Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions

Cliﬀord Algebras, Cliﬀord Groups, and a Generalization of the Quaternions: The Pin and Spin Groups Jean Galli er Department of Computer and Informa tion Science Universit y of P ennsylv ania Phila delphia, P A 1 9 104, USA e-mail : jean@cis .upe nn.edu September 30, 201 4 2 3 Cliﬀord Algebras, Cliﬀord Groups, and a Generalization of the Quaternions: The Pin and Spin Groups Jean Gallier Abstract: One of the main goals of these notes is to explain how rotations in R n are induced b y the a ctio n of a certain group Spin ( n ) o n R n , in a wa y tha t generalizes the action of the unit complex n um b ers U (1) on R 2 , and the action of the unit quaternions SU (2) on R 3 ( i.e. , the action is deﬁned in terms of m ultiplication in a larger algebra con taining b oth the group Spin ( n ) and R n ). The group Spin ( n ), called a spinor gr oup , is deﬁned as a certain subgroup of units of an algebra Cl n , the Cliﬀor d algebr a asso ciated with R n . Since the spinor gr oups are certain w ell c hosen subgroups of units of Cliﬀord a lgebras, it is necessary to inv estigate C liﬀord algebras to g et a ﬁrm understanding of spinor gro ups. These notes pro vide a tutoria l on Cliﬀord algebra and the groups Spin a nd Pin , including a study of the s tructure of the Cliﬀord algebra Cl p,q asso ciated with a nondegene rate symmetric bilinear form of signat ure ( p , q ) a nd culminating in the b eautif ul “8- p erio dicit y theorem” of Elie Carta n and Raoul Bott (with pro ofs). 4 Con ten ts 1 Cliﬀord Algebras, Cliﬀord Groups, P in and Spin 7 1.1 In t r o duction: Rota tions As Group Actions . . . . . . . . . . . . . . . . . . . 7 1.2 Cliﬀord Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Cliﬀord Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.4 The Gr oups P in ( n ) and Spin ( n ) . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5 The Gr oups P in ( p, q ) and Spin ( p, q ) . . . . . . . . . . . . . . . . . . . . . . 31 1.6 P erio dicit y of the Cliﬀord Algebras C l p,q . . . . . . . . . . . . . . . . . . . . 33 1.7 The Complex Cliﬀord Algebras Cl( n, C ) . . . . . . . . . . . . . . . . . . . . 37 1.8 The Gr oups P in ( p, q ) and Spin ( p, q ) as double cov ers . . . . . . . . . . . . . 38 1.9 More on the T op o lo gy o f O ( p, q ) and SO ( p, q ) . . . . . . . . . . . . . . . . . 42 5 6 CONTENTS Chapter 1 Cliﬀord Algebras, Cliﬀord Groups, and the Groups Pin ( n ) and Spin ( n ) 1.1 In tr o du c t ion: Rotations As Grou p Actio n s The main goal of this chapter is to explain how r otations in R n are induced b y the actio n of a ce rtain group Spin ( n ) on R n , in a w ay that g eneralizes the a ction of the unit complex n um b ers U (1) on R 2 , and the actio n of the unit quaternions SU (2) on R 3 ( i.e. , the action is deﬁned in terms of m ultiplication in a larger algebra containing b ot h the group Spin ( n ) and R n ). The group Spin ( n ), called a spino r gr oup , is deﬁned as a certain subgroup o f units of an algebra Cl n , the Cliﬀor d algebr a asso ciated with R n . F urthermore, for n ≥ 3, w e are luc ky , b ecause the group Spin ( n ) is to p ologically simpler than the gro up SO ( n ). Indeed, for n ≥ 3, the group Spin ( n ) is simply connected (a fa ct tha t it not s o easy to pro ve without some machine ry), whereas SO ( n ) is not simply connected. In tuitiv ely sp eaking, SO ( n ) is more t wisted than Spin ( n ). In fact, we will see that Spin ( n ) is a double co v er o f SO ( n ). Since t he spinor gr oups are certain we ll c hosen subroups of units of Cliﬀord algebras, it is necessary to in v estigate Cliﬀord alg ebras to get a ﬁrm understanding of spinor groups. This c ha pter pro vides a tutorial on Cliﬀord algebra and the gr o ups Spin and Pin , including a study of the s tructure of the Cliﬀord algebra Cl p,q asso ciated with a nondegene rate symmetric bilinear form of signat ure ( p , q ) a nd culminating in the b eautif ul “8- p erio dicit y theorem” of Elie Cartan and Raoul Bo tt (with pro ofs). W e a lso explain when Spin ( p, q ) is a do uble- co v er of SO ( p, q ). The reader should be w arned that a certain amoun t of algebraic (and top ological) bac kground is exp ected. This being said, p ersev erant readers will be rew arded b y b eing expo sed to some beautiful and nontrivial concepts and results, including Elie Carta n and Raoul Bott “8-p erio dicit y theorem.” Going back to rotations as transformations induced by gr o up a ctions, recall that if V is a v ector space, a line ar action (on the left) of a gr oup G on V is a map α : G × V → V satisfying the follow ing conditions, where, for simplicit y of notation, w e denote α ( g , v ) by g · v : 7 8 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN (1) g · ( h · v ) = ( g h ) · v , f or all g , h ∈ G and v ∈ V ; (2) 1 · v = v , for all v ∈ V , where 1 is the iden tit y of the group G ; (3) The map v 7→ g · v is a linear isomorphism o f V for ev ery g ∈ G . F or example, the (multiplicativ e) g roup U (1) of unit complex n um b ers acts on R 2 (b y iden tif ying R 2 and C ) via complex m ultiplication: F or ev ery z = a + ib (with a 2 + b 2 = 1), for every ( x, y ) ∈ R 2 (viewing ( x, y ) as the complex num b er x + iy ), z · ( x, y ) = ( ax − by , ay + bx ) . No w, ev ery unit complex n um b er is of the form cos θ + i sin θ , and th us the abov e a ction of z = cos θ + i sin θ on R 2 corresp onds to the rota t ion of angle θ around the orig in. In the case n = 2, the groups U (1) and SO (2) are isomorphic, but this is an exception. T o represen t ro tations in R 3 and R 4 , we need the quaternions. F or our purp oses, it is con v enient to deﬁne the quaternions as ce rtain 2 × 2 complex matrices. Let 1 , i , j , k b e the matrices 1 =  1 0 0 1  , i =  i 0 0 − i  , j =  0 1 − 1 0  , k =  0 i i 0  , and let H b e the set of all ma t r ices o f the form X = a 1 + b i + c j + d k , a, b, c, d ∈ R . Th us, ev ery matrix in H is of the for m X =  a + ib c + id − ( c − id ) a − ib  , a, b, c, d ∈ R . The quaternions 1 , i , j , k satisfy the famous iden tities disc ov ered by Hamilton: i 2 = j 2 = k 2 = ijk = − 1 , ij = − ji = k , jk = − kj = i , ki = − ik = j . As a consequence, it can b e v eriﬁed that H is a sk ew ﬁeld (a no ncomm utative ﬁeld) called the quaternions . It is also a real v ector space o f dimension 4 with basis ( 1 , i , j , k ); thus as a v ector space, H is isomorphic to R 4 . The unit quaternions are the quaternions suc h that det( X ) = a 2 + b 2 + c 2 + d 2 = 1 . Giv en an y quaternion X = a 1 + b i + c j + d k , the c onjugate X of X is giv en by X = a 1 − b i − c j − d k . 1.2. CLIFFORD ALGEBRAS 9 It is easy to c hec k that the mat rices asso ciated with the unit quaternions are exactly the matrices in SU (2). Th us, w e call SU (2) the group of unit quaternions. No w w e can deﬁne an action of t he g roup of unit quaternions SU ( 2) on R 3 . F or this, w e use the fact that R 3 can b e iden tiﬁed with the pure quaternions in H , namely , the quaternions of the form x 1 i + x 2 j + x 3 k , where ( x 1 , x 2 , x 3 ) ∈ R 3 . Then, we deﬁne the a ction of SU (2) o v er R 3 b y Z · X = Z X Z − 1 = Z X Z , where Z ∈ SU (2) and X is any pure quaternion. No w, it turns out that the map ρ Z (where ρ Z ( X ) = Z X Z ) is indeed a rota t ion, and that the map ρ : Z 7→ ρ Z is a surjectiv e homomor- phism ρ : SU (2) → SO (3) wh ose k ernel is {− 1 , 1 } , where 1 denotes the multiplic ativ e unit quaternion. (F or details, see Gallier [1 6], Chapter 8). W e can also deﬁne an action of the gr o up SU ( 2) × SU (2) ov er R 4 , b y iden tifying R 4 with the quaternions. In this case, ( Y , Z ) · X = Y X Z , where ( Y , Z ) ∈ SU (2) × SU (2) and X ∈ H is an y quaternion. Then, t he map ρ Y , Z is a rota tion (where ρ Y , Z ( X ) = Y X Z ), a nd the map ρ : ( Y , Z ) 7→ ρ Y , Z is a surjectiv e homomorphism ρ : SU (2) × SU (2) → SO (4) whose ke rnel is { ( 1 , 1 ) , ( − 1 , − 1 ) } . ( F or details, see Ga llier [16], Chapter 8 ). Th us, we o bserv e that for n = 2 , 3 , 4, the rotations in SO ( n ) can b e realized via the linear action of some group (the case n = 1 is trivial, since SO (1) = { 1 , − 1 } ). It is also the case that the action of each group can b e someho w b e described in terms of m ultiplication in some lar g er algebra “ containing” the original v ector space R n ( C fo r n = 2 , H for n = 3 , 4). Ho w ev er, these groups app ear to hav e b een discov ered in a n ad ho c fashion, and there do es not app ear to b e an y univ ersal w ay to deﬁne the action of these groups o n R n . It would certainly b e nice if t he action w as alwa ys of the form Z · X = Z X Z − 1 (= Z X Z ) . A systematic w ay o f constructing groups realizing rotations in terms of linear action, using a uniform notion of action, do es exist. Suc h groups are t he spinor groups, to b e des crib ed in the follo wing sections. 1.2 Cliﬀord Alg e bras W e explained in Section 1.1 how the ro t ations in SO (3) can b e realized by the linear a ction o f the group of unit quaternions SU (2) on R 3 , and ho w the rota tions in SO (4) can b e realized b y the linear action of the group SU (2) × SU (2) on R 4 . The main reasons why the r o tations in SO (3) can b e represen ted b y unit quaternions are the follo wing: 10 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN (1) F or ev ery nonzero v ector u ∈ R 3 , the reﬂection s u ab out the hyperplane p erp endicular to u is represen ted b y the map v 7→ − uv u − 1 , where u and v are view ed as pure quaternions in H ( i . e . , if u = ( u 1 , u 2 , u 2 ), then view u a s u 1 i + u 2 j + u 3 k , and similarly for v ). (2) The group SO (3) is generated b y the reﬂections. As one can imagine, a successful generalization of the quaternions, i.e. , the disco very of a g r o up G inducing the rot a tions in SO ( n ) via a linear action, dep ends o n the abilit y to generalize prop erties (1) and (2 ) ab o v e. F ortuna t ely , it is true that the group SO ( n ) is generated b y t he hyperplane reﬂections. In fact, this is also true for the orthogonal group O ( n ), and more generally fo r the gro up of direct isometries O (Φ) of any nondegenerate quadratic fo rm Φ, b y t he Cartan-D i e udonn´ e the or em (for instance, see Bourbaki [6], or Gallier [16], Chapter 7, Theorem 7.2 .1). In order to generalize (2), we ne ed to understand ho w the group G acts on R n . No w, the case n = 3 is sp ecial, b ecause the underlying space R 3 on whic h the rotations act can b e embedded as the pure quaternions in H . The case n = 4 is also special, b ecause R 4 is the underlyin g space of H . The generalization to n ≥ 5 requires mo r e machinery , namely , the notions of Cliﬀord groups and Cliﬀord algebras. As w e