Clifford Algebras I Let E Be an N-Dimensional Real Vector Space

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Clifford Algebras I Let E be an n-dimensional real vector space with an inner product q. Definition. The Clifford algebra Cl(V,q) over E is the algebra which is generated by V and satisfies the following relations: u · v + v · u = −2q(v, w)hu, vi where u, v are elements in V and hu, vi is the inner product q(u, v). • In particular, we have v2 = −q(v, v)1.

Proposition 1 (universal property of Cl(V,q)). Let A0 be any associative algebra over R with unit, containing V as a linear subset, generated by V , such that the anticommutation relation (1) vw + wv = −2hv, wi·1, holds for all v, w in A0. Then there is a natural surjective homomorphism

(2) α : Cl(V,q) → A0.

Proposition 2. Choose an orthonormal basis e1, ··· ,en of V . Each w ∈Cl(V,q) has a unique representation in the form i1 ··· in (3) w = X ai1···in e1 en , iν =0 or1

with real coeﬃcients ai1···in . Thus

i1 i2 in {e1 e2 ···en }, where i1,i1, ··· ,in =0, 1. gives a basis of Cl(V ) and dim Cl(V )=2n.

Proof. If {e1, ··· ,en} is an orthonormal basis of E, any element of Cl(V,q) can be written as a polynomial in the ej . Since ej ei = −eiej −2hei,eji·1, and in particular 2 ej = −hej ,eji·1=−1, we can, starting with terms of highest order, rearranging each monomial in such a polynomial so that ej appears with j is ascending order, and no exponent greater than 1 occurs on ant ej . In other words, each element w ∈ Cl(V,g) can be written

in the form (3) with real coeﬃcients ai1···in . • Denote by A the set of formal expressions of the form (3), which is a real vector space of dimension 2n. We have a natural inclusion V ⊂ A. • We then deﬁne a “product” A ⊗ A → A in which a product of monomials

i1 in j1 jn e1 ···en · e1 ···en

with each iν and each jµ equal to 0 or 1, is a linear combination of monomials jµ iν of such a form, by pushing each eµ past the eν for ν>µand invoking the anticommutation relations. – It is routine to verify that this gives A the structure of an associative algebra, generated by V . – The universal property implies that A is isomorphic to Cl(V,q). Thus each w ∈Cl(V,q) has a unique representation in the form (3), and thus dim Cl(V,q)=2n if dim V = n.

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Unitary Representations of the Cliﬀord Algebra

Deﬁnition A. A unitary representation of the Cliﬀord algebra Cl(E) is a pair (W, cW ) of a complex vector space W and a real-linear mapping cW : E → EndCW satisfying the following conditions:

(1) cW extends to a homomorphism Cl(E) → EndCW as W-algebras. (2) For each element v in E, cW (v) is skew-hermitian.

The mapping cW is called the Clifford multiplication on R. We may also denote by cW the R-algebra homomorphism Cl(E) → EndCR. Definition A*. Let K ⊂ k be a field containing k. Then a K-representation of the Clifford algebra Cl(V,q) is a k-algebra homomorphism

ρ : Cl(V,q) → HomK (W, W ) into the algebra of linear transformations of a ﬁnite dimensional vector space W over K. The space W is called a Cl(V,q)-module over K. – We shall often simplify notation by writing ρ(ϕ)(w) ≡ ϕ · w, ∀ϕ ∈ Cl(V,q),w∈ W. The product ϕ · w is called Cliﬀord multiplication. • We shall be interested in K representations of Cl(V,g), where K = R, C or H. • Note that a complex vector space is a real vector space W together with a real linear map J : W → W such that J 2 = −Id. - A complex representation of Cl(V,g) is a real representation ρ : Cl(V,g) → HomR(W, W ) such that ρ(ϕ) ◦ J = J ◦ ρ(ϕ), ∀ϕ ∈ Cl(V,g). Thus the image of ρ commutes with the subalgebra span{Id,J} =∼ C. • Analogously a quaterion vector space is a real vector space W together witha three real linear transformations I, J and K such that I2 = J 2 = K2 = −Id. IJ = −JI = K, JK = −KJ = I, KI = −IK = J. This makes W into an H-module. - A quaterion representation of Cl(V,g) is a real representation ρ : Cl(V,g) → HomR(W, W ) such that ρ(ϕ) ◦ I = I ◦ ρ(ϕ),ρ(ϕ) ◦ J = J ◦ ρ(ϕ),ρ(ϕ) ◦ K = K ◦ ρ(ϕ), ∀ϕ ∈ Cl(V,g). Thus the image of ρ commutes with the subalgebra span{Id,I,J,K} =∼ H.

