Spheres with more than 7 vector fields: all the fault of Spin(9)
SSPHERES WITH MORE THAN 7 VECTOR FIELDS:ALL THE FAULT OF Spin (9)
MAURIZIO PARTON AND PAOLO PICCINNI
Abstract.
We give an interpretation of the maximal number of linearly independent vector fields on spheres interms of the Spin(9) representation on R . This casts an insight on the role of Spin(9) as a subgroup of SO(16)on the existence of vector fields on spheres, parallel to the one played by complex, quaternionic and octonionicstructures on R , R and R , respectively. Contents
1. Introduction 12. Preliminaries 33. The sphere S
64. Spheres S p − , for p = 1 , ,
3, and S
75. Higher dimensional examples: S and S Introduction
The existence of a nowhere zero vector field on odd dimensional spheres S n − ⊂ R n is an elementary con-sequence of the identification R n = C n and of the action of the complex imaginary unit i on the normal vectorfield N . Similarly, on spheres S n − ⊂ R n and S n − ⊂ R n , one gets 3 and 7 tangent orthonormal vector fieldsfrom the identification R n = H n and R n = O n . Here the 3 imaginary units i, j, k of quaternions H and the 7imaginary units i, j, k, e, f, g, h of octonions O are used. These numbers 1 , , S m − ⊂ R m , provided the (even) dimension m of the ambient space is not divisibleby 16.The maximal number σ ( m ) of linearly independent vector fields on S m − is expressed as σ ( m ) = 2 p + 8 q − , where σ ( m ) + 1 = 2 p + 8 q is the Hurwitz-Radon number , referring to the decomposition(1.1) m = (2 k + 1)2 p q , where 0 ≤ p ≤ . See [15], [21] for the original Hurwitz-Radon proof, obtained in the framework of compositions of quadratic forms.See also [10] for a simplified proof, using representation theory of finite groups. Next, [1], [2] and [3] contain theJ. F. Adams’ celebrated theorem stating that σ ( m ) is maximal. Also, [23] is an overview on related problems, [16,Chapters 11 and 15] and [17, Chapter V] are standard references.The much more recent paper [19] contains a combinatorial construction of a maximal system of orthonormalvector fields on spheres and an updated bibliography on the subject. In [19] a method of construction based on permutations of coordinates is developed, generating tangent vector fields by acting on the normal vector throughsuitable monomial matrices , that is, permutations and reflections of the coordinates. We will also proceed throughpermutations of coordinates and monomial matrices, although our main point is, as suggested in the title, to pointout the role of the group Spin(9) in all the dimensions m that allow more than 7 linearly independent vector fieldson S m − .In Table A we list some of the lowest dimensional spheres S m − ⊂ R m admitting a maximal number σ ( m ) > Date : October 19, 2018.2010
Mathematics Subject Classification.
Primary 15B33, 53C27, 57R25.
Key words and phrases.
Spin(9), octonions, vector fields on spheres.Both authors were supported by the MIUR under the PRIN Project “Geometria Differenziale e Analisi Globale”. a r X i v : . [ m a t h . DG ] A p r MAURIZIO PARTON AND PAOLO PICCINNI
Table A.
Some spheres S m − with more than 7 vector fields m − σ ( m ) 8 9 8 11 8 9 8 15 . . . 16 . . . 17 . . . 19 . . . 23 . . .The first of them is S ⊂ R , that turns out to be a homogeneous space of the Lie group Spin(9). The uniqueHopf fibration related to octonions can be written (cf. [12]) in either of the two ways(1.2) S S −→ S , Spin(9)Spin(7)
Spin(8)Spin(7) −→ Spin(9)Spin(8) . Indeed, a construction of 8 orthonormal tangent vector fields on S ⊂ R from the Spin representation ofSpin(9) has been our starting point, although it was not completely clear for a while how such a constructionextends to the next significative case, namely S ⊂ R · . Accordingly, in writing the present paper we choseto postpone the proofs referring to arbitrary dimension after dealing with some “low dimensional” spheres, i.e. S up to S . However, Section 6, which contains the proof of the main statements for arbitrary dimension, isindependent of the previous sections.We have to mention that the framework we are going to use comes from Riemannian geometry in dimension16, that often refers to both the division algebra O of octonions and the Lie group Spin(9). Just to give a coupleof examples, we quote the study of Spin(9) as a weak holonomy group on Riemannian manifolds M [11], andthe construction of exotic manifolds in the Cayley hyperbolic setting [4].Thus, we will consider Spin(9) as a subgroup of the rotation group SO(16), acting on R = O . It is generatedby the block transformations(1.3) (cid:18) xy (cid:19) −→ (cid:18) r R O u R O u − r (cid:19) (cid:18) xy (cid:19) , where ( x, y ) ∈ O , ( r, u ) ∈ S ⊂ R × O = R and R O u , R O u are the right multiplication on the octonions by u, u ,respectively (cf. Section 2 and [13, page 288]).This approach focuses on the set of the nine self-dual involutions I , . . . , I : R −→ R , defined by the nine choices ( r, u ) = (1 , , (0 , , (0 , i ) , . . . (0 , h ) in Formula (1.3). These involutions satisfy thecondition I α I β = −I β I α , ≤ α < β ≤ , (see [13, pages 287–289] and [11], [20]), so that the 36 compositions I α I β are complex structures on R .We will see how the eight complex structures J , . . . , J on R defined by J α def = I α I : R −→ R , α = 1 , . . . , R , R and R , respectively. Indeed, we use these four kind of actions as fundamentalingredients in generating a set of vector fields on spheres of any dimension, see Theorems 6.6, 6.10 and 6.11. Toour knowledge, this role of Spin(9) was never observed before.Our results are briefly collected in Table B, where C t and C are linear operators defined in Remark 6.8, and L i , . . . , L h are left multiplications. It is worth remarking that, as in the permutation of coordinates method, C t ( J · )and C( L · ) are monomial matrices. Table B.