will see, for eve ry n ≥ 2, there is a compact, connected (and simply connected when n ≥ 3) group Spin ( n ), the “spinor group,” and a surjectiv e homo mo r phism ρ : Spin ( n ) → SO ( n ) whose k ernel is {− 1 , 1 } . This time, Spin ( n ) acts directly o n R n , b ecause Spin ( n ) is a certain subgroup o f the group of units of the Cliﬀor d algebr a Cl n , and R n is naturally a subspace of Cl n . The group of unit quaternions SU (2) turns out to b e isomor phic to the spinor group Spin (3). Because Spin (3) a cts directly on R 3 , the represen tation of rotations in SO (3) b y elemen ts o f Spin (3) may b e view ed as more natural than the represen tation by unit quaternions. The group SU (2) × SU (2) turns out to b e isomorphic to the spinor group Spin (4), but this isomorphism is less ob vious. In summary , w e are going to deﬁne a gr o up Spin ( n ) represen ting the rotations in SO ( n ), for an y n ≥ 1, in the sense that there is a linear action of Spin ( n ) on R n whic h induces a surjectiv e homomorphism ρ : Spin ( n ) → SO ( n ) whose k ernel is {− 1 , 1 } . F urthermore, the action of Spin ( n ) on R n is g iv en in terms of m ultiplication in an algebra Cl n con taining Spin ( n ), and in w hic h R n is a lso em b edded. It turns out that as a b onus, for n ≥ 3, t he group Spin ( n ) is top olog ically simpler than SO ( n ), since Spin ( n ) is simply connected, but SO ( n ) is not. By b eing astute, w e can also construct a group Pin ( n ) and a linear action of Pin ( n ) on R n that induces a surjectiv e homomorphism ρ : Pin ( n ) → O ( n ) whose k ernel is {− 1 , 1 } . The diﬃcult y here is the presence of the negative sign in (2). W e will see how A tiy ah, Bott and Shapiro circum ve n t this problem by using a “ twisted adjoint action,” as opp osed to the usual adjoint action (where v 7→ uv u − 1 ). 1.2. CLIFFORD ALGEBRAS 11 Our presen ta t ion is hea vily inﬂuenced b y Br¨ ock er and tom D iec k [7] (Chapter 1 , Section 6), where most details can b e found. This Chapter is almost en tirely ta k en from the ﬁrst 11 pages of the b eautiful and seminal pap er b y A tiy ah, Bot t and Shapiro [3], Cliﬀord Mo dules, and w e highly recommend it. Another excellen t (but concise) exp osition can b e found in Kirillo v [18]. A ve ry tho r o ugh exp osition can b e found in t w o places: 1. Lawson and Mic helsohn [20 ], where the material on Pin ( p, q ) and Spin ( p, q ) can b e found in Chapter I. 2. Lo unesto’s excellen t bo ok [21]. One may also w a n t to consult Ba k er [4], Curtis [12], P o rteous [24], F ulton and Harris (Lecture 20) [15], Cho quet-Bruhat [11], Bourba ki [6], and Chev alley [10], a classic. The original source is Elie Cartan’s b o o k (1937) whose translation in English app ears in [8 ]. W e begin b y recalling what is an algebra o ve r a ﬁeld. Let K denote an y (comm utative ) ﬁeld, although fo r our purp oses w e ma y assume that K = R (and o ccasionally , K = C ). Since we will only b e dealing with a sso ciative algebras with a m ultiplicativ e unit, w e only deﬁne algebras of this kind. Deﬁnition 1.1. Giv en a ﬁeld K , a K -algebr a is a K -v ector space A together with a bilinear op eration · : A × A → A , called m ultiplic ation , whic h makes A in to a ring with unit y 1 (or 1 A , when we w ant to b e v ery precise). This means that · is asso ciativ e and that there is a multiplicativ e iden t ity elemen t 1 so that 1 · a = a · 1 = a , for all a ∈ A . G iven t w o K -algebras A and B , a K -algebr a homomorphism h : A → B is a linear map that is also a ring homomorphism, with h (1 A ) = 1 B . F or example, the ring M n ( K ) o f all n × n mat rices o v er a ﬁeld K is a K -algebra. There is an obvious notion of ide al o f a K -algebra: An ideal A ⊆ A is a linear subspace of A that is also a t w o-sided ideal with resp ect to m ultiplication in A . If the ﬁeld K is understo o d, w e usually simply s ay an algebra instead of a K -algebra. W e will also need a quic k review of tensor pro ducts. The basic idea is that tensor pro ducts allo w us to view mu ltilinear maps as linear maps. The maps b ecome simpler, but the spaces (pro duct spaces) become more complicated (tensor pro ducts). F or more details, see A tiy ah and Macdonald [2]. Deﬁnition 1.2. Give n t w o K -v ector spaces E and F , a tenso r pr o duct of E and F is a pair ( E ⊗ F , ⊗ ), where E ⊗ F is a K -v ector space and ⊗ : E × F → E ⊗ F is a bilinear map, so that for ev ery K -ve ctor space G and ev ery bilinear map f : E × F → G , there is a unique linear map f ⊗ : E ⊗ F → G with f ( u, v ) = f ⊗ ( u ⊗ v ) for a ll u ∈ E and all v ∈ V , 12 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN as in the diagram b elo w: E × F ⊗ / / f % % ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ E ⊗ F f ⊗   G The ve ctor space E ⊗ F is deﬁned up to isomorphism. The vectors u ⊗ v , where u ∈ E and v ∈ F , g enerate E ⊗ F . Remark: W e should really denote the tensor pro duct of E and F b y E ⊗ K F , since it dep ends o n the ﬁeld K . Since w e usually deal with a ﬁxed ﬁeld K , we use the simpler notation E ⊗ F . W e hav e na t ur a l isomorphisms ( E ⊗ F ) ⊗ G ≈ E ⊗ ( F ⊗ G ) and E ⊗ F ≈ F ⊗ E . Giv en t w o linear maps f : E → F and g : E ′ → F ′ , w e hav e a unique bilinear map f × g : E × E ′ → F × F ′ so that ( f × g ) ( a, a ′ ) = ( f ( a ) , g ( a ′ )) for all a ∈ E and all a ′ ∈ E ′ . Th us, we ha v e the bilinear map ⊗ ◦ ( f × g ) : E × E ′ → F ⊗ F ′ , and so, there is a unique linear map f ⊗ g : E ⊗ E ′ → F ⊗ F ′ so that ( f ⊗ g ) ( a ⊗ a ′ ) = f ( a ) ⊗ g ( a ′ ) for all a ∈ E and all a ′ ∈ E ′ . Let us now assume that E a nd F ar e K -alg ebras. W e w a n t to mak e E ⊗ F into a K - algebra. Since the m ultiplication op erations m E : E × E → E and m F : F × F → F are bilinear, w e get linear maps m ′ E : E ⊗ E → E and m ′ F : F ⊗ F → F , and th us the linear map m ′ E ⊗ m ′ F : ( E ⊗ E ) ⊗ ( F ⊗ F ) → E ⊗ F . Using t he isomorphism τ : ( E ⊗ E ) ⊗ ( F ⊗ F ) → ( E ⊗ F ) ⊗ ( E ⊗ F ), w e get a linear map m E ⊗ F : ( E ⊗ F ) ⊗ ( E ⊗ F ) → E ⊗ F , whic h deﬁnes a multiplic ation m on E ⊗ F (namely , m ( u, v ) = m E ⊗ F ( u ⊗ v )). It is easily c heck ed that E ⊗ F is indeed a K -algebra under the m ultiplication m . Using the simpler notation · f or m , we hav e ( a ⊗ a ′ ) · ( b ⊗ b ′ ) = ( ab ) ⊗ ( a ′ b ′ ) for a ll a, b ∈ E and a ll a ′ , b ′ ∈ F . Giv en any v ector space V ov er a ﬁeld K , there is a sp ecial K -algebra T ( V ) together with a linear map i : V → T ( V ), with the following univ ersal mapping prop ert y: Giv en an y 1.2. CLIFFORD ALGEBRAS 13 K -algebra A , for an y linear map f : V → A , there is a unique K - a lgebra homomorphism f : T ( V ) → A so tha t f = f ◦ i, as in the diagram b elo w: V i / / f " " ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ T ( V ) f   A The algebra T ( V ) is the tensor algebr a of V . The algebra T ( V ) may b e constructed as the direct sum T ( V ) = M i ≥ 0 V ⊗ i , where V 0 = K , and V ⊗ i is t he i -fold tensor pro duct of V with itself. F or ev ery i ≥ 0, there is a natural injection ι n : V ⊗ n → T ( V ), and in particular, an injection ι 0 : K → T ( V ). The m ult iplicative unit 1 of T ( V ) is the image ι 0 (1) in T ( V ) of the unit 1 o f the ﬁe ld K . Since ev ery v ∈ T ( V ) can b e expressed a s a ﬁnite sum v = v 1 + · · · + v k , where v i ∈ V ⊗ n i and the n i are natural n um b ers with n i 6 = n j if i 6 = j , to deﬁne m ultiplication in T ( V ), using bilinearity , it is enough to deﬁne the m ultiplication V ⊗ m × V ⊗ n − → V ⊗ ( m + n ) . Of course, this is deﬁned b y ( v 1 ⊗ · · · ⊗ v m ) · ( w 1 ⊗ · · · ⊗ w n ) = v 1 ⊗ · · · ⊗ v m ⊗ w 1 ⊗ · · · ⊗ w n . (This has to b e made rig orous b y using isomorphisms inv olving the asso ciativit y of tensor pro ducts; for details, see see A tiyah and Macdonald [2].) The algebra T ( V ) is an ex ample of a gr ad e d alge br a , where t he ho mo gene ous ele m ents of r an k n are t he elemen ts in V ⊗ n . Remark: It is imp ortant to no t e t ha t m ultiplication in T ( V ) is not comm utative . Also, in all rigor, the unit 1 of T ( V ) is not equal to 1, the unit of the ﬁeld K . How ev er, in view of the injection ι 0 : K → T ( V ), for the sak e of notational simplicit y , w e will denote 1 by 1. More generally , in view of the injections ι n : V ⊗ n → T ( V ), we identify elemen ts of V ⊗ n with their images in T ( V ). Most algebras of intere st arise a s well-c hosen quotien ts of the tensor algebra T ( V ). This is true for the exterior algebr a V • V (also called Gr assmann algebr a ), where w e tak e the quotien t of T ( V ) mo dulo the ideal generated by all elemen ts of the form v ⊗ v , where v ∈ V , and for the s ymm etric algebr a Sym V , where w e take the quotien t of T ( V ) mo dulo the ideal generated by a ll elemen ts o f the form v ⊗ w − w ⊗ v , where v , w ∈ V . A Cliﬀord algebra ma y b e view ed as a reﬁnemen t of t he exterior algebra, in whic h w e tak e the quotien t of T ( V ) mo dulo the ideal g enerated b y all elemen ts of the form v ⊗ v − Φ( v ) · 1, where Φ is the quadratic form asso ciated with a symmetric bilinear form ϕ : V × V → K , 14 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN and · : K × T ( V ) → T ( V ) denotes the scalar pro duct of the algebra T ( V ). F or simplicit y , let us a ssume that w e are no w dealing with real algebras. Deﬁnition 1.3. Let V b e a real ﬁnite-dimensional v ector space to gether with a sy mmetric bilinear form ϕ : V × V → R and asso ciated quadratic form Φ( v ) = ϕ ( v , v ). A C l i ﬀ or d algebr a asso c i a te d w i th V and Φ is a real a lgebra Cl( V , Φ) together with a linear map i Φ : V → Cl( V , Φ) satisfying the condition ( i Φ ( v )) 2 = Φ( v ) · 1 for all v ∈ V , and so that for ev ery real algebra A and ev ery linear map f : V → A with ( f ( v )) 2 = Φ( v ) · 1 for all v ∈ V , there is a uniq ue alg ebra homomorphism f : Cl( V , Φ) → A so tha t f = f ◦ i Φ , as in the diagram b elo w: V i Φ / / f $ $ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ Cl( V , Φ) f   A W e use the nota tion λ · u for the pro duct of a scalar λ ∈ R a nd of an elemen t u in the algebra Cl( V , Φ), and juxtap osition u v for the multiplication of tw o elemen ts u and v in the alg ebra Cl( V , Φ). By a familiar arg ument, any tw o Cliﬀord algebras asso ciated with V and Φ are isomorphic. W e o ften denote i Φ b y i . T o sho w the existence of Cl( V , Φ), observ e that T ( V ) / A do es the job, where A is the ideal of T ( V ) generated by a ll elemen ts of the form v ⊗ v − Φ( v ) · 1, where v ∈ V . The map i Φ : V → Cl ( V , Φ) is the comp osition V ι 1 − → T ( V ) π − → T ( V ) / A , where π is the natural quotien t map. W e often denote t he Cliﬀord algebra Cl( V , Φ) simply b y Cl(Φ). Remark: Observ e that Deﬁnition 1.3 do es not assert that i Φ is injectiv e or that there is an injection of R in to Cl( V , Φ), but w e will prov e la t er that b oth facts are true when V is ﬁnite-dimensional. Also, as in t he case o f the tensor algebra, the unit of the algebra Cl( V , Φ) and the unit of the ﬁeld R are not equal . Since Φ( u + v ) − Φ( u ) − Φ( v ) = 2 ϕ ( u , v ) and ( i ( u + v )) 2 = ( i ( u )) 2 + ( i ( v )) 2 + i ( u ) i ( v ) + i ( v ) i ( u ) , 1.2. CLIFFORD ALGEBRAS 15 using t he fa ct that i ( u ) 2 = Φ( u ) · 1 , w e get i ( u ) i ( v ) + i ( v ) i ( u ) = 2 ϕ ( u, v ) · 1 . As a consequence, if ( u 1 , . . . , u n ) is an o r t hogonal ba sis w.r.t. ϕ (whic h means that ϕ ( u j , u k ) = 0 for all j 6 = k ), we ha v e i ( u j ) i ( u k ) + i ( u k ) i ( u j ) = 0 for all j 6 = k . Remark: Certain authors drop the unit 1 of the Cliﬀord algebra Cl( V , Φ) when writing the iden tities i ( u ) 2 = Φ( u ) · 1 and 2 ϕ ( u, v ) · 1 = i ( u ) i ( v ) + i ( v ) i ( u ) , where the second iden tit y is often written as ϕ ( u, v ) = 1 2 ( i ( u ) i ( v ) + i ( v ) i ( u )) . This is v ery confusing and tec hnically wrong, b ecause we only hav e an injection o f R into Cl( V , Φ), but R is not a subset of Cl( V , Φ).  