spaces C 4,~ and C 2,2, it is shown that complex 8-dimensional ternary composition algebras can be constructed out of a ~-module K 4, where is a certain ...
IL NUOVO CIMENT0
VOL. 103 B, N. 2
Febbr~io 1989
Ternary Composition Algebras: 8 Dimensions Out of 4? R. SHAW
School of Mathematics, University of Hull - Hull, HU6 7RX, UK (ricevuto il 26 Luglio 1988)
Summary. - - Just as real 8-dimensional ternary composition algebras, of signatures (8, 0) and (4, 4), can be constructed out of 4-dimensional complex spaces C4,~ and C2,2, it is shown that complex 8-dimensional ternary composition algebras can be constructed out of a ~ - m o d u l e K 4, where is a certain commutative algebra of dimension 4 over R. The neutral signature real form of the K4-construction is associated with the symmetrybreaking chain S0(4, 4))Spin+(3, 4)~SL(4; R), as compared to the chain S0(4, 4) ~ Spin§ 4) ~SU(2, 2) associated with the C2'2-construction. On breaking the symmetry further, by passing to the binary octonionic multiplication, the ternary viewpoint is seen to throw light on both the Zorn and the Giinaydin forms of octonionic multiplication and leads to severaI unorthodox forms as well. PACS 02.10 - Algebra, set theory and graph theory. PACS 02.20 - Group theory.
1. -
Introduction.
L e t E denote a real n-dimensional vector space equipped with a positive definite inner product ( , ) . Suppose that t h e r e exists a map { }: E 3--~ E which satisfies the axioms C1
( } is trilinear,
C2
{aac} = (a, a) c ={caa},
C3
({abe}, {abe}} = (a,a}(b, b}(c, c)
for all a, b, c e E . In such circumstances we r e f e r to the triple (E, ( , }, ( } ) as a 3Cn algebra, or as a (n-dimensional, real) ternary composition algebra. In view of axiom C2, the last name should perhaps receive the qualifying ~of a special 161
162
R. SHAW
alternative kind,. However, in the light of the work of McCrimmon (1), it turns out that the omission of the simplifying ,,alternative axiom~, C2 does not, up to isotopy and permutation of variables, lead to anything new. For n>~3 a ternary vector cross product for (E, ( , ) ) is defined to be a map X : E 3---) E which satisfies the axioms (cf. (2,3)) X1
X is trilinear;
X2
X(a, b, c) is orthogonal to each of a, b, c;
X3
(X(al, a2, as), X(al, a2, as)) = d e t ( ( a , a3}).
The triple (E, ( , ) , X) will then be termed a 3Xn algebra. It can be shown (~,~) that, for n ~ 3, 3Cn and 3Xn algebras are in the one-one correspondence given by { } ~ X , where (1.1)
{abc} = X(a, b, c) + (a, b) c + (b, c) a - (a, c) b.
(In a previous study (6) of 3Xn algebras the author made use of the ternary composition algebra property C3, but not the full equivalence, for n >13, of the two axiom systems C1-C3 and X1-X3.) In establishing this equivalence of the two axiom systems the following identity is made use of:
(1.2)
terms a and e give rise to the , 1,
x 9162
w h e r e : ~/,• ~r is a nondegenerate symmetric bilinear form on .~(. Upon polarizing (A.1) we obtain (A.2)
x -~= 2 1 - x.
The Cayley-Dickson process constructs out of the pair ( ~ , ~) a similar pair (P3, 0 but with dim • = 2n, such that ~ contains ~d as a subalgebra (with t73 having the same 1 as .tr The construction is as follows: g~3is defined to consist of elements x + yj, with x, y 9 .4", which multiply according to the law (A.3)
(u + vj)(x + yj) = ( u x + ~y~v) + ( y u + vx~')j ,
w h e r e ~ denotes some fixed nonzero element of F. The algebra t~9 is seen to possess an involution ~ which is given by
(A.4)
(x + yj): = x ~ - y j
and which satisfies (A.5)
zz ~= (z, z> 1,
TERNARY COMPOSITION ALGEBRAS: 8 DIMENSIONS OUT OF 4?
179
where (A.6)
( u + vj, x + yj} = (u, x} - ~(v, y} .
