The aim of this section is to introduce the complex Clifford algebra of then-dimensional Euclidean space, in which the Spinc group is constructed. Then, we will restrict the irreducible representations of the complex Clifford algebra to the complex spin group to get the complex spinor representation. For basic notions on Clifford algebras and Spinc geometry, we refer to [LM89, Fri00, Hij01, Mon05, BHMM].
1.1.1 The complex Clifford algebra
We consider Rn (resp. Cn) equipped with the canonical scalar product given by hv, wi:=
n
X
j=1
vjwj, for any v, w∈Rn (resp. Cn).
Definition 1.1.1. The real Clifford algebraCln (resp. the complex Clifford algebraCln) is the unitary algebra generated by Rn (resp. Cn) subject to the relations
v·w+w·v =−2hv, wi, (1.1) for any v, w∈Rn (resp. Cn).
It is easy to check that if (ej)16j6n is an orthonormal basis of Rn (resp. Cn), then {1, ei1 ·...·eik, 16i1 < ... < ik6n, 06k 6n}
is a basis of Cln (resp. Cln), thus dimRCln = 2n (resp. dimCCln = 2n). The algebra Cln can be also viewed as the exterior algebra of Rn by the following isomorphism of
47
48 CHAPTER 1. INTRODUCTION TO COMPLEX SPIN GEOMETRY vector spaces
Λ∗Rn −→ Cln
ei1 ∧...∧eik 7−→ ei1 ·...·eik.
Similary Λ∗Cn =Cln. But, it is well known that Λ∗Cn'Λ∗Rn⊗RC, hence the complex Clifford algebra is isomorphic to the complexification of the real one, i.e. Cln'Cln⊗RC. The complex Clifford algebra is periodic with period 2 (Cln+2 ' Cln ⊗Cl2) and the real Clifford algebra is also periodic but with period 8 (Cln+8 ' Cln⊗Cl8). For simplicity, we restrict ourselves to the study of the complex Clifford algebras. The algebra Cln decomposes into the direct sum of odd and even elements. In fact, the following map
β:Cln −→ Cln
ei1 ·...·eik 7−→ (−1)kei1 ·...·eik satisfies β2 = Id. Hence, it gives rise to the decomposition
Cln =Cl0n⊕Cl1n,
where Cljn = {u ∈ Cln, β(u) = (−1)ju}, j = 0 or 1. The subspace Cl0n is called the even part andCl1nis called the odd part of Cln. It is easy to see thatCl0nis a subalgebra of Cln, but not Cl1n. The classification of complex Clifford algebras is given by:
Proposition 1.1.1. The complex Clifford algebras are either isomorphic to C(2m), or to C(2m)⊕C(2m). Indeed,
Cl2m ' C(2m)'End(Σ2m), (1.2)
Cl2m+1 ' C(2m)⊕C(2m)'End(Σ2m)⊕End(Σ2m), (1.3) where C(2m) denotes the ring of 2m×2m complex matrices and Σ2m 'C2
m.
The above isomorphisms can be given explicitly in terms of the Pauli matrices defined by:
The isomorphism (1.2) is then given by
ej −→iE⊗...⊗E⊗gγ(j)⊗T ⊗...⊗T
1.1. THE COMPLEX SPIN GROUP AND THE SPINOR REPRESENTATION 49 Theorem 1.1.1. When n = 2m is even, Cl2m has a unique irreducible complex repre-sentation χ2m of complex dimension 2m,
χ2m :Cl2m −→End(Σ2m).
If n = 2m+ 1 is odd, Cl2m+1 has two inequivalent irreducible representations both of complex dimension 2m,
χj2m+1 :Cl2m+1 −→End(Σj2m) for j = 0 or 1,
where Σj2m ={σ∈Σ2m, χj2m+1(ωC)σ = (−1)jσ} andωC is the complex volume element ωC =
im e1·...·en if n = 2m, im−1 e1·...·en if n= 2m+ 1.
Finally, we give the following isomorphism α, which is of particular importance to study the irreducibility of the spinor representation and for the identification of the Spinc bundles in the context of immersions of hypersurfaces:
α:Cln −→ Cl0n+1
ej 7−→ ej ·ν, (1.4)
where Cn is embedded in Cn+1 such that (Cn)⊥ is spanned by a unit inner vectorν.
1.1.2 The complex spin group and the spinor representation
The real spin group Spinn is the multplicative subgroup of Cl∗n generated by even prod-ucts of vectors of length 1, i.e.
Spinn:={v1·...·v2k∈Cln | vj ∈Rn such that hvj, vji= 1} ⊂Cl0n.
The spin group is a compact, connected, simply connected (for n>3) Lie group of real dimension n(n−1)2 . The complex Clifford algebra contains the group Spinn as well as the groupS1 of all unit complex numbers. Together, they generate a group which we denote by Spincn. Since S1 ∩Spinn={±1}, the complex spin group Spincn is given by
Spincn = Spinn×Z2 S1, where
[a, z] = [a0, z0]⇐⇒
(a, z) = (a0, z0) or
(a, z) = (−a0,−z0) .