W e w arn the readers that Law son and Mic helsohn [20] adopt the opp osite of our sign con v ention in deﬁning Cliﬀord algebras, i.e. , they use the condition ( f ( v )) 2 = − Φ( v ) · 1 fo r all v ∈ V . The most confusing consequence of this is that their Cl( p, q ) is o ur Cl( q , p ). Observ e that when Φ ≡ 0 is the quadratic form iden tically zero ev erywhere, then the Cliﬀord algebra Cl( V , 0) is just the exterior a lgebra V • V . Example 1.1. Let V = R , e 1 = 1, and a ssume that Φ( x 1 e 1 ) = − x 2 1 . Then, Cl(Φ) is spanned b y the ba sis (1 , e 1 ). W e ha v e e 2 1 = − 1 . Under the bijection e 1 7→ i, the Cliﬀord algebra Cl( Φ), also denoted by Cl 1 , is isomorphic t o the algebra of complex n um b ers C . No w, let V = R 2 , ( e 1 , e 2 ) b e the cano nical basis, and assume that Φ( x 1 e 1 + x 2 e 2 ) = − ( x 2 1 + x 2 2 ). Then, Cl(Φ) is spanned by the basis (1 , e 1 , e 2 , e 1 e 2 ). F urthermore, we ha v e e 2 e 1 = − e 1 e 2 , e 2 1 = − 1 , e 2 2 = − 1 , ( e 1 e 2 ) 2 = − 1 . 16 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN Under the bijection e 1 7→ i , e 2 7→ j , e 1 e 2 7→ k , it is easily chec k ed that the quaternion iden tities i 2 = j 2 = k 2 = − 1 , ij = − ji = k , jk = − kj = i , ki = − ik = j , hold, and th us the Cliﬀord algebra Cl(Φ), also denoted by Cl 2 , is isomorphic t o the alg ebra of quaternions H . Our prime goal is t o deﬁne an action of Cl(Φ) on V in suc h a w ay that b y restricting this action to some suitably c hosen m ultiplicative subgroups of Cl(Φ), we get surjectiv e homomorphisms on to O (Φ) and SO (Φ), resp ectiv ely . The key point is that a reﬂec tion in V ab out a h yp erplane H orthogonal to a v ector w can b e deﬁned b y suc h an a ctio n, but some negativ e sign shows up. A correct handling of signs is a bit subtle and require s t he in tro duction o f a canonical an ti-automor phism t , and of a canonical automorphism α , deﬁned as follo ws: Prop osition 1.1. Every Cli ﬀ or d algebr a Cl(Φ) p ossesses a c anonic al anti-automorp h ism t : Cl(Φ) → Cl (Φ) sa tisfying the pr op erties t ( xy ) = t ( y ) t ( x ) , t ◦ t = id , and t ( i ( v )) = i ( v ) , for al l x, y ∈ Cl(Φ) and al l v ∈ V . F urthermor e, such an anti-automorph ism is unique. Pr o of. Consider t he opp osite alg ebra Cl(Φ) o , in whic h the pro duct of x a nd y is giv en b y y x . It has the univ ersal mapping prop erty . Thus , w e get a unique isomorphism t , as in the diagram b elo w: V i / / i # # ● ● ● ● ● ● ● ● ● ● Cl( V , Φ) t   Cl(Φ) o W e also denote t ( x ) b y x t . When V is ﬁnite-dimensional, fo r a mor e palat a ble description of t in terms of a basis of V , see t he para graph fo llo wing Theorem 1.4. The canonical automorphism α is deﬁned using the prop osition Prop osition 1.2. Every Cliﬀor d alge br a Cl( Φ) has a uniq ue c anonic a l automorphism α : Cl ( Φ) → Cl(Φ) satisfying the pr op erties α ◦ α = id , and α ( i ( v )) = − i ( v ) , for al l v ∈ V . 1.2. CLIFFORD ALGEBRAS 17 Pr o of. Consider t he linear map α 0 : V → Cl(Φ) deﬁned b y α 0 ( v ) = − i ( v ), for all v ∈ V . W e get a unique homomorphism α as in the diagram b elo w: V i / / α 0 # # ● ● ● ● ● ● ● ● ● ● Cl( V , Φ) α   Cl(Φ) F urthermore, ev ery x ∈ Cl(Φ) can b e written as x = x 1 · · · x m , with x j ∈ i ( V ), and since α ( x j ) = − x j , we get α ◦ α = id . It is clear that α is bijectiv e. Again, when V is ﬁnite-dimensional, a more palatable description of α in terms of a basis of V can b e g iven. If ( e 1 , . . . , e n ) is a ba sis of V , then the Cliﬀord algebra Cl(Φ) consists of certain kinds of “ p olynomials,” linear com binat ions of monomials of the f o rm P J λ J e J , where J = { i 1 , i 2 , . . . , i k } is a n y subset (p ossibly empt y) o f { 1 , . . . , n } , with 1 ≤ i 1 < i 2 · · · < i k ≤ n , and the monomial e J is the “pro duct” e i 1 e i 2 · · · e i k . The map α is the linear map deﬁned on monomials b y α ( e i 1 e i 2 · · · e i k ) = ( − 1) k e i 1 e i 2 · · · e i k . F or a more rig orous explanation, se e t he par agraph f o llo wing Theorem 1.4 . W e now show that if V has dimension n , then i is injectiv e and Cl(Φ) has dimension 2 n . A clev er w a y of doing this is to introduce a graded tensor pro duct. First, observ e that Cl(Φ) = Cl 0 (Φ) ⊕ Cl 1 (Φ) , where Cl i (Φ) = { x ∈ Cl( Φ) | α ( x ) = ( − 1) i x } , where i = 0 , 1 . W e sa y that w e ha v e a Z / 2 -gr ading , whic h means that if x ∈ Cl i (Φ) and y ∈ Cl j (Φ), then xy ∈ Cl i + j (mo d 2) (Φ). When V is ﬁnite-dimensional, since ev ery elemen t o f Cl (Φ) is a linear com binat io n of the form P J λ J e J as e xplained earlier, in view of the desc ription of α giv en ab o v e, w e s ee that the elemen ts of Cl 0 (Φ) ar e those fo r whic h the monomials e J are pro ducts of an ev en n umber of factors, and t he elemen ts of Cl 1 (Φ) are those fo r whic h the mono mials e J are pro ducts of an o dd n um b er of factors. Remark: Observ e that Cl 0 (Φ) is a subalgebra of Cl(Φ), whereas Cl 1 (Φ) is not. Giv en tw o Z / 2-graded algebras A = A 0 ⊕ A 1 and B = B 0 ⊕ B 1 , t heir gr ade d tensor pr o duct A b ⊗ B is deﬁned by ( A b ⊗ B ) 0 = ( A 0 ⊗ B 0 ) ⊕ ( A 1 ⊗ B 1 ) , ( A b ⊗ B ) 1 = ( A 0 ⊗ B 1 ) ⊕ ( A 1 ⊗ B 0 ) , 18 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN with multiplication ( a ′ ⊗ b )( a ⊗ b ′ ) = ( − 1) ij ( a ′ a ) ⊗ ( bb ′ ) , for a ∈ A i and b ∈ B j . The reader should ch ec k that A b ⊗ B is indeed Z / 2- graded. Prop osition 1.3. L e t V and W b e ﬁnite dimensiona l ve c tor sp ac es with quadr atic form s Φ and Ψ . T hen, ther e is a quadr atic form Φ ⊕ Ψ on V ⊕ W deﬁne d by (Φ + Ψ)( v , w ) = Φ( v ) + Ψ( w ) . If w e write i : V → Cl(Φ) and j : W → Cl(Ψ) , we c an deﬁne a l i n e ar map f : V ⊕ W → Cl (Φ) b ⊗ Cl(Ψ) by f ( v , w ) = i ( v ) ⊗ 1 + 1 ⊗ j ( w ) . F urthermor e, the map f induc es an isomorphism (also denote d by f ) f : Cl( V ⊕ W ) → Cl(Φ) b ⊗ Cl(Ψ) . Pr o of. See Br ¨ o c ke r and tom Diec k [7], Chapter 1 , Section 6, pag e 57. As a corollary , w e obtain the fo llo wing result: Theorem 1.4. F or every ve c tor sp ac e V of ﬁnite dimension n , the map i : V → Cl(Φ) is inje ctive. Given a b asis ( e 1 , . . . , e n ) of V , the 2 n − 1 pr o ducts i ( e i 1 ) i ( e i 2 ) · · · i ( e i k ) , 1 ≤ i 1 < i 2 · · · < i k ≤ n, and 1 form a b asis of Cl(Φ) . Thus, Cl(Φ) has dimension 2 n . Pr o of. The pro of is by induction on n = dim( V ). F o r n = 1, the t ensor a lg ebra T ( V ) is just the p olynomial ring R [ X ], where i ( e 1 ) = X . Th us, Cl( Φ) = R [ X ] / ( X 2 − Φ( e 1 )), a nd the result is ob vious. Since i ( e j ) i ( e k ) + i ( e k ) i ( e j ) = 2 ϕ ( e i , e j ) · 1 , it is clear that the pro ducts i ( e i 1 ) i ( e i 2 ) · · · i ( e i k ) , 1 ≤ i 1 < i 2 · · · < i k ≤ n, and 1 generate Cl(Φ). Now, there is alw a ys a basis that is orthog onal with resp ect to ϕ (fo r example, see Artin [1], Chapter 7, or Gallier [16], Chapter 6, Problem 6.14), and th us, w e ha v e a splitting ( V , Φ) = n M k =1 ( V k , Φ k ) , where V k has dimension 1. Choosing a ba sis so that e k ∈ V k , t he theorem follo ws by induction from Prop osition 1.3. 1.3. CLIFFORD GROUPS 19 Since i is injectiv e, fo r simplicit y of nota tion, from now on w e write u for i ( u ) . Theorem 1.4 implies that if ( e 1 , . . . , e n ) is an ort hogonal basis of V , then Cl (Φ) is the a lg ebra presen ted b y the g enerators ( e 1 , . . . , e n ) and the relat io ns e 2 j = Φ( e j ) · 1 , 1 ≤ j ≤ n, and e j e k = − e k e j , 1 ≤ j, k ≤ n, j 6 = k . If V has ﬁnite dimension n and ( e 1 , . . . , e n ) is a basis of V , b y Theorem 1.4, the ma ps t and α are completely determined b y their action on the ba sis elemen ts. Na mely , t is deﬁned b y t ( e i ) = e i t ( e i 1 e i 2 · · · e i k ) = e i k e i k − 1 · · · e i 1 , where 1 ≤ i 1 < i 2 · · · < i k ≤ n , and of course, t (1) = 1. The map α is deﬁned by α ( e i ) = − e i α ( e i 1 e i 2 · · · e i k ) = ( − 1) k e i 1 e i 2 · · · e i k where 1 ≤ i 1 < i 2 · · · < i k ≤ n , and of course, α (1) = 1. F urt hermore, the ev en-graded elemen ts (the elemen ts of Cl 0 (Φ)) ar e those generated by 1 and the basis elemen ts consisting of an ev en n um b er of f a ctors e i 1 e i 2 · · · e i 2 k , a nd the o dd-graded elemen ts (the elemen ts of Cl 1 (Φ)) are those generated b y the basis eleme n ts consisting of an o dd num b er of factors e i 1 e i 2 · · · e i 2 k +1 . W e a re now ready to deﬁne the Cliﬀord group a nd in v estigate some of its prop erties. 1.3 Cliﬀord Grou ps First, w e deﬁne c onjugation on a Cliﬀord algebra Cl(Φ) as the map x 7→ x = t ( α ( x )) for all x ∈ Cl(Φ) . Observ e that t ◦ α = α ◦ t. If V has ﬁnite dimension n a nd ( e 1 , . . . , e n ) is a ba sis of V , in view of previous remarks, conjugation is deﬁned b y e i = − e i e i 1 e i 2 · · · e i k = ( − 1) k e i k e i k − 1 · · · e i 1 where 1 ≤ i 1 < i 2 · · · < i k ≤ n , and of course, 1 = 1. Conjugatio n is an an ti-auto morphism. 