Applied Mathematics and Computation 115 (2000) 77±88 www.elsevier.com/locate/amc

Sedenions: algebra and analysis K. Imaeda a, M. Imaeda b,*

a 507-150, Yokoikami, Okayamashi, 701-1145, Japan b IFUP, 24201, Rue De Cezanne, Laguna Niguel, CA 92677, USA

Abstract A 16-dimensional Cayley±Dickson algebra is presented. Its unique algebraic prop- erties, its zero-divisors, and the solutions to a general linear equation are found. A theory of function is developed in terms of the regularity (monogenicity) conditions and some such functions are constructed. Ó 2000 Elsevier Science Inc. All rights reserved.

Keywords: Cayley±Dickson algebra; Sedenions; Monogenicity; Quaternions Octonions

1. Introduction

What we mean here by a sedenion algebra is a non-commutative, non-as- sociative, non-alternative, but power-associative 16-dimensional Cayley± Dickson algebra with a quadratic norm and whose elements are constructed from real numbers, R, by iterations of Cayley±Dickson process [1,2], where 1 is chosen as the ®eld parameter at each step of the construction. In such process 2N -dimensional algebras of complex numbers, C (N ˆ 1), quaternions, Q (N ˆ 2), octonions, O (N ˆ 3), sedenions, S (N ˆ 4), and other power-associa- tive hypercomplex numbers are successively obtained. The fact that sedenions do not necessarily constitute a composition nor a division algebra, the reason for which can be traced to the non-associativity of O, and possessing properties that are not obvious within our current physical theories, may have been the motivation for introducing concepts and modi®- cations to achieve or eliminate some speci®c metric and algebraic properties. Some parts of our work paralells that of other authors [3±5]; but unlike the

* Corresponding author. E-mail: [email protected]

0096-3003/00/$ - see front matter Ó 2000 Elsevier Science Inc. All rights reserved. PII: S0096-3003(99)00140-X 78 K. Imaeda, M. Imaeda / Appl. Math. Comput. 115 (2000) 77±88 sedenions discussed in [3,4] our sedenions are non-alternative, non-modular and pocess real norm. In [5] alternativity is restored by introducing ternary products. Should Hurwitz theorem be one possible indication of non-confor- mity of sedenions to current mathematical methods in physics? It is clear from Muses [6] and Gursey [7], and other authors that some new algebraic and analytical structures, such that might be o€ered by a 16-dimensional algebra, may hold a key to better formulation of physical theories. Furthermore, its relation to the Lie group G2 remains compelling under any circumstances. We had found that the zero-divisors of sedenions are con®ned to some hypersur- faces and that the linear equations are solvable under certain conditions. Our ®rst task is to list some basic algebraic properties of sedenions in Section 2. Find the zero-divisors and the conditions for their existence, in Section 3. The non-trivial task of solving a linear equation is treated in Section 4. We also develop a theory of function of a sedenion variable based on Fueter's Morera- type regularity (monogenicity) conditions and construct intrinsic, polynomial, and exponential functions, in Section 5, as the special cases of more general 2N - dimensional theory studied earlier [8,9]. A brief account of octonions that we refer to in the main text is found in Appendix A.

2. Basic algebra

In this paper we denote sedenions by S, T, Z, in bold capital letters, and octonions by capital letters, A, B, X, Y, etc... We de®ne a sedenion,

S :ˆ A; B†2S; A; B 2 O; as an ordered pair of two real octonions, A and B. The conjugation of S is then de®ned in terms of A, the conjugate of A, and of B by S :ˆ A; B†: As with octonions this implies that S ˆ S, and that the automorphism is im- proper. The product of two sedenions, S :ˆ A; B† and T :ˆ C; D†, under Cayley±Dickson process, is ST ˆ A; B† C; D† :ˆ AC ‡ qDB ; BC ‡ DA†: 1†

We choose the ®eld parameter, q: ˆ1.