The pair (•, ~) is referred to as the z-double of the pair (.~, ~). Note, as particular cases of the foregoing, the properties (A.7)
f=sl
and
j~=-j
and also (A.8)
j x = x~'j
and
x ~= x s ,
for x 9 . ~'r ~ .
We can start the doubling process with the choice ~ d = F with ~ the identity involution and so with (x, y} = xy. We then find, see for example(~), that i) the doubles of g (of dimension 2 and called quadratic algebras over F) are commutative and associative, with nontrivial involution; ii) the doubles of quadratic algebras (of dimension 4 and called generalized quaternion algebras over F) are associative but not commutative; iii) the doubles of quaternion algebras (of dimension 8 and called generalized octonion, or Cayley, algebras over F) are neither associative nor commutative, but are alternative--satisfying, that is, (A.9)
x(xy) = x ~y
and
(xy) y = x y 2 .
All of these algebras are in fact (binary) composition algebras with respect to the appropriate (, }, satisfying, that is, (A.10)
(xy, xy} = (x, x} (y, y } .
However, the doubles of an octonion algebra are not alternative and hence (see, e.g., (~)) not composition algebras. In fact, in this appendix, we are only interested in the cases g = R and F = C, and for these cases we m a y as well restrict our choice of ~ to be + 1. Moreover, when F = C the choices s = + 1 and s = - 1 yield isomorphic algebras. Starting out from ~ d = R suppose that we make the choice z = - 1 at each of the above doubling stages i), ii) and iii). We obtain t h e r e b y the well-known real division algebras (A.11)
i) C,
ii) H,
iii) 0 ,
of dimensions 2, 4 and 8 over R. However if at any stage of the doubling we choose s to be + I then, see (A.6), we obtain an algebra whose scalar product ( , } has neutral signature; moreover once the fly of neutral signature gets into the ointment it sticks throughout any subsequent doublings. If already at stage i) we make the choice e = + 1 we obtain the ,split complex numbers,, C = (1, ~}, ('3) N. JACOBSON: Basic Algebra I (Freeman, San Francisco, Cal., 1974), subsect. 7"6.
180
R. SHAW
where ~2 = + 1. By introducing the idempotents (1 ___~)/2 we see that ~ is isomorphic to the (,double field,, 2R(= R(~ R) (which is not of course a field). Stages ii) and iii), for any choices of sign, now yield algebras isomorphic to the (real) split quaternions f / a n d the (real) split octonions O. If instead we start out from LJ~"= C then, whatever our choices of signs, we obtain, at the successive stages i), ii) and iii) of doubling, complex algebras, of dimensions 2, 4 and 8, which are isomorphic, respectively, to (A.12)
i) CC ~- C~',
ii) C H = Cfl,
iii) C ~ = CoO.
Here C0, for example, denotes (~octonions with complex coefficients,,. We may view C ~ as the tensor product C | 5~ of the real algebras C and O made into a complex algebra by means of the definition (A.13)
a(fl|174
a, fle C, a e O .
Similarly for the other complex algebras in (A.12).
APPENDIX B Zorn's vector matrices.
Over C there is, up to isomorphism, a unique (") quaternion algebra. This can be taken to be the algebra 5"(2) of all 2 • 2 complex matrices, with involution A ~/i~ given by (B.1)
It is worth noting that _;I is not only the metrical adjoint of A with respect to symplectic geometry on C 2, but is also the (nonmetrical) classical adjoint adj A. (For a discussion of the ((the peculiarities of dimension 2~,, see(45)). Only in dimension 2 is adj A linear in A, and only in dimension 2 does the determinant define a quadratic form, thereby allowing us to realize the composition algebra property (A.10) in the manner (B.2)
det (AC) = (detA) (det C).
The scalar product (, } on C(2), arising from the quadratic form A ~ det A, can be expressed as (B.3)
(A, C} = l t r ( A C ) .
(") J. R. FAULKNERand J. C. FERRAR: Bull. London Math. Soc., 9, 1 (1977). (4~) R. SHAW: Linear Algebra and Group Representations, Vol. 1 (Academic Press, London, 1982), subsect. 5"6.
TERNARY COMPOSITION ALGEBRAS: 8 DIMENSIONS OUT OF 4?
