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Lecture 11: Clifford algebras

In this lecture we introduce Clifford algebras, which will play an important role in the rest of the class. The link with K-theory is the Atiyah-Bott-Shapiro construction [ABS], which implements the K-theory of suspensions via Clifford modules. We will begin the next lecture with this ABS construction.

An algebra from the

The orthogonal group On is a subset of an algebra: the algebra MnR of n n matrices. The ˆ Clifford algebra plays a similar role for a double cover group of the orthogonal group.

(11.1) Heuristic motivation. Orthogonal transformations are products of reflections. For a norm vector ξ Rn define P

(11.2) ρξ η η 2 η,ξ ξ, p q “ ´ x y where , is the standard inner product. x´ ´y Theorem 11.3 (Sylvester). Any g On is the composition of n reflections. P ď Proof. The statement is trivial for n 1. Proceed by induction: if g On fixes a unit norm vector ξ “ K P then it fixes the orthogonal complement R ξ , and we are reduced to the theorem for On´1. If p ¨ q there are no fixed unit norm vectors, then for any unit norm vector ζ set ξ gpζq´ζ . The “ |gpζq´ζ| composition ρξ g fixes ζ and again we reduce to the n 1 -dimensional orthogonal complement.  ˝ p ´ q Now generate an algebra from the unit norm vectors, with relations inspired by those of reflec- tions. Note immediately that the vectors ξ both correspond to the same reflection ρξ ρ ξ. ˘ “ ´ Therefore, we expect from the beginning that the Clifford algebra “double counts” orthogonal transformations. Now since the square of a reflection is the identity, we impose the relation

(11.4) ξ2 1, ξ 1. “ ˘ | | “

The sign ambiguity is that described above, and we choose a sign independent of ξ. It follows that

(11.5) ξ2 ξ 2 “ ˘| |

n for any ξ R . Now if ξ1,ξ2 0, then ξ1 ξ2 ?2 has unit norm and from P x y “ p ` q{

2 2 2 ξ1 ξ2 ξ ξ ξ1ξ2 ξ2ξ1 2 ξ1ξ2 ξ2ξ1 (11.6) 1 ` 1 ` 2 ` p ` q ˘ ` p ` q ˘ “ ˆ ?2 ˙ “ 2 “ 2

K-Theory (M392C, Fall ’15), Dan Freed, October 12, 2015 1 2 D. S. Freed we deduce

(11.7) ξ1ξ2 ξ2ξ1 0, ξ1,ξ2 0. ` “ x y “ Equations (11.4) and (11.7) are the defining relations for the Clifford algebra. Check that the reflection (11.2) is given by

´1 (11.8) ρξ η ξηξ p q “ ´ in the Clifford algebra. By composition using Theorem 11.3 we obtain the action of any orthogonal transformation on η Rn. P Definition 11.9. For n Z define the real Clifford algebra Cℓn as the unital associative real algebra P generated by e1,...,e|n| subject to the relations

2 ei 1, i 1,...,n (11.10) “ ˘ “ eiej ejei 0, i j. ` “ ­“ C The complex Clifford algebra Cℓn is the complex algebra with the same generators and same relations. C Note Cℓ0 R and Cℓ C. “ 0 “ C C Example 11.11. There is an Cℓ Cℓ obtained by multiplying each generator ei ´n – n by ? 1. ´ Example 11.12. Cℓ´1 can be embedded in the algebra M2R by setting 0 1 (11.13) e1 ´ . “ ˆ1 0 ˙ C C The same equation embeds Cℓ´1 in M2 . Example 11.14. We identify CℓC with End C2 by setting ´2 p q 0 1 0 i (11.15) e1 ´ , e2 , “ ˆ1 0 ˙ “ ˆi 0˙ where i ? 1. This does not work over the reals. The product “ ´ i 0 (11.16) e1e2 ´ “ ˆ 0 i˙ is i times a grading operator on C2. ´ Example 11.17. The real Clifford algebras Cℓ1 and Cℓ´1 are not isomorphic. We embed in M2R in the former case by setting 0 1 (11.18) e1 “ ˆ1 0˙ and in the latter using (11.13). Note that the doubled orthogonal group 1, e1 is different in t˘ ˘ u the two cases: in Cℓ1 it is isomorphic to the Klein group Z 2Z Z 2Z whereas in Cℓ´1 it is cyclic { ˆ { of order four. Topics in and Physics: K-Theory (Lecture 11) 3

(11.19) Spin and Pin. For n 0 let S Rn Rn denote the sphere of unit norm vectors. Since Rn ą n p q Ă embeds in Cℓ˘n, so too does S R . We assert without proof that the group it generates is a Lie p q group Pin˘n Cℓ˘n. It follows from Theorem 11.3 that there is a surjection Pin˘n On Ă Ñ defined by composing the reflections (11.8). The inverse Spin˘n of the special orthogonal n group SOn consists of products of an even number of elements in S R . There is an isomorphism p q Spin Spin , but as we saw in Example 11.14 this is not true in general for Pin. n – ´n (11.20) The . The Clifford algebra arises from the following question, posed by Dirac: Find a square root of the Laplace operator. We work on flat Euclidean space En, which is the affine space An endowed with the translation-invariant metric constructed from the standard inner product on the underlying Rn of translations. Let x1,...,xn be the standard affine coordinates on En. The Laplace operator is

n 2 (11.21) ∆ B . “ ´ xi 2 iÿ“1 pB q

A first-order operator

(11.22) D γi B “ xi B satisfies D2 ∆ if and only if γi satisfy the Clifford relation “