In constructing the sedenion basis, el; l ˆ 0; ...; 15†, from the octonions basis, il; l ˆ 0; ...; 7†, we choose the generator, e8 ˆ (0;1), and de®ne

el :ˆ il l ˆ 0; 1; ...; 7†;

e8‡l :ˆ ile8 i0 ˆ e0 ˆ 1†: K. Imaeda, M. Imaeda / Appl. Math. Comput. 115 (2000) 77±88 79

The multiplication of the two basal elements, ej and ek (the Greek (Latin) in- dices take some or all values of 0; 1; ...; 15 1; ...; 15†), satis®es

ejek ˆdjk ‡ ejkmem; 2† where the value of the structure constant, ejkm, which is totally antisymmetric in its indices, is given by the usual permutation rule for a 3-indexed antisymmetric tensor with values 0 or ‹1, and determined by the appropriate 35 sedenion cycles j; k; m†. In terms of the basis a real sedenion, S, may be written com- ponentwise as

X15 S :ˆ s0 ‡ sjej; sl 2 R; jˆ1 P15 where s :ˆ jˆ1 sjej is termed as the ``pure sedenion''. The structure constant, ejkm, is associated with the non-commutativity of sedenions by X S; T† :ˆ ST TS ˆ 2 sjtkejkmem; 3† jkm† where the summation is over all possible permutations of sedenion cy- cles j; k; m†. The non-associativity of three sedenions, S, T, and U, is measured by the non-zero (not necessarily zero) associator, (S, T, U):ˆ (ST) U S(T U). Although the multiplication of basal elements is strictly alternative (true for the basis of any 2N -dimensional Cayley±Dickson algebra with ®eld parameter 1) it is not necessarily true for arbitrary sedenions. For two sedenions, S and T, the associator S; S; T† ˆ A; B; D†; A; B; C†† ˆ S; S; T† 4† is not necessarily zero. Furthermore, unlike octonions,

ejek†em ‡ ejem†ek m 6ˆ k† does not always vanish. This translates to the fact that the second structure constants, which deals with triple products of sedenions, are non-antisym- metric in its indices. The norm of a sedenion S, de®ned as

X15   2 n S† :ˆ SS ˆ SS ˆ n A†‡n B†ˆ sl 2 R; lˆ0 is a non-degenerate quadratic form associated with a symmetric bilinear form 1 S Á T ˆ n S ‡ T†n S†n T††: 2 Sedenions do not form a composition algebra. Indeed, n(ST) n(S) n(T) does not vanish in general. Furthermore, sedenions do not constitute a division 80 K. Imaeda, M. Imaeda / Appl. Math. Comput. 115 (2000) 77±88 algebra. For a sedenion S ˆ A; B†, which satis®es certain norm and ``vector- closure'' conditions, there exist a sedenion, Z, which is both left and right zero- divisor of S, ZS ˆ 0 ˆ SZ:

3. Zero-divisors

One cannot solve a general linear equation of sedenion in a usual manner due to the non-alternativity of the algebra. However, we can write a sedenion as an ordered pair of octonions. We ®rst deal with the case of ®nding the zero- divisors of sedenions. For the right zero-divisor, Z,ofS, SZ ˆ 0: 5† Representing S and Z by A; B† and X ; Y †, respectively, where A; B; X ; Y are non-zero octonions, for the non-trivial case, we have

AX YB ˆ 0; BX ‡ YA ˆ 0; 6† from which we obtain 1 1 A ˆ B; X ; Y †; B ˆ A; X ; Y †; n Z† n Z† 7† 1 1 X ˆ Y ; A; B†; Y ˆ X ; A; B†: n S† n S† We can now prove the following.

Proposition (Zero-divisor conditions). Eq. (6) implies the norm condition: (I) n A†ˆn B†,andn X †ˆn Y †. Eq. (7) implies the following \vector-closure" conditions: (II) A; B; X ; Y are pure octonions; thus the left and the right zero-divisors are same for a given sedenion, S. (III) Any product of A, B, X, Y, such as AB, AX, AY, are pure sedenions, (IV) The triple products arising from A, B, X, Y, such as X(AB), Y(AB), A(XY), and B(XY), are pure octonions. (V) Non-vanishing of a determinant, as discussed below.