A maximal system of vector fields on S m − ( m = (2 k + 1)2 p q , k ≥ p = 0 , , , q ≥ k, p, q ) Sphere σ ( m ) Vector fields( k, , q ) S (2 k +1)16 q − q { C t ( J α ) } t =1 ,...,qα =1 ,..., ( k, , q ) S k +1)16 q − q + 1 { C t ( J α ) } t =1 ,...,qα =1 ,..., C( L i )( k, , q ) S k +1)16 q − q + 3 { C t ( J α ) } t =1 ,...,qα =1 ,..., C( L i ) , C( L j ) , C( L k )( k, , q ) S k +1)16 q − q + 7 { C t ( J α ) } t =1 ,...,qα =1 ,..., C( L i ) , . . . , C( L h )For simplicity, we write C t ( J ) for C t ( J ) N and C( L ) for C( L ) N , where N is a normal unit vector field on S m − The computations for S and up to S were first made with the help of the software Mathematica , whichwas the heuristic tool to formulate the correct form of the conjectures that became Theorems 6.6, 6.10 and 6.11.Note that in our construction the sphere S ⊂ R plays a basic role. Indeed, S is the lowest dimensionalsphere which admits more than 7 tangent orthonormal vector fields. Also, S is the total space of the three Hopffibrations S S −→ C P , S S −→ H P , S S −→ S . Recall that the first two of them are not subfibrations of the third [18]. However, by writing down the vectorfields tangent to the fibers in the three cases, one sees that no combination of them allows to get the maximalnumber 8 of the orthonormal tangent vector fields on S . Table B shows how the responsibility of such a maximalsystem on S can be ascribed to the Spin(9) structure of R , and more generally it shows also how the same Liegroup Spin(9) produces q − q .On the other hand, one can observe that the space of complex structures on R splits, under the Spin(9) action:Λ ( R ) = Λ ⊕ Λ = spin (9) ⊕ Λ , and that our use in Table B of the particular complex structures J , . . . , J on R is just a possible choice, amongthe many ones, by suitable selections of 8 complex structures in the component spin (9) ⊂ Λ ( R ), the Lie algebraof Spin(9).In Section 2 we introduce specific notations to deal with low-dimensional cases. In Section 3 we explain the S situation as starting point for higher dimensions. In Section 4 we show how S , S , S and S can beseen in this respect as a combination of what obtained on S and of the standard actions of C , H , O . Section 5introduces an iterative construction associated with the decomposition 1.1 of the dimension of the sphere in themost elementary case, that is, S . In Section 6 we introduce the general notation, then we state and prove ourmain statements, Theorems 6.6, 6.10 and 6.11. Acknowledgements.
The authors wish to thank Rosa Gini for her help in developing the argument in Section 6,and the referee for the careful reading of a first draft and for useful comments that lead us to revised proofs of themain statements. 2.
Preliminaries
We briefly recall the Cayley-Dickson process, used to construct new algebras from old ones. Let A be a ∗ -algebra ,namely a real algebra equipped with a linear map ∗ : A → A , called conjugation , satisfying a ∗∗ = a , ( ab ) ∗ = b ∗ a ∗ for all a, b ∈ A . Then a new ∗ -algebra A (cid:48) is defined by A (cid:48) def = { ( a, b ); a, b ∈ A} , ( a, b )( c, d ) def = ( ac − d ∗ b, da + bc ∗ ) and ( a, b ) ∗ def = ( a ∗ , − b ) . This construction produces the algebra C from R (the linear map ∗ on the latter being the identity), then H from C and O from H . We choose the standard canonical bases { , i } , { , i, j, k } and { , i, j, k, e, f, g, h } for C , H and O respectively. In particular, we use the following multiplication table in O , where the left factor is in the firstcolumn. Table C.
Multiplication in O i j k e f g hi − k − j f − e − h gj − k − i g h − e − fk j − i − h − g f − ee − f − g − h − i j kf e − h g − i − − k jg h e − f − j k − − ih − g f e − k − j i − x = h + h e , y = k + k e ∈ O can be also viewed through the multiplication and the conjugation in H by the formula(2.1) xy = ( h k − k h ) + ( k h + h k ) e . Note also that the conjugation in O , defined as x def = h − h e , gives the non-commutativity law xy = ¯ y ¯ x . Lemma 2.1.
Let A n be the Cayley-Dickson algebra obtained inductively from A = R through the Cayley-Dicksonprocess. Denote by a ∗ and by (cid:60) ( a ) def = ( a + a ∗ ) the conjugate and the real part of elements a ∈ A n , respectively. MAURIZIO PARTON AND PAOLO PICCINNI
Denote then by [ a, b, c ] def = ( ab ) c − a ( bc ) the associator of a, b, c ∈ A n , and by < a, b > the scalar product in A n = R n . Then the following formulas hold good for all a, b, c ∈ A n : (2.2) ( ab ) ∗ = b ∗ a ∗ , (cid:60) ([ a, b, c ]) = 0 , < a, b > = (cid:60) ( ab ∗ ) . Proof.
The first and third formulas are verified by induction on n , the number of steps in the Cayley-Dicksonprocess.To check the second formula, note that by linearity one can assume any of a, b, c to be of the form ( x,
0) or (0 , x ),with x ∈ A n − . Thus there are eight cases to be verified. Four of these cases lead to both products ( ab ) c, a ( bc )of type (0 , x ), hence with zero real part. The case a = ( x, , b = ( y, , c = ( z,
0) is done by induction on n .The remaining three cases are when two among a, b, c are of the form ( x,
0) and the third of the form (0 , x ). Forexample, when a = ( x, , b = (0 , y ) , c = (0 , z ) one gets (cid:60) ([ a, b, c ]) = (cid:60) ( x ( z ∗ y ) − z ∗ ( yx )) = (cid:60) (( z ∗ y ) x − z ∗ ( yx )) andan inductive argument gives the conclusion. (cid:3) We denote by L H , R H and L O , R O the left and right multiplication in H and O , respectively. Explicit matrixrepresentations for the right and left multiplication by i, j, k in H are R H i = − − , R H j = − −
11 0 0 00 1 0 0 , R H k = −
10 0 1 00 − , (2.3) L H i = − −
10 0 1 0 , L H j = − − , L H k = −
10 0 − . (2.4)Although, as well-known, C , H and O are the only normed algebras over R , we will use for our first examplesalso the algebra S of sedenions, obtained from O through the Cayley-Dickson process. Some characterization of S has been given very recently in the context of locally complex algebras , cf. [8]. For further informations on S , seealso [6] and [7].Denoting by 1 , e , . . . , e the canonical basis of S over R , we can write the multiplication table D, where itappears the existence of divisors of zeroes in S : for example ( e − e )( e + e ) = 0. Table D.
