Lecture 18 - Clifford Algebras and Spin groups

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1 Lecture 18 - Clifford Algebras and Spin groups April 5, 2013 Reference: Lawson and Michelsohn, Spin Geometry. 1 Universal Property If V is a vector space over R or C, let q be any quadratic form, meaning a map q : V W with q(αv) = α 2 q(v) for scalars α. Given a pair (V, q) we can construct the Clifford algebra Cl(V, q) as the quotient of the tensor algebra V by the ideal generated by v v + q(v), as v ranges over V. Note that if q = 0, then Cl(V, q) = V. Ordinarily we consider the case that q is nondegenerate. Over the complex numbers, nondegenerate quadratic forms are all equivalent up to change of basis, so the only invariant in the dimension of the space V. Setting dim C (V ) = n we obtain the algebras Cl n. In the real setting, besides the dimension of the vector space dim R (V ) = n, the only other invariant of a quadratic form is its signature, which is characterized by a pair of non-negative integers (r, s) where r + s = n. We obtain the Clifford algebras Cl r,s. Clifford algebras have the following universal property. If (V, q) is a vector space (over R or C) with quadratic form q and ϕ : V GL(W, W ) is a linear map so that ϕ(v)ϕ(v) = q(v) 2, then ϕ extends uniquely to a Clifford representation map ϕ : Cl(V, q) GL(W, W ). 2 Reason for Studying Clifford Algebras Let D be a normed division algebra over reals, with norm. We obtain an inner product x, y = 1 ( x + y 2 x 2 y 2). (1) 2 Since ax = a x, we have that L a is an orthogonal transformation whenever a = 1. 1

2 Consider the subspace of vectors in D orthogonal to 1; we shall call this Im(D). The unit sphere in D is a group of orthogonal transformation of D, and the tangent space to the unit sphere at 1, its Lie algebra, can be identified with Im(D). Therefore L a for a Im(D) is skew-symmetric. If a is also a unit vector, then 1, a 2 = a, a = 1 (2) so that a 2 = 1. If dim(d) = n, this gives a faithful representation of Cl n 1 on D. 3 Periodicity Proposition 3.1 We have the following algebra isomorphisms Cl n,0 Cl 0,2 Cl 0,n+2 Cl 0,n Cl 2,0 Cl n+2,0 (3) Cl r,s Cl 1,1 Cl r+1,s+1 Pf. We do only the first isomorphism; the rest are entirely analogous. Consider the vector space R n+2 with orthonormal basis e 1,..., e n+2, with negative definite inner product. Also let f 1,..., f n and g n+1, g n+2 be orthonormal bases of R n with positive definite inner product, and R 2 with negative definite inner product. Let F : R n+2 Cl n,0 Cl 0,2 be the map { f i g n+1 g n+2, if i {1,..., n} F (e i ) = (4) 0 g i, if i {n + 1, n + 2} Noting that F (e i )F (e j ) = F (e j )F (e i ) when i j and F (e i ) 2 = 1, this extends to the desired algebra isomorphism. Theorem 3.2 ( Bott periodicity ) We have the following isomorphisms Cl n+8,0 Cl n,0 R Cl 8,0 Cl n,0 R R(16) Cl 0,n+8 Cl 0,n R Cl 0,8 Cl 0,n R R(16) Cl n+2 Cl n C Cl 2 Cl n C C(2) Pf. Noting that H 2 R(4), we can use the previous lemma to obtain Cl n+8,0 The other cases are similar. Cl n,0 (Cl 0,2 Cl 2,0 ) 2 Cl n,0 H H R(2) R(2) Cl n,0 R(4) R(4) Cl n,0 R(16) (5) (6) 2

4 where α is the parity involution. We have that Ãd v is a representation (as Ãd ϕãd η = Ãd ϕη) and that when v V. Ãd v = σ v. (15) Theorem 4.2 The Ãd maps P in r,s O(r, s) Spin r,s SO(r, s) (16) are surjective. Their kernels are Z 2 {±1}. Except in the cases (r, s) = (1, 1), (1, 0), (0, 1), these two-sheeted coverings are connected over any connected component of O(r, s) (resp. SO(r, s)). Pf. The only thing to prove is that 1 and 1 can be joined by a path in Spin r,s P in r,s. Chosing orthogonal unit vectors e 1, e 2 with q(e 1 ) = q(e 2 ) (either both +1 or both 1), we have that e 1 cos(t) + e 2 sin(t) and e 1 cos(t) + e 2 sin(t) are unit vectors and γ(t) = (e 1 cos(t) + e 2 sin(t)) ( e 1 cos(t) + e 2 sin(t)) = ± ( cos 2 (t) sin 2 (t) ) + 2e 1 e 2 cos(t) sin(t) = ± cos(2t) + e 1 e 2 sin(2t) is a path from 1 to 1 in Spin r,s. (17) 5 Spin Representations 5.1 Complete Reducibility A basic theme in the study of Clifford representation is the reduction of the representation to a Pin or Spin representation. For instance, any representations of a Clifford algebra is completely reducible. To see this, note that any Clifford representation V is a Pin representation, so V is completely reducible as a Pin representation. But P in r,s contains a basis of Cl r,s, so the action of the Clifford algebra preserves the factors. 4