Deﬁnition. A vector space E is a Z2-graded vector space if it is equipped with 2 an element E ∈ End E satisfying E =1. 0 1 Denote by E and E the eigenspaces of E with eigenvalues 1 and −1, respectively. Then the assignment of E ie equivalent to the direct sum decomposition E = E0 ⊕ E1. We call the superscript 0, 1 the degree. For an element a of Ek, we write the degree of a by k = degree a. 3

Deﬁnition B. A Z2-graed unitary representation of the Cliﬀord algebra Cl(E) is a pair (W, cW ) of a Z2-graded complex vector space W , equipped with 2 (1) an element E ∈ End E satisfying E =1, and (2) a Hermitian metric which is invarianr under the action E, and (3) an R-algebra homomorphism cW : Cl(E) → EndCW as W-algebras

such that for each element v in E, cW (v) is of degree 1 and skew-hermitian. The mapping cW is called the Cliﬀord multiplication on R. We may also denote by cW the R-algebra homomorphism Cl(E) → EndCR.

• The element vol, which is deﬁned by

vol = e1 · e2 ···en

is invariant under the change of a basis by the action of SO(n) and only depends on the orientation of E. – Reversing the orientation, the element, the sign of vol is reversed. – If the orientation is not speciﬁed, vol is an element, unique up to sign, in Cl(E). 0 0 – Indeed, if e1, ···en is another positively orientes, orthonormal basis, then

0 ei = X gij ej , for g =(gij ) ∈SO(n). j

Since eiej + ejei = −2δij , we have

0 0 e1 ···en = (det g)e1 ···en = e1 ···en.

Lemma 3. Let ej , 1 ≤ j ≤ n, be an oriented, orthonormal basis of V , and define the charity operator n  , if n is even p ···  2 Γ=i e1 en, where p = n +1 . , if n is odd  2 Then Γ ∈ Cl(V ) ⊗ C does not depend on the basis of V used in the definition. It satisfies Γ2 =1and supercommutes with ν for ν ∈ V ; in other words,

−νΓ, if n is even Γν = νΓ, if n is odd.

Corollary 4. If n is odd, Γ belongs to the center of C(V ) ⊗ C , while if n is even and E is a complex Clifford module, we can define a Z2-grading on E by E± = {ν ∈ E| Γν = ±ν}. Furthermore, if dim V =4m, m ∈ N, then Γ belongs to the real Clifford algebra Cl(V ), so that real Clifford modulus are also Z2-graded. 4

Deﬁnition C. A unitary repesentation is called irreducible, if it does not contain non-trivial sub-representation.

Deﬁnition C*. Let V , q ,k ⊆ K be as in Deﬁnition A.AK-representation ρ : Cl(V,q) → HomK (W, W ) is said to be reducible if the vector space can be written as a non-trivial direct sum over K

W = W1 ⊕ W2

such that ρ(ϕ)(Wj ) ⊂ Wj , for j =1, 2 and ∀ϕ ∈ Cl(V,g). A representation is called irreducible if it is not irreducible.

Example 1. Let E = R, and V = C ⊕ C, where V 0 = V 1 = C. Deﬁne a real linear mapping ρ : E → End V by

0 −1 ρ(e )= . 1 10

Here e1 = 1. Then V is an irreducible Z2-graded unitary representation of Cl(R). Example 2. Let E = R2, and V = C ⊕ C, where V 0 = V 1 = C. Deﬁne a real linear mapping ρ : E → End V by