Multiplication in S e e e e e e e e e e e e e e e e − e − e e − e − e e e − e − e e − e e e − e e − e − e e e − e − e e e − e − e − e − e e e e e − e − e − e e − e e − e e − e − e e − e e e − e − e − e − e e e e e e e − e − e − e − e e e − e e − e − − e e e − e e − e e − e e − e e e e − e − e e − − e e − e − e e e − e − e e e − e e e − e − e e − e e − e − e e e − e − e e − e − e − e − e − e − e − e − e e e e e e e e e − e e − e e e − e − e − − e e − e e e − e e e e − e − e − e e e − e e − − e − e − e e e e − e e e − e e − e e − e − e e − − e e − e e e e e e e − e − e − e − e e e e − − e − e − e e − e e − e e e e − e − e − e e − e e − e − e e − e − e e e − e e e − e − e − e e e − e − e e e − e − e e e − e e − e e − e − e e e − e − S m − ⊂ R m , and decompose m as m = (2 k + 1)2 p q , where p ∈ { , , , } . First, weobserve that a vector field B tangent to the sphere S p q − ⊂ R p q induces a vector field(2.5) ( B, . . . , B ) (cid:124) (cid:123)(cid:122) (cid:125) k +1 times tangent to the sphere S (2 k +1)2 p q − . For this reason, we will assume in what follows that m = 2 p q . Wheneverwe extend a vector field in this way, we call the vector field given by (2.5) the diagonal extension of B .If q = 0, that is, if m is not divisible by 16, the vector fields on S m − are given by the complex, quaternionic oroctonionic multiplication for p = 1 , q ≥ m = 16 l . In this case, it is convenient to denote the coordinates in R l by ( s , . . . , s l ) where each s α , for α = 1 , . . . , l , belongs to S , and can thus be identified with a pair ( x α , y α ) of octonions. PHERES WITH MORE THAN 7 VECTOR FIELDS: ALL THE FAULT OF Spin(9) 5
The unit (outward) normal vector field N of S m − is still denoted by N def = ( s , . . . , s l ) where (cid:107) s (cid:107) + · · · + (cid:107) s l (cid:107) = 1 . Therefore, we can think of N as an element of S l = ( O ) l = R l .Whenever l = 2 , S l = ( O ) l given by(2.6) D : (( x , y ) , . . . , ( x l , y l )) −→ (( x , − y ) , . . . , ( x l , − y l )) . We will refer to D as a conjugation , due to its similarity with that in ∗ -algebras.Moreover, it is convenient to use formal notations as: N = ( s , s ) def = s + is ∈ S , (2.7) N = ( s , s , s , s ) def = s + is + js + ks ∈ S , (2.8) N = ( s , s , s , s , s , s , s , s ) def = s + is + js + ks + es + f s + gs + hs ∈ S , (2.9)allowing to define left multiplications L in sedenionic spaces S , S and S (like in C , H and O ) as follows.If l = 2 the left multiplication is(2.10) L i ( s , s ) def = − s + is , whereas if l = 4 we define L i ( s , . . . , s ) def = − s + is − js + ks ,L j ( s , . . . , s ) def = − s + is + js − ks ,L k ( s , . . . , s ) def = − s − is + js + ks , (2.11)and finally if l = 8 we define L i ( s , . . . , s ) def = − s + is − js + ks − es + f s + gs − hs ,L j ( s , . . . , s ) def = − s + is + js − ks − es − f s + gs + hs ,L k ( s , . . . , s ) def = − s − is + js + ks − es + f s − gs + hs ,L e ( s , . . . , s ) def = − s + is + js + ks + es − f s − gs − hs ,L f ( s , . . . , s ) def = − s − is + js − ks + es + f s + gs − hs ,L g ( s , . . . , s ) def = − s − is − js + ks + es − f s + gs + hs ,L h ( s , . . . , s ) def = − s + is − js − ks + es + f s − gs + hs . (2.12)Note that in all three cases l = 2 , L i ( N ) , . . . , L h ( N ) are tangent to S , S and S ,respectively.In accordance with its role in the Hopf fibration (1.2), Spin(9) can be also defined as the subgroup of SO(16)preserving the decomposition of O in octonionic lines l c = { ( x, cx ) } and l ∞ = { (0 , y ) } , where x, c, y ∈ O (see [12]).In the framework of G -structures, a Spin(9) -structure on a Riemannian manifold M can be defined as a rank 9vector subbundle V ⊂ End(
T M ), locally spanned by nine endomorphisms I α satisfying the following conditions:(2.13) I α = Id , I ∗ α = I α , I α I β = −I β I α if α (cid:54) = β , where I ∗ α denotes the adjoint of I α (cf. the Introduction as well as [11]).For M = R , I , . . . , I are generators of the Clifford algebra Cl(9), considered as endomorphisms of its 16-dimensional real representation ∆ = R = O . Accordingly, unit vectors v ∈ S ⊂ R can be looked at, via theClifford multiplication, as symmetric endomorphisms v : ∆ → ∆ . As seen in the Introduction, the explicit wayto describe this is by v = r + u ∈ S ⊂ R × O (that is, r ∈ R , u ∈ O and r + u ¯ u = 1) acting on pairs ( x, y ) ∈ O by(2.14) (cid:18) xy (cid:19) −→ (cid:18) r R O u R O u − r (cid:19) (cid:18) xy (cid:19) , where R O u and R O u denote the right multiplication on the octonions by u and u , respectively (cf. [13, page 280]). MAURIZIO PARTON AND PAOLO PICCINNI
A basis of the standard Spin(9) structure on S = O can thus be written by looking at (2.14) and at the ninevectors (0 , , (0 , i ) , . . . , (0 , h ) , (1 , ∈ S ⊂ R × O . It consists of the following symmetric endomorphisms:(2.15) I = (cid:18) (cid:19) , I = (cid:18) − R O i R O i (cid:19) , I = (cid:32) − R O j R O j (cid:33) , I = (cid:18) − R O k R O k (cid:19) , I = (cid:18) − R O e R O e (cid:19) , I = (cid:32) − R O f R O f (cid:33) , I = (cid:18) − R O g R O g (cid:19) , I = (cid:18) − R O h R O h (cid:19) , I = (cid:18) Id 00 − Id (cid:19) . The right octonionic multiplications R O i , . . . , R O h can be written as 8 × R O i = (cid:18) R H i − R H i (cid:19) , R O j = (cid:18) R H j − R H j (cid:19) , R O k = (cid:18) R H k − R H k (cid:19) ,R O e = (cid:18) − IdId 0 (cid:19) , R O f = (cid:18) L H i L H i (cid:19) ,R O g = (cid:18) L H j L H j (cid:19) , R O h = (cid:18) L H k L H k (cid:19) . The space Λ R of 2-forms in R decomposes under Spin(9) asΛ R = Λ ⊕ Λ (cf. [11, page 146]), where Λ ∼ = spin (9) and Λ is an orthogonal complement in Λ R ∼ = so (16). Explicitgenerators of both subspaces can be written by looking at the 36 compositions J αβ def = I α I β , for α < β , and at the84 compositions J αβγ def = I α I β I γ , for α < β < γ , all complex structures on R .Among the 36 complex structures J αβ , in this paper we use only the eight J α def = J α , whose matrix form is:(2.17) J = (cid:18) − IdId 0 (cid:19) , J = (cid:18) R O i R O i (cid:19) , J = (cid:32) R O j R O j (cid:33) , J = (cid:18) R O k R O k (cid:19) ,J = (cid:18) R O e R O e (cid:19) , J = (cid:32) R O f R O f (cid:33) , J = (cid:18) R O g R O g (cid:19) , J = (cid:18) R O h R O h (cid:19) . For the matrices associated with the remaining J αβ , where 1 ≤ α < β ≤
8, see [20].