5 5.2 Fundamental Irreducible Clifford Representations We have the following table n Cl n Cln 0 0 R 1 C R 2 H C 3 H H H 4 H(2) H H 5 C(4) H(2) 6 R(8) C(4) 7 R(8) R(8) R(8) 8 R(16) R(8) R(8) (18) This table can be used to give the irreducible representations of Cl n. In particular, with n 3 (mod 4) we have a single irreducible representation, and with n = 3 mod 4, we have two representations. For n = 3 mod 8, these representations are inequivalent. However, we proceed more theoretically. Consider the Clifford group, the finite group E n generated by an orthonormal basis e 1,..., e n of R n. We have that the group algebra RE n is nearly the Clifford algebra: in particular Cl n RE n / {( 1) + 1 = 0}. (19) Note that E n = { ± m i=1 e ki i } ki = ±1 (20) is a subgroup of Cl n with 2 n+1 elements. Set En 0 = E n Cl 0 n, so that En 0 has 2 n elements. Finally, if n is even, then the center of E n is {±1} and if n is odd, its center is {±1, ±ω} where ω = e 1... e n. Now consider that each element of E n acts on a 1-dimensional subspace, so we have a 2 n+1 -dimensional representation of RE n, which passes to a 2 n dimensional representation of Cl n, which we shall call W n. Proposition 5.1 (Clifford Representations) In the cases n 0, 1, 2 mod 4, we the Clifford algebra Cl n has a single irreducible representation W n of dimension 2 n. In the case that n 3 mod 4, the Clifford algebra Cl n has two inequivalent irreducible representations W n + and Wn of dimensions 2 n 1. Pf. We consider the case of n 3 mod 4. Let W n be the algebra constructed above. We know that ω is central and ω 2 = 1. Therefore π ± = 1 2 (1 ± ω) splits the representation: 5

6 W ± n = π ± W n. To see that theses representations are inequivalent, notice that ω acts as 1 on W + n and 1 on W n. Notice that if n = 4m, then ω 2 = 1 and W n also splits as a vector space: W n = W + n W n where W ± n = π ± W. However π ± is not central, so the factors are exchanged by elements of Cl 1 n. But Cl 0 n preserves the factors. We can restrict these representations Cl n on W n to Spin n Cl 0 n Cl n ; we denote the resulting representation S n. Theorem 5.2 (Fundamental Spin Representations) If n 1, 2, 3 mod 4 then S n is the direct sum of two identical representations, which we call n. If n 0 mod 4 then S n is the direct sum of two different, non-isomorphic representations; we denote S n = n = + n n where ± n = π n 1 ± S n. Pf. Note that the parity involution α preserves Spin n Cl 0 n Cl n, and that the projectors π 0 = 1 2 (1 + α), π1 = 1 2 (1 α) commute and sum to unity. Thus the representation of Spin n on Cl n splits into representations on Cl 0 n and Cl 1 n. Case n = 2m + 1 is odd. We have that ω / spin(n). If m is even (n 1 mod 4) then the representation on Cl 0 n is irreducible because Cl 0 n Cl n 1 is irreducible. Further, the factors Cl 0 n and Cl 1 n are exchanged by ω, which is central; hence the representations on Cl 0 n and Cl 1 n are equivalent. If m is odd (n 3 mod 4) then we have a secondary decomposition Cl n = Cl + n Cl n, where Cl 0 n sits in the sum diagonally. However Cl 0 n Cl n 1 is irreducible, and projects onto either factor isomorphically. Thus again we obtain two equivalent representations. Case n = 2m is even. We have that ω spin(n), and that ω is central in Spin(n) (although it is not central in Cl n ). If m is odd so n 2 mod 4 then we get the usual even-odd splitting. If m is even so n 0 mod 4 then ω 2 = 1 and we can decompose Cl n into two factors, corresponding to the ±1 eigenvalues on ω. These factors are clearly inequivalent representations, and ω Spin(n) acts differently. Also, An element of a fundamental spin representation v n is called a spinor. 6

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