By suitable substitution and by using octonion properties (see Appendix A) we obtain the following coupled equations from (7):

xX A; B†‡ X ; A; B†ˆ0; xY A; B†‡ Y ; A; B†ˆ0; 8† K. Imaeda, M. Imaeda / Appl. Math. Comput. 115 (2000) 77±88 81 where n S†2 x :ˆ 2 R: n A; B† One can show that x ˆ 1, which implies that Eq. (8) for X reduces to

XA†B X BA†ˆ0: 9†

Componentwise this equation is written as X7 k0jxj ˆ 0; 10† jˆ1 ! ! X7 X7 xjaj br xkbk ar ˆ 0 8r ˆ 1; ...; 7; jˆ1 kˆ1 P where k0j :ˆ km† akbmejkm: The vector-closure conditions for zero-divisors imply

x0 ˆ 0; X7 X7 AX †0 ˆ ajxj ˆ 0; BX †0 ˆ bjxj ˆ 0; 11† jˆ1 jˆ1 X7 X AB††0 ˆ k0jxj ˆ 0; jˆ1 and thus Eq. (9) is automatically satis®ed.

We proceed to solve for X in a usual manner: when aj and bj j ˆ 1; ...; 7† are given xj are determined only by the three independent equations (11). By re-indexing the equations (for notational purposes only) so that x1; x2; x3 are the three independent unknown, and xk 4 6 k 6 7† are the free parameters, such that the determinant,

k01 k02 k03

D :ˆ a1 a2 a3

b1 b2 b3 does not vanish, we ®nd P k x k k f :p 0k k 02 03 P 1 a x a a x1 ˆ D f :p k k 2 3 : P

f :p bkxk b2 b3 82 K. Imaeda, M. Imaeda / Appl. Math. Comput. 115 (2000) 77±88

Similarly for x2 and x3. The corresponding solution for Y can now be obtained using the appropriate equation or simply from X A; B† Y ˆ : n S† We may de®ne a left or a right inverse of S as S S1 :ˆ ‡ Z; n S† however, this does not automatically imply that S1†1 ˆ S.

4. Linear equations

We now solve for Z, in terms of S and T, in a linear equation, SZ ˆ A; B† X ; Y †ˆT; A 6ˆ 0 6ˆ B: Using the identity (4) and de®ning octonions, F and G,byST :ˆ F ; G†,we obtain n S†X A; B; Y †ˆF ; 12† n S†Y ‡ A; B; X †ˆG:

By substitution we have for X,

xX A; B†‡ A; B; X †ˆ2K; 13† where ‰n S†F ‡ A; B; G†Š A; B† n S†2 K :ˆ ; x :ˆ : 2n A; B† n A; B† Writing this equation componentwise we have the following simultaneous equations: X7 x k0jxj ˆ k0; jˆ1 14† X7 X xk0kx0 kjkxj ‡ 1 x† ejkmk0mxj ˆ kk; jˆ1 jm† where we had used the fact that for each ®xed j and k, X X apbqejkpq ˆ k0mejkm kjk; kjk :ˆ aj; bk†: pq† m K. Imaeda, M. Imaeda / Appl. Math. Comput. 115 (2000) 77±88 83

These equations may be written in a matrix form as

almxm ˆ kl; 15† where alm is a skew-symmetric matrix with

all ˆ 0; a0j ˆxk0j ˆaj0; ajk ˆ kjk 1 x†ejkmk0m; which satis®es X ajka0k ˆ 0; j ˆ 1; ...; 7: k

Both xm and kl are column matrices. Through a tedious but straightforward calculation we obtain the determinant

!4 X7 4 4 2 jalmjˆx 1 x† k0j : 16† jˆ1 From Eq. (16) we have the following cases: Case 1: x ¹ 1. 1. K ˆ 0: then X ˆ 0 ) Z ˆ 0.

2. K ¹ 0: the solution for xl l ˆ 0; ...; 7† is uniquely determined by A; B; F and G. Case 2: x ˆ 1. 1. K ˆ 0: the case is that of the zero-divisor discussed in Section 3. 2. K ¹ 0: from Eq. (14) we have

k0kx0 bkv ‡ akw ˆ kk; 17† where X7 X7 X7 v :ˆ ajxj; w :ˆ bjxj; k0jxj ˆk0: jˆ1 jˆ1 jˆ1 The equations are now solved in a conventional manner as follows. Note that the seven equations of (17) are imposed on x0, v,andw, and that the rank of D and M, where 0 1 0 1 k01 a1 b1 k01 a1 b1 k1 @ k02 a2 b2 A @ k02 a2 b2 k2 A L :ˆ ÁÁÁ; M :ˆ ÁÁÁÁ; k07 a7 b7 k07 a7 b7 k7 should be equal to or be less than three.

