Ternary Composition Algebras. I. Structure Theorems: Definite and Neutral Signatures

Ternary Composition Algebras. I. Structure Theorems: Definite and Neutral SignaturesAuthor(s): Ronald ShawSource: Proceedings: Mathematical and Physical Sciences, Vol. 431, No. 1881 (Oct. 8, 1990), pp.1-19Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/79935 .

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Ternary composition algebras. I. Structure theorems: definite and neutral signatures

BY RONALD SHAW

School of Mathematics, University of Hull, Hull HU6 7RX. U.K.

Previous work on ternary composition algebras of definite signature is generalized to the only other case allowed by the axioms, namely that of neutral signature. One new feature is the presence, in the neutral case, of 'counter-automorphisms'. The structure of the 'exceptional' eight-dimensional algebras is probed, and, in various

guises. a fundamental dichotomy is unearthed. Canonical forms are obtained, and various further properties are derived.

1. Introduction

Let E = Ers denote throughout a real n-dimensional vector space which is equipped with O(r. s)-geometry by means of a scalar product <( > of signature (r,s). where r + s = n. Suppose that E is also equipped with a bilinear multiplication E x E- E: (a.b) ab for which there exists an identity element eEE:

ae = a = ea, for all aEE. (1.1)

and for which the (binary) composition property

<ab, ab> = <a, a>) b b> (1.2)

holds for all a. bE E. The resulting real algebra E is then termed a composition algebra. Bv Hurwitz's theorem (see Jacobson 1974) composition algebras only exist in dimensions n = 1. 2. 4 and 8. In our present case of real algebras there exist (see Faulkner & Ferrar 1977). in addition to R itself (and up to isomorphism), just six

composition algebras. Three of these are the real division algebras

C,H 0I(= quaternions), 0 (= octonions), (1.3)

associated with the positive definite signatures (2.0), (4,0). (8.0), respectively, and the remaining three are the split composition algebras

C.I Hl, (1.4)

associated with the neutral signatures (1. 1), (2,2). (4.4). respectively. The automorphism groups of the division algebras (1.3) are well known:

Aut C Z Z2. Aut H-1 SO0(3). Aut 0 G2. (1.5)

as are those of the non-division algebras (1.4):

AutC-Z2. Aut0 SO(1 2), AutO G2 (1.6)

where G2 denotes the non-compact form of the exceptional group G2 (of dimension 14). It is known (see Freudenthal 1964: Faulkner & Ferrar 1977) that the octonions are also intimately connected with the other exceptional groups F4, E6. E7, E8.

Proc. R. Soc. Lond. A (1990) 431, 1-19

Printed in Great Britain

I 1 Vol. 431. A (8 October 1990)



Despite the fact that, unlike the other composition algebras, the octonion algebras are not associative, nevertheless octonions had a very early introduction into

quantum physics, by Jordan et al. (1934). Moreover, during the past quarter of a

century there have been numerous further attempts to incorporate octonionic considerations into physics. A selection of physical octonionic references can be found in Shaw (1989).

The present paper is devoted to the study of a certain generalization of the above notion of a (real) composition algebra, namely that of a (real) ternary composition algebra S (= Sr'S) = (E, <, >,{ }), where E (= E' ) and <, > are as previously and { } is a ternary product E3 -E which satisfies the axioms (Shaw 1988c)

C1 {} is trilinear,

C2 {aac} = <a, a> c = {caa}, (1.7)

C3 <{abc}, {abc}> = <a, a> <b, b> <c, c>,

for all a, b, c eE. Such an algebra will be referred to also as a 3C (r, s) algebra. From C1-C3 it is clear that ternary composition algebras in fact come along in quadruples

= (E, <,>,{), {} = (E, <,)>,o{}), gneg = (E, <, )neg,}neg)

oneg = (E,<, neg o{}neg),j (1.8)

where ?{abc}={cba}, {abc}neg =-{abc} and <a, b>ne = -<a, b>. If S is a 3C(r,s) algebra, then so is the opposite algebra O?; however, gneg and ogneg are then 3C (s, r) algebras.

Now there is a close connection between ternary composition algebras and the

(binary) composition algebras which we first discussed (and which we now refer to as 2C (r, s) algebras). In the one direction, given a 3C (r, s) algebra f = (E, <, >,{ ) let us choose any 'unit' vector e E:

<e,e) = =?1 (1.9)

and define an associated binary multiplication upon E by

ac = r{aec}. (1.10)

Then axioms C1-C3 entail that, under this multiplication, (E, <,)) becomes a 2C (r, s) algebra in the case y = + 1, while (E, <, >neg) becomes a 2C (s, r) algebra in the case y =- 1. In either case the chosen unit vector e is the identity element of the

composition algebra. In the other direction we can construct a 3C (r, s) algebra out of a 2C (r, s) algebra by defining, for example,

{abc}= (ab)c, (1.11)

where a - a- denotes the usual conjugation (or involution) K for the (binary) composition algebra:

= Ka = 2a, e) e-a, (1.12)

which is known (Jacobson 1974) to satisfy

ab = ba (1.13)

and aa = <a, a>e = aa. (1.14)

Nevertheless there is good reason to study ternary composition algebras in their own right without appeal to the properties of the binary composition algebras, Proc. R. Soc. Lond. A (1990)

R. Shaw 2



Ternary composition algebras. I

because the former have more symmetry than the latter. Indeed the automorphism groups of the former turn out, in the cases n = 2, 4, 8, to be given by

Auts2' 0 (2), Aut 4, 80 SO (4), Aut8 _ Spin(7) (1.15) and

Aute11 - O(1, ), Aut22 _ SO0(2,2), Auta, 4 Spin+(3,4) (1.16)

as compared with those of the latter, as given in (1.5) and (1.6). At any rate this will be the point of view of the present paper, to study the more symmetrical before the less symmetrical.

The present article is the first of two whose intention is to provide a reasonably comprehensive study of ternary composition algebras. These articles generalize some earlier results (Shaw 1987, 1988c), which dealt only with the case of positive-definite signature, and will include derivations of many further results. In this first article we commence by showing that the study of ternary composition algebras is, in dimension n > 2, completely equivalent to the study of ternary vector cross products. Previous authors (Eckmann 1942; Brown & Gray 1967; Gray 1969; Harvey & Lawson 1982; Diinderer et al. 1984) studied the latter, for dimension n = 8, by relating their properties to those of the octonions. In studying these ternary algebras in their own right our chief mode of attack will be the use of various multiplication operators as defined in ? 2. Some first properties are obtained there (see also the Appendix) and further properties of these multiplication operators are uncovered from time to time in the ensuing sections. In this first article we spell out what is meant by an automorphism of our algebra &, and point out, in the case of neutral signature, that counter-automorphisms are also of relevance. However, at least in the eight-dimensional cases which are our chief concern, we delay until Part II the detailed study of automorphisms, and of related Lie algebra and Clifford algebra aspects.

