphism and right Cli¬ord multiplication of §N n+ by §n+ is

just §N n+ , all the elements of of §+ n+ yielding 0. So we have the vector space

isomorphism

nC(g) = §N n+ — §b’ .

In other words,

¦(U (g))nC(g)

162CHAPTER 9. CLIFFORD ALGEBRAS AND SPIN REPRESENTATIONS.

is a direct sum of irreducible modules all isomorphic to Vρ with multiplicity

equal to

dim §b’ = 2s+N

where s = dim h and N = dim n’ = dim n+ . Let us compute the dimension

of Vρ using the Weyl dimension formula which asserts that for any irreducible

¬nite dimensional representation V» with highest weight » we have

φ∈¦+ (ρ + », φ)

dim V» = .

φ∈¦+ (ρ, φ)

If we plug in » = ρ we see that each factor in the numerator is twice the

corresponding factor in the denominator so

dim Vρ = 2N . (9.27)