Remark . Of course, one can see the complex structures J α as defined by (2.17) on the algebra S of sedenions.In the following Sections 3, 4 and 5 we use the same symbol J α to denote their diagonal extensions to S l , as in(2.5), so that if N = ( s , . . . , s l ) we have J α N def = ( J α s , . . . , J α s l ). (cid:3) Remark . As a matter of notations, it is worth to mention that, throughout the paper, letters i, j, k, e, f, g, h denote only units in C , H and O . Instead, indexes are denoted by greek letters α, β, . . . . (cid:3) The sphere S Denote by N def = ( x, y ) def = ( x , . . . , x , y , . . . , y )the (outward) unit normal vector field of S ⊂ R = S . We point out that the existence of zero divisors in thealgebra S of sedenions infers that left multiplications by the sedenions unities e α , for α = 1 , . . . ,
15, give rise to15 vector fields that are not linearly independent . For example, consider the normal vector (cid:126)b = √ e + √ e : bylooking at Table D one sees that e (cid:126)b = e (cid:126)b .On the other hand, the identification R = O gives on S the 7 tangent vector fields L O i B def = ( L O i x, L O i y ) = ( − x , x , − x , x , − x , x , x , − x , − y , y , − y , y , − y , y , y , − y ) ,L O j B def = ( L O j x, L O j y ) = ( − x , x , x , − x , − x , − x , x , x , − y , y , y , − y , − y , − y , y , y ) ,L O k B def = ( L O k x, L O k y ) = ( − x , − x , x , x , − x , x , − x , x , − y , − y , y , y , − y , y , − y , y ) ,L O e B def = ( L O e x, L O e y ) = ( − x , x , x , x , x , − x , − x , − x , − y , y , y , y , y , − y , − y , − y ) ,L O f B def = ( L O f x, L O f y ) = ( − x , − x , x , − x , x , x , x , − x , − y , − y , y , − y , y , y , y , − y ) ,L O g B def = ( L O g x, L O g y ) = ( − x , − x , − x , x , x , − x , x , x , − y , − y , − y , y , y , − y , y , y ) ,L O h B def = ( L O h x, L O h y ) = ( − x , x , − x , − x , x , x , − x , x , − y , y , − y , − y , y , y , − y , y ) , (3.1) PHERES WITH MORE THAN 7 VECTOR FIELDS: ALL THE FAULT OF Spin(9) 7 spanning the distribution of vertical leaves in the Hopf fibration S → S . Thus, any other vector field on S ⊂ R orthogonal to the 7 listed in (3.1) should belong to the horizontal distribution of the Hopf fibration.To work out a construction of 8 orthonormal vector fields on S , consider (as a possible choice among the 36complex structures I α I β , α < β ) the eight complex structures given in (2.17): J , . . . , J : S −→ S . Proposition 3.1.
The vector fields J N =( − y , − y , − y , − y , − y , − y , − y , − y , x , x , x , x , x , x , x , x ) ,J N =( − y , y , y , − y , y , − y , − y , y , − x , x , x , − x , x , − x , − x , x ) ,J N =( − y , − y , y , y , y , y , − y , − y , − x , − x , x , x , x , x , − x , − x ) ,J N =( − y , y , − y , y , y , − y , y , − y , − x , x , − x , x , x , − x , x , − x ) ,J N =( − y , − y , − y , − y , y , y , y , y , − x , − x , − x , − x , x , x , x , x ) ,J N =( − y , y , − y , y , − y , y , − y , y , − x , x , − x , x , − x , x , − x , x ) ,J N =( − y , y , y , − y , − y , y , y , − y , − x , x , x , − x , − x , x , x , − x ) ,J N =( − y , − y , y , y , − y , − y , y , y , − x , − x , x , x , − x , − x , x , x )(3.2) are tangent to S and orthonormal. This comes indeed as a special case of the following slightly more general proposition.
Proposition 3.2.
Fix any β , ≤ β ≤ , and consider the complex structures I α I β , with α (cid:54) = β . Then the vector fields I α I β N are tangent to S and orthonormal.Proof. It is sufficient to use properties (2.13) of the symmetric endomorphisms I α . (cid:3) Spheres S p − , for p = 1 , , , and S In this Section, we write explicitly maximal systems of vector fields for S , S , S . The proof that any ofthese systems is orthonormal is straightforward. We will also explain the S case. Case p = The next sphere having more than 7 tangent vector fields is S , whose maximal number is 9. In thiscase, we obtain 8 vector fields by writing the unit normal vector field as N = ( s , s ) = ( x , y , x , y ) ∈ S ⊂ S ,where x , y , x , y ∈ O , and repeating Formulas (3.2) for each pair ( x , y ) , ( x , y ): J N = ( J s , J s )= ( − y , − y , . . . , − y , − y , x , x , . . . , x , x , − y , − y , . . . , − y , − y , x , x , . . . , x , x ) ,J N = ( J s , J s )= ( − y , y , . . . , − y , y , − x , x , · · · − x , x , − y , y , . . . , − y , y , − x , x , . . . , − x , x ) ,J N = ( J s , J s )= ( − y , − y , . . . , − y , − y , − x , − x , . . . , − x , − x , − y , − y , . . . , − y , − y , − x , − x , . . . , − x , − x ) ,J N = ( J s , J s )= ( − y , y , . . . , y , − y , − x , x , . . . , x , − x , − y , y , . . . , y , − y , − x , x , . . . , x , − x ) ,J N = ( J s , J s )= ( − y , − y , . . . , y , y , − x , − x , . . . , x , x , − y , − y , . . . , y , y , − x , − x , . . . , x , x ) ,J N = ( J s , J s )= ( − y , y , . . . , − y , y , − x , x , . . . , − x , x , − y , y , . . . , − y , y , − x , x , . . . , − x , x ) ,J N = ( J s , J s )= ( − y , y , . . . , y , − y , − x , x , . . . , x , − x , − y , y , . . . , y , − y , − x , x , . . . , x , − x ) ,J N = ( J s , J s )= ( − y , − y , . . . , y , y , − x , − x , . . . , x , x , − y , − y , . . . , y , y , − x , − x , . . . , x , x ) . (4.1)A ninth orthonormal vector field, completing the maximal system, is found using the formal left multiplica-tion (2.10) and the automorphism D (2.6):(4.2) D( L i N ) = D( − s , s ) = ( − x , y , x , − y ) . MAURIZIO PARTON AND PAOLO PICCINNI
Case p = The sphere S has a maximal number of 11 orthonormal vector fields. The normal vector fieldis in this case given by N = ( s , . . . , s ) = ( x , y , . . . , x , y ) ∈ S ⊂ S , and 8 vector fields arise as J α N , for α = 1 , . . . ,
8. Three other vector fields are again given by the formal left multiplications 2.11 and the automorphismD (2.6): D( L i N ) = ( − x , y , x , − y , − x , y , x , − y ) , D( L j N ) = ( − x , y , x , − y , x , − y , − x , y ) , D( L k N ) = ( − x , y , − x , y , x , − y , x , − y ) . (4.3) Case p = The sphere S has a maximal number of 15 orthonormal vector fields. Eight of them are still givenby J α N , for α = 1 , . . . ,