Remark 1.1. The author was led to the choice C1-C3 of axioms as a natural development arising out of a previous study (Shaw 1987) of a well-known (Brown & Gray 1967) axiom system X1-X3 (see (2.1) below) for a ternary vector cross product. If instead we drop axiom C2, leaving only C1 and C3, then we have what McCrimmon (1983) terms a composition triple. However, the end result of MeCrimmon's ingenious investigation is that, up to isotopy and permutation of arguments, nothing new arises from the omission of axiom C2. If one could discover a reasonably simple direct proof that a composition triple is isotopic to a permutation of one which obeys axiom C2 then the methods of the present paper, or indeed (via Theorem 2.6) those of Brown & Gray (1967), could be used to give a simpler derivation of MeCrimmon's results.

2. Ternary composition algebras and ternary vector cross product algebras In dimensions n > 3 a ternary vector cross product for (E, <(, ) is a map X:E3-> E

which satisfies the axioms (cf. Eckmann 1942; Brown & Gray 1967), Xl X is trilinear, X2 X(a, b, c) is orthogonal to each of a, b, c, (2.1) X3 <X(a,b,c),X(a,b,c)> = (aAbAclaAbAc),

for all a, b, ceE. Here the scalar product < I on A3E is taken to be such that

<al A a A a3 i b b A 2b3) = det (<a, bj)). (2.2) Proc. R. Soc. Lond. A (1990)

3



In such circumstances, with E = Er 8, we refer to the triple ? (-= _r, ) = (E, <, >,X) as a 3X (r, s) algebra. Associated with the 3X (r, s) algebra 1 is the scalar quadruple product Q0 defined by

?0(a,b,c,d)= <a,X(b,c,d)) (2.3) and also a family {Ta, ; a, b E} of linear operators on E, where

Ta, bc = X(a, b, c). (2.4) Observe that 01 and T are, respectively, quadrilinear and bilinear functions of their arguments.

Lemma 2.1. <P, X, and T are alternating functions of their arguments. Moreover, Ta , eSk (E, E), for each a, beE.

Proof. As in Shaw (1988c).

Returning now to a 3C(r,s) algebra g we can define, for a, beE, three different kinds of multiplication operators on E by

7a,bC = {abc}, a,bc= {acb} and a,b c= {cba}. (2.5) Each of Ya,b, /a,b and ca,b is a linear operator, and depends bilinearly upon its arguments a, b e E. Some of the properties of these multiplication operators are listed in the Appendix. Further properties are as stated in the next two lemmas. In the proof of Lemma 2.2 we denote by A the adjoint of a linear operator A on E:

<u,Av> = (Au, v>.

Also we will throughout this paper use the term counter-isometry for a linear operator A on E which satisfies

<Au,Av) =-(u,v), for all u,v E. (2.6) Of course counter-isometries exist if and only if the signature (r, s) is neutral (that is if and only if r = s).

Lemma 2.2. (a) If <a,a> = <b, b> = 1, then a, b t, ab and a, b all belong to the isometry group O(E) - O(r, s) of E = Er .

(b) If a, a> =- <b, b> = ? i, then y7a, a, b and a, b are all counter-isometries ofE. (c) The signature of E is either (positive, or negative) definite or neutral.

Proof. By (A 3) we have a, b 7a,b = +I or )a, b a, b =-I according as we are in case (a) or case (b) of the Lemma. Similarly for a, b and ra, b. Finally, since an indefinite signature entails the existence of a, b such that <a, a> = -b, b> = I, part (c) follows from part (b).

In the next lemma, {A,B} denotes the anticommutator AB+BA.

Lemma 2.3. If a, b, c, d are mutually orthogonal vectors of E, and if u E is any vector, then

Ya,b =-b,aESk(E,E), ca,b =-ob, aSk(E,E). (2.7)

7b, u7u, c= -Yc, uYu, b, b,u u,c =- ,u u,b (2.8a)

Yu, bYc,u -Yu, cYb, u, u,b c,u =-u, cOb,u (2.8b)

{Ya, b a, c}= 0= {a,b,a,c, (2.9)

{Ya,b, c,d} =

{Yb,c,, Ya,d} =

{7c, aYb, d}, (2.10a)

{ea, b c, re,= {(b, c, =

, d} {= {c, a, b, d}. (2.10b) Proc. R. Soc. Lond. A (1990)

4 R. Shaw



Ternary composition algebras. 1 5

Proof. Equations (2.7), (2.8) are special cases of properties in the Appendix. Equation (2.9) then follows. Equation (2.10) follows from (2.9) after a suitable linearization in the vector a.

Lemma 2.4. If {a, b} is an orthonormal basis for a non-singular two-dimensional subspace of E, then

2 ( 2_ \ -1, if <a, a> = b, b>, (Ta'b)2--(~a ' b2-+, if (a, a)

-a (b, b).

(^Ya,b) (,b) = 1+1, if <a,a>=-<b,b>.

Proof. Given <a, b> = 0 we have from (2.7) and (A 3)

(7a, )2 = (Oa, b)2= -<a,a> <b, b>I. (2.11)

We now show that the two axiom systems C1-C3 and X1-X3 are (for n > 2) just two facets of the same area of mathematics. To this end define Z:E3 -E by

Z(a, b, c) = <a, b>c+ b, c> a-<a, c> b (2.12) and observe that it is trilinear and also satisfies C2:

Z(a, a, c) = <a, a) c = Z(c, a, a). We now set up a one-one correspondence between maps { }: E3 - E and X: E3 - E by the relation

{abc} =X(a, b, c) + Z(a, b, c). (2.13)

Theorem 2.5. Under the correspondence {} +-X defined by (2.13), r"sS = (E, <, >,{}) is, for n > 2, a 3C (r, s) algebra if and only if 4r,s = (E, <, ),X) is a 3X(r, s) algebra.

Proof. The proof given in Shaw (1988c) applies whether or not the signature is definite. The tie-up between axioms C3 and X3 is by way of the identity

<al, al> a2, a2><a3, a3> = det(<ai, a>) +<Z(al, a2, a3), Z(al, a2, a3)). (2.14) Remark 2.6. Under the correspondence g +-+ of Theorem 2.5 we have,

corresponding to the quadruples , of, gneg, ogneg of ternary composition algebras in (1.8), a quadruple , O , neg osneg of ternary vector cross product algebras, and associated quadruples X, -X, -X, X and 0, - 0, 0, - of ternary vector cross products and of scalar quadruple products.