181
As a first attempt to generalize the foregoing to vector matrices of the form ~,~eG,
a, b e G 3,
let us define (B.5)
detA = ~ - a. b
and (B.6) and make the obvious definition of multiplication (B.7)
L aa+ c
Then our attempt fails because the r.h.s, of (B.2) acquires the addition of the extra term (B.8)
(a . d)(b
. c) -
(a . b)(c
. d) .
Now, in terms of the standard vector cross product on C 3, this last can be expressed as (B.9)
- (a x c). (b x d).
Consequently we will succeed in our attempt to generalize the composition algebra property (B.2) to vector matrices provided that we adopt the following modification to the multiplication law (B.7): (B.10)
a
c
aa+~c+b•
~],+a.d
"
(The fact that u x v is orthogonal to u and v also enters into the confirmation that (B.2) holds under the modified rule (B.10) for matrix multiplication.) The resulting realization of the (unique, up to isomorphism) complex octonion algebra is due to Zorn. R e m a r k . - Actually Zorn's multiplication law (46.47)differs slightly from that in (B.10). The choice (B.10) agrees with that made in(15'",48) and is easily seen to yield an algebra which is isomorphic to that of Zorn.
(46) M. ZORN: Abh. Math. Sem. Hamburg, 9, 395 (1932). C) N. JACOBSON:Lie Algebras (Interscience, New York, N.Y., 1962). (4s) G. B. SELIGMAN: Trans. Am. Math. Soc., 94, 452 (1960).
182
R. SHAW
Over R there are, up to isomorphism, just twooctonion algebras, namely the division algebra dl and the nondivision algebra O. These can be obtained from the above Zorn form of C O by imposing two different reality conditions. To obtain O, impose the reality conditionA = (A) t, whereA is obtained from A by taking the complex conjugate of each entry: (B.11)
[fia ab]----I~
~]'
where ~ denotes the complex conjugate of ~ 9 C, etc. Thus O is realized in terms of vector matrices of the form (B. 12)
A --
,
~e
C,
a 9
C ~.
To obtain the algebra ~ of the real split octonions, impose instead the reality condition A = A, that is simply restrict a, fi in (B.4) to be real scalars, and a and b to be real triples. R e m a r k . - Since a (binary) vector cross product a x c exists also in dimension 7, can we not generalize Zorn's construction by replacing C ~ in (B.4) by C 7, to obtain thereby an algebra of ,(sedenions~,, cf.(~), of dimension 16 over C? However the terms (B.8) and (B.9) are in general no longer equal for the sevendimensional vector cross product. In fact, their difference is--see theorem A of(8)---C(a, c, b, d), where ~ is the dual *~ of the scalar triple product ~. Consequently, in conformity of course with Hurwitz's theorem, our sedenion algebra is not a composition algebra, in that (B.2) acquires an extra term ~(a, c, b, d) on its 1.h.s.
(49) L. SORGSEPPand J. LOHMUS:Had,onic J., 4, 327 (1981).
9
R I A S S U N T O (*)
Proprio come algebre a composizione ternaria a otto dimensioni reali, di segnature (8, 0) e (4, 4), possono essere costruite da spazi complessi a quattro dimensioni C4,~ e C2,2, si mostra che algebre a composizione ternaria a otto dimensioni complesse possono essere formulate da un K4 con modulo ~ , dove ~ e una terra algebra commutativa di dimensione 4 su R. La forma reale della segnatura neutra della costruzione K4 ~ associata alla catena della rottura di simmetria SO(4, 4)~ Spin+(3, 4)~ SL(4; R), in confronto alla catena S O ( 4 , 4 ) ~ S p i n + ( 3 , 4 ) ~ S U ( 2 , 2 ) associata alla costruzione C2'2. Nel rompere ulteriormente la simmetria, passando alla moltiplicazione ottonionica, si vede che il punto di vista del ternario fa luce sia sulle forme di Zorn che di G~inaydin della moltiplicazio~e ottonionica e porta anche a parecchie forme non ortodosse. (*) Traduzione a cura della Redazione.
TERNARY COMPOSITION ALGEBRAS: 8 DIMENSIONS OUT OF 4.9
183
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