A Note on Milner's E-numbers

A Note on Milner's E-numbersAuthor(s): C. W. KilmisterSource: Proceedings of the Royal Society of London. Series A, Mathematical and PhysicalSciences, Vol. 218, No. 1132 (Jun. 9, 1953), pp. 144-148Published by: The Royal SocietyStable URL: http://www.jstor.org/stable/99359 .

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A note on Mimer's i-numbers

BY C. W. KILMISTER

King's College, London

(Communicated by G. Temple, F.R.S.-Received 12 March 1953)

Recent results of Milner, which appeared to require an extension of the tensor calculus, are extended and generalized. It is shown that Milner's suggested extension is, in fact, unnecessary.

1. INTRODUCTION

In two recent papers (Milner 1952 a, b), referred to here as A, B respectively, Milner has suggested that certain transformations require an extension of tensor theory. Sir Edmund Whittaker has drawn attention to Milner's work (Whittaker I95I, pp. 1, 14) and supported his suggestion. The basis of Milner's argument is thatwhen the transformation of a seconrd-rank tensor is expressed as an equivalence transformation (T --T' = PTQ) in the obvious matric algebra, a certain linear function of the elements of the tensor is not transformed under an equivalence transformation. One of the results of this paper is that the transformation of the derived set of elements is merely an equivalence transformation in a new matric algebra, so that only the interpretation of the standard tensor theory is altered.

The principal results of Milner's two papers appear to be the following: (a) the conversion of matrices to E-numbers (A, 3.9), and the properties of this

conversion (A, 3'10; A, 3.11); (b) the introduction of summed transformations (A, 3.14), and the connexion of

summed and unsummed transformations; (c) the group property and tensor property of summed transformations (B, I 1;

B, 1P14, 1.15); (d) the product formula (A, 3.16); (e) the invariant scalar (A, 3 27). In ? 2 of the present paper, the generalization of Milner's results (b), (c) and (e)

is found for a general algebra, and for general linear functions. In the particular case when the algebra is that of E-numbers (or quaternion linear functions of quaternions), it is shown in ? 3 that all Milner's results may be deduced very shortly by quaternion methods. Similar results are true for any even Clifford algebra, and in ? 3 the explicit form of them for the EF-number algebra is given.

2. PROPERTIES OF A GENERAL ATLGEBRA

Let A be an algebra, with a unit quantity e, over the (real or complex) number field, K, and let L be any non-singular linear mapping of the elements of A on to themselves. Thus La, L-la are defined for all a of A. We define a new binary com- position a o b in the set of elements of A, by the equation

L(ab) = Lao Lb. (2.1) [ 1443



A note on Milner's s-numbers 145

Hence a o b = L(L-1a. L-lb),A

and ab = L1(LaoLb). (2J2)

It is obvious that, if A, 4a are in K,

ao (Ab +uc) = Aaob+ ,aoc,

(Aa+Ib)oc = Aaoc +boc, (2.3)

(aob)oc = ao(boc).

We have thus a new algebra A * (say) with the same elements as A, but compositions + and o. Under an equivalence transformation of A,

a-+a' = bac, (2.4)

we have La' = Lb oLaoLc, (2.5)

so that La and La' are related by an equivalence transformation of A*. Suppose, in particular, that the linear function L may be written in the form

La = Ep P aq. (Pm, qm in A), m N (2*6)

with, say, L-la = E rmasm. m

For instance, if A is an even Clifford algebra, any linear function has this form. We then have La' = Y,pm bacqm

m

- E Pm br Las cq. m,n

- E bmnLa.Cmn (say), m,n

where bmn Pm brn, Cmn = 8cqm

Hence, in this case, any transformation (2 5) is a 'transformation of summed type' in Milner's sense. This establishes the results corresponding to Milner's results (b) for a general algebra, and any non-singular transformation of the form (2.6). Since the transformations (2.5) are the equivalence group of A* and, further, the zero element is its own transform under L, the properties corresponding to Milner's (c) follow at once. Our deduction of Milner's result has in fact proved the extension, that the derived group of summed transformations is actually isomorphic with the group of equivalence transformations. It is apparent that Milner's suggested extension of tensor theory is in fact only a difference in interpretation; and that the new interpretation is, in the cases usually considered, much less simple than the old. In a different context, however, this might not be so.

The remaining general result of Milner is (e), the existence of a quadratic invariant of the transformation L. We now find conditions on L for the existence of such an invariant in any finite algebra.

Let L be a general mapping, not restricted by (2-6). We may obviously write L as a square matrix, with an element a of A as a column matrix. Milner's result is that for a certain L there exists a matrix G such that, if La = a*, then

a'*Ga* = a'Ga,



146 C. W. Kilmister

where dashes denote transposition. It is equally easy to consider the condition that

b'*Ga* b'Ga, (2.7)

for all a, b in A. We call this an 'invariant of Milner's type'. Since La = a*, we have a'L' = a'*, so that (2-7) becomes

b'L'GLa = b'Ga

for all a, b, and hence a = L'L or L' = OL-W-1.

Thus a necessary and sufficient condition for an invariant of Milner's type is that the transposed and reciprocal of L should be equivalent under automorphisms. As a particular case, if L is orthogonal, G 1 and the condition is well known.