If the multiplication operators associated with the opposite algebra O? are denoted by 7a, b, etc., then for future reference note that

Ya,b a,b, ? ta,b = tb,a, 0a-Ya, ab =-Ta b (2.15)

3. Automorphisms and counter-automorphisms From now on, by Theorem 2.6, we can for n > 2 study interchangeably a 3C (r, s)

algebra gr,s or a 3X (r, s) algebra _r,s. Since

Z(a,b,c) = ((a,b)I-Ja,b)c = Z(c,b,a) (3.1)

it follows from (2.13) that we have the following relations between the various multiplication operators:

Ya,b = a, b>I + Ta,b-Ja, b (3.2a)

O'a,b = (a, b)-Ta, Ja, b, (3.2b)

Ya,b = (b,a- 2a,b (3.2c)

Proc. R. Soc. Lond. A (1990)



Here Ja, be Sk (E, E) = so (E) is defined by

J, c= <a,c> b -b,c>a.

Of course Ja, , for linearly independent a, b, generates by exponentiation 'rotations' in the plane -<a, b > spanned by a, b.

Notice that (3.2c) implies that Ya, and b, a agree with each other when acting upon -<a,b>-:

{abp} = {pba}, if <p,a> = p, b> = 0. (3.3)

Notice also that Ja,b Ta, b = - = T,a, b,

and hence, from (3.2), that for given a, b the operators

7a, b, 'a,b, Ta, b, Ja, b

mutually commute. Incidentally the commutativity of ya , and o a,e yields, via (1.10), the flexible law a(ca) = (ac)a for the binary multiplication.

In what follows we will make the standard (see, for example, Shaw 1983) identifications

2E = A (E) = Sk (E,E)

(where the second one is relative to a fixed choice of <, >) by means of

2a A b +-a ? b-b? a -a -J ,b. (3.4)

In a similar vein we will sometimes view 0 E Sk (E4, R) as an element of A4E, and X as a map A3E- E. Associated with ?0 is a linear operator T on A2 E defined by

<T(a A b), c A d) = 0(a, b, c, d). (3.5)

(We use two scalar products < I and <(, on APE - cf. (2.2) and (3.5) - related by )> = p!, >. The scalar product (,) is that one inherited from ?PE upon

identifying A E with the subspace AP(E) consisting of skew symmetric tensors.) Using the skew symmetry of P, observe that

T= T, that is TeS(A2E, A2E). (3.6)

Using also the fact that under the identification (3.4) we have

<B, c Ad> = c,Bd>, for B A2E- Sk(E,E). (3.7)

observe further that T(a A b) = Ta,b. (3.8)

We will denote by Y-- Ay the natural left action of Ae GL(E) upon any of the objects Y under consideration. Thus in particular we have

A<a, b> = <A-la,A-lb>, (3.9a)

(AX) (a, b, c) = AX(A-la,A-lb,A-lc), (3.9b)

{abc} = A{(A-1a) (A-lb) (A-lc)}, (3.9c) AT= (A2A)

o To (A A-1), (3.9d)

Aq(a,b,c,d) = 0(A-a,A-lb,A-c,A-1d), or A5 = (A4A)<P. (3.9e)

For A E O (E) we may write (3.9d) equally well as

AT = AdA o To AdA-l (3.9f) Proc. R. Soc. Lond. A (1990)

R. Shaw 6



Ternary composition algebras. 1 7

since AdA:Sk(E,E)->Sk(E,E) with affect B-ABA-1, coincides with A2A: A\2E A2E under our identification of Sk (E, E) with A2E. Equation (3.9f) applies

equally if A is a counter-isometry, for which we have A =-A- and so AdA coincides with - A2A.

By an automorphism of the 3C (r, s) algebra r, s = (E, (,, { }) we mean an element A e GL (E) which keeps fixed both , > and { }. The group Aut gr sS consisting of all such automorphisms is thus that subgroup of O(E) defined by

Aut S r,s = {A E O(E): Aabc} = {(Aa) (Ab) (Ac)}}. (3.10) The group Aut 8r, is similarly defined, but in fact (for n > 3) is seen to coincide with Aut gr, S under the correspondence in Theorem 2.6. Equally well Aut r, s consists of those A E O(E) which leave the scalar quadruple product 0 invariant, or, in terms of T satisfy

AdAoT= ToAdA. (3.11)

By a counter-automorphism of fr 8 we will mean an element A e GL(E) which sends <, > to <, >neg and {} to { }neg (see after (1.8)). In other words a counter-automorphism is a counter-isometry A which satisfies

A{abc} = -{(Aa)(Ab) (Ac)}. (3.12) Of course counter-automorphisms can exist only in the case r = s of neutral signature. In this case, note that the composition of two counter-automorphisms is an automorphism. We will denote by *Autr, r the group consisting of all the automorphisms and counter-automorphisms of gr, r, n = 2r. It is a subgroup of the group *O(E) consisting of all the isometries and counter-isometries of E = Er,. Again *Aut ar, r (defined analogously) coincides, for n > 3, with *Aut r,.

Lemma 3.1. Counter-automorphisms of r,,r leave (P invariant.

Proof. If A is a counter-automorphism then

'(Aa,Ab,Ac,Ad) = (Aa,X(Ab,Ac,Ad)> = <Aa, -AX(b, c, d)> = a,X(b, c, d)> = 0(a, b, c, d).

Remark 3.2. We will see later that counter-automorphisms do exist whenever our algebra is of neutral signature. Consequently, in these cases at least, the invariance group G (0) of the scalar quadruple product (P is larger (being at least *Aut o) than the automorphism group of the algebra S. (For n = 4 we will see in ?4 that G(qb) is much larger than Aut 1.)

Incidentally virtually the same proof as in the Lemma shows that if A is a counter- isometry which leaves '5 invariant, then A is a counter-automorphism. Thus

*Aut Trr = *O(E) n G(1). (3.13) Remark 3.3. An equivalent definition of a counter-automorphism A of g is that A

is an isomorphism of g with gneg (see (1.8)). We can also introduce the notion of an opposite automorphism of 4, meaning an isomorphism of I with its opposite ?S (see (1.8)). Thus an opposite automorphism of g is an element A O(E) which satisfies

A{abc}= {(Ac)(Ab)(Aa)}. (3.14)

By Lemma 2.2 we know that the signature of E is either definite or neutral. Properties of the negative definite algebras can be trivially obtained from those of




positive definite algebras by way of (1.8). So we shall be concerned only with the two cases of positive definite and of neutral signature. For n = I or 2, ternary composition algebras are given by setting X = 0 in (2.13). To obtain 3C (2,0), or 3C(1, 1), algebras, set E = C, or E = C, and <a,a> = aa, whence {abc} = qbe. One easily sees that the automorphism groups are as given in (1.15) and (1.16), and that, in the neutral signature case,

*Aut T l=0 O(E)U J O(E), (3.15)

where, for E = C, J is the counter-isometry defined by a ia, where i2 = +1.