20 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN The m ultiplicativ e group of in v ertible elemen ts of Cl(Φ) is de noted b y Cl(Φ) ∗ . Observ e that f o r any x ∈ V , if Φ( x ) 6 = 0, then x is in v ertible because x 2 = Φ( x ); that is, x ∈ Cl(Φ) ∗ . W e would like Cl(Φ) ∗ to act on V via x · v = α ( x ) v x − 1 , where x ∈ Cl(Φ) ∗ and v ∈ V . In general, there is no reason why α ( x ) v x − 1 should b e in V or wh y this action deﬁnes an automorphism of V , so w e restrict this map to the subset Γ(Φ) of Cl (Φ) ∗ as follo ws. Deﬁnition 1.4. Giv en a ﬁnite dimensional v ector space V and a quadratic form Φ on V , the Cliﬀor d gr oup of Φ is the group Γ(Φ) = { x ∈ Cl ( Φ) ∗ | α ( x ) v x − 1 ∈ V for all v ∈ V } . The map N : C l( Q ) → Cl( Q ) giv en by N ( x ) = x x is called the no rm of Cl(Φ). F or an y x ∈ Γ ( Φ), let ρ x : V → V b e the map deﬁned by v 7→ α ( x ) v x − 1 , v ∈ V . It is no t entirely obvious wh y the map ρ : Γ(Φ) → GL ( V ) giv en by x 7→ ρ x is a linear action, and for that matter, wh y Γ(Φ) is a group. This is b ecause V is ﬁnite-dimensional and α is an auto morphism. Pr o of. F o r any x ∈ Γ(Φ), the map ρ x from V to V deﬁned by v 7→ α ( x ) v x − 1 is clearly linear. If α ( x ) v x − 1 = 0, since by h yp othesis x is in v ertible a nd since α is an automorphism α ( x ) is also in v ertible, so v = 0. Th us our linear map is injectiv e, and since V has ﬁnite dimension, it is bijectiv e. T o prov e that x − 1 ∈ Γ ( Φ), pic k any v ∈ V . Since the linear map ρ x is bijective , there is some w ∈ V suc h that ρ x ( w ) = v , whic h means that α ( x ) w x − 1 = v . Since x is in v ertible and α is an automorphism, w e get α ( x − 1 ) v x = w , so α ( x − 1 ) v x ∈ V ; since this ho lds for any v ∈ V , we ha v e x − 1 ∈ Γ(Φ). Since α is an automorphism, if x, y ∈ Γ(Φ), for any v ∈ V w e ha v e ρ y ( ρ x ( v )) = α ( y ) α ( x ) v x − 1 y − 1 = α ( y x ) v ( y x ) − 1 = ρ y x ( v ) , whic h sho ws that ρ y x is a linear automo r phism o f V , so y x ∈ Γ(Φ) and ρ is a homomorphism. Therefore, Γ(Φ) is a group a nd ρ is a linear represen tation. 1.3. CLIFFORD GROUPS 21 W e a lso deﬁne the group Γ + (Φ), called t he sp e cial Cliﬀor d gr oup , by Γ + (Φ) = Γ(Φ) ∩ Cl 0 (Φ) . Observ e that N ( v ) = − Φ( v ) · 1 for a ll v ∈ V . Also, if ( e 1 , . . . , e n ) is a basis of V , we leav e it as an exercise to che c k that N ( e i 1 e i 2 · · · e i k ) = ( − 1) k Φ( e i 1 )Φ( e i 2 ) · · · Φ( e i k ) · 1 . Remark: The map ρ : Γ(Φ) → GL ( V ) given b y x 7→ ρ x is called the twiste d adjoint r ep- r esentation . It was in tro duced b y A tiy ah, Bott and Shapiro [3]. It has t he adv an tage o f not introducing a spurious negativ e sign, i.e. , w hen v ∈ V and Φ( v ) 6 = 0, the map ρ v is the reﬂection s v ab out the hy p erplane orthogonal to v (see Prop osition 1.6). F urthermore, when Φ is nondegenerate, the k ernel Ker ( ρ ) of the represen tation ρ is given b y Ker ( ρ ) = R ∗ · 1, where R ∗ = R − { 0 } . The earlier adjo int r epr esentation (used b y Chev alley [10] and o t hers) is g iv en b y v 7→ xv x − 1 . Unfortunately , in this case, ρ x represen ts − s v , where s v is the reﬂection ab o ut the h yp erplane orthogonal to v . F urthermore, the k ernel of the represen tation ρ is generally big g er than R ∗ · 1. This is the reason wh y the t wisted adjoint represen tation is preferred (and mus t b e used for a prop er treatmen t of the Pin group). Prop osition 1.5. T h e maps α and t induc e an automorphism and an anti-automorphis m of the C liﬀor d g r oup, Γ(Φ) . Pr o of. It is not v ery instructiv e; see Br¨ oc k er and to m Diec k [7], Chapter 1, Section 6, page 58. The follow ing prop o sition sho ws wh y Cliﬀord groups generalize the quaternions. Prop osition 1.6. L et V b e a ﬁnite dimension a l ve ctor sp ac e and Φ a quadr a tic form on V . F or every ele ment x of the Cliﬀor d gr oup Γ(Φ) , if x ∈ V and Φ( x ) 6 = 0 , then the map ρ x : V → V give n by v 7→ α ( x ) v x − 1 for al l v ∈ V is the r eﬂe ction ab out the hyp erplane H ortho gona l to the ve ctor x . Pr o of. Recall that the reﬂec tion s ab out the h yp erplane H orthog onal to the v ector x is giv en b y s ( u ) = u − 2 ϕ ( u, x ) Φ( x ) · x. Ho w ev er, w e ha v e x 2 = Φ( x ) · 1 and u x + xu = 2 ϕ ( u, x ) · 1 . 22 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN Th us, w e hav e s ( u ) = u − 2 ϕ ( u, x ) Φ( x ) · x = u − 2 ϕ ( u, x ) ·  1 Φ( x ) · x  = u − 2 ϕ ( u, x ) · x − 1 = u − 2 ϕ ( u, x ) · (1 x − 1 ) = u − (2 ϕ ( u, x ) · 1) x − 1 = u − ( ux + xu ) x − 1 = − xux − 1 = α ( x ) ux − 1 , since α ( x ) = − x , f or x ∈ V . Recall that the linear represen t a tion ρ : Γ(Φ) → GL ( V ) is g iv en b y ρ ( x )( v ) = α ( x ) v x − 1 , for a ll x ∈ Γ ( Φ) and all v ∈ V . W e would lik e to sho w that ρ is a surjectiv e homomorphism from Γ(Φ) onto O ( ϕ ) , and a surjectiv e homomorphism from Γ + (Φ) on to SO ( ϕ ). F or this, w e will need to assume that ϕ is nondegenerate, whic h means that for ev ery v ∈ V , if ϕ ( v , w ) = 0 for all w ∈ V , then v = 0. F or simplicit y of ex p osition, we ﬁrst ass ume that Φ is t he quadrat ic f o rm on R n deﬁned b y Φ( x 1 , . . . , x n ) = − ( x 2 1 + · · · + x 2 n ) . Let Cl n denote the Cliﬀord a lgebra Cl(Φ) and Γ n denote the Cliﬀord group Γ(Φ). The follo wing lemma play s a crucial r o le: Lemma 1.7. The kern e l of the map ρ : Γ n → GL ( n ) is R ∗ · 1 , the multiplic ative gr oup of nonzer o sc a l a r multiples of 1 ∈ Cl n . Pr o of. If ρ ( x ) = id, then α ( x ) v = v x for all v ∈ R n . (1) Since Cl n = Cl 0 n ⊕ Cl 1 n , w e can w rite x = x 0 + x 1 , with x i ∈ Cl i n for i = 0 , 1. Then, equation (1) b ecomes x 0 v = v x 0 and − x 1 v = v x 1 for a ll v ∈ R n . (2) Using Theorem 1.4, w e can express x 0 as a linear com bination o f monomials in t he canonical basis ( e 1 , . . . , e n ), so that x 0 = a 0 + e 1 b 1 , with a 0 ∈ Cl 0 n , b 1 ∈ Cl 1 n , 1.3. CLIFFORD GROUPS 23 where neither a 0 nor b 1 con tains a summand w ith a factor e 1 . Applying the ﬁrst relation in (2) t o v = e 1 , w e g et e 1 a 0 + e 2 1 b 1 = a 0 e 1 + e 1 b 1 e 1 . (3) No w, the basis ( e 1 , . . . , e n ) is orthogonal w.r.t. Φ, whic h implie s that e j e k = − e k e j for a ll j 6 = k . Since eac h monomial in a 0 is o f ev en degree and con tains no factor e 1 , w e get a 0 e 1 = e 1 a 0 . Similarly , since b 1 is o f o dd degree and contains no facto r e 1 , we get e 1 b 1 e 1 = − e 2 1 b 1 . But then, from ( 3 ), w e g et e 1 a 0 + e 2 1 b 1 = a 0 e 1 + e 1 b 1 e 1 = e 1 a 0 − e 2 1 b 1 , and so, e 2 1 b 1 = 0. How ev er, e 2 1 = − 1, and so, b 1 = 0. Therefore, x 0 con tains no monomial with a factor e 1 . W e can a pply the same argumen t to the other basis elemen ts e 2 , . . . , e n , and th us, w e just prov ed that x 0 ∈ R · 1. A similar arg ument applying to the second equation in (2), with x 1 = a 1 + e 1 b 0 and v = e 1 sho ws that b 0 = 0. W e also conclude that x 1 ∈ R · 1 . How ev er, R · 1 ⊆ Cl 0 n , and so x 1 = 0. Finally , x = x 0 ∈ ( R · 1) ∩ Γ n = R ∗ · 1. Remark: If Φ is any nondegenerate quadratic form, w e know (for instance, see Ar t in [1], Chapter 7, or Gallier [16], Chapter 6, Problem 6.14 ) that there is an ort ho gonal ba sis ( e 1 , . . . , e n ) with respect to ϕ ( i. e . ϕ ( e j , e k ) = 0 for all j 6 = k ) . Thus , the comm utation relations e 2 j = Φ( e j ) · 1 , with Φ( e j ) 6 = 0 , 1 ≤ j ≤ n, and e j e k = − e k e j , 1 ≤ j, k ≤ n, j 6 = k hold, and since the pro of only rests on these facts, Lemma 1.7 holds for any no ndegenerate quadratic form.  Ho w ev er, Lemma 1.7 ma y fail fo r degenerate quadratic forms. F or example, if Φ ≡ 0, then Cl( V , 0) = V • V . Consider the elem en t x = 1 + e 1 e 2 . Clearly , x − 1 = 1 − e 1 e 2 . But no w, for an y v ∈ V , w e hav e α (1 + e 1 e 2 ) v (1 + e 1 e 2 ) − 1 = ( 1 + e 1 e 2 ) v (1 − e 1 e 2 ) = v . Y et, 1 + e 1 e 2 is no t a scalar mu ltiple o f 1. 24 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN The follow ing prop o sition sho ws that the notion of norm is w ell-b eha v ed. Prop osition 1.8. If x ∈ Γ n , then N ( x ) ∈ R ∗ · 1 . Pr o of. The tr ic k is to sho w that N ( x ) is in the k ernel of ρ . T o say that x ∈ Γ n means that α ( x ) v x − 1 ∈ R n for a ll v ∈ R n . Applying t , w e get t ( x ) − 1 v t ( α ( x )) = α ( x ) v x − 1 , since t is the identit y o n R n . Th us, w e hav e v = t ( x ) α ( x ) v ( t ( α ( x )) x ) − 1 = α ( xx ) v ( xx ) − 1 , so xx ∈ Ker ( ρ ). By Prop osition 1.5, w e ha v e x ∈ Γ n , and so, xx = x x ∈ Ker ( ρ ). Remark: Again, the pro of also holds for the Cliﬀord gro up Γ(Φ) asso ciated with any non- degenerate quadratic form Φ. When Φ( v ) = − k v k 2 , where k v k is the standard Euclidean norm of v , we ha v e N ( v ) = k v k 2 · 1 for all v ∈ V . Ho w eve r, for other quadratic f orms, it is p ossible that N ( x ) = λ · 1 where λ < 0, and this is a diﬃcult y tha t needs to b e ov ercome. Prop osition 1.9. The r e s triction of the norm N to Γ n is a homomorphi s m N : Γ n → R ∗ · 1 , and N ( α ( x )) = N ( x ) for al l x ∈ Γ n . Pr o of. W e hav e N ( xy ) = xy y x = xN ( y ) x = xxN ( y ) = N ( x ) N ( y ) , where the third equalit y holds b ecause N ( x ) ∈ R ∗ · 1. W e also hav e N ( α ( x )) = α ( x ) α ( x ) = α ( xx ) = α ( N ( x )) = N ( x ) . Remark: The pro of also holds fo r the Cliﬀord gr oup Γ ( Φ) associated with an y nondegen- erate quadratic form Φ. Prop osition 1.10. We have R n − { 0 } ⊆ Γ n and ρ (Γ n ) ⊆ O ( n ) . Pr o of. Let x ∈ Γ n and v ∈ R n , with v 6 = 0. W e ha v e N ( ρ ( x )( v )) = N ( α ( x ) v x − 1 ) = N ( α ( x )) N ( v ) N ( x − 1 ) = N ( x ) N ( v ) N ( x ) − 1 = N ( v ) , since N : Γ n → R ∗ · 1. How ev er, for v ∈ R n , we kno w tha t N ( v ) = − Φ( v ) · 1 . Th us, ρ ( x ) is norm-preserving, a nd so, ρ ( x ) ∈ O ( n ). Remark: The pro of that ρ (Γ(Φ)) ⊆ O (Φ) a lso holds for the Cliﬀor d gro up Γ(Φ) a sso ciated with any nondegenerate quadratic form Φ. The ﬁrst stateme n t needs to b e replace d b y the fact that ev ery non-isotropic v ector in R n (a vec tor is non-isotropic if Φ( x ) 6 = 0) b elongs to Γ(Φ). Indeed, x 2 = Φ( x ) · 1, whic h implies that x is in v ertible. W e a re ﬁnally ready fo r the in tro duction of the groups Pin ( n ) and Spin ( n ). 