The complex spin group is not simply connected sinceπ1(Spincn) =Z. Moreover, it can be identified with a subgroup of Cl∗n. In fact, we consider the map
: Spinn×S1 −→ Cl∗n (a, z) 7−→ ı(a)z,
where ı : Cln ,→ Cln is the canonical injection. We remark that ker = Z2, hence Spincn 'Im, and it is a subgroup of Cl∗n.
50 CHAPTER 1. INTRODUCTION TO COMPLEX SPIN GEOMETRY Example 1.1.1. The groups Spinc4 and Spinc2. We know that
Spin2 ⊂Cl02 'Cl1 'C.
So Spin2 ⊂C∗. But, it is a commutative connected Lie group of dimension 1. The only commutative connected Lie groups of dimension 1 are S1 or R. Finally, Spin2 =S1 and Spinc2 'S1×Z2 S1. Now, recall the following identifications:
Since the Euclidean scalar product on H'R4 is left invariant by ξ(a, b), i.e.
(X,X) = detX = detx= (x,x),
it follows that the surjective map ξ : Spin4 −→ SO4 has kernel Z2. By the above identifications, the group Spin4 can be realized as:
Spin4 =n and the complex spin group as
Spinc4 =n
Now, we define the adjoint map Ad by Ad :Cl∗n −→ End(Rn)
w 7−→ Adw :v −→Adw(v) = w·v·w−1.
It is straighforward to check that, for all x ∈ Rn with kxk = 1, the map Adx is an endomorphism of Rn and −Adx is the reflection across x⊥, hence an element of SOn. By the Cartan-Dieudonn´e theorem, every element of SOnis a product of an even number of reflections. We thus have the proposition:
Proposition 1.1.2. The linear map ξ := Ad|Spinn : Spinn −→ SOn is surjective and Spinn is the double cover of SOn, i.e. the following sequence is exact
1−→Z2 −→Spinn−→ξ SOn−→1.
1.1. THE COMPLEX SPIN GROUP AND THE SPINOR REPRESENTATION 51 If n > 3, Spinn is the universal cover of SOn. Since S1 is the double cover of S1, the complex spin group is the double cover of SOn×S1, this yields to the exact sequence
1−→Z2 −→Spincn ξ
c
−→SOn×S1 −→1, where ξc= (ξ,Id2).
Example 1.1.2. From Example 1.1.1, we have:
ξc: Spinc4 −→ SO4×S1 ⊂GL(H)×S1 the Lie algebra son of SOn by the following isomorphism
ξ∗ :spinn −→ son'Λ2Rn ej·ek 7−→ 2 ej∧ek.
The Lie algebra spincn =spinn⊕iR of the group Spincn is then isomorphic to son⊕iR. (ξc)∗ :spincn 'spinn⊕iR −→ son⊕iR
ej ·ek+it 7−→ 2 ej ∧ek+ 2it.
Using Theorem 1.1.1 and the isomorphism (1.4), we have:
Proposition 1.1.3. Forn = 2m, the restriction ofχ2m toCl0n 'Cln−1 splits intoΣ2m = Σ+2m⊕Σ−2m, where Σ+2m andΣ−2m are two complex inequivalent irreducible representations of Cl0n. For n= 2m+ 1, the restrictions of χ02m+1 and χ12m+1 to Cl0n are irreducible and equivalent.
We define the complex spin representation ρcn (resp. ρn) by the restriction of an irreducible representation of Cln to Spincn (resp. Spinn):
ρcn := Cl0n ofCln.Hence, two representations ofCl0n are irreducible or equivalent if and only if it is the case for their restrictions to Spincn. By Proposition 1.1.3, we have:
52 CHAPTER 1. INTRODUCTION TO COMPLEX SPIN GEOMETRY Theorem 1.1.2. When n= 2m, ρcn decomposes into two inequivalent irreducible repre-sentations (ρcn)+ and (ρcn)−, i.e.
ρcn= (ρcn)++ (ρcn)− : Spincn→End(Σ2m).
The space Σ2m decomposes intoΣ2m = Σ+2m⊕Σ−2m, where ωC acts on Σ+2m as the identity and minus the identity on Σ−2m. If n = 2m + 1, when restricted to Spincn, the two representations χ02m+1|
Spinc n
and χ12m+1|
Spinc n
are equivalent and we simply choose Σ2m :=
Σ02m. The above facts are also true for ρn.
The natural inclusion i : Spinn ,→ Spincn, given by i(a) = [a,1] relates the spinor representations ρn and ρcn. In fact, we have:
ρcn([a, z]) =zρn(a).
It is well known that det(ρn(a)) = 1, thus detρcn([a, z]) =z2[
n2]
.
The complex vector space Σ2m carries a natural Hermitian scalar producth·,·i. This scalar product satisfies
hv·σ1, σ2i=− hσ1, v·σ2i for all σ1, σ2 ∈Σ2m and v ∈Rn. Moreover, for this scalar product, the spinor representation ρcn is unitary, i.e.
h[a, z]·σ1,[a, z]·σ2i=hσ1, σ2i for all σ1, σ2 ∈Σ2m and [a, z]∈Spincn.