4. Algebras of dimension n = 4

The possibility n = 3 is clearly inconsistent with axioms X2 and X3, and so we now turn to the possibility n = 4. In this case P, being alternating, can only be + A or - where z denotes a normalized determinant function for E. Let {a, b, c, d} be any ordered orthonormal basis for E which is positive in the sense that

0(a, b,c,d) = +1. (4.1) Then the non-zero values of X upon this basis are forced by X1-X3 to be those obtained by permutation from the values

X(b,c,d)= a,a>a, X(a,c,d)=-<b,b)b,

X(a,b, d)= <c,c> c, X(a,b,c)= - <d,d>d. (4.2)

Assuming that the signature is definite or neutral we confirm that (4.2) does define a ternary vector cross product X. Indeed equations (4.2) imply that

X(a, b, c) = *(a A b A c) (4.3) for a suitably signed star operator, and as is well known (see, for example, Shaw 1983, eq. 11.2.33), the star operator is an isometry in the cases of signatures (4, 0), (2, 2) and (0, 4), whence axiom X3 is satisfied. On the other hand, in agreement with Lemma 2.2c, signatures (1, 3) and (3, 1) are not allowed. For X3 applied to (4.2) yields

<a,a> <b, b> c,c> <d,d> = + 1. (4.4)

(Equivalently, the star operator is not an isometry in the case of signatures (1, 3) and (3, ).)

Since AA = (detA)-1 A, the automorphism group for our four-dimensional algebra is

Auto = SL(E) nO(E) = SO(E) (n = 4), (4.5) as already announced in (1.15) and (1.16). Notice that (in the case n = 4) the invariance group of the scalar quadruple product 0, namely SL(E), is much larger than the automorphism group of the ternary algebra! In the neutral signature case we also have, from (3.13),

*Aut 2 2 = SL(E2' ) n *O(E) = SO (E2' ) U J SO(E2 2) (4.6) for a suitable counter-isometry J. For example if the basis {a,b,c,d} in (4.2) has signature (+ + - -) then we could take J to be the multiplication operator yc a, with effect

Ja = c, Jb = d, Jc = a, Jd = b. (4.7) Proc. R. Soc. Lond. A (1990)

8 R. Shaw




Lemma 4.1. Let i be a four-dimensional ternary composition algebra, and let a, beE be unit vectors. Then (a) if (a, a) = <b, b> the multiplication operators ya b and a, b

are automorphisms of i; (b) if <a,a> = <-b,b> then Ya, b and ca, b are counter-

automorphisms of f.

Proof. The Lemma follows from (4.5), (4.6) in view of Lemma 2.2.

Lemma 4.2. If {a, b, c, d} is any positive orthonormal basis for the four-dimensional algebra i then

-Ya,b = Ja,b + a, a>) b, b> J,, (4.8a) -

Ca, = J,-<a, a> b, b> Jc,d, (4.8b)

Ya,bYb,c = <(bb>)7a,, 0a, b e = <b,b>a, c (4.9)

Ya,b = + <a, a b, b)y, a,a,b = -<a, a> <b, b>) o,, (4.10)

Ya, Yc,d = -, o ta, bc = +1, (4.11)

Yb, a Yc, a d,a = -<a, a> I= - -b, C e,a d ,a (4.12)

Proof. By (3.5) we have <T(a A b) I c A d> = 20(a, b, c, d), so that }T equals the star

operator *: A2E-> A2E (defined using the orientation provided by 0P). Consequently the multiplication operator Tab (= T(a A b), see (3.8)) is given, under our identification (3.4), by

Ta,b 2 * (aAb) = - J,. (4.13)

The multiplication operators Ya,, a, b are, from (3.2), accordingly given by the self- dual and anti-self-dual combinations

- Ya, b = Ja, b * (Ja, ), -a,b= Ja,b * (Ja, b), (4.14)

of the infinitesimal generators Ja, of the orthogonal group SO(E), whence we have

(4.8). Equations (4.9) and (4.10) follow from the explicit expressions (4.8), bearing in

mind the fact, (4.4), that the signature is either definite or neutral. Equations (4.11), (4.12) then follow.

Lemma 4.3. In any four-dimensional ternary composition algebra the properties

Yu,v ew,z = C'w,z7 (4.15)

Yu,v v, w = <v, 7u,v w r TV,w = <v> , u, w (4v.16)

Yu,vYv, wYw, u = <(, > <v, v> w,w> I = ou, v ,,v, w,, (4.17)

hold for arbitrary choices of the vectors u, v, w, z.

Proof. The commutativity property (4.15) is a well-known feature of the combinations (4.14) of the S0(4), or S0(2,2), infinitesimal generations Ja,b, associated of course with the group isomorphisms S0(4) (SU(2) x SU(2))/Z2 and

SO+(2,2) (SU(, 1)xSU(, 1))/Z2. By using (A 1), (A3) and (A 6) we see that

(4.16) holds whenever u = v, or v = w, or u = w. On the other hand we have just seen in (4.9) that (4.16) holds whenever u,v,w are three distinct elements of an orthonormal basis. It then follows that (4.16) holds for u, v, w subject only to the restriction (v, v> # 0, and hence - for example, by continuity - holds for general u, v, w. Finally (4.17) follows from (4.16), after using (A 3) and (A 6).

Finally we give a two-dimensional 'complex' construction of a real four- dimensional ternary composition algebra i. (This will prove of use in Part II of the


9



present series, when we will obtain a four-dimensional 'complex' construction of a real eight-dimensional ternary composition algebra.) In the case of a 3C (4, 0) algebra we use a complex vector space C2 equipped with U(2)-geometry, while in the case of a 3C (2,2) algebra we use a C-module C2 equipped with 'U(2; C)-geometry' (cf. Shaw 1989). That is in either case we deal with a 'vector space' C2(j), say, over C or C, where the 'imaginary' unit j satisfies

*2 = whee + , in the case of C, j _, where = _ -i, in the case of, (4.18)

the space C2(j) being equipped with a non-degenerate scalar product (u, v), linear in u antilinear in v and hermitian symmetric:

(Au, v) = A(u, v), (u, Av) = *(u,v), (u,v)* = (v, ). (4.19) Here ' ' denotes 'complex' conjugation:

(at+fj)* = -fplj, o,fleR. We lay down that C2(j) is of dimension (rank) 2, with orthonormal basis {eo, el}:

(eo, eo) = 1 - (el, el), (e0, el) = 0.