(i) The rank of L is three: by re-indexing the ak and bk (strictly for notational purposes only), so that

k01 a1 b1

D ˆ k02 a2 b2 6ˆ 0;

k03 a3 b3 84 K. Imaeda, M. Imaeda / Appl. Math. Comput. 115 (2000) 77±88 we ®nd D D D x ˆ 1 ; v ˆ 2 ; w ˆ 3 ; 0 D D D where Dj j ˆ 1; 2; 3† is the determinant in which the three entries of the jth column in D are replaced by k1, k2 and k3; respectively. Now to solve for xk we have the following system of equations: X7 k0kxk ˆk0; kˆ1

X7 D X7 D a x ˆ 2 ; b x ˆ 3 : j j D j j D jˆ1 jˆ1 The solution is given by, D x ˆ j 8j ˆ 1; 2; 3; j D where

k01 k02 k03

D :ˆ a1 a2 a3 ;

b1 b2 b3 and Dj is the determinant D in which the three entrices in the jth column are replaced by X7 D X7 D X7 k k x ; 2 a x ; 3 b x ; 0 0j j D j j D j j jˆ1 jˆ1 jˆ1 where xj j ˆ 4; ...; 7† denote the free parameters. (ii) The rank of L is two: we consider B ˆ aA; a 2 R, and conclude from Eq. (12) that F G X ˆ ; Y ˆ : n S† n S† Thus, Z is uniquely determined by S as ST Z ˆ :: n S†

5. Theory of functions

As found by many authors both past and present the usual Cauchy± Riemann type di€erentiability conditions, when extended to functions of a hypercomplex variable, are too restrictive to yield interesting, if not useful, K. Imaeda, M. Imaeda / Appl. Math. Comput. 115 (2000) 77±88 85 functions. Our path is to follow one of the most successful approach in con- structing functions, that of regular functions; a theory originally developed for quaternions by Fueter [10], which was then re®ned, generalized, and extended to the cases of a variable in other Cayley±Dickson algebras [11±16] as well as in Cli€ord algebras, and which are successfully used in physics by authors too many to list them here. We de®ne the regularity (monogenicity) condition for sedenions in terms of the variation of an integral of a function in a 16- dimensionalP Euclidean space. For a function, F(Z), of a sedenion variable, 15 15 Z: ˆ zlel; zl 2 R†, consider an integral over a closed hypersurface S , lˆ0 Z J‰S15Š :ˆ dZ F Z†: S15

The surface element dZ can be written in terms of the di€erentials, dzl,orof 15 the directional cosines, cl, and the volume element, dr,ofS as

X15 X15 dZ ˆ ozlel ˆ cl drel: lˆ0 lˆ0 Using Stoke's theorem we can write J[S15] as a volume integral over V16 en- closed in S15 as Z J‰S15Šˆ DF Z††dV ; V 16 where X15 Y15 D :ˆ olel; dV :ˆ dzl: lˆ0 lˆ0

15 15 Now consider an in®nitesimal variation of the surface SA to SB in 15 the neighbourhood of a point Z on SA . The volume enclosed by the two surfaces is Z dV :ˆ dV 16 2 R:

SASB  De®ne R DF† dV dJ‰SŠ ‰J‰S15ŠJ‰S15ŠŠ dV :ˆ lim A B ˆ lim ˆ DF Z†: dS dV !0 dV dV !0 dV 15 Then the function F(Z) is left regular at Z0 on S if J[S] is stationary under the in®nitesimal variation of the surface at Z0,

dJ‰SŠ ˆ DF Z† ˆ 0: dS ZˆZ0 ZˆZ0 86 K. Imaeda, M. Imaeda / Appl. Math. Comput. 115 (2000) 77±88

The de®nition is independent of the choice of S15 for DF(Z) ˆ 0. The right D- regularity as well as D-regularity conditions can be de®ned in a similar manner. We now outline the construction of some regular functions of a sedenion variable. (I) Intrinsic functions: Starting with an analytic function,

g z†ˆu x0; x†‡iv x0; x†; p of a complex variable, z ˆ x0 ‡ ix, and replacing i ˆ 1 by a pure sedenion P 15 x e i ) jˆ1 j j ; x where v u uX15 t 2 x :ˆ xj ; jˆ1 the function assumes the form P 15 v x ; x†x e G X†ˆu x ; x†‡ jˆ1 0 j j ; 0 x where X is a sedenion variable. Then the function