1.4. THE GROUPS PIN ( N ) AND SP IN ( N ) 25 1.4 The Grou p s Pin ( n ) and Spi n ( n ) Deﬁnition 1.5. W e deﬁne the pinor gr oup P in ( n ) as t he k ernel Ker ( N ) of the homomo r - phism N : Γ n → R ∗ · 1, and the spinor gr oup Spin ( n ) a s P in ( n ) ∩ Γ + n . Observ e that if N ( x ) = 1, t hen x is in v ertible, and x − 1 = x since xx = N ( x ) = 1. Thus , w e can write Pin ( n ) = { x ∈ Cl n | α ( x ) v x − 1 ∈ R n for a ll v ∈ R n , N ( x ) = 1 } = { x ∈ Cl n | α ( x ) v x ∈ R n for a ll v ∈ R n , xx = 1 } , and Spin ( n ) = { x ∈ Cl 0 n | xv x − 1 ∈ R n for a ll v ∈ R n , N ( x ) = 1 } = { x ∈ Cl 0 n | xv x ∈ R n for a ll v ∈ R n , xx = 1 } Remark: According to Atiy ah, Bott and Shapiro, the use of the name Pin ( k ) is a jok e due to Jean- Pierre Serre (A tiy ah, Bo tt and Shapiro [3], pa ge 1). Theorem 1.11. The r estriction of ρ : Γ n → O ( n ) to the pinor gr oup Pin ( n ) is a surje ctive homomorphism ρ : Pin ( n ) → O ( n ) who s e kernel is {− 1 , 1 } , and the r estriction of ρ to the spinor gr oup Spin ( n ) is a surje ctive homomorphis m ρ : Spin ( n ) → SO ( n ) whose kernel is {− 1 , 1 } . Pr o of. By Prop o sition 1.10 , we ha v e a map ρ : Pin ( n ) → O ( n ). The reader can easily c hec k that ρ is a homomorphism. By the Carta n-Dieudonn ´ e theorem (see Bourbaki [6], or Gallier [16], Chapter 7 , Theorem 7.2 .1 ), eve ry isometry f ∈ SO ( n ) is the comp osition f = s 1 ◦ · · · ◦ s k of hyperplane reﬂections s j . If w e assume that s j is a reﬂection ab out the h yp erplane H j orthogonal to the nonzero v ector w j , by Prop osition 1 .6, ρ ( w j ) = s j . Since N ( w j ) = k w j k 2 · 1, w e can replace w j b y w j / k w j k , so that N ( w 1 · · · w k ) = 1, and then f = ρ ( w 1 · · · w k ) , and ρ is surjectiv e. Note that Ker ( ρ | Pin ( n )) = Ker ( ρ ) ∩ ker( N ) = { t ∈ R ∗ · 1 | N ( t ) = 1 } = {− 1 , 1 } . As to Spin ( n ), we just need to sho w that the restriction of ρ to Spin ( n ) maps Γ n in to SO ( n ). If this w as not the case, t here w ould b e s ome improp er isometry f ∈ O ( n ) so that ρ ( x ) = f , where x ∈ Γ n ∩ Cl 0 n . Ho w ever, w e can express f a s the comp osition of an o dd n um b er of reﬂections, sa y f = ρ ( w 1 · · · w 2 k + 1 ) . Since ρ ( w 1 · · · w 2 k + 1 ) = ρ ( x ) , 26 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN w e ha v e x − 1 w 1 · · · w 2 k + 1 ∈ Ker ( ρ ). By Lemma 1.7, w e m ust hav e x − 1 w 1 · · · w 2 k + 1 = λ · 1 for some λ ∈ R ∗ , and thus w 1 · · · w 2 k + 1 = λ · x, where x has ev en degree and w 1 · · · w 2 k + 1 has o dd degree, whic h is imp ossible. Let us denote t he set o f elemen ts v ∈ R n with N ( v ) = 1 (with norm 1 ) by S n − 1 . W e hav e the follo wing corollary of Theorem 1.1 1: Corollary 1.12. The g r oup P in ( n ) is gener ate d by S n − 1 , and every element of Spin ( n ) c an b e written as the pr o duct of an even numb er of elem ents of S n − 1 . Example 1.2. The r eader should verify that Pin (1) ≈ Z / 4 Z , Spin (1) = { − 1 , 1 } ≈ Z / 2 Z , and also tha t Pin (2) ≈ { ae 1 + be 2 | a 2 + b 2 = 1 } ∪ { c 1 + de 1 e 2 | c 2 + d 2 = 1 } , Spin (2) = U (1) . W e ma y also write Pin (2) = U (1) + U (1), where U (1) is the group of complex n um b ers of mo dulus 1 (the unit circle in R 2 ). It can a lso b e show n that Spin (3) ≈ SU (2) and Spin (4) ≈ SU (2) × SU (2). The g roup Spin ( 5 ) is isomorphic t o the symplectic gro up Sp (2), and Spin (6 ) is isomorphic t o SU (4) (see Curtis [12] or P orteous [24]). Let us tak e a closer lo ok at Spin (2). The Cliﬀord algebra Cl 2 is generated b y t he four elemen ts 1 , e 1 , e 2 , , e 1 e 2 , and they satisfy the relations e 2 1 = − 1 , e 2 2 = − 1 , e 1 e 2 = − e 2 e 1 . The group Spin (2) consists of a ll pro ducts 2 k Y i =1 ( a i e 1 + b i e 2 ) consisting of an ev en n umber of factors and suc h that a 2 i + b 2 i = 1. In view o f the ab ov e relations, ev ery suc h elemen t can b e written as x = a 1 + be 1 e 2 , 1.4. THE GROUPS PIN ( N ) AND SP IN ( N ) 27 where x satisﬁes the conditions that xv x − 1 ∈ R 2 for a ll v ∈ R 2 , and N ( x ) = 1. Since X = a 1 − be 1 e 2 , w e get N ( x ) = a 2 + b 2 , and the condition N ( x ) = 1 is simply a 2 + b 2 = 1. W e claim that if x ∈ Cl 0 2 , then xv x − 1 ∈ R 2 . Indeed, sinc e x ∈ Cl 0 2 and v ∈ Cl 1 2 , w e ha v e xv x − 1 ∈ Cl 1 2 , whic h implies that xv x − 1 ∈ R 2 , since the only elemen ts of Cl 1 2 are those in R 2 . Then, Spin (2) cons ists of those elemen ts x = a 1 + be 1 e 2 so that a 2 + b 2 = 1. If w e let i = e 1 e 2 , we observ e that i 2 = − 1 , e 1 i = − i e 1 = − e 2 , e 2 i = − i e 2 = e 1 . Th us, Spi n (2 ) is isomorphic to U (1). Also note that e 1 ( a 1 + b i ) = ( a 1 − b i ) e 1 . Let us ﬁnd out explicitly what is the action of Spin (2 ) on R 2 . Give n X = a 1 + b i , with a 2 + b 2 = 1 , for any v = v 1 e 1 + v 2 e 2 , w e hav e α ( X ) v X − 1 = X ( v 1 e 1 + v 2 e 2 ) X − 1 = X ( v 1 e 1 + v 2 e 2 )( − e 1 e 1 ) X = X ( v 1 e 1 + v 2 e 2 )( − e 1 )( e 1 X ) = X ( v 1 1 + v 2 i ) X e 1 = X 2 ( v 1 1 + v 2 i ) e 1 = ((( a 2 − b 2 ) v 1 − 2 abv 2 )1 + ( a 2 − b 2 ) v 2 + 2 abv 1 ) i ) e 1 = (( a 2 − b 2 ) v 1 − 2 abv 2 ) e 1 + ( a 2 − b 2 ) v 2 + 2 abv 1 ) e 2 . Since a 2 + b 2 = 1 , we can write X = a 1 + b i = (cos θ )1 + (sin θ ) i , and the ab o v e deriv ation sho ws that α ( X ) v X − 1 = (cos 2 θ v 1 − sin 2 θ v 2 ) e 1 + (cos 2 θ v 2 + sin 2 θ v 1 ) e 2 . This means that the rot a tion ρ X induced by X ∈ Spin (2) is t he rotatio n of angle 2 θ around the origin. Observ e that the maps v 7→ v ( − e 1 ) , X 7→ X e 1 establish bijections b et w een R 2 and Spin (2 ) ≃ U (1). Also, note that the action of X = cos θ + i sin θ view ed as a complex num b er yields the rotation o f ang le θ , whereas the action 28 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN of X = (cos θ )1 + ( sin θ ) i view ed as a member of Spin (2) yields the rotatio n o f angle 2 θ . There is nothing wrong. In general, Spin ( n ) is a tw o–to–one cov er of SO ( n ). Next, let us tak e a closer lo ok at Spin (3). The C liﬀord algebra Cl 3 is g enerated b y the eigh t elemen ts 1 , e 1 , e 2 , , e 3 , , e 1 e 2 , e 2 e 3 , e 3 e 1 , e 1 e 2 e 3 , and they satisfy the relations e 2 i = − 1 , e j e j = − e j e i , 1 ≤ i, j ≤ 3 , i 6 = j. The group Spin (3) consists of a ll pro ducts 2 k Y i =1 ( a i e 1 + b i e 2 + c i e 3 ) consisting of an ev en n um b er o f factors and suc h that a 2 i + b 2 i + c 2 i = 1. In view of the ab ov e relations, ev ery suc h elemen t can b e written as x = a 1 + be 2 e 3 + ce 3 e 1 + de 1 e 2 , where x satisﬁes the conditions that xv x − 1 ∈ R 3 for a ll v ∈ R 3 , and N ( x ) = 1. Since X = a 1 − be 2 e 3 − ce 3 e 1 − de 1 e 2 , w e get N ( x ) = a 2 + b 2 + c 2 + d 2 , and the condition N ( x ) = 1 is simply a 2 + b 2 + c 2 + d 2 = 1. It turns out that the conditions x ∈ Cl 0 3 and N ( x ) = 1 imply that xv x − 1 ∈ R 3 for a ll v ∈ R 3 . T o prov e this, ﬁrst observ e that N ( x ) = 1 implies tha t x − 1 = ± x , a nd t ha t v = − v for a n y v ∈ R 3 , and so, xv x − 1 = − xv x − 1 . Also, since x ∈ Cl 0 3 and v ∈ Cl 1 3 , w e hav e xv x − 1 ∈ Cl 1 3 . Th us, w e can write xv x − 1 = u + λe 1 e 2 e 3 , for some u ∈ R 3 and some λ ∈ R . But e 1 e 2 e 3 = − e 3 e 2 e 1 = e 1 e 2 e 3 , and so, xv x − 1 = − u + λe 1 e 2 e 3 = − xv x − 1 = − u − λe 1 e 2 e 3 , whic h implies that λ = 0. Th us, xv x − 1 ∈ R 3 , as claimed. Then, Spin (3 ) consists of t ho se elemen ts x = a 1 + be 2 e 3 + ce 3 e 1 + de 1 e 2 so that a 2 + b 2 + c 2 + d 2 = 1. Under t he bijection i 7→ e 2 e 3 , j 7→ e 3 e 1 , k 7→ e 1 e 2 , 1.4. THE GROUPS PIN ( N ) AND SP IN ( N ) 29 w e can che c k that we hav e an isomorphism b etw een the group SU (2) of unit quaternions and Spin (3). If X = a 1 + be 2 e 3 + ce 3 e 1 + de 1 e 2 ∈ Spin (3) , obse rv e tha t X − 1 = X = a 1 − be 2 e 3 − ce 3 e 1 − de 1 e 2 . No w, using the iden tiﬁcation i 7→ e 2 e 3 , j 7→ e 3 e 1 , k 7→ e 1 e 2 , w e can easily c hec k that ( e 1 e 2 e 3 ) 2 = 1 , ( e 1 e 2 e 3 ) i = i ( e 1 e 2 e 3 ) = − e 1 , ( e 1 e 2 e 3 ) j = j ( e 1 e 2 e 3 ) = − e 2 , ( e 1 e 2 e 3 ) k = k ( e 1 e 2 e 3 ) = − e 3 , ( e 1 e 2 e 3 ) e 1 = − i , ( e 1 e 2 e 3 ) e 2 = − j , ( e 1 e 2 e 3 ) e 3 = − k . Then, if X = a 1 + b i + c j + d k ∈ Spin ( 3), for ev ery v = v 1 e 1 + v 2 e 2 + v 3 e 3 , we hav e α ( X ) v X − 1 = X ( v 1 e 1 + v 2 e 2 + v 3 e 3 ) X − 1 = X ( e 1 e 2 e 3 ) 2 ( v 1 e 1 + v 2 e 2 + v 3 e 3 ) X − 1 = ( e 1 e 2 e 3 ) X ( e 1 e 2 e 3 )( v 1 e 1 + v 2 e 2 + v 3 e 3 ) X − 1 = − ( e 1 e 2 e 3 ) X ( v 1 i + v 2 j + v 3 k ) X − 1 . This sho ws that the ro t a tion ρ X ∈ SO (3) induced b y X ∈ Spin (3) can b e view ed as the rotation induced by the quaternion a 1 + b i + c j + d k on the pure quaternions, using t he maps v 7→ − ( e 1 e 2 e 3 ) v , X 7→ − ( e 1 e 2 e 3 ) X to g o from a v ector v = v 1 e 1 + v 2 e 2 + v 3 e 3 to the pure quaternion v 1 i + v 2 j + v 3 k , and bac k. W e close this se ction b y taking a close r look at Spin (4). The group Spin (4) cons ists of all pro ducts 2 k Y i =1 ( a i e 1 + b i e 2 + c i e 3 + d i e 4 ) consisting of an ev en num b er of factors and suc h that a 2 i + b 2 i + c 2 i + d 2 i = 1. Using the relations e 2 i = − 1 , e j e j = − e j e i , 1 ≤ i, j ≤ 4 , i 6 = j, ev ery e lemen t of Spin (4) can b e written as x = a 1 1 + a 2 e 1 e 2 + a 3 e 2 e 3 + a 4 e 3 e 1 + a 5 e 4 e 3 + a 6 e 4 e 1 + a 7 e 4 e 2 + a 8 e 1 e 2 e 3 e 4 , 30 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN where x satisﬁes the conditions that xv x − 1 ∈ R 4 for a ll v ∈ R 4 , and N ( x ) = 1. Let i = e 1 e 2 , j = e 2 e 3 , k = 3 3 e 1 , i ′ = e 4 e 3 , j ′ = e 4 e 1 , k ′ = e 4 e 2 , and I = e 1 e 2 e 3 e 4 . The reader will easily v erify that ij = k jk = i ki = j i 2 = − 1 , j 2 = − 1 , k 2 = − 1 i I = I i = i ′ j I = I j = j ′ k I = I k = k ′ I 2 = 1 , I = I . Then, ev ery x ∈ Spin (4) can b e written as x = u + I v , with u = a 1 + b i + c j + d k and v = a ′ 1 + b ′ i + c ′ j + d ′ k , with t he extra conditio ns stated ab ov e. Using the ab ov e iden t it ies, w e hav e ( u + I v )( u ′ + I v ′ ) = uu ′ + v v ′ + I ( uv ′ + v u ′ ) . As a consequence, N ( u + I v ) = ( u + I v )( u + I v ) = u u + v v + I ( u v + v u ) , and th us, N ( u + I v ) = 1 is equiv alen t to u u + v v = 1 and uv + v u = 0 . As in the case n = 3, it turns out that the conditions x ∈ Cl 0 4 and N ( x ) = 1 imply that xv x − 1 ∈ R 4 for all v ∈ R 4 . The only change to the pro of is that xv x − 1 ∈ Cl 1 4 can b e written as xv x − 1 = u + X i,j,k λ i,j,k e i e j e k , for some u ∈ R 4 , with { i, j, k } ⊆ { 1 , 2 , 3 , 4 } . As in the previous proof, w e g et λ i,j,k = 0 . Then, S pin (4) cons ists of those elemen ts u + I v so that u u + v v = 1 and uv + v u = 0 , with u and v of the form a 1 + b i + c j + d k . Finally , we see that Spin (4) is isomorphic to Spin (3) × Spin (3) under the isomorphism u + v I 7→ ( u + v , u − v ) . 1.5. THE GROUPS PIN ( P , Q ) AND SPIN ( P , Q ) 31 Indeed, w e ha ve N ( u + v ) = ( u + v )( u + v ) = 1 , and N ( u − v ) = ( u − v )( u − v ) = 1 , since u u + v v = 1 and uv + v u = 0 , and ( u + v , u − v )( u ′ + v ′ , u ′ − v ′ ) = ( uu ′ + v v ′ + uv ′ + v u ′ , uu ′ + v v ′ − ( uv ′ + v u ′ )) . Remark: It can b e show n t ha t the assertion if x ∈ Cl 0 n and N ( x ) = 1, t hen xv x − 1 ∈ R n for all v ∈ R n , is true up to n = 5 (see Porteous [24], Chapter 13, Prop osition 13 .58). Ho w ev er, this is a lr eady fa lse for n = 6. F or example, if X = 1 / √ 2(1 + e 1 e 2 e 3 e 4 e 5 e 6 ), it is easy to see that N ( X ) = 1, and y et, X e 1 X − 1 / ∈ R 6 . 