8, whereas the formal left multiplications given in 2.12 yield the 7 tangent vector fieldsD( L α N ), for α ∈ { i, . . . , h } . Remark . Up to dimension 127 the vector fields were built through the following construction: the first 8 werethe (diagonal extension of the) J α , and the next 1 , C , H or O actions by leftmultiplications on blocks of 16 coordinates, as in Formulas (2.10), (2.11), (2.12). We call those actions the blockextensions of the original actions, and in what follows we denote the extension of A by block( A ). (cid:3) The sphere S . To write a system of 16 orthonormal vector fields on S ⊂ R consider the decomposition(4.4) R = R ⊕ · · · ⊕ R , and observe that both the number and the dimension of components are sixteen. The unit outward normal vectorfield can be written as N = ( s , . . . , s ) , where s , . . . , s are sedenions.Consider now the 16 ×
16 matrices giving the complex structures J , . . . , J , and listed as (2.17). They act on N not only on the 16-dimensional components of (4.4), but also formally on the (column) 16-ples of sedenions( s , . . . , s ) T . According to which of the two actions of the same matrices are considered in R , we use thenotations J , . . . , J or block( J ) , . . . , block( J )for the obtained complex structures on R . The following 16 vector fields are obtained: J N , . . . , J N , (4.5) D(block( J ) N ) , . . . , D(block( J ) N ) , (4.6)where D has been defined in Formula (2.6). We call level vector fields and level vector fields the ones given by(4.5) and (4.6) respectively.We checked by Mathematica that (4.5) and (4.6) give an orthonormal (maximal) system, but we give now anelementary algebraic proof.
Proposition 4.2.
Formulas (4.5) and (4.6) give a maximal system of orthonormal tangent vector fields on S .Proof. As in Section 2, we denote sedenions as pairs s α def = ( x α , y α ) of octonions. The unit normal vector field is(4.7) N = ( s , . . . , s ) = ( x , y , . . . , x , y ) ∈ S , and one gets the tangent vectors: J N = ( J s , . . . , J s ) = ( − y , x , . . . , − y , x ) ,J N = ( J s , . . . , J s ) = ( R O i y , R O i x , . . . , R O i y , R O i x ) ,J N = ( J s , . . . , J s ) = ( R O j y , R O j x , . . . , R O j y , R O j x ) ,J N = ( J s , . . . , J s ) = ( R O k y , R O k x , . . . , R O k y , R O k x ) ,J N = ( J s , . . . , J s ) = ( R O e y , R O e x , . . . , R O e y , R O e x ) ,J N = ( J s , . . . , J s ) = ( R O f y , R O f x , . . . , R O f y , R O f x ) ,J N = ( J s , . . . , J s ) = ( R O g y , R O g x , . . . , R O g y , R O g x ) ,J N = ( J s , . . . , J s ) = ( R O h y , R O h x , . . . , R O h y , R O h x ) , (4.8)that are easily checked to be orthonormal. PHERES WITH MORE THAN 7 VECTOR FIELDS: ALL THE FAULT OF Spin(9) 9
Moreover, one obtains eight further vector fields:D(block( J ) N ) = D( − s , − s , − s , − s , − s , − s , − s , − s , s , s , s , s , s , s , s , s )= ( − x , y , − x , y , − x , y , − x , y , − x , y , − x , y , − x , y , − x , y ,x , − y , x , − y , x , − y , x , − y , x , − y , x , − y , x , − y , x , − y ) , D(block( J ) N ) = D( − s , s , s , − s , s , − s , − s , s , − s , s , s , − s , s , − s , − s , s )= ( − x , y , x , − y , x , − y , − x , y , x , − y , − x , y , − x , y , x , − y , − x , y , x , − y , x , − y , − x , y , x , − y , − x , y , − x , y , x , − y ) , D(block( J ) N ) = D( − s , − s , s , s , s , s , − s , − s , − s , − s , s , s , s , s , − s , − s )= ( − x , y , − x , y , x , − y , x , − y , x , − y , x , − y , − x , y , − x , y , − x , y , − x , y , x , − y , x , − y , x , − y , x , − y , − x , y , − x , y ) , D(block ( J ) N ) = D( − s , s , − s , s , s , − s , s , − s , − s , s , − s , s , s , − s , s , − s )= ( − x , y , x , − y , − x , y , x , − y , x , − y , − x , y , x , − y , − x , y , − x , y , x , − y , − x , y , x , − y , x , − y , − x , y , x , − y , − x , y ) , D(block( J ) N ) = D( − s , − s , − s , − s , s , s , s , s , − s , − s , − s , − s , s , s , s , s )= ( − x , y , − x , y , − x , y , − x , y , x , − y , x , − y , x , − y , x , − y , − x , y , − x , y , − x , y , − x , y , x , − y , x , − y , x , − y , x , − y ) , D(block( J ) N ) = D( − s , s , − s , s , − s , s , − s , s , − s , s , − s , s , − s , s , − s , s )= ( − x , y , x , − y , − x , y , x , − y , − x , y , x , − y , − x , y , x , − y , − x , y , x , − y , − x , y , x , − y , − x , y , x , − y , − x , y , x , − y ) , D(block( J ) N ) = D( − s , s , s , − s , − s , s , s , − s , − s , s , s , − s , − s , s , s , − s )= ( − x , y , x , − y , x , − y , − x , y , − x , y , x , − y , x , − y , − x , y , − x , y , x , − y , x , − y , − x , y , − x , y , x , − y , x , − y , − x , y ) , D(block( J ) N ) = D( − s , − s , s , s , − s , − s , s , s , − s , − s , s , s , − s , − s , s , s )= ( − x , y , − x , y , x , − y , x , − y , − x , y , − x , y , x , − y , x , − y , − x , y , − x , y , x , − y , x , − y , − x , y , − x , y , x , − y , x , − y ) , (4.9)similarly vefified to be orthonormal.To see that each vector J α N is orthogonal to each D(block( J β ) N ), for α, β = 1 , . . . ,
8, look at Formulas (2.16)for R O i , . . . , R O h and write the octonionic coordinates as x λ = h λ + h λ e , y µ = k µ + k µ e . Then the scalar product < J α N, D(block( J β ) N ) > can be computed by using Formula (2.1) for product of octonions. For example, recallfrom Formulas (4.8) that J N = ( y h, x h, . . . , y h, x h, y h, x h, . . . , y h, x h ) , so that the computation of < J N, D(block( J ) N ) > gives rise to pairs of terms like in < J N, D(block( J ) N ) > = (cid:60) ( − ( R O h y ) x − ( R O h x ) y + . . . ) = (cid:60) ( − kk h − h kk − kh k − k kh + . . . ) . To conclude, observe that the real part (cid:60) of the sums of each of the corresponding underlined terms is zero. Thisis due to the identity (cid:60) ( hh (cid:48) h (cid:48)(cid:48) ) = (cid:60) ( h (cid:48) h (cid:48)(cid:48) h ), that holds for all h, h (cid:48) , h (cid:48)(cid:48) ∈ H . (cid:3) Remark . One can recognize that the final argument in the above proof uses the last formula in 2.1. The secondformula in the same Lemma is also useful to check orthogonality in higher dimensions, although in Section 6 wewill follow a different procedure.The same argument shows in fact a slightly more general statement:
Proposition 4.4.