X15 7  2 F X† :ˆ à G X†; à :ˆ DD ˆ ol; lˆ0 is left-regular. (II) Polynomial functions: Consider a function " #n ‰ XT ‡ TX†Šn X15 F X; T†ˆ ˆ x e x †t ; n 2 0 j j j jˆ1 where X15 X15 X :ˆ xlel; T :ˆ tjej 2 S: lˆ0 jˆ1

The power associativity ensures without ambiguity that Fn(X, T) can be written as a power series in tk, X n1 n2 n15 Fn X; T†ˆ n!Pn1n2 ...n15 x0; x1; ...; x15†t1 t2 ...t15 ; nr† P where n :ˆ 15 n , and P is a polynomial function of a sedenion variable jˆ1 j n1n2...n15 1 X P x ; x ; ...; x †ˆ x e x † ... x e x †: n1n2...n15 0 1 15 n! 0 k1 k1 0 kn kn kr† K. Imaeda, M. Imaeda / Appl. Math. Comput. 115 (2000) 77±88 87

The summation is over all possible permutations of the series kr†ˆ k1; ...; kn†; fk1; ...; kngˆf1; 2; ...; 15g, and the number of times each of 1; 2; ...; 15 appears in the series is n1; n2; ...; n15, respectively. It can be easily shown that F (X, T) and, henceforth, the polynomials P , are both side n n1n2...n15 D-regular. (III) Exponential functions: By again considering the hypercomplex variable P15 F X† :ˆ jˆ1 x0ej xj†, we may de®ne exponential function of S as i†n Exp‰iF X; T†Š :ˆ Fn X; T†: n! This function is both side D-regular. (IV) Cauchy-type kernel function: For a function X H X† :ˆ ; n X† of a variable X,ina2N ( ˆ m)-dimensional Cayley±Dickson algebra, one can readily show that, C H X† Ãn H X†ˆ n ; n 2 Z‡; ‰ n X†Šn where Yn X15 o2 C ˆ 1†n2nn! m 2k†; à :ˆ DD ˆ ; n ox2 kˆ1 lˆ0 l and that ÃnH X† is a both side regular function of an m-dimensional Cayley± Dickson algebra if n ˆ m=2 1. Thus, one can construct a Cauchy-type kernel function of a sedenion variable which is both side regular by choosing n ˆ 7. However, a general integral formula for sedenions based on this Cauchy-type kernel function cannot be constructed due to the non-antisymmetric nature of the second structure constant.

Acknowledgements

The author K. Imaeda would like to thank Mr. Susumu Ohta for his stimulating talks on the relations of hypercomplex numbers to the theories of superstrings and supergravity.

Appendix A. Octonions

We now list octonion properties and relations that are relevant to our main work. Let A, B, C 2 O. Componentwise the octonion A, the conjugate, the norm, and the inverse may be written, respectively, as 88 K. Imaeda, M. Imaeda / Appl. Math. Comput. 115 (2000) 77±88

X7 X15  A :ˆ a0 ‡ ajij; A :ˆ a0 ajij; jˆ1 jˆ1 A:1† X7   2 1  n A† :ˆ AA ˆ AA ˆ al; A :ˆ A=n A†; lˆ0 where the multiplication of basal elements, ij j ˆ 1; ...7†, is governed by the same relations as that of the sedenion basis, ej j ˆ 1; ...; 7†, de®ned in Eq. (2). The non-zero (not necessarily zero) associator, which is a pure octonion, is de®ned by X A; B; C† :ˆ AB†C A BC†ˆ2 ajbkcmejkmnin; jkmn† where the second structure constant, ejkmn, is antisymmetric in its indices, and is determined from the seven distinct associator cycles, j; k; m; n† by the usual 4- indexed tensor rule: ejkmn ˆ (+1, 1, or 0), depending on the (even, odd, or otherwise) permutation of the octonion associator cycle. The following identities hold for octonion associators: A; B; C†ˆ A; B; C†ˆ B; A; C†; ÁÁÁ A; B; A; B; C†† ˆ A; B† A; B; C†ˆ A; B; C† A; B†; A:2† A; B† :ˆ AB BA; A; AB; C†ˆ A; B; C†A:

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