1.5 The Grou p s Pin ( p, q ) and Sp in ( p, q ) F or ev ery nondegenerate quadratic form Φ ov er R , t here is an orthogonal basis with resp ect to which Φ is giv en by Φ( x 1 , . . . , x p + q ) = x 2 1 + · · · + x 2 p − ( x 2 p +1 + · · · + x 2 p + q ) , where p and q only dep end on Φ. The quadratic form corr esp onding to ( p, q ) is denoted Φ p,q and w e call ( p, q ) the signatur e of Φ p,q . L et n = p + q . W e deﬁne the group O ( p, q ) as the group o f isometries w.r.t. Φ p,q , i.e. , the group of linear maps f so that Φ p,q ( f ( v )) = Φ p,q ( v ) fo r all v ∈ R n and the group SO ( p , q ) as the subgroup of O ( p, q ) consisting of the isometrie s f ∈ O ( p, q ) with det( f ) = 1. W e den ote the Cliﬀord a lg ebra Cl(Φ p,q ) where Φ p,q has signature ( p, q ) by Cl p,q , the corresp onding Cliﬀord group by Γ p,q , and t he sp ecial Cliﬀord group Γ p,q ∩ Cl 0 p,q b y Γ + p,q . Note that with this new notatio n, Cl n = Cl 0 ,n .  As w e mentioned earlier, since Lawson and Mic helsohn [20] ado pt the o pp osite o f o ur sign con v ention in deﬁning Cliﬀord algebras; their Cl ( p, q ) is our Cl( q , p ). As w e mentioned in Section 1.3, w e hav e the problem that N ( v ) = − Φ( v ) · 1, but − Φ( v ) is not necessarily p ositiv e (where v ∈ R n ). The ﬁx is simple: Allo w elemen ts x ∈ Γ p,q with N ( x ) = ± 1. Deﬁnition 1.6. W e deﬁne t he p i n or gr oup Pin ( p, q ) as the group Pin ( p, q ) = { x ∈ Γ p,q | N ( x ) = ± 1 } , and the spinor gr oup Spin ( p, q ) as Pin ( p, q ) ∩ Γ + p,q . 32 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN Remarks: (1) It is easily c hec k ed that the group Spin ( p, q ) is also g iv en b y Spin ( p, q ) = { x ∈ Cl 0 p,q | xv x ∈ R n for a ll v ∈ R n , N ( x ) = 1 } . This is b ecause Spin ( p, q ) consists of elemen ts of ev en degree . (2) One can chec k that if N ( x ) 6 = 0 , then α ( x ) v x − 1 = xv t ( x ) / N ( x ) . Th us, w e hav e Pin ( p, q ) = { x ∈ Cl p,q | xv t ( x ) N ( x ) ∈ R n for a ll v ∈ R n , N ( x ) = ± 1 } . When Φ( x ) = − k x k 2 , w e hav e N ( x ) = k x k 2 , and Pin ( n ) = { x ∈ Cl n | xv t ( x ) ∈ R n for a ll v ∈ R n , N ( x ) = 1 } . Theorem 1.11 generalizes as follow s: Theorem 1.13. The r estriction of ρ : Γ p,q → GL ( n ) to the pinor gr oup Pin ( p, q ) is a surje ctive homo morphism ρ : Pin ( p, q ) → O ( p, q ) whose kernel i s {− 1 , 1 } , and the r estriction of ρ to the s p inor gr oup Spin ( p, q ) is a surje ctive homomorph i s m ρ : Spin ( p, q ) → SO ( p, q ) whose kernel is {− 1 , 1 } . Pr o of. The Cartan-D ieudonn´ e also holds for an y nondegenerate quadratic form Φ, in the sense that ev ery isometry in O (Φ) is the comp osition o f reﬂe ctions deﬁned b y h yp erplanes orthogonal to non- isotropic v ectors (see Dieudonn ´ e [13], Chev a lley [1 0], Bourbaki [6], or Gallier [16 ], Chapter 7, Problem 7.14 ). Thu s, Theorem 1.11 a lso holds for an y nondegenerate quadratic form Φ. The only change to the pro o f is the following: Since N ( w j ) = − Φ( w j ) · 1, w e can replace w j b y w j / p | Φ( w j ) | , so t hat N ( w 1 · · · w k ) = ± 1, and then f = ρ ( w 1 · · · w k ) , and ρ is surjectiv e. If we consider R n equipped with t he quadratic form Φ p,q (with n = p + q ), w e denote the set of elemen ts v ∈ R n with N ( v ) = 1 b y S n − 1 p,q . W e hav e t he fo llowing corollary of Theorem 1.13 (generalizing Corollary 1.14): Corollary 1.14. The gr o up Pin ( p, q ) is gener ate d b y S n − 1 p,q , and every element of Spin ( p, q ) c an b e written a s the pr o duct of an even numb er of elemen ts of S n − 1 p,q . 1.6. PERIODICITY OF THE CLIFF ORD ALGEBRAS CL P ,Q 33 Example 1.3. The r eader should chec k that Cl 0 , 1 ≈ C , Cl 1 , 0 ≈ R ⊕ R . W e a lso hav e Pin (0 , 1) ≈ Z / 4 Z , Pin (1 , 0) ≈ Z / 2 Z × Z / 2 Z , from which w e g et Spin ( 0 , 1) = Spin (1 , 0) ≈ Z / 2 Z . Also, sho w that Cl 0 , 2 ≈ H , Cl 1 , 1 ≈ M 2 ( R ) , Cl 2 , 0 ≈ M 2 ( R ) , where M n ( R ) denotes the algebra of n × n matrices. One can also w ork out what are Pin (2 , 0), P in (1 , 1), and Pin (0 , 2); see Cho quet-Bruhat [11], Chapter I, Section 7, page 26. Sho w that Spin (0 , 2) = Spin (2 , 0 ) ≈ U (1) , and Spin (1 , 1) = { a 1 + be 1 e 2 | a 2 − b 2 = 1 } . Observ e that Spin (1 , 1) is not connected. More generally , it can b e show n that Cl 0 p,q and Cl 0 q ,p are isomorphic, f r om whic h it follo ws that Spin ( p, q ) and Spin ( q , p ) are isomorphic, but Pin ( p, q ) and Pin ( q , p ) are not isomorphic in g eneral, and in part icular, Pin ( p, 0 ) and P in (0 , p ) ar e not isomorphic in general (see Cho quet-Bruhat [11], Chapter I, Section 7). Ho we v er, due to the “8 - p erio dicit y” of the Cliﬀord algebras (to b e discussed in the next section), it follow s that Cl p,q and Cl q ,p are isomorphic when | p − q | = 0 mo d 4. 1.6 P erio dic i t y of th e Cliﬀo r d A l g ebras Cl p,q It turns out that the real algebras Cl p,q can b e build up as tensor pro ducts o f the basic algebras R , C , and H . As p oin ted out b y Lo unesto (Section 2 3 .16 [21]), the description of the real a lg ebras Cl p,q as matrix a lg ebras and the 8- p erio dicit y was ﬁrst observ ed b y Elie Cartan in 190 8 ; see Cartan’s art icle, Nombr e s Complexes , based on the or ig inal article in German b y E. Study , in Molk [23], article I-5 (fasc. 3), pages 329-468 . These algebras are deﬁned in Section 36 under t he name “‘Systems of Cliﬀord and Lipsc hitz,” page 463- 466. Of course, Cartan used a v ery diﬀeren t notation; see page 464 in the article cited ab o v e. These fa cts w ere redisco v ered indep enden tly b y Raoul Bot t in the 1960’s (see R a oul Bott’s commen ts in V olume 2 of his Collected pap ers.). W e will use the nota tion R ( n ) (resp. C ( n )) for the algebra M n ( R ) of all n × n real matrices ( resp. the algebra M n ( C ) of all n × n complex matrices). As men tio ned in Example 1.3, it is not hard to sho w that Cl 0 , 1 = C Cl 1 , 0 = R ⊕ R Cl 0 , 2 = H Cl 2 , 0 = R ( 2) , 34 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN and Cl 1 , 1 = R ( 2) . The k ey to the classiﬁcation is the followin g lemma: Lemma 1.15. We h a ve the isomorphisms Cl 0 ,n +2 ≈ Cl n, 0 ⊗ Cl 0 , 2 Cl n +2 , 0 ≈ Cl 0 ,n ⊗ Cl 2 , 0 Cl p +1 ,q +1 ≈ Cl p,q ⊗ Cl 1 , 1 , for al l n, p, q ≥ 0 . Pr o of. Let Φ 0 ,n ( x ) = − k x k 2 , where k x k is the standard Euclidean norm on R n +2 , and let ( e 1 , . . . , e n +2 ) b e an ortho no rmal basis for R n +2 under the standard Euclidean inner pro duct. W e also let ( e ′ 1 , . . . , e ′ n ) b e a set of generators for Cl n, 0 and ( e ′′ 1 , e ′′ 2 ) b e a set of generators for Cl 0 , 2 . W e can deﬁne a linear map f : R n +2 → Cl n, 0 ⊗ Cl 0 , 2 b y its action on the basis ( e 1 , . . . , e n +2 ) as follow s: f ( e i ) =  e ′ i ⊗ e ′′ 1 e ′′ 2 for 1 ≤ i ≤ n 1 ⊗ e ′′ i − n for n + 1 ≤ i ≤ n + 2. Observ e that for 1 ≤ i, j ≤ n , w e ha v e f ( e i ) f ( e j ) + f ( e j ) f ( e i ) = ( e ′ i e ′ j + e ′ j e ′ i ) ⊗ ( e ′′ 1 e ′′ e ) 2 = − 2 δ ij 1 ⊗ 1 , since e ′′ 1 e ′′ 2 = − e ′′ 2 e ′′ 1 , ( e ′′ 1 ) 2 = − 1, and ( e ′′ 2 ) 2 = − 1, and e ′ i e ′ j = − e ′ j e ′ i , f o r all i 6 = j , and ( e ′ i ) 2 = 1 , for all i with 1 ≤ i ≤ n . Also, for n + 1 ≤ i, j ≤ n + 2, w e ha v e f ( e i ) f ( e j ) + f ( e j ) f ( e i ) = 1 ⊗ ( e ′′ i − n e ′′ j − n + e ′′ j − n e ′′ i − n ) = − 2 δ ij 1 ⊗ 1 , and f ( e i ) f ( e k ) + f ( e k ) f ( e i ) = 2 e ′ i ⊗ ( e ′′ 1 e ′′ 2 e ′′ n − k + e ′′ n − k e ′′ 1 e ′′ 2 ) = 0 , for 1 ≤ i, j ≤ n and n + 1 ≤ k ≤ n + 2 (since e ′′ n − k = e ′′ 1 or e ′′ n − k = e ′′ 2 ). Th us, w e hav e f ( x ) 2 = − k x k 2 · 1 ⊗ 1 for all x ∈ R n +2 , and b y the univ ersal mapping prop ert y of Cl 0 ,n +2 , we get an algebra map e f : Cl 0 ,n +2 → Cl n, 0 ⊗ Cl 0 , 2 . Since e f maps o n to a set of g enerators, it is surjectiv e. How ev er dim(Cl 0 ,n +2 ) = 2 n +2 = 2 n · 2 = dim (Cl n, 0 )dim(Cl 0 , 2 ) = dim(Cl n, 0 ⊗ Cl 0 , 2 ) , and e f is a n isomorphism. 1.6. PERIODICITY OF THE CLIFF ORD ALGEBRAS CL P ,Q 35 The pro of of the second iden tit y is ana lo gous. F or t he third identit y , we hav e Φ p,q ( x 1 , . . . , x p + q ) = x 2 1 + · · · + x 2 p − ( x 2 p +1 + · · · + x 2 p + q ) , and let ( e 1 , . . . , e p +1 , ǫ 1 , . . . , ǫ q +1 ) b e an orthog onal basis for R p + q +2 so that Φ p +1 ,q +1 ( e i ) = +1 and Φ p +1 ,q +1 ( ǫ j ) = − 1 for i = 1 , . . . , p + 1 a nd j = 1 , . . . , q + 1. Also, let ( e ′ 1 , . . . , e ′ p , ǫ ′ 1 , . . . , ǫ ′ q ) b e a set of generators for Cl p,q and ( e ′′ 1 , ǫ ′′ 1 ) b e a set of generators for Cl 1 , 1 . W e deﬁne a linear map f : R p + q +2 → Cl p,q ⊗ Cl 1 , 1 b y its action on the basis as follow s: f ( e i ) =  e ′ i ⊗ e ′′ 1 ǫ ′′ 1 for 1 ≤ i ≤ p 1 ⊗ e ′′ 1 for i = p + 1, and f ( ǫ j ) =  ǫ ′ j ⊗ e ′′ 1 ǫ ′′ 1 for 1 ≤ j ≤ q 1 ⊗ ǫ ′′ 1 for j = q + 1. W e can c hec k that f ( x ) 2 = Φ p +1 ,q +1 ( x ) · 1 ⊗ 1 for all x ∈ R p + q +2 , and w e ﬁnish the pro of as in the ﬁrst case. T o a pply this lemma, w e need some further isomorphisms among v arious matrix alg ebras. Prop osition 1.16. The fol lowing isom orphisms hold: R ( m ) ⊗ R ( n ) ≈ R ( mn ) for al l m, n ≥ 0 R ( n ) ⊗ R K ≈ K ( n ) for K = C or K = H and a l l n ≥ 0 C ⊗ R C ≈ C ⊕ C C ⊗ R H ≈ C (2) H ⊗ R H ≈ R (4) . Pr o of. D eta ils can b e found in Lawson and Mic helsohn [2 0]. The ﬁrst t w o isomorphisms are quite ob vious. The third isomor phism C ⊕ C → C ⊗ C is obtained by sending (1 , 0) 7→ 1 2 (1 ⊗ 1 + i ⊗ i ) , (0 , 1) 7→ 1 2 (1 ⊗ 1 − i ⊗ i ) . The ﬁeld C is isomorphic to the subring of H generated by i . Th us, w e can view H a s a C -v ector space under left scalar m ultiplication. Consider the R -bilinear map π : C × H → Hom C ( H , H ) giv en by π y , z ( x ) = y x z , where y ∈ C and x, z ∈ H . Th us, w e get an R - linear map π : C ⊗ R H → Hom C ( H , H ). Ho w ev er, w e ha v e Hom C ( H , H ) ≈ C (2). F urt hermore, since π y , z ◦ π y ′ ,z ′ = π y y ′ ,z z ′ , 36 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN the map π is an alg ebra homomorphism π : C × H → C (2) . W e can c hec k on a basis that π is injectiv e, and since dim R ( C × H ) = dim R ( C (2)) = 8 , the map π is an isomorphism. The last isomorphism is prov ed in a similar fashion. W e now ha v e the main p erio dicit y theorem. Theorem 1.17. (Cartan/Bott) F or a l l n ≥ 0 , w e have the fol lowin g isomorphisms: Cl 0 ,n +8 ≈ Cl 0 ,n ⊗ Cl 0 , 8 Cl n +8 , 0 ≈ Cl n, 0 ⊗ Cl 8 , 0 . F urthermor e, Cl 0 , 8 = Cl 8 , 0 = R ( 16) . Pr o of. By Lemma 1.15 w e ha v e the isomorphisms Cl 0 ,n +2 ≈ Cl n, 0 ⊗ Cl 0 , 2 Cl n +2 , 0 ≈ Cl 0 ,n ⊗ Cl 2 , 0 , and th us, Cl 0 ,n +8 ≈ Cl n +6 , 0 ⊗ Cl 0 , 2 ≈ Cl 0 ,n +4 ⊗ Cl 2 , 0 ⊗ Cl 0 , 2 ≈ · · · ≈ Cl 0 ,n ⊗ Cl 2 , 0 ⊗ Cl 0 , 2 ⊗ Cl 2 , 0 ⊗ Cl 0 , 2 . Since Cl 0 , 2 = H a nd Cl 2 , 0 = R ( 2), by Prop osition 1.16 , w e g et Cl 2 , 0 ⊗ Cl 0 , 2 ⊗ Cl 2 , 0 ⊗ Cl 0 , 2 ≈ H ⊗ H ⊗ R (2 ) ⊗ R (2) ≈ R (4 ) ⊗ R (4) ≈ R (1 6) . The second isomorphism is prov ed in a similar fashion. F rom a ll this, we can deduce the fo llo wing table: n 0 1 2 3 4 5 6 7 8 Cl 0 ,n R C H H ⊕ H H (2) C (4) R (8) R (8) ⊕ R (8) R (16) Cl n, 0 R R ⊕ R R (2) C (2) H (2) H (2) ⊕ H (2) H (4) C (8) R (16 ) A table of the Cliﬀord g roups Cl p,q for 0 ≤ p, q ≤ 7 can b e fo und in Kirillov [18], and for 0 ≤ p, q ≤ 8, in Lawson and Mic helsohn [20] (but b ew are t hat their Cl p,q is our Cl q ,p ). It can also b e shown that Cl p +1 ,q ≈ Cl q +1 ,p Cl p,q ≈ Cl p − 4 ,q +4 1.7. THE COMPLEX CLIFF ORD ALGEBRAS CL( N , C ) 37 with p ≥ 4 in the second iden tit y (see Lounesto [21], Chapter 16, Sections 16.3 and 16.4). Using the second iden tity , if | p − q | = 4 k , it is easily show n b y induction on k that Cl p,q ≈ Cl q ,p , as claimed in t he previous s ection. W e a lso hav e t he isomorphisms Cl p,q ≈ Cl 0 p,q +1 , fro w whic h it follows that Spin ( p, q ) ≈ Spin ( q , p ) (see Cho quet-Bruhat [1 1], Chapter I, Sections 4 and 7) . How ev er, in general, P in ( p, q ) a nd Pin ( q , p ) are not isomorphic. In fact, Pin (0 , n ) and P in ( n, 0 ) are not isomorphic if n 6 = 4 k , with k ∈ N (se e Cho quet-Bruhat [11 ], C hapter I, Section 7, pag e 27). 