Consider the particular case when L2 = 1, as in Milner's paper. Then !( 1- L) is idempotent, and hence I

2(l1- L) r

(Albert 1937, P. 88), where - denotes equivalence under automorphisms. It follows at once that I( 1 - L) (I - L'), so that

LrczL = L-1,

and the condition is certainly satisfied. We have thus proved the following extension of Milner's result (e):

If A is a finite algebra over K and L a linear self-reciprocal mapping of A on to itself, then there exists a quadratic form (or a bilinear functional) invariant under L. This form is uniquely determined by L.

3. PROPERTIES OF LINEAR FUNCTIONS

In this section the particular case considered by Milner is discussed in more detail, using quaternions. References are made to the theory developed in papers by the author (Kilmister i949, I95I), referred to as I, II respectively. Suffixes i,j, k, 1, and Greek suffixes, take the values 0, ..., 3, and a summation convention applies to them.

It is well known (Conway 1937, P. 146) that the algebra of quaternion linear functions of quaternions is the same as E-number algebra. Consider a general quaternion linear function of a quaternion defined by I (3.8)

fq = ajS(a?q), (3.1)

and define f*q =a qa~. (3.2)

This definition is obviously independent of the particular sets aj,aq chosen to expressf. Since, by I (3-11),

f*q = (j)ajejS(qa,e,),

(where (j) does not count as a suffix), we have, using I (3.13),

f*q = a( )aiejqa'e1

_ ajS(atq).



A note on Milner's g-numbers 147

Hence (3. 1) and (3.2) define a linear transformation

f>f*Lf, L2=1. (3.3)

Further,byIl (4.2), (4.21) and (4.22) andEddington (1946, ?83) thetransformation L corresponds, in the linear function algebra, to Eddington's cross-dual in the EF-algebra, as is shown in detail below. Accordingly we call f* the cross-dual off.

Using (3.1) and (3 2) in reverse gives, in the notation of I, ?3,

K41q q 2ejS(ejq) = 2PIjq (say). (3.4)

Applying 1 (3 .12), we have K*. q (k) et ek qej ek,

so that K* (k)Kik jk, (3.5)

where we use the notation ej ej = ej (3.6)

for the multiplication of the quaternion units. For example, 01 = 1, 23 -32 = 1.

(We suppose the summation convention to apply to suffixes in such positions as k in (3.5).) Taking the cross-dual of each side of (3.5) we have conversely

j =(k)X*i*k jk (3.7)

On the other hand, if we write f* fjK* =fiK.j,

we have at once = }(k)fik j(3

fij = I (k)fik ikj, (3-8)

and fq-2fjPijq. (3.9)

Since Pij q (j) es q1,

where q -= qi ej,

it follows from (3.4) and (3.9) that 2(j)fw are the matrix elements off in the matrix representation of I, ?4. Hence (3.8) is closely connected with Eddington's result (I 936, ? 5.7) and is equivalent to Nikolsky's formula (I 935). Moreover, the connexion between (3.9) and I (3.15) is that between Milner's E-numbers and g-numbers, (a). The equations (3-8) are in a most suitable form for explicit calcu]ation.

Since we may rewrite (3e5) in the notation of II (4.5) as

L -2 (k) Kok, ko, (3.10)

it follows that the cross-dual transformation is of the form (2.6), and the theory of ? 2 applies to the E-number algebra. It remains only to find the matrix a in (2 7). Using IT (4.14), the equation (3.10) shows L to be self-transposed. Hence, in this case, L is orthogonal, and G-1.

The extension of the results of this paragraph to a general even Clifford algebra is obvious. For simplicity, we give the results for the EF-number algebra, which is the only case which appears at present to be of importance. Corresponding to (3.1) we have (11 (4.2)) 15

Ff- E gwqs( ), f(3.11) p=O



148 C. W. Kilmister

and we define the cross-dual, F*, by

F*f gpfgp (3.12)

Take the matrix representation of f to be {fO}, and similarly for gp, g. Then, in the notation of II, 1 5

(Ff ) a,8- E =p, pjay Faflya

* 15

4 pO = fl

Thus aF*a ,8,, which proves our definition to be equivalent to Eddington's (I946, ? 83). From the matrix form, F -> F* is obviously self-reciprocal, and thus

K* = 4Pj,kl l3 iklk- ih' (3.13) where P Kiklf = qs(Kf ) j

analogous to (3.4). Using 1 (4.9) gives the results

Kn,pq 4 W(i)K?nj, p,i jq'

Krrlnap = 4(ij) K* .. (3.14)

corresponding to (3.5) and (3.7). In view of Eddington's definition of the cross-dual as a permutation of matrix suffixes, (3.14) is a remarkable result. Finally, the quadratic invariant is obvious when written in matrix form.

Most of the results of ? 3 were previously given by the author in a Ph.D. thesis (Kilmister 1950, ? 12).

REFERENCES

Albert, A. A. 1937 fodern higher algebra. University of Chicago Press. Conway, A. W. 1937 Proc. Roy. Soc. A, 162, 145. Eddington, Sir A. S. I936 Relativity theory of protons and electrons. Cambridge University

Press. Eddington, Sir A. S. I946 Fundamental theory. Cambridge University Press. Kilmister, C. W. I949 Proc. Roy. Soc. A, 199, 517 (I). Kilmister, C. W. 1950 Ph.D. thesis, University of London. Kilmister, C. W. I95I Proc. Roy. Soc. A, 207, 402 (II). Milner, S. R. I95za Proc. Roy. Soc. A, 214, 292 (A). Milner, S. R. I95zb Proc. Roy. Soc. A, 214, 312 (B). Nikolsky, N. 1935 Proc. Roy. Soc. A, 150, 411.



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A Note on Milner's E-numbers