Fix any β , ≤ β ≤ , and consider the complex structures I α I β , with α (cid:54) = β , defined on R = R ⊕ · · · ⊕ R by acting with the corresponding matrices on the listed -dimensional components, thatis, by the diagonal extension of I α I β . Consider also the further complex structures D(block( I α I β )) , for α (cid:54) = β ,defined by the same matrices, now acting on the column matrix ( s , . . . s ) T of sedenions. Then {I α I β N, D(block( I α I β ) N ) } α (cid:54) = β is a maximal system of orthonormal tangent vector fields on S . Higher dimensional examples: S and S The dimension m = 2 · , that is, S , is the lowest case where a last ingredient of our general constructionenters the scene. When q increases by 1 in the decomposition m = (2 k + 1)2 p q , in fact, the conjugation issomehow twisted, as we will see in the following.Imitating what we have done up to now leads to consider the diagonal extensions of J α N (that is, level 1 vectorfields) and the diagonal extensions of D(block( J α ) N ) (that is, level 2 vector fields), together with one more vectorfield.To define this additional vector field we need to extend the formal left multiplication defined by Formula (2.10).To this aim, consider the decomposition R · = R ⊕ R and denote now by s , s elements in R . Then use the formal notation N = ( s , s ) def = s + is ∈ S · − , (5.1)and define a formal left multiplication L i in R · using Formula (2.10). One could then expect that D( L i N ) beorthogonal to { J α N, D(block( J α ) N ) } α =1 ,..., , but this is not the case. In fact, D( L i N ) appears to be orthogonalto level 1 vector fields, but not to level 2 vector fields. Why this happens, will be clear in Section 6.To make everything work, we need to extend not only L i , but also the conjugation D . To this aim, split elements s α ∈ R as ( x α , y α ) where x α , y α ∈ R / , and define a conjugation D on R using Formula (2.6):(5.2) D : (( x , y ) , ( x , y )) −→ (( x , − y ) , ( x , − y )) . The additional vector field we were looking for turns out to be D(D ( L i N )): Theorem 5.1.
A maximal orthonormal system of tangent vector fields on S · − is given by the following · vector fields: J N , . . . , J N ,
D(block( J ) N ) , . . . , D(block( J ) N ) , D(D ( L i N ))(5.3) Proof.
This statement was first verified through a
Mathematica computation, like the statements for higher di-mensions entering in the following Remark. For the mathematical proof, we refer to the more general argumentsin Section 6. (cid:3)
Remark . In a completely similar way, we obtain a maximal orthonormal system of tangent vector fields on S − , S − and S − . For instance, the maximal system on S − is given by: J N , . . . , J N (level 1) , D(block( J ) N ) , . . . , D(block( J ) N ) (level 2) , D(D (block( J ) N ) , . . . , D(D (block( J ) N ) (level 3) , (5.4)where diagonal extensions have been used for D and D .6. The general case
In this Section, we will give a (maximal) orthonormal system of vector fields on any odd-dimensional sphere.To this aim, for any even m ∈ N , we identify (linear) vector fields on S m − with skew-symmetric m × m matrices,that is, with elements of so ( m ). The orthogonality condition between vector fields A, B ∈ so ( m ) turns then into AB + BA = 0, and a vector field A ∈ so ( m ) has length 1 if and only if A = − Id m . In this way, we get rid of thenormal vector field N used in Sections up to 5.Let A = ( a αβ ) α,β =1 ,...,m ∈ Mat m be an m × m matrix, and let diag m,n , block m,n : Mat m → Mat mn be givenrespectively by diag m,n ( A ) def = A . . . A , block m,n ( A ) def = ( a αβ Id n ) α,β =1 ,...,m Thus, diag m,n ( A ) is defined by n blocks m × m , whereas block m,n ( A ) is defined by m blocks n × n . The diag m,n and block m,n operators formalize the blockwise extension of the action of A on R m to R mn , seen as ( R m ) n and( R n ) m respectively. In particular diag m,n is what we have called the diagonal extension, and block m,n is what wehave called the block extension in the previous sections.For instance, if J is the first matrix given in Formula (2.17), thendiag , ( J ) = (cid:18) J J (cid:19) and block , (cid:18) −
11 0 (cid:19) = (cid:18) − Id Id (cid:19) are nothing but the matrix of the linear operators J defined as first member in Formula (4.1) and L i defined inFormula (2.10) respectively. PHERES WITH MORE THAN 7 VECTOR FIELDS: ALL THE FAULT OF Spin(9) 11
Remark . In this section, J , . . . , J denote always the 16 dimensional matrices given by Formulas (2.17).Observe also that any A ∈ so ( m ) induces two vector fields in any dimension multiple of m : diag m,n ( A ) andblock m,n , both in so ( mn ).Next Lemma collects properties of diag and block that we will need. Lemma 6.2. (1) If m ≤ m (cid:48) and mn = m (cid:48) n (cid:48) , then diag m,n = diag m (cid:48) ,n (cid:48) ◦ diag m, m (cid:48) m . (2) diag and block are algebra homomorphisms. (3) Let A = diag m,n ( A (cid:48) ) and B = block n,m ( B (cid:48) ) . Then AB = BA . (4) diag and block commute: diag lm,n ◦ block l,m = block ln,m ◦ diag l,n . Proof.