1.7 The Compl e x Cliﬀord Al gebras C l( n, C ) One can also consider Cliﬀord alg ebras ov er the complex ﬁeld C . In this case, it is we ll-know n that eve ry nondegenerate quadrat ic f o rm can b e expressed by Φ C n ( x 1 , . . . , x n ) = x 2 1 + · · · + x 2 n in some or thonormal basis. Also, it is eas ily sho wn that the comple xiﬁcation C ⊗ R Cl p,q of the real Cliﬀord algebra Cl p,q is isomorphic to Cl(Φ C n ). Th us, all these complex a lgebras are isomorphic for p + q = n , and w e denote them by Cl( n, C ). Theorem 1.15 yields the f ollo wing p erio dicit y theorem: Theorem 1.18. The fol lowing isom orphisms hold: Cl( n + 2 , C ) ≈ Cl ( n, C ) ⊗ C Cl(2 , C ) , with Cl(2 , C ) = C (2) . Pr o of. Since Cl ( n, C ) = C ⊗ R Cl 0 ,n = C ⊗ R Cl n, 0 , b y Lemma 1.15, w e hav e Cl( n + 2 , C ) = C ⊗ R Cl 0 ,n +2 ≈ C ⊗ R (Cl n, 0 ⊗ R Cl 0 , 2 ) ≈ ( C ⊗ R Cl n, 0 ) ⊗ C ( C ⊗ R Cl 0 , 2 ) . Ho w ev er, Cl 0 , 2 = H , Cl( n, C ) = C ⊗ R Cl n, 0 , and C ⊗ R H ≈ C (2), so w e get C l(2 , C ) = C (2) and Cl( n + 2 , C ) ≈ Cl ( n, C ) ⊗ C C (2) , and the theorem is pro v ed. As a corollary of Theorem 1.18, w e obtain the fa ct that Cl(2 k , C ) ≈ C (2 k ) and Cl(2 k + 1 , C ) ≈ C (2 k ) ⊕ C (2 k ) . 38 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN The table o f the previous sec tion can also b e completed as fo llo ws: n 0 1 2 3 4 5 6 7 8 Cl 0 ,n R C H H ⊕ H H (2) C (4) R (8) R (8) ⊕ R (8) R (16) Cl n, 0 R R ⊕ R R (2) C (2) H (2) H (2) ⊕ H (2) H (4) C (8) R (16) Cl( n, C ) C 2 C C (2) 2 C (2) C (4) 2 C (4) C (8) 2 C (8) C (16) . where 2 C ( k ) is an abbrev ation for C ( k ) ⊕ C ( k ). 1.8 The Groups Pin ( p, q ) and Spin ( p , q ) as d ouble co v ers of O ( p, q ) and S O ( p, q ) It turns out t hat the groups P in ( p, q ) a nd Spin ( p, q ) hav e nice to p ological prop erties w.r.t. the groups O ( p, q ) and SO ( p, q ). T o explain this, w e review the deﬁnition of cov ering maps and cov ering spac es (for details, se e F ulton [14], Chapter 11). Another intere sting source is Chev alley [9], where is is pro v ed that Spin ( n ) is a univ ersal double co ver of SO ( n ) for all n ≥ 3. Since C p,q is an algebra of dimens ion 2 p + q , it is a top ological space as a v ector space isomorphic to V = R 2 p + q . Now, the group C ∗ p,q of units of C p,q is op en in C p,q , b ecause x ∈ C p,q is a unit if the linear map y 7→ xy is an isomorphism, and GL ( V ) is op en in End( V ), the space of endomorphisms of V . Th us, C ∗ p,q is a Lie group, and since Pin ( p, q ) and Spin ( p, q ) are clearly closed subgroups of C ∗ p,q , they a re a lso L ie g roups. Deﬁnition 1.7. Giv en t w o top ological spaces X and Y , a c overing map is a con tinuous surjectiv e map p : Y → X with the prop ert y that for ev ery x ∈ X , there is some op en subset U ⊆ X with x ∈ U , so that p − 1 ( U ) is the disjoin t union of op en subsets V α ⊆ Y , and the restriction of p to eac h V α is a homeomorphism on to U . W e sa y that U is evenly c over e d by p . W e a lso sa y that Y is a c overing sp ac e of X . A co v ering map p : Y → X is called trivial if X itself is eve nly cov ered by p ( i.e. , Y is the disjoint union of op en subsets Y α eac h homeomorphic to X ), and nontrivia l otherwise. When eac h ﬁb er p − 1 ( x ) has the same ﬁnite cardinaly n for all x ∈ X , w e say that p is an n -c overing (or n -she ete d c overing ). Note that a co v ering map p : Y → X is not alw a ys trivial, but alw a ys lo c al ly trivial ( i.e. , for eve ry x ∈ X , it is trivial in some op en neigh b orho o d of x ). A cov ering is trivial iﬀ Y is isomorphic to a pro duct space of X × T , where T is an y set with the discrete t o p ology . Also, if Y is connected, then the co v ering map is nontrivial. Deﬁnition 1.8. An isomorphism ϕ b et w een co v ering maps p : Y → X and p ′ : Y ′ → X is a homeomorphism ϕ : Y → Y ′ so that p = p ′ ◦ ϕ . T ypically , the space X is connected, in whic h case it can b e shown that all the ﬁb ers p − 1 ( x ) ha v e the same cardinality . One of the most imp ortan t prop erties of co vering spaces is the path–lif t ing prop ert y , a prop ert y that we will use to sho w that Spin ( n ) is path-connected. 1.8. THE GROUPS PIN ( P , Q ) AND SPIN ( P , Q ) AS DOUBLE CO VERS 39 Prop osition 1.19. (Path lifting) L et p : Y → X b e a c ov e ring map, and let γ : [ a, b ] → X b e any c ontinuous curve fr om x a = γ ( a ) to x b = γ ( b ) in X . If y ∈ Y i s any p oint so that p ( y ) = x a , then ther e is a unique c urve e γ : [ a, b ] → Y s o that y = e γ ( a ) and p ◦ e γ ( t ) = γ ( t ) for al l t ∈ [ a, b ] . Pr o of. See F ulton [15], Chapter 1 1, Lemma 11.6 . Man y imp ortant co v ering maps arise from the actio n of a group G on a space Y . If Y is a top ological space, an action (on the left) of a gr oup G on Y is a map α : G × Y → Y satisfying t he following conditions, where fo r simplicit y of notation, w e denote α ( g , y ) b y g · y : (1) g · ( h · y ) = ( g h ) · y , for all g , h ∈ G and y ∈ Y ; (2) 1 · y = y , fo r all ∈ Y , where 1 is t he identit y of the group G ; (3) The map y 7→ g · y is a homeomorphism of Y for ev ery g ∈ G . W e deﬁne an equiv alence relation on Y as follows : x ≡ y iﬀ y = g · x fo r some g ∈ G (c heck that this is an equiv alence relatio n). The equiv alence class G · x = { g · x | g ∈ G } of an y x ∈ Y is called the orbit of x . W e o btain the quotien t space Y /G and the pro jection map p : Y → Y /G sending ev ery y ∈ Y to its orbit. The space Y /G is giv en t he quotien t top ology (a subset U of Y /G is o p en iﬀ p − 1 ( U ) is op en in Y ). Giv en a subset V of Y and an y g ∈ G , w e let g · V = { g · y | y ∈ V } . W e sa y that G acts eve n ly on Y if for ev ery y ∈ Y , there is an o p en subset V containing y so that g · V and h · V are disjoin t for an y tw o distinct elemen ts g , h ∈ G . The imp ortance o f the notio n a group acting ev enly is that suc h a ctions induce a co v ering map. Prop osition 1.20. If G is a gr oup acting evenly on a sp ac e Y , then the pr oje ction map p : Y → Y /G is a c ov e ring map. Pr o of. See F ulton [15], Chapter 1 1, Lemma 11.1 7. The following propo sition sho ws that Pin ( p, q ) and Spin ( p, q ) are nontrivial co vering spaces, unless p = q = 1 . Prop osition 1.21. F or al l p, q ≥ 0 , the gr o ups Pin ( p, q ) and Spin ( p, q ) ar e d o uble c overs of O ( p, q ) and SO ( p, q ) , r esp e ctively. F urthermo r e, they ar e nontrivial c ove rs unless p = q = 1 . 40 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN Pr o of. W e kno w that kernel of the homomorphism ρ : Pin ( p, q ) → O ( p, q ) is Z 2 = {− 1 , 1 } . If w e let Z 2 act on Pin ( p, q ) in the na tural w a y , then O ( p, q ) ≈ Pin ( p, q ) / Z 2 , and the reader can easily chec k that Z 2 acts eve nly . By Prop osition 1.20 , w e get a double cov er. The argumen t for ρ : Spin ( p, q ) → SO ( p , q ) is similar. Let us no w assume tha t p 6 = 1 or q 6 = 1. In o rder to prov e that w e hav e non trivial co v ers, it is enough to sho w that − 1 and 1 ar e connected by a path in Pin ( p, q ) (If we had Pin ( p, q ) = U 1 ∪ U 2 with U 1 and U 2 op en, disjoin t, and homeomorphic to O ( p, q ), then − 1 and 1 w ould not b e in the same U i , and so, they would b e in disjoin t connected comp onen ts. Th us, − 1 and 1 can’t b e path– connected, and similarly with Spin ( p, q ) and SO ( p, q ) .) Since ( p, q ) 6 = (1 , 1), w e can ﬁnd t w o or t hogonal ve ctors e 1 and e 2 so that Φ p,q ( e 1 ) = Φ p,q ( e 2 ) = ± 1. Then, γ ( t ) = ± cos(2 t ) 1 + sin ( 2 t ) e 1 e 2 = ( cos t e 1 + sin t e 2 )(sin t e 2 − cos t e 1 ) , for 0 ≤ t ≤ π , deﬁnes a path in Spin ( p, q ), since ( ± cos(2 t ) 1 + sin( 2 t ) e 1 e 2 ) − 1 = ± cos(2 t ) 1 − sin(2 t ) e 1 e 2 , as desired. In particular, if n ≥ 2, since the group SO ( n ) is path-connected, the group Spin ( n ) is also path-connected. Indeed, giv en any t w o p oin ts x a and x b in Spin ( n ), there is a path γ f rom ρ ( x a ) to ρ ( x b ) in SO ( n ) (where ρ : Spin ( n ) → SO ( n ) is the co v ering map). By Prop osition 1.19, t here is a path e γ in Spin ( n ) with origin x a and some origin e x b so that ρ ( e x b ) = ρ ( x b ). Ho w ever, ρ − 1 ( ρ ( x b )) = {− x b , x b } , and so e x b = ± x b . The argumen t used in the pro of of Propo sition 1.21 sho ws that x b and − x b are path- connected, and so, there is a path from x a to x b , and Spin ( n ) is path-connected. In fact, for n ≥ 3, it turns o ut that Spin ( n ) is simp ly connected. Suc h a cov ering space is called a universal c over (for instance, see Chev alley [9]). This last fact req uires more algebraic top ology than w e are willing to ex plain in detail, and we only sk etc h t he pro of. The notions of ﬁbre bundle, ﬁbrat io n, and homotopy sequence asso ciated with a ﬁbratio n are needed in the pro of. W e refer the p ersev erant readers to Bott and T u [5] ( Chapter 1 and Chapter 3 , Sections 16 –17) o r Rotman [25] (Chapter 11 ) for a detailed explanation of these concepts. Recall that a top ological space is simpl y c onne cte d if it is path connected and if π 1 ( X ) = (0), whic h means that ev ery closed path in X is homotopic to a p oin t. Since we just prov ed that Spin ( n ) is path connected for n ≥ 2, w e j ust need to pro v e that π 1 ( Spin ( n )) = (0) for all n ≥ 3. The following facts are needed to prov e the ab ov e assertion: (1) The sphere S n − 1 is simply connected for all n ≥ 3. (2) The group Spin (3) ≃ SU (2) is homeomorphic to S 3 , and thus, Spin (3) is simply connected. 