(11.23) γiγj γjγi 2δij, 1 i,j n, ` “ ´ ď ď as in (11.10). If we let (11.21), (11.22) act on the space C8 En; S of functions with values in a p q vector space S, then we conclude that S is a Cℓ´n-.

(11.24) Z 2Z-gradings. So far we have not emphasized the Z 2Z-grading evident in the examples: { { odd products of generators such as (11.13), (11.15), (11.18) are represented by block off-diagonal matrices whereas even products of generators (11.16) are represented by block diagonal matrices.

Superalgebra For a more systematic treatment, see [DM, §1]. We use ‘super’ synonymously with ‘Z 2Z-’. { (11.25) Super vector spaces. Let k be a field, which in our application will always be R or C. A S S0 S1 is a pair S,ǫ of a vector space over k and an operator ǫ with “ ‘ p q ǫ2 idS. The subspaces S0, S1 are the 1, 1-eigenspaces, respectively. Eigenvectors are called “ ` ´ even, odd. The product S1 S2 of super vector spaces carries the grading ǫ1 ǫ2. The main b b new point is the Koszul sign rule, which is the symmetry of the :

S1 S2 S2 S1 (11.26) b ÝÑ b 1 2 s1 s2 1 |s ||s |s2 s1, b ÞÝÑ p´ q b 4 D. S. Freed

(11.27) . Let A A0 A1 be a super algebra, an algebra with a compatible grading: “ ‘ Ai Aj Ai`j, where the degree is taken in Z 2Z. A homogeneous element z in its center satisfies ¨ Ă { za 1 |z||a|az for all homogeneous a A. The center is itself a super algebra, which is of course “ p´ q P commutative (in the Z 2Z-graded sense). The opposite super algebra Aop to a super algebra A is { |a1||a2| the same underlying vector space with product a1 a2 1 a2a1 on homogeneous elements. ¨ “ p´ q All algebras are assumed unital. Tensor products of super algebras are taken in the graded sense: the multiplication in A1 A2 is b

2 1 (11.28) a1 a2 a1 a2 1 |a1||a2|a1 a1 a2a2. p 1 b 1qp 2 b 2q “ p´ q 1 2 b 1 2

Undecorated tensor products are over the ground field. Unless otherwise stated a module is a left module. An I A in a super algebra is graded if I I A0 I A1 . Ă “ p X q ‘ p X q (11.29) Super matrix algebras. Let S S0 S1 be a finite dimensional super vector space over k. “ ‘ Then End S is a central simple super algebra. which preserve the grading on S are even, those which reverse it are odd. A super algebra isomorphic to End S is called a super matrix algebra.

Clifford algebras For more details see [ABS, Part I], [De1, §2]. A on a vector space V is a function Q: V k such that Ñ

(11.30) B ξ1,ξ2 Q ξ1 ξ2 Q ξ1 Q ξ2 , ξ1,ξ2 V, p q “ p ` q ´ p q ´ p q P is bilinear and Q nξ n2Q ξ . p q “ p q Definition 11.31. The Clifford algebra Cℓ V,Q Cℓ V of a quadratic vector space is an algebra p q “ p q equipped with a i: V Cℓ V,Q which satisfies the following : If Ñ p q ϕ: V A is a linear map to an algebra A such that Ñ

2 (11.32) ϕ ξ Q ξ 1A, ξ V, p q “ p q ¨ P then there exists a unique algebra homomorphismϕ ˜: Cℓ V,Q A such that ϕ ϕ˜ i. p q Ñ “ ˝ We leave the reader to prove that i is injective and that Cℓ V,Q is unique up to unique isomor- p q phism. Furthermore, there is a surjection

(11.33) V Cℓ V,Q b ÝÑ p q from the , as follows from its universal property. This gives an explicit construction 2 of Cℓ V,Q as the quotient of V by the 2-sided ideal generated by ξ Q ξ 1 V , ξ V . The p q b ´ p q ¨ b P tensor algebra is Z-graded, and since the ideal sits in even degree the quotient Clifford algebra is Topics in Geometry and Physics: K-Theory (Lecture 11) 5

Z 2Z-graded. The increasing filtration 0V ď1V ď2V induces an increasing filtration { b Ă b Ă b Ă ¨ ¨ ¨ on Cℓ V,Q whose associated graded is isomorphic to the (Z-graded) ‚V . There p q is a canonical isomorphism Ź