Proof of points 1 and 4 reduce to the definition of diag and block. Point 2 reduces to the fact that sumand multiplication of block matrices respect the block decomposition. As for 3, if B (cid:48) = ( b αβ ) α,β =1 ,...,n , we have: AB = A (cid:48) . . . A (cid:48) b Id m . . . b n Id m ... · · · ... b n Id m . . . b nn Id m = A (cid:48) b Id m . . . A (cid:48) b n Id m ... · · · ... A (cid:48) b n Id m . . . A (cid:48) b nn Id m = b Id m A (cid:48) . . . b n Id m A (cid:48) ... · · · ... b n Id m A (cid:48) . . . b nn Id m A (cid:48) = BA (cid:3) Remark that for commutation to work in Lemma 6.2(3), m and n must be swapped.Using the diag and block operators, we can now formalize the conjugation used in the previous sections. Definition 6.3.
Let s ∈ N . ThenC def = (cid:18) − (cid:19) ∈ Mat , D s def = block , s (C) ∈ Mat s . Remark . The matrix D s swaps the signs of the last s coordinates of a vector in R s . Using this notation,the matrix of the conjugation D defined in Formula (2.6) becomes diag ,l (D ), and the matrix of the conjugationD defined in Formula (5.2) becomes diag , (D ). (cid:3) The basic block of our construction is the case m = 16 q , and this is done in the next subsection (Theorem 6.6).The case m = 2 p q for p = 1 , , m = (2 k + 1)2 p q will follow (Theorem 6.11). All the lemmas used in these 3 theorems are collected at the endof the paper. The case S q − , for q ≥ Let s, t ∈ N , where t ≥ s = 1 , . . . , t −
1. ThenD t,s def = diag s , t − s (D s ) ∈ Mat t . With this notation, the cases q = 1 , , • If q = 1, a maximal system is given by vector fields of level 1, that is { J , . . . , J } . • If q = 2, we have vector fields of level 1 given by diag , ( { J , . . . , J } ), and vector fields of level 2 givenby D , block , ( { J , . . . , J } ). • If q = 3, we have level 1 vector fields given by diag , ( { J , . . . , J } ) and level 2 vector fields givenby diag , (D , block , ( { J , . . . , J } )). Moreover, we have 8 further level 3 vector fields given byD , D , block , ( { J , . . . , J } ).Denoting the product of conjugations by(6.1) Mat t (cid:51) C t def = (cid:40) Id if t = 1 , (cid:81) t − s =1 D t,s if t ≥ , we can state the general theorem for S q − . Theorem 6.6.
For any q ≥ , the q vector fields on S q − given by { B q ( t, J α ) def = diag t , q − t (C t block , t − ( J α )) } t =1 ,...,qα =1 ,..., are a maximal orthonormal set. Proof. If q = 1, then B q ( t, J α ) = J α and the statement reduces to Proposition 3.2. Thus, assume that q ≥ B q ( t, J ) , B q ( t (cid:48) , J (cid:48) ), and assume with no loss of generality that t ≤ t (cid:48) . Then, observe that B q ( t, J ) B q ( t (cid:48) , J (cid:48) ) = diag t , q − t (C t block , t − ( J ))diag t (cid:48) , q − t (cid:48) (C t (cid:48) block , t (cid:48)− ( J (cid:48) )) = diag t (cid:48) , q − t (cid:48) (diag t , t (cid:48)− t (C t block , t − ( J ))C t (cid:48) block , t (cid:48)− ( J (cid:48) )) . Thus, it is enough to consider the case t (cid:48) = q . We are then reduced to show that, for any q ≥ J, J (cid:48) ∈{ J , . . . , J } :(1) B q ( t, J ) B q ( q, J (cid:48) ) + B q ( q, J (cid:48) ) B q ( t, J ) = 0 for 1 ≤ t ≤ q and ( t, J ) (cid:54) = ( q, J (cid:48) );(2) B q ( q, J ) is an almost complex structure.We divide the proof of (1) in the cases 1 ≤ t ≤ q − t = q .If 1 ≤ t ≤ q −
1, we have B q ( t, J ) B q ( q, J (cid:48) ) = B q ( t, J )C q block , q − ( J (cid:48) )= B q ( t, J ) q − (cid:89) s =1 D q,s block , q − ( J (cid:48) ) = q − (cid:89) s =1 s (cid:54) = t D q,s B q ( t, J )D q,t block , q − ( J (cid:48) ) = − q − (cid:89) s =1 D q,s B q ( t, J )block , q − ( J (cid:48) ) = − q − (cid:89) s =1 D q,s block , q − ( J (cid:48) ) B q ( t, J ) = − B q ( q, J (cid:48) ) B q ( t, J ) . If t = q and J (cid:54) = J (cid:48) , we have B q ( q, J ) B q ( q, J (cid:48) ) =C q block , q − ( J )C q block , q − ( J (cid:48) ) = C q C q block , q − ( J )block , q − ( J (cid:48) ) = C q C q block , q − ( JJ (cid:48) ) J ⊥ J (cid:48) = C q C q block , q − ( − J (cid:48) J ) = − C q block , q − ( J (cid:48) )C q block , q − ( J ) = − B q ( q, J (cid:48) ) B q ( q, J ) . Finally, the case t = q and J = J (cid:48) . We have to show that B q ( q, J ) is an almost complex structure. We have B q ( q, J ) B q ( q, J ) = C q block , q − ( J )C q block , q − ( J ) = C q C q block , q − ( J ) = − Id q . (cid:3) Remark . From the proof, it appears that for any t = 1 , . . . , q −
1, the level t conjugation D q,t makes level q vector fields B ( q, J α ) orthogonal to level t vector fields B ( t, J β ). (cid:3) Remark . Overloading the symbols C t and C, the linear operators used in Table B in the Introduction can nowbe defined by C t : Mat −→ Mat (2 k +1)2 p q A (cid:55)−→ diag t , (2 k +1)2 p q − t (C t block , t − ( A ))and by C : Mat p −→ Mat (2 k +1)2 p q A (cid:55)−→ diag p q , k +1 (diag q , p (C q )block p , q ( A )) . (cid:3) The case S p q , for p = , , When p = 1 , , L C , L H or L O respectively. Thus, to state the Theoremwe give the following definition. Definition 6.9. G def = { L C i } ⊂ Mat , G def = { L H i , L H j , L H k } ⊂ Mat , G def = { L O i , L O j , L O k , L O e , L O f , L O g , L O h } ⊂ Mat . PHERES WITH MORE THAN 7 VECTOR FIELDS: ALL THE FAULT OF Spin(9) 13
Theorem 6.10.
