A 3-Fold Vector Product in R ^8

A 3-Fold Vector Product in R '

By P E T E R

ZVENGROWSKI

An r-fold vector p r o d u c t in R ~ is, according to ECKMANN map X:R ~'=R "• R "• • ~-~R" satisfying

(i)

X(al,

a 2.....

(ii)

[I X ( a l ,

as

.....

[2], a

continuous

ar)" ai = 0, 1 g i g n , ar) ll 2 = lai " a i l .

I t can be deduced from [2] and [4] t h a t R n admits an r-fold v e c t o r p r o d u c t in precisely the following cases: a) r = 1 and n is even, b)

r~-n--l,

c) r := 2 a n d n =

3,7,

d) r = 3 a n d n =

8.

Explicit formulas for these vector p r o d u c t s are k n o w n in all b u t the last case, and the purpose of this n o t e is to give a simple formula for the last case, taking a d v a n t a g e of the CAYLEY numbers. Such a formula was useful to the a u t h o r in examining h o m o t o p y classes of v e c t o r fields on S 7 (cf. [5]).

1. The Cayley Numbers We t a k e the CAYLEY n u m b e r s C to be the 8-dimensional algebra over the reals having basis vectors ex, ea . . . . . es and suitably defined products. I n particular, ex-----1 is the unit and for i , j >

1 one has

eie ~ =

{-1, i--j

--ejei, i Cj. The subspace o f C o r t h o g o n a l to ea is called the pure CAYLEV numbers. Conjugation is defined b y writing a n y CAYLEY n u m b e r x as x = re~ q- x', where x' is pure, and setting x = r e 1 - - x ' . T h e CAYLSY n u m b e r s inherit the n o r m and scalar p r o d u c t of R s, a n d

=

Ixl

150

l~r~

Zrs~oRowsxI

Lemma 1 . 1 . For any CAYLEY numbers a, b, one has a : a , and a b + ba ~ 2(a. b).

ab:

,

Proo/: By linearity it suffices to prove these when a, b are the units e 1 . . . . , es, and these cases follow from the above remarks on multiplication. Corollary: -ab + ba : 2 ( a . b-) : 2(a. b). The multiplication in C can be explicitly defined by regarding C as a 2dimensional algebra over the quaternions H , with basis 1, It, and setting f

(q~ + qsk)(q~ + qJc) :

t

-!

f

(q~ql --q2q2) + (q2ql + q2q~)k.

We shall need this formula to prove the next lemma. Lemma 1.2. For any CAYLEY numbers a, b, c, a(bc) + b ( a c ) :

2(a. b)c.

Proo/: Let a -~ a 1 + ask, b ~ b 1 + bJc, c ~- c 1 ~- C2k , where a 1, a2, bl, b2,

c~, cs are quaternions. B y direct computation, a(-bc) + b ( a c ) :

(bla ~ + albl)C 1 + c~(a~b2 + b-2a.,) +

+ [cs(albl + blab) + (b2a2 + asb-2)c2]k.

Applying Lemma 1.1 and its corollary, this equals 2[(b1- al)c 1 + cl(a ~. b~) + (c 2 (a 1. bl) + (b2. a2)c2)k] --~ 2(a~ .bl + as .bs)(Cl + c2k) -~ 2 ( a . b ) c .

According to [1], the subalgebra of C generated b y any two elements (and their conjugates) is associative. In particular, for any CAYLEY numbers a, b, c, the product a ( b c ) a is well defined, and according to [3], we have a ( b c ) a -~ = (ab)(ac). Lemma 1 . 3 .

For any CAYL~r numbers a, b, c, one has

(i)

a . (a(bc)) = l a l 2 ( b . c ) ,

(ii)

c. (a (b-c)) ---- Ic I~(a. b), and

(iii)

b.(a(bc)) = --]b]2(c.a) + 2(a.b)(b.c).

A 3-Fold Vector P r o d u c t in R s

151

P r o o f of(i): 2 a . ( a ( b c ) ) ~ a(a(bc)) ~- ( a ( b c ) ) a z a ( ( ~ b ) a ) ~ (a(-bc))-a. Since a, a , c b, a n d c b ~- bc generate an associative subalgebra, this equals

a(-cb)-a ~- a ( b c ) a z a(~b -~ -bc)a : 2a(b . c ) a : 21a]2(b .c). P r o o f of (ii): 2 c . (a(-b-c)) : c ( ( ~ b ) a ) + (a(bc)) c -~ 2r, say, where r m u s t be a real number. T h e n making use of the preceding results w i t h o u t specific mention, we have rc : c((cb)-a)c + (a(bc))[c[ 2

~- (c(-cb))(-ac) + (a(bc))[c[ ~ :

Icl~(b(-dc) -[- a(bc))

: 21c[2(a.b)c. P r o o f of (iii):

2 5 . ( a ( b c ) ) ~-- b ( ( v b ) a ) ~- (a(bc))b

: b [ ( - - b c -[- 2 b . c ) a ] + [ a ( - - c b -t- 2b.c)]b ~-- 2 ( 5 . c ) ( b a + a~b) - - b ( ( b c ) a ) = 4(b .c)(a .b) - - 2 r ,

- - (a(cb))-b

for some real n u m b e r r.