0 −1 0 i ρ(e )= ,ρ(e )= . 1 10 2 i 0

2 Here e1, e2 is an orthonormal basis of R . Then V is an irreducible Z2-graded unitary representation of Cl(R2). Example 3. Let E = R4, and V = H ⊕ H, where V 0 = V 1 = H. Take {i, j} as basis of H. and write H = C + jC. Then the left multiplication by i, j ,k is represented by the matrices

i 0 0 −1 0 −i i = ,j= ,k= . 0 −i 10 −i 0

4 Denote by e1, e2, e3, e4 an orthonormal basis of R . Deﬁne a real linear mapping ρ : E → EndC V by

0 −1 0 i ρ(e )= ,ρ(e )= , 0 10 1 i 0

0 j 0 k ρ(e )= ,ρ(e )= . 2 j 0 3 k 0

4 Then V is an irreducible Z2-graded unitary representation of Cl(R ). – This V is not a complexiﬁcation of any Z2-graded unitary representation. 5

Example 4. On a vector space V with a positive-deﬁnite inner product, if we j j+1 deﬁne Ev :Λ V → Λ V by

Ev(v1 ∧···∧vj )=v ∧ v1 ∧···∧vj ,

j+1 j then iθ] :Λ V → Λ V is its adjoint, and

M(v)=Ev − iv] deﬁnes a linear map from V into End(Λ∗V ) which extends to an algebra homomorphism M : Cl(V,g) → End(Λ∗V ). The anticommutation relation M(v)M(w)+M(w)M(v)=−2hv, wiI is equivalent to Eviw] + iw] Ev = −hv, wiI.

Theorem 5. (1) If n =2m, Cl(E) has a unique irreducible unitary repesentation of dimension 2m. (2) If n =2m +1, Cl(E) has exactly two irreducible unitary repesentation, each of which is of dimension 2m. These two√ representations are distinguished by the action of vol, which is one of ±( −1)m+1. In other words, there are two representations according to the orientations of E.

n+1 i1 i2 in Proof. Denote by G the ﬁnite group of order 2 , generated by {±e1 e2 ···en }. – Unitary representations of Cl(E) are exactly unitary representations of G such that −1 ∈ G acts by the multiplication by −1. – Note that the number of conjugate classes equals the number of irreducible unitary representations. –Ifn is even, the center of G is {±1}; if n is odd, the center of G is {±1, ±vol}, and the identity vol1 =(−1)m+1 holds. When n is odd, the outer automorphism of G, induced by ei 7→ −ei interchanges two unitay representations. – Thus they have the same dimension and vol acts on them diﬀerently by sign.

Theorem 6. (1) If n =2m − 1, Cl(E) has a unique irreducible Z2-graded unitary repesentation of dimension 2m. (2) If n =2m, Cl(E) has exactly two Z2-graded irreducible unitary repesentation, each of which is of dimension 2m. These two√ representations are distinguished by the action of vol, which is m one of ±( −1) R. In other words, there are two representations according to the orientations of E. They are exchanged by shifting the degree. 6

Proof. Z2-graded representations of Cl(E) are in one-to-one correspondence with unitary representations of cl(Re0 ⊕ E). Hence the theorem reduces to Theorem 5.

• We next mention that a grading can be put on Cl(V,g). Namely, 0 let Cl (V,g) denote the set of sums of the form (3) with i1 + ···+ in even, 1 and let Cl (V,g) denote the set of sums of the form (3) with i1 + ···+ in odd. – This speciﬁcation is independent of the choice of the basis {ej}. • We have the direct sum decomposition

Cl(V )=Cl0(V ) ⊕ Cl1(V ).

The superscript 0, 1 is regarded as an element in Z2 and is called the degree. – Also, we clearly have

u ∈ Clj (V,g),w∈ Clk(V,g) → uw ∈ Clj+k(V,g),

and thus the product in Cl(E)

Clj(E) ⊗ Clk(E) → Clj+k(E).

where j and k are either 0 or 1, and we compute j + k mod 2. That is, Cl(E) carries a structure of a Z2-graded R-algebra. –If(V,g)isRn with its standard Euclidean metric, we denote Clj (V,g)byClj(n), j = 0 or 1. • We note that there is an isomorphism

j :Cl(2k − 1) → Cl0(2k)

uniquely speciﬁed by the property that, for v ∈ R2k−1,

1 j(v)=ve2k, if v ∈Cl (2k − 1)

2k−1 where {e1, ··· ,e2k−1} denotes the standard basis of R , with e2k added to form a basis of R2k. – This will be useful for constructing spinors on odd-dimensional space.