For any q ≥ and p = 1 , or , the q + 2 p − vector fields on S p q − given by { B p,q ( t, J α ) def = diag q , p ( B q ( t, J α )) } t =1 ,...,qα =1 ,..., { L p,q ( G ) def = diag q , p (C q D q )block p , q ( G ) } G ∈G p are a maximal orthonormal system.Proof. The orthonormality for { L p,q ( G ) } G ∈G p is a direct consequence of Lemma 6.2(3)(2), the orthonormality of G p , and the fact that diag q , p (C q D q )diag q , p (C q D q ) = Id p q .The orthonormality for { B p,q ( t, J α ) } t =1 ,...,qα =1 ,..., follows from Theorem 6.6.We are then left to show that B p,q ( t, J α ) L p,q ( G ) + L p,q ( G ) B p,q ( t, J α ) = 0, for t = 1 , . . . , q and α = 1 , . . . , B p,q ( t, J α ) L p,q ( G ) = diag q , p ( B q ( t, J α ))diag q , p (C q D q )block p , q ( G ) = diag q , p ( B q ( t, J α )C q D q )block p , q ( G ) = − diag q , p (C q D q B q ( t, J α ))block p , q ( G ) = − diag q , p (C q D q )diag q , p ( B q ( t, J α ))block p , q ( G ) = − diag q , p (C q D q )block p , q ( G )diag q , p ( B q ( t, J α )) = − L p,q ( G ) B p,q ( t, J α ) . (cid:3) The general case: S m − for any even m. Defining G def = ∅ , we can state the general case S m − , m =(2 k + 1)2 p q , in one single Theorem: Theorem 6.11.
For any k ≥ , q ≥ and p = 0 , , or , the q + 2 p − vector fields on S (2 k +1)2 p q − given by { B k,p,q ( t, J α ) def = diag t , (2 k +1)2 p q − t (C t block , t − ( J α )) } t =1 ,...,qα =1 ,..., { L k,p,q ( G ) def = diag p q , k +1 (diag q , p (C q )block p , q ( G )) } G ∈G p are a maximal orthonormal set.Proof. Follows from Theorems 6.6, 6.10 and property (2) of diag in Lemma 6.2. (cid:3)
Lemmas.
In this last subsection, we collect all the lemmas appearing as references in the equalities to proveTheorems 6.6, 6.10 and 6.11.
Lemma 6.12.
Let q ≥ and ≤ t ≤ q − . Then B q ( t, J ) and block , q − ( J (cid:48) ) commute.Proof. Follows from the fact that B q ( t, J ) ∈ Im(diag q − , ) (Lemma 6.2(1)) and Lemma 6.2(3). (cid:3) Lemma 6.13.
Let q ≥ and ≤ t ≤ q − . Then D q,t and block , q − ( J ) commute.Proof. Follows from the fact that D q,t ∈ Im(diag q − , ) (Lemma 6.2(1)) and Lemma 6.2(3). (cid:3) Lemma 6.14.
Let q ≥ , ≤ t, s ≤ q − and s (cid:54) = t . Then D q,s and B q ( t, J ) commute.Proof. Assume s > t . Then the claim follows from Lemma 6.2(3), writing D q,s = block · q − s , s (diag , q − s ( C ))and B q ( t, J ) ∈ Im(diag s , · q − s ) (Lemma 6.2(1)).If s < t , we use Lemma 6.2(1) to write D q,s = diag t , q − t (D t,s ), so thatD q,s B q ( t, J ) = diag t , q − t (D t,s C t block , t − ( J ))= diag t , q − t (C t D t,s block , t − ( J )) = diag t , q − t (C t block , t − ( J )D t,s )= B q ( t, J )D q,s , where C t and D t,s commutes since they are both diagonal matrices. (cid:3) Lemma 6.15.
Let q ≥ and ≤ t ≤ q . Then D q,t and B q ( t, J ) anticommute.Proof. D q,t B q ( t, J ) = diag t , q − t (D t C t block , t − ( J ))= diag t , q − t (C t D t block , t − ( J )) where C t and D t commutes since they are both diagonal matrices. We are thus reduced to show that D t andblock , t − ( J ) anticommutes. But using the explicit expressions given in Formulas 2.17 for the J s, we can writeblock , t − ( J ) asblock , t − ( J ) = (cid:32) , t − ( R O )block , t − ( R O ) 0 (cid:33) if J ∈ { J , . . . , J } , (cid:32) − Id t Id t (cid:33) if J = J , and in both cases a block multiplication by hand shows that D t block , t − ( J ) = − block , t − ( J )D t . (cid:3) Corollary 6.16.
Let q ≥ and ≤ t ≤ q − . Then C q and B q ( t, J ) anticommute.Proof. Follows from the Definition (6.1) of C q and Lemmas 6.14, 6.15. (cid:3) Lemma 6.17.
Let q ≥ and ≤ t ≤ q − . Then D q,t D q,t = Id q .Proof. Follows from Lemma 6.2(2) and from C = Id . (cid:3) Lemma 6.18.
Let q ≥ and ≤ t ≤ q . Then B q ( t, J ) and C q D q anticommute.Proof. If q = 1, then B q ( t, J ) = J and C q D q = D anticommute, as in proof of Lemma 6.15 with t = 1.If q ≥
2, then split the proof in cases 1 ≤ t ≤ q − t = q .If q ≥ ≤ t ≤ q −
1, use Corollary 6.16 to show that B q ( t, J ) and C q anticommute. Then, write B q ( t, J ) ∈ Im(diag q , ) and use Lemma 6.2(3) to show that B q ( t, J ) and D q commute.If q ≥ t = q , use Lemma 6.13 to show that B q ( q, J ) and C q commute, and finally apply Lemma 6.15 with t = q to show that B q ( q, J ) and D q anticommute. (cid:3) Remark . It would be interesting, but we were not able, to compare the maximal systems of vector fieldsconstructed in the present paper with the ones appearing in previous constructions, in particular with the vectorfields obtained in [19].
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