We denote by <u, v> and [u, v] the real and imaginary parts of (u, v):

(u,v) = <u, v) + j[u, v]. (4.20) Note therefore that

[u,v] = -e ju, v> = e(u, j> = -[v, u]. (4.21)

Considering C2(j) as a four-dimensional real space, E(j) say, observe that E(j) is equipped with real orthogonal geometry via <,> and with Sp(4; i)-geometry via [, ]. In the cases e = + l, e =- 1 the scalar product <, > has signature (4,0), (2,2) respectively. We also define the following 'complex' version W of Z in (2.12):

W(u, v, w) = (u, v)w+ (v, w)u-(u, w)v. (4.22) Lemma 4.4. Let X(u, v, w) be obtained from W(u, v, w) by retaining only the purely

imaginary parts of the scalar products:

X(u, v, w) = Cycu, v, wU, v] jw, (4.23) where Cyc,,b ,f(a,b,c) =f(a,b,c)+f(b,c,a)+f(c,a,b). Then (E(j),<,),X) is a 3X (4,0) or 3X(2,2) algebra according as e=+1 or e=-1, with scalar quadruple product po(z, u, v, w) = e Cyc,v,w [z, u] [v, w]. (4.24) The associated 3C (4, 0) or 3C (2, 2) algebra (E(j), < , { }) is given by

{uvw} = W(u,v, w). (4.25)

Proof. From its definition in (4.24), we see that 0 is alternating and moreover assumes the values + 1 upon the ordered orthonormal basis

{eO, e1, eO, e1}, where eO = jeO, e, = je1. (4.26)

Upon factoring this normalized determinant function 0 through , ) as in (2.3), we will accordingly obtain a ternary vector cross product X for E(j). Using (4.21) we see that X is indeed as stated in (4.23). Finally, since W = Z+X, where as usual Z(u, v, w) = u, v> w + <v, w>u- (u, w> v, the associated ternary composition {} is indeed W.


10 R. Shaw




5. The ternary Hurwitz theorem

We now study the possibilities for the existence of ternary composition algebras in dimension n > 4. The first thing to note is that if such an algebra exists, for a

particular n, then it possesses a plentiful supply of four-dimensional subalgebras.

Lemma 5.1. Let {b, c, d} be any ordered orthonormal triad of vectors of E, dimE = n > 4, set

a= <b,b><c,c><d,d>{bcd} (= <b,b)>c,c> d,d)X(b,c,d)), (5.1)

and let H = -<a, b, c, d>- (= the subspace spanned by a, b, c, d). Then H is a four- dimensional subalgebra having {a, b, c, d} as positive - see (4.1) - orthonormal basis.

Proof. An obvious slight modification of that in Shaw (1988c).

Remark 5.2. By a subalgebra of a ternary composition algebra we in general mean not only a subspace H such that {hkl} eH for all h, k, l H, but require also that H be a non-singular subspace.

Lemma 5.3. Let H be any four-dimensional subalgebra, and let {a, b, c, d} be any positive orthonormal basis for H. Then

(a) if h, k are any non-null vectors of H((h, h> # 0 (<k, k>), the operators y, k and

Orh,k map H onto H and H' onto H'; (b) for non-null heH and non-null pe H the operators v, h and or, h inject H into

H';

(c) Yb, aYc, aa = -<a,a>) = -b a,oac, aOda,, (5.2)

where HH O(E) denotes the involution which is + 1 upon H and -1 upon H'.

Proof. Again the proof given in Shaw (1988c) for an algebra of definite signature needs only obvious and slight modifications for it to go through generally. The proof of (5.2) starts out of course from the n = 4 result (4.12).

Note incidentally the following consequence of Lemma 5.3a:

for all h, k H, Yh, and rh,k commute with /H. (5.3)

Theorem 5.4. (The 'ternary Hurwitz theorem'.) The dimension n of a ternary composition algebra i must equal 1, 2, 4 or 8. Moreover the signature must be definite or neutral.

Proof. By our previous results, all we have to show is that n > 4 implies n = 8.

Now, given n > 4, let us choose any four-dimensional subalgebra H = -<a, b, c, d>- as in Lemma 5.1 and let p be any non-null element of H'. Then Ya,p is invertible, by (A 3), and moreover, by (2.8) and (5.2), it anticommutes with nH. Hence HH has zero

trace, whence dim H = dim H = 4, and so n = 4 + 4 = 8.

6. Algebras of dimension n = 8

We now study the structure of the real ternary composition algebras of chief interest - those of dimension 8. Bearing in mind (1.8) and Lemma 2.2c, from now on f = (E, , ), { }) will denote either a 3C (8, 0) algebra or a 3C (4, 4) algebra. First of all

we will take note of several pleasing decompositions of the algebra into an orthogonal direct sum of subalgebras. Dimensionally the possibilities are 8 = 4 4 and


11



8 = 2 G 2 @ 2 ( 2, but in the neutral case each possibility subdivides further upon taking into account signature aspects. In particular the next theorem entails that both of the possibilities,

3C (4,4) = 3C (4, 0)0 3C (0, 4), (6. a) 3C (4,4) = 3C(2,2) 3C (2,2), (6.1b)

are readily achieved.

Theorem 6.1. If H is a four-dimensional subalgebra, then so is H'.

Proof. An obvious slight modification of that in Shaw (1988c). Let now Zo c E denote any non-singular two-dimensional subspace of E. If {u, v}

is an orthonormal basis for Z0, set J = y~,. (Any other choice of orthonormal basis for ZO which defines the same orientation as {u, v) gives rise to the same operator J.) If (v, v> = e<u, u) then J has the properties previously noted in (4.28). Consequently E readily admits J-invariant direct sum decompositions of the kind

E = ZO Z1 Z2Z 3, (6.2) where the subspaces Z. are mutually orthogonal and are each of dimension 2. (Being non-singular, each Z. is in fact a two-dimensional subalgebra.) For a, ,f = 0, , 2, 3 let us also define the four-dimensional subspaces

Ha = Za0 Z (ac Z /). (6.3) Lemma 6.2. (a) Each Ht, is a four-dimensional subalgebra. (b) If (aflyS) is any

permutation of (0123) then {Z, Z,}) Z ZC . (6.4)

Proof. (a) By Lemma 5.1 each of H01, H02 and H03 is a four-dimensional subalgebra. By Theorem 6.1 so are their orthogonal complements H23, H31 and H12.

(b) If aeZ,, beZ., ceZ,, then for distinct c, fi, y we have, by Lemma 5.3a in conjunction with part (a) of the present lemma, not only {abc} = y , c e H, but also {abc} = y, , a H,, whence {abc} E H,, n H,, = Z8.

Next we investigate the problem of finding, under the natural action of 0 (E), a set of canonical forms for our eight-dimensional algebras based upon the space E. We will show that, for a given signature, the algebra & can be cast into one or the other of just two mutually exclusive canonical forms, referred to below as types I and II.