Computing rb as in (ii), one finds r ----[bl2(c.a).

2. Applications to Vector Products A 2-fold vector p r o d u c t in R 7 is o b t a i n e d (cf. [2]) b y identifying R 7 with the pure CAYLEY numbers, a n d for a n y two vectors b, c in R 7 p u t t i n g X2 (b, c) ----b c ~- b . c . We shall now define a 3-fold v e c t o r p r o d u c t in R 8, which in a certain sense generalizes X 2. Theorem 2 . 1 . L e t a , b, c be vectors in R s, regarded as the CAYLEY numbers. Then X s ( a , b, c) ~- - - a(bc) + a(b .c) - - b(c .a) -~ c(a .b) defines a 3-fold v e c t o r p r o d u c t in R%

Proo/: L e m m a 1.3 will be f r e q u e n t l y used w i t h o u t specific mention, and we write X -~ Xa(a, b, c). The c o n t i n u i t y o f X3 is obvious. T h u s a . X ~ la]~(b.c) + la]2(b.c) - - ( a . b ) ( c . a ) + ( a . c ) ( a . b ) -~ O, and the proofs t h a t b . X ~ O, c . X -~ 0, are entirely similar. Finally, writing d --~ (a(bc)), have

we

152

I~R

IXI ~ = X.X

= Idl a -}- [al~(b.c) ~ ~- I b l ~ ( a . c ) ~ ~- I c l 2 ( a . b ) 2 - - 2 ( a . d ) ( b . c )

-}- 2 (b . d) (c . a) - - 2 ( c . d ) ( a . b ) :

Zv~GRowa~

-}- ( - - 2 -~ 2 - - 2) (a . b) (b . c) (c . a)

lal2]b[2]c[ ~ -~ ]a[2(b'c) ~ -b I b [ ~ ( a ' c ) ~ ~- [ c l 2 ( a ' b ) 2 - - 2 [ a l 2 ( b . c ) 2 - - 21b[~(a.e) 2 ~- 4 ( a . b ) ( b . c ) ( c . a )

- - 2[c{2(a.b) ~

- - 2 (a- b) (b. c) (c. a) ~-- ]a]21bl21c] 2 - - ] a l ~ ( b . c ) 2 - - [b[2(a.c) ~ - - [ c l ~ ( a . 5 ) 2 + 2 ( a . b ) ( b . c ) ( c . a ) [a[ ~ a . b Iillc :

b.a

[bl ~ b . c

c.a

c.b

W h e n r e s t r i c t e d t o t h e STIEFEL m a n i f o l d VT,~, t h e 2-fold p r o d u c t X2 gives a cross section X~ : V~,2 -~ V~,8. S i m i l a r l y , X 3 gives a cross section X3 : Vs,3 -> -~ Vs,4. T h e n t h e f a c t t h a t X3(el, b , c ) -~ bc ~- b . c = X 2 ( b , c ) , where (el, b, c) 9 Vs,a (and h e n c e b, c a r e pure), i m p l i e s t h a t t h e following d i a g r a m commutes: i

VT,~

* Vs.3

i' 9

F u r t h e r d e t a i l s on t h e s e v e c t o r p r o d u c t s c a n be f o u n d in [5]. The University o/Illinois REFERENCES [1] DICXSON L. E. : Linear Algebra~, Cambridge Univ. Press (1914). [2] EC~rUA~rNB. : Ste~ige L~sungen linearer Gleichunyasy~em~, Comment. Math. Holy. 15, 358-366 (1942). 13] TODA H., S~rro Y. a n d YOKOTA I. : Note on the Generator of ~ (SO (n)), Kyoto Univ. Memoirs Vol. X X X Math., No. 3, 227-230 (1957). 14] W m ' ~ A D G. W. : N o ~ on Cross Sections of Stiefd Manifolds, Comment. Math. Helv. 37, 239-240 (1963). [5] ZV~NOROWSXI P.: Vector Fields and Vector Products, Chapter 4: Vector Products, Thesis, Univ. of Chicago (1965).

( R e c e i v e d April 5, 1965)