(11.34) Cℓ V 1 V 2,Q1 Q2 Cℓ V 1,Q1 Cℓ V 2,Q2 , p ‘ ‘ q – p q b p q deduced from the universal property. The standard Clifford algebras in Definition 11.9 have the form Cℓ V,Q for V Rn, Cn and Q the positive or negative definite standard quadratic form p q “ on V . The Clifford algebras are central simple as Z 2Z-graded algebras. I will leave the simplicity { (there are no nontrivial 2-sided homogeneous ideals) as an exercise and here prove the centrality. Proposition 11.35. Cℓ V,Q has center k. p q 0 1 Proof. Suppose x x x is a central element. Fix an orthonormal e1,...,en of V . Then “ ` for every i 1,...,n we have “ 0 0 x ei eix “ (11.36) 1 1 x ei eix “ ´ 0 0 1 0 1 There is a unique decomposition x a eib where a , b belong to the Clifford algebra generated “ ` by the basis elements excluding ei. Then

0 0 1 0 2 1 x ei a ei eib ei eia ei b “ ` “ ´ p q (11.37) 0 0 2 1 eix eia ei b . “ ` p q 0 0 0 1 Since x is central we have x ei eix , and so (11.37) implies that b 0. Since this holds for 0 “ 1 1 0 “ every i, we conclude that x is a scalar. Similarly, write x a eib so that “ ` 1 1 0 1 2 0 x ei a ei eib ei eia ei b “ ` “ ´ ` p q (11.38) 1 1 2 0 eix eia ei b ´ “ ´ ´ p q from which x1 0.  “ ˚ For a vector space L and θ L let ǫθ denote exterior multiplication by θ, which is an endo- P of the exterior algebra ‚L˚. For ℓ L the adjoint of exterior multiplication by ℓ is ‚ ˚ P contraction ιℓ, an ofŹ L of degree 1. ´ Ź Proposition 11.39. Suppose V L L˚ with the split quadratic form Q ℓ θ θ ℓ , ℓ L, “ ‘ p ` q “ p q P θ L˚. Set S ‚L˚ with its Z 2Z-grading by the parity of the degree. Then the map V End S P “ { Ñ Ź ℓ ιℓ (11.40) ÞÝÑ θ ǫθ ÞÝÑ extends to an isomorphism Cℓ V – End S of the Clifford algebra with a super matrix algebra. p q ÝÝÑ Proof. Using (11.34) we reduce to the case dim L 1 which can be checked by hand; it is essentially “ Example 11.14.  6 D. S. Freed

(11.41) Algebraic Bott periodicity. We may in the future discuss basic Morita theory, in which we will see that super matrix algebras are in some sense trivial. That is the spirit of the following theorem. We say the of a finite dimensional super vector space S S0 S1 is d0 d1 if “ ‘ | dim Si di. “ Theorem 11.42. There are of superalgebras

C Cℓ – End S , dim S 1 1, ´2 ÝÝÑ p q “ | (11.43) – Cℓ´8 End SR , dim SR 8 8. ÝÝÑ p q “ |

1|1 Proof. The complex case is Example 11.14. For the real case we let Cℓ´2 act on W C via the “ formulas in (11.15). This action commutes (in the graded sense) with the odd real structure

(11.44) J z0,z1 z1, z0 . p q “ p q

That is, J : W W is antilinear, odd, and squares to idW. Set S W b4. It carries an action b4 Ñ b4 ´ “ of Cℓ Cℓ´8 which commutes with J . The latter is antilinear, even, and squares to idS, so is ´2 – a real structure.  As stated in the proof, Wb2 carries a quaternionic structure J b2: the Koszul sign rule (11.26) implies that J b2 squares to minus the identity. (Check that sign! It will test your understanding of the sign rule.)

(11.45) Spin and Pin redux. Sitting inside the Clifford algebra Cℓ V,Q is the Pin V,Q p q p q generated by S V and its even subgroup Spin V,Q Pin V,Q Cℓ V,Q 0. When V is real and p q p q “ p q X p q Q is definite these are compact Lie groups. In that case we can average a metric over a real or complex Clifford module S S0 S1 so that Pin V,Q acts orthogonally (unitarily in the complex “ ‘ p q case). It follows that e S V is self- or skew-adjoint, according as Q is positive or negative P p q definite.

Remark 11.46. There is a tricky sign in the proper definition of ‘self-adjoint’ and ‘skew-adjoint’ in the super world. There is a way around that sign to a more standard convention, which is the one we use; see [DM, §4.4], [De2, §4].

References [ABS] M. F. Atiyah, R. Bott, and A. Shapiro, Clifford modules, Topology 3 (1964), no. suppl. 1, 3–38. [De1] Pierre Deligne, Notes on , Quantum Fields and Strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI, 1999, pp. 99–135. [De2] , Note on quantization, Quantum fields and strings: a course for mathematicians 1 (1999), 367. [DM] Pierre Deligne and John W. Morgan, Notes on (following Joseph Bernstein), Quantum fields and strings: a course for mathematicians, Vol. 1, 2 (Princeton, NJ, 1996/1997), Amer. Math. Soc., Providence, RI, 1999, pp. 41–97.