To this end choose any ordered orthogonal triad {e, e2, e2, such that <ei, ei> = + 1, i 1, 2,3, and set eO = (e e, e3) and H = <e e, el, , e3>-. Then, by Lemma 5.1, H is a 3C (4, 0) subalgebra having {e0, el, e2, e3} as positive basis:

00123 -(e0, e2, e23) = +1. (6.5) Choose any unit vector el H'. Thus we have

+l1, for a 3C (8, 0) algebra, e >=

e = {- 1, for a 3C (4, 4) algebra. (6.6)

Define also ei = {e0, e e}, i = , 2, 3. Then in fact we have

ea = o'o0ea, a = 0,1,2,3. (6.7) (Here, and below, we abbreviate y,, as yat in the case when u = ea, v = e,; a similar interpretation applies to cra and Ja, for a, b = 0, 1,2,3, 0', 1', 2', 3'.) By Lemmas 2.2 Proc. R. Soc. Lond. A (1990)

12 R. Shaw




and 5.3, {eo, el, e2, e3,} is an orthonormal basis for H'. Of course, by Theorem 6.1, H' is itself either a 3C (4, 0) or a 3C (0, 4) subalgebra, according as e = +1 or e =-1,

equivalently according as yo0' is an isometry or a counter-isometry. It follows from the foregoing that the basis vector eo belongs to the following seven

four-dimensional subalgebras: H = e0, el, eg, e> (=H), H/i = -<O, e,e, e, e>-, i= 1,2,3, (6.8)

Hai = <eo,ei, ek,,e,>-, ijk = 123,231,312.)

(The labelling of the seven HA is chosen so as to agree with that in Shaw (1988a).) Moreover we have

00123 = + 1 Oii''O'= Oitk'j' (6.9)

The assertions concerning -<e, ei, ei, e, >- and 00ii,o follow immediately from (6.7). On the other hand from yioej = ek (for ijk = 123, 231, 312) follows

--7io o'o0 ej = 7oo ek,

since, by (2.9), y00o anticommutes with yio. Consequently

yio ej = - ek, (6.10)

whence -<e, ei, ek,, ej>- is a subalgebra and k'ij = - (ek, ek> = -c. We now use the fact that if {a, b, c, d} is a positive orthonormal basis for any four-

dimensional subalgebra then, by (4.8), we have

Yb,al = J, bl+(a,>a>b,b> J, (6.11)

where the vertical strokes denote restriction to -a, b, c, d>-.

By using (6.11) in conjunction with the subalgebras HA in (6.8) we can immediately read off simultaneous canonical forms for the seven multiplication operators:

yp0; p= 1,2,3,0',1',2',3'. (6.12)

For example (6.11) asserts (after taking into account (6.9)) that y00 agrees with

J00'+ Ji' when acting upon HAi, whence we have the fourth of the following seven canonical forms:

Yio = + -- Jjk -(j'k' + Joit,

Yo'o + J00+'+?Jil +22' +33' (6.13)

i'o J0,, + Jk'j + Jkj, + J,0,i

where, as usual, ijk runs through the values 123, 231, 312. Canonical forms for the seven multiplication operators crp follow from (6.13) on account of (3.2c):

?o o = 2Jop - Yp (6.14)

However, when proceeding further to consider the corresponding simultaneous canonical forms for the remaining multiplication operators we find that our path divides into two according as the ordered orthonormal basis {e0,, el, e2,, e3,} for (H)I is positive or negative. In the former case the operator y0 0 maps a positive basis for HO onto a positive basis for (HW)i; hence, in this case, y0,0 gives rise to a mapping HO -? (Hw)I of four-dimensional algebras which is for e = + 1 an algebra isomorphism, and for e = -1 an algebra counter-isomorphism. In the latter case we see instead -

bearing in mind that [+e,, ifa = 0,

?? ea -e, if a=1,2,3,


13



- that it is the operator 0, 0 which induces an algebra isomorphism (if e = + 1) or an algebra counter-isomorphism (if e =-l) of the four-dimensional algebras H" and (H).)I

By continuously varying the orthonormal triad {el, e2, e3} in the foregoing, and also the unit vector e0, subject to it being orthogonal to the four-dimensional subalgebra H generated by e1, e2 and e3, we see that we have lighted upon a fundamental dichotomy concerning our eight-dimensional algebra g. Let H be any 3C(4,0) subalgebra of g and choose any unit vector h eH and any unit vector p E H'; we will say that & is of type I if y, h induces an algebra isomorphism (if g is a 3C (8, 0) algebra) or an algebra counter-isomorphism (if g is a 3C (4,4) algebra) of H onto Hl, and we will say that g is of type II if the same applies but with P, h instead of yp h

(Incidentally, it turns out that if o is of type I then yp,h *Aut , for any orthonormal pair {p, h}, while if ( is of type II then rp h^ *Aut e. This will be proved in Part II of the series.)

From now on we will concentrate on the cases of a 3C (8, 0) (e = + 1) or a 3C (4,4) (e = -1) algebra & of type I. Information concerning type II algebras follows simply by interchanging the roles of ya b and ra,b. In other words, by (2.15),

S & type I<> ? E type II. (6.16) In particular, see Remark 2.6, a ternary vector cross product X for (E, <, )) is of type I if and only if the ternary vector cross product -X for (E, <, )) if of type II.

Granted that our algebra & is of type I then from (6.9) we obtain the further values

o0'1'2'3' = + 1, (jk'j' 6- -

O'i'cj (6.17) For example, knowing now (for type I) that yio' ej = eek it follows, after acting upon both sides with yoo, that y,o, ej --ee. Similarly, applying yj0 to both sides of yj,'k e0, = - e we obtain yj,k ej = sek. Observe from (6.9) and (6.17) that 0, considered as an element of A4E, enjoys a self-duality property.

We can now write down canonical forms for all of the remaining multiplication operators, by arguing as we did in the derivation of (6.13). We find that the complete canonical form for a ternary composition { } of type I is given by the canonical forms (6.13) together with the following seven sets of equations relating the multiplication operators yab for distinct a, b:

Yoi = Yjk = - 6Yj'k' --o'i',

7oo' = Y11'= 722' 733', (6.18)

0i' = 7k'j 7kj' = 'i

Here, as in (6.9), (6.13) and (6.17), ijk runs through the three values 123, 231 and 312 and e = + 1 or -1 according as the signature is definite or neutral. (Of course the values y,, for a = b are determined by Axiom C2 to be simply Yaa, = <ea, ea> , and so are omitted from the above canonical form.) Certain symmetry and Clifford algebra aspects of our canonical form are discussed further in Shaw (1988a).

Lemma 6.3. Let H be any four-dimensional subalgebra of &, and let {a, b, c, d) be any orthogonal basis for H. Then the following results hold.

(a) ya, b commutes with 7y, d, and oa,b commutes with rc,d (b) ya, b 7,d = 0(a, b,c, d)GH, where the linear operator GH is independent of the

choice of basis for H and has, relative to the direct sum decomposition E = H G H', the form

GH = (-I) gH.


14 R. Shaw




(c) ca, b cd, = =((a, b, c, d)SH, where the operator SH has the form SH =I gH

(d) (GH)2 = I = (SH)2, whence (gH)2 = .

(e) For h, k H the restrictions to H' of the multiplication operators h, k and rh, k

commute with gH, and hence leave invariant the + 1 and -1 eigenspaces of gH. (f) Either (type I) gH =--I, whence

7a, b c, d = (a, b, c, d)I, and oa,b (O'c d = (a, b,c,d)1HH, (6.19)

or else (type II) gH = +1, whence

7a, b c, d = -q(a,b, c, d)HH, and ca, b c,d = 0(a,b,c,d)I. (6.20)

Proof. As in Shaw (1988d).