1.8. THE GROUPS PIN ( P , Q ) AND SPIN ( P , Q ) AS DOUBLE CO VERS 41 (3) The group Spin ( n ) acts on S n − 1 in suc h a w a y that w e ha ve a ﬁbre bundle with ﬁbre Spin ( n − 1): Spin ( n − 1) − → Spin ( n ) − → S n − 1 . F act (1) is a standard propo sition of algebraic topo logy , and a pro o f can found in man y b o oks. A particularly elegan t and y et sim ple argumen t cons ists in showing that an y closed curv e on S n − 1 is homotopic to a curve that omits some p oint. First, it is easy to see that in R n , any closed curv e is homotopic to a piecewise linear curv e (a p olygonal curv e), and the r a dial pro jection of suc h a curv e on S n − 1 pro vides the desired curv e. Then, w e use the stereographic pro jection o f S n − 1 from any p oint o mitted by that curv e to g et another closed curv e in R n − 1 . Since R n − 1 is simply connected, that curv e is homotopic t o a p o in t, and so is its preimage curv e on S n − 1 . Another simple pro of uses a sp ecial v ersion of the Seifert—v an Kamp en’s theorem (see Gramain [17]). F act (2 ) is easy to es tablish directly , using ( 1 ). T o prov e (3), we let Spin ( n ) a ct on S n − 1 via the standard action: x · v = xv x − 1 . Because SO ( n ) acts transitive ly on S n − 1 and there is a surjection Spin ( n ) − → SO ( n ), the g roup Spin ( n ) also a cts tra nsitive ly on S n − 1 . Now, w e ha v e to show that the stabilizer of any elemen t of S n − 1 is Spin ( n − 1). F or example, we can do t his for e 1 . This amoun ts to some simple c alculations taking into account the iden tities among ba sis elemen ts. Details of this pro of can b e f o und in Mneimn ´ e and T estard [22], Chapter 4. It is still necessary to prov e t ha t Spin ( n ) is a ﬁbre bundle o v er S n − 1 with ﬁbre Spin ( n − 1 ) . F or this, w e use the follo wing results whose pro of can b e found in Mneimn ´ e and T estard [22], Chapter 4: Lemma 1.22. Given any top olo gic al gr oup G , if H is a close d sub gr oup of G and the pr o- je ction π : G → G/H has a lo c al s e ction a t every p oin t of G/H , then H − → G − → G/H is a ﬁbr e bund le with ﬁ b r e H . Lemma 1.22 implies the following k ey prop osition: Prop osition 1.23. Given any lin e ar Lie gr oup G , if H is a close d sub gr oup of G , then H − → G − → G/H is a ﬁbr e bund le with ﬁ b r e H . No w, a ﬁbre bundle is a ﬁbration (as deﬁned in Bott and T u [5], Chapter 3, Section 1 6 , or in R otman [25], Chapter 11). F or a pro of of this fact, see Rotman [25], Chapter 11, or Mneimn ´ e and T estard [22], Chapter 4. So, there is a homoto p y sequen ce associat ed with the ﬁbration (Bott and T u [5], Chapter 3 , Section 17 , or Rotman [25], Chapter 11, Theorem 11.48), and in particular, w e hav e the exact sequence π 1 ( Spin ( n − 1)) − → π 1 ( Spin ( n )) − → π 1 ( S n − 1 ) . 42 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN Since π 1 ( S n − 1 ) = (0 ) for n ≥ 3 , w e g et a surjection π 1 ( Spin ( n − 1)) − → π 1 ( Spin ( n )) , and so, b y induction and (2), we get π 1 ( Spin ( n )) ≈ π 1 ( Spin (3)) = (0) , pro ving that Spin ( n ) is simply connected fo r n ≥ 3. W e can also show that π 1 ( SO ( n )) = Z / 2 Z for all n ≥ 3 . F o r this, we use Theorem 1.11 and Prop osition 1.21, whic h imply that Spin ( n ) is a ﬁbre bundle o v er SO ( n ) with ﬁbre {− 1 , 1 } , for n ≥ 2: {− 1 , 1 } − → Spin ( n ) − → SO ( n ) . Again, the homotopy sequence of the ﬁbration exists, and in particular w e get the exact sequence π 1 ( Spin ( n )) − → π 1 ( SO ( n )) − → π 0 ( {− 1 , +1 } ) − → π 0 ( SO ( n )) . Since π 0 ( {− 1 , +1 } ) = Z / 2 Z , π 0 ( SO ( n )) = (0), a nd π 1 ( Spin ( n )) = (0), wh en n ≥ 3, w e get the exact sequence (0) − → π 1 ( SO ( n )) − → Z / 2 Z − → (0) , and so, π 1 ( SO ( n )) = Z / 2 Z . Therefore, SO ( n ) is not simply connected for n ≥ 3. Remark: Of course, w e hav e been rather ca v alier in our pres en tation. Giv en a top ological space X , the g roup π 1 ( X ) is the fund a m ental gr oup of X , i.e. t he g roup o f homotopy classes of closed paths in X ( under comp osition of lo ops). But π 0 ( X ) is generally not a group! Instead, π 0 ( X ) is the set of path-connected comp onents of X . Ho w ev er, when X is a Lie group, π 0 ( X ) is indeed a group. Also, w e hav e to mak e sense o f what it means for the sequence to b e exact. All this can b e made rigorous (see Bott a nd T u [5], Chapter 3, Section 17, o r Rotma n [25 ], Chapter 11). 1.9 More on the T op ology of O ( p, q ) and SO ( p , q ) It turns out that the top olog y of the group O ( p, q ) is completely determined by the top o logy of O ( p ) and O ( q ). This result can b e obtained as a simple consequence of some standa r d Lie group t heory . The k ey notion is tha t of a pseudo-algebraic g roup. Consider the group GL ( n, C ) of in ve rtible n × n matrices with complex co eﬃcien ts. If A = ( a k l ) is suc h a matrix, denote by x k l the real part (res p. y k l , the imaginary part) of a k l (so, a k l = x k l + iy k l ). Deﬁnition 1.9. A subgroup G of GL ( n, C ) is pseudo-algebr aic iﬀ there is a ﬁnite set of p olynomials in 2 n 2 v ariables with real co eﬃcien ts { P i ( X 1 , . . . , X n 2 , Y 1 , . . . , Y n 2 ) } t i =1 , so that A = ( x k l + iy k l ) ∈ G iﬀ P i ( x 11 , . . . , x nn , y 11 , . . . , y nn ) = 0 , for i = 1 , . . . , t. 1.9. MORE ON THE TOPOLOGY OF O ( P , Q ) AND SO ( P , Q ) 43 Recall that if A is a complex n × n -ma t r ix, its a d joint A ∗ is deﬁned by A ∗ = ( A ) ⊤ . Also, U ( n ) denotes the gro up of unitary matrices, i.e. , those matrices A ∈ GL ( n, C ) so that AA ∗ = A ∗ A = I , and H ( n ) de notes the vector space of Hermitian matrices, i.e. , those matrices A so that A ∗ = A . Then, we hav e the follo wing theorem whic h is essen tially a reﬁned v ersion of the p olar decomp osition of matrices: Theorem 1.24. L et G b e a pseudo-algebr a ic sub gr o up of GL ( n, C ) stable under adjunction (i.e., we hav e A ∗ ∈ G whe never A ∈ G ). T hen, ther e is some inte ger d ∈ N so that G is home omorph i c to ( G ∩ U ( n )) × R d . Mor e over, if g is the Lie alge b r a of G , the map ( U ( n ) ∩ G ) × ( H ( n ) ∩ g ) − → G, given by ( U, H ) 7→ U e H , is a home omorph i s m onto G . Pr o of. A pro of can b e found in Kna pp [1 9], Chapter 1, o r Mneimn ´ e and T estard [2 2], Chapter 3. W e no w apply Theorem 1 .2 4 to determine the structure of the space O ( p, q ). Let J p,q b e the matrix J p,q =  I p 0 0 − I q  . W e know that O ( p, q ) consists of the matrices A in GL ( p + q , R ) suc h that A ⊤ J p,q A = J p,q , and so O ( p, q ) is clearly pseudo-algebraic. Using the ab ov e equation, it is easy to determine the Lie algebra, o ( p, q ), of O ( p, q ). W e ﬁnd that o ( p, q ) is give n b y o ( p, q ) =   X 1 X 2 X ⊤ 2 X 3      X ⊤ 1 = − X 1 , X ⊤ 3 = − X 3 , X 2 arbitrary  , where X 1 is a p × p matrix, X 3 is a q × q matrix, a nd X 2 is a p × q matrix. Consequen tly , it immediately follo ws that o ( p, q ) ∩ H ( p + q ) =   0 X 2 X ⊤ 2 0      X 2 arbitrary  , a vec tor space of dimension pq . Some simple calcu lations also sho w that O ( p, q ) ∩ U ( p + q ) =   X 1 0 0 X 2      X 1 ∈ O ( p ) , X 2 ∈ O ( q )  ∼ = O ( p ) × O ( q ) . Therefore, w e obtain the structure of O ( p, q ): 44 CHAPTER 1 . CLIFF ORD ALGEBRAS, CLIFF ORD GROUPS, PIN AND SPIN Prop osition 1.25. The top olo gic al s p ac e O ( p, q ) is home omo rp h ic to O ( p ) × O ( q ) × R pq . Since O ( p ) has tw o connected comp onents when p ≥ 1, w e see that O ( p, q ) has four connected comp onen ts when p, q ≥ 1. It is also o b vious that SO ( p, q ) ∩ U ( p + q ) =   X 1 0 0 X 2      X 1 ∈ O ( p ) , X 2 ∈ O ( q ) , det( X 1 ) det( X 2 ) = 1  . This is a subgroup of O ( p ) × O ( q ) that w e denote S ( O ( p ) × O ( q )). F urthermore, it is easy to show that so ( p, q ) = o ( p, q ). Th us, w e a lso ha v e Prop osition 1.26. The top olo gic al sp ac e SO ( p, q ) is home omorp h ic to S ( O ( p ) × O ( q )) × R pq . Note t ha t SO ( p, q ) ha s tw o connected comp o nen ts when p, q ≥ 1 . The connected comp onen t o f I p + q is a gro up denoted SO 0 ( p, q ). This lat t er space is homeomorphic to SO ( p ) × SO ( q ) × R pq . As a closing remark observ e that the dimension of all these spaces dep ends only on p + q : It is ( p + q )( p + q − 1) / 2. Ac kno wledgmen t s . I thank Eric King whose incis iv e questions and relen tless quest for the “essence” of rotations ev en tua lly caused a lev el of discomfort high enough to force me to impro ve the clarit y of these notes. R o tations are elusiv e! I also thank: F red Lunnon for man y insighful commen ts and for catc hing a num b er of t yp os; and T roy W o o who also rep orted some t yp os. Bibliograph y [1] Michael Artin. Algebr a . Pren tice Hall, ﬁrst edition, 199 1. [2] M. F. A t iyah and I. G. Macdonald. Intr o duction to Commutative Algebr a . Addison W esley , third e dition, 19 6 9. [3] Michael F A tiyah, Raoul Bott, and Arnold Shapiro . Cliﬀord mo dules. T op olo gy , 3, Suppl. 1 :3–38, 1964. [4] Andrew Bak er. Matrix Gr oups. An Intr o d uction to Lie Gr oup The ory . SUMS. Springer, 2002. [5] Ra oul Bott and T u Loring W. Diﬀ e r ential F o rms in A lg ebr aic T op olo gy . GTM No. 82. Springer V erlag, ﬁrst edition, 1986. [6] Nicolas Bourba ki. Alg ` ebr e, Chapitr e 9 . El ´ emen ts de Math ´ ematiques. Hermann, 1968. [7] T. Br¨ oc k er and T . tom Diec k. R epr esentation of C o m p act Lie Gr oups . GTM, V ol. 98. Springer V erlag, ﬁrst edition, 1985. [8] ´ Elie Carta n. T he ory of Spinors . Dov er, ﬁrst edition, 1966 . [9] Claude Chev alley . The ory of Lie Gr oups I . Princeton Mathematical Series, No. 8. Princeton Unive rsit y Press, ﬁrst edition, 1 946. Eigh th printing. [10] Claude Chev alley . The Algebr aic The o ry of Spinors and Cliﬀor d Algebr as. Col le c te d Works, V ol. 2 . 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T ec hnical rep ort, Univ ersity of P ennsylv a nia , Math. Departmen t, Philadelphia, P A 19104, 2001. Course Notes for Math 654 . [19] An thony W. Knapp. Lie Gr oups Beyond an Intr o duction . Progress in Mathematics, V ol. 14 0 . Birkh¨ auser, second edition, 2 002. [20] Blaine H. Law son and Mar ie- L ouise Miche lsohn. Spin Ge ometry . Princeton Math. Series, No. 38 . Princeton Unive rsit y Press , 1 989. [21] P ertti Lounesto. Cliﬀor d A lgeb r as and Spinors . LMS No. 286. Cambridge Univ ersit y Press, second ed ition, 20 0 1. [22] R. Mneimn ´ e and F. T estard. I ntr o duction ` a la T h´ e orie des Gr o up es de Lie C l a ssiques . Hermann, ﬁrst edition, 1997. [23] Jules Molk. Enc yclop´ e die des Sci e n c es Math´ ematiques Pur es et Appli q u ´ ees. T o m e I (pr emier volume), Arithm ´ etique . Gauthier-Villars, ﬁrst edition, 1916. [24] Ian R . Porteous. T op olo gic al Ge om etry . Cambridge Univ ersit y Press, second edition, 1981. [25] Joseph J. Ro tman. 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Clifford Algebras, Clifford Groups, and a Generalization of the Quaternions

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