Our results so far are summarized in parts (a)-(d) of the next theorem. (To obtain

(6.22), (6.23) use (2.11) and (5.2) in conjunction with (6.21) = (6.19).) For the proof of the final part (e), together with more details concerning the SO(E) orbits, the reader is again referred to Shaw (1988d). In particular one sees, for a fixed choice of star operator on A4E, that the algebra g can be allocated to type IR or type IL

according as the scalar quadruple product satisfies *.0 = + 0 or * =- 0.

Theorem 6.4. (a) The ternary composition algebras based upon the eight-dimensional real scalar product space (E, <, >) fall into two O(E)-orbits, say type I U type II.

(b) e type I= >?0e type II.

(c) If g is of type I, and if H is any four-dimensional subalgebra having {a, b, c, d} as

positive orthonormal basis, then

Ya,bYc,d =-I, and oa, be,d = HH. (6.21)

We also have ya,b = <a, a> b, b>yd (6.22)

and yc,aa caa, d = (a, a)c, d H, oc,a , = <a, a> , d. (6.23)

Moreover if p is any unit vector e H' then yp, agives rise to a mapping H- H' which for <a, a> = <p, p> is an algebra isomorphism, and for <a, a> = -<p, p is an algebra

counter-isomorphism. (d) If 6 is of type I these exist (a 21-parameter family of) orthonormal bases of the kind

introduced around (6.5), (6.6) relative to which the algebra assumes the canonical form

displayed in (6.13), (6.18) (supplemented of course by ya,a = <a,a>I). The non-zero

values of the scalar quadruple product (P on such a canonical basis are those determined

by (6.9) and (6.17) (and by the skew symmetry of P). Consequently Pe A4E is self-dual.

(e) Each O(E)-orbit splits into two SO(E)-orbits, say IRU IL and II U IIL. If 0

belongs to type IR then Kq, - P, - K provide examples of type IL, IIR, IIL,

respectively, for any choice of K O_(E). Choosing K to be the 'conjugation' defined in

terms of a canonical basis for P by Ke = eo, Kea = -ea a = 1,2,3,0', 1', 2', 3', then, in

an appropriate (1 + 7)-dimensional notation applied to the canonical form (6.9), (6.17),

examples of the four SO(E)-types can be displayed as follows:

type IR: 0 = + eo A + * ,

type IL: Kq =--eo A + *7 0'

type II: - =-e 0A-*7-, (6.4)

type 1l: K = +eo A0 q-*7 .

Here 0 A 3E',E' = -eo>-', and *7 denotes a star operator A3E'-> AE' c A 4E.


15



Remark 6.5. When invoking part (b) of the theorem to relate a type I algebra & to a type II algebra oe, a cautionary remark may be in order: if {a, b, c, d} is a positive orthonormal basis for a subalgebra H of I, then take note that {a, b, c, d} is a negative orthonormal basis when H is considered as a subalgebra of the opposite algebr. 0o.

7. Further properties of 3C (8, 0) and 3C (4, 4) algebras In this section we assume that our 3C (8, 0) or 3C (4,4) algebra 6" is of type I, except

when we explicitly state otherwise.

Theorem 7.1. For all u, v, w eE we have the 'factorization property'

cU, v =, w =u <v, v> cr, ,w (7.1) and also the property

7{uvw},w =<, w )> Yu, v (7.2)

Proof. Just as we proved (4.16) starting out from (4.9), so we can prove (7.1) starting out from (6.23). Applying the identity (7.1) to zeE we have

{{zwv} vu} = (v, > {zwu}, (7.3)

which immediately yields (7.2) after changing z, w, v, u to u, v, w, x, respectively. Incidentally, the operator equation (7.1) may conceivably merit study in its own

right. Certainly it possesses solutions, namely metrical dyads of the form u)> <l, quite different from the ones currently under consideration.

Corollary 7.2. Choose eeE such that <e,e> = 1 and let E' = <e>-L. For u, veE' define

X'(u, v) = X(u, e, v), (7.4)

so that (E',<,>,X') is seen to be a 2X(7,0) or 2X(3,4) algebra. Then this seven- dimensional binary vector cross product X' satisfies the identity

(X'(a,b),X'(u,v)> = <a A bl u A v)+ 0(a,b, u,v) (7.5)

for all a, b, u, v E.

Proof. The a, u matrix element of the operator identity ob, er, e= (e, e) b, vyields

<{abe}, {uve}> = <e, e> <a, {uvb}>, (7.6)

whence we have the required result

<X(a, b, e), X(u, v, e)> = <e, e> [<a A b I u A v) + (a, b, u, v)] (77) valid for all a, b, u, v orthogonal to e.

Remark 7.3. On account of its self-duality the scalar quadruple product 0, viewed as an element of A4 E, can be expressed in the form

5 = eAA+*,70 (7.8)

for some element E A3 E', and for a (suitably signed) star operator : A3 E'-> A4 E'. Consequently in the right-hand side of the identity (7.5) we have

5(a, b, u, v) = (* 0) (a, b, u, v) (7.9) Proc. R. Soc. Lond. A (1990)

16 R. Shaw




since a, b, u, v eE'. In this form the identity (7.5) appeared as Theorem A in Shaw

(1988b), where it was given a different proof. Incidentally, viewed as an element of A3 (E), 0 in (7.8) is the scalar triple product associated with the binary vector cross

product X': ((u,v,w) = <u,X'(v,w)>, u,v,weE'. (7.10)

For (7.8) yields (u, v, w) = d(e, u, v, w)

= u,X(v, e, w)> = <u,X'(v, w)).

Lemma 7.4. If a, b, c, u, v, w are distinct elements chosen from an orthonormal basis for E, then

<X(a,b,c), X(u,v,w)) = 0. (7.11)

Proof. Set d=X(a,b,c); then, by Lemma 5.1, H= -a,b,c,d>- is a four- dimensional subalgebra. Since u, v, w are orthogonal to a, b, c we have

u = u' + ad, v = v'+ fid, w = w' + yd

for some u',v',w' eH' and some a, fl, ye R. Now, by Theorem 6.1, X(u',v',w')eHH. Consequently, using axioms Xl and X2, we have

<X(a,b,c), X(u,v,w)> = -d,X(u',v',w')) = 0.

Theorem 7.5. For all a, b, c, u, v, w E E the following identity holds:

<X(a,b,c), X(u,v,w) =a A b A cu A v A w>+F(a,b,c,u,v,w), (7.12)

where, writing f(a, b, c) +f(b, c, a) +f(c, a, b) as Cyc,, cf(a, b, c),

F(a, b, c, u, v, w) = Cyca, b,Cyc, a, )> 0 (b,c, v, w). (7.13)

Proof. Since X is alternating we may as well assume that a, b,c are mutually orthogonal, and that u, v, w are mutually orthogonal. Let the difference between the two sides of (7.12) be denoted S(a, b, c, u, v, w). According to (7.7) S = 0 when c = w.

Consequently S is alternating in all six of its variables. So all we need to do is to check that S = 0 when a, b, c, u, v, w take distinct values from an orthonormal basis for E.

By Lemma 7.4 this is indeed the case.

Remark 7.6. In terms of the tensor components (5abcd = p(ea, eb, ec, ed) of 0, the

identity (7.12) has the coordinate expression (cf. DeWit & Nicolai 1984; Diindarer et al. 1984),

jabcs 68_ a +[bc 9[a(bc] (7.14) uvws uvw lu Vw]*

For an algebra of type II the identities corresponding to (7.12), (7.14) are obtained

simply by replacing 0 by - P.

Appendix. Properties of multiplication operators

It follows from their definitions in (2.5) that the multiplication operators Y, b and

oa,b are bilinear functions of their arguments a, beE. By (1.7b) they satisfy

a, a a, a>I= (oa,a. (A l)

By appropriate polarizations (linearizations) of (A 1) and (1.7c) we deduce further


17



18 R. Shaw

properties as listed below. The proofs are almost the same as for the case of definite signature as given in an earlier paper (Shaw 1988c), and so are omitted. (In the case of (A 5), this is initially derived under the proviso that <a, a) is 7 0. However, such non-null vectors span E, so (A 5) holds in general.)

7a, b + b, a = 2(a,b>)I = ra, b + b,, (A 2)

'a, b a, b = a,a> <b, b I = a, b ca,b (A 3)

Ya, b Ya, c + Ya, cYa b 2a, b a <b,c> I = Ta, b O,c + (ac, -a, b (A 4)

Ya, b + a, b = 2<a,b)I= boa, b + a, b(, (A 5)

Ya,b = Yb,a and a,b =

b,, a (A 6) Yb, a a,b = <a, a> <b, b>I = - b,a O'a,b, (A 3')

Yb, a Ya, c +Yc, a Ya, b = 2 <a, ) (, c) I = ob a (a, c + c, a a,, b (A 7a)

Ya, b c, a + , c b, a = 2 a, a> b, c> = a,,b c, + ca,c'b,. (A 7b)

The analogue of (A 3) holds for the middle multiplication operator a, b of (2.5) (a similar consequence of (1.7c)):

a,bfia,b = (a, a>) b, b>I (A 8) and hence so does the analogue of (A 4). One can show that, for even n, a, b lies in O_(E) whenever <a, a = b, b) = +1. One also sees that

aa,b = tb, a (A 9)

Let us now make a choice of unit vector e E to act as fixed "base point'. The associated conjugation Ke = <e, e>)- e, will be written K: a ha. Thus

Ka = = -a+2<e, e>-1 <e, a> e. (A 10)

First of all note the properties

7e, a= -a,e = Ya,e, e, a = 0a, e = 0'a,e, (A 11) which follow from (A 5), (A 6). Next we show that we can use K to establish simple communications between the three families of operators {Y, e}, e{/} , e} {a, e} which are parametrized by the variable vector acE:

Ya,eK = K,, = Koe, (=Koa,e). (A 12) In particular the family {a, e} is the similarity transform of the family {ya e} by the fixed conjugation K = K,:

Ka, eK = (ta, e' (A 1.3)

Let us prove (A 12) in a more general setting. We start out from the properties

#a,a{aacb} = <a, a>{abc}, ta,a^{bca}= <a, a>{cba}, (A 14) which in turn follow, via (2.13) from

, aX(a, b, c) =-< a> X(a, b, c) (A 15) and a, a Z(a, b, c) = a, a) Z(a, c, b). (A 16) Of course (A 15) holds because X(a, b, c) is perpendicular to the vector a. Similarly (A 16) holds because Z(a, b, c) consists of a part b, c> a parallel to a and a part J c, a = <a, b> c - <a, c> b perpendicular to a.





Various properties result from (A 14). In particular we have

a,a#a,b = (a, a)>a, b, / a, Ya,= a, a>/a,c (A 17) and ,a,a = a, a>) a, b, /la,a Ca,c = <a, a>,. (A 18)

We also have the adjoints of these equations; in particular from (A 18) we obtain, after the interchange of a and b,

b, ai b,b <b,b) aa,b. (A 19)

Finally note also that (A 17) and (A 19) yield, after using (A 8), (A 9),

Ya, b a, b = /4a, a (= a, Ya, b)' (A 20)

Loosely stated this last result states that {abcba} is unambiguous as it stands, without the need for further braces:

{ab{cba}} = {a{bcb} a) = {{ab} ba}. (A 21)

With the particular choice a = e in (the adjoint of) (A 17) and in (A 18) we obtain

(A 12), and hence (A 13), as desired. Setting b = e in (A 20) we obtain also

Ya, e Oa, e = <e, e)> ia, aK = Oa,eYae (A 22)

Incidentally property (A 13) can be seen also from (3.2), upon noting that Ka commutes with T, b and anticommutes with Ja,b

Some further properties of the operators a ,b can be found in Shaw (1988d, Appendix).

References Brown, R. B. & Gray, A. 1967 Comment. Math. Helv. 42, 222-236. De Wit, B. & Nicolai, H. 1984 Nucl. Phys. B 231, 506-532.

Diinderer, R., Giirsey, F. & Tze, C.-H. 1984 J. math. Phys. 25, 1496-1506.

Eckmann, B. 1942 Comment. Math. Helv. 15, 318-339.

Faulkner, J. R. & Ferrar, J. C. 1977 Bull. Lond. math. Soc. 9, 1-35.

Freudenthal, H. 1964 Adv. Math. 1, 145-190.

Gray, A. 1969 Trans. Am. math. Soc. 141, 465-504.

Harvey, R. & Lawson, H. B. 1982 Acta math., Stockh. 148, 47-157.

Jacobson, N. 1974 Basic algebra I. San Francisco: Freeman.

Jordan, P., Von Neumann, J. & Wigner, E. P. 1934 Ann. Math. 35, 29-64.

McCrimmon, K. 1983 Trans. Am. math. Soc. 275, 107-130.

Shaw, R. 1983 Linear algebra and group representations. London: Academic Press.

Shaw, R. 1987 J. Phys. A 20, L689-L694.

Shaw, R. 1988a J. Phys. A 21, 7-16.

Shaw, R. 1988b J. Phys. A 21, 593-597.

Shaw, R. 1988c J. math. Phys. 29, 2329-2333.

Shaw, R. 1988d Hull Math. Res. Rep. 1(1). Shaw, R. 1989 Nuovo Cim. B 103, 161-183.

Received 9 May 1989; revised 18 April 1990; accepted 24 April 1990


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Ternary Composition Algebras. I. Structure Theorems: Definite and Neutral Signatures

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