Aboutf/ K° P NONASSOCIATIVITY in PHYSICS and CAYLEY- GRAVES9 OCTONIONS

Home , Alternative algebra, Octonion

ACADEMY OF SCIENCES OF THE ESTONIAN S.S.R. Division of Physical, Mathematical and Technical Sciences

LSorgsepp, J.Lohmus ABOUTf/ K° p NONASSOCIATIVITY IN PHYSICS AND CAYLEY- GRAVES9 OCTONIONS

TARTU 1978 Academy of Sciences of Estonian SSR Department of Physical, Mathematical and Technical Sciences

Preprint F-7 (1978)

L.Sorgsepp Initltutt o& A&£KophyAic& and KtmoiphVilc ?ky&

J.Lohmus Institute, ol

ABOUT NONARSOCIATIVITY IN PHYPICS AND CAYLEY-GRA.VFS' OCTONIONS

Mathematical preliminaries

Tartu 1978 CONTENTS

1. INTRODUCTION 3 1.1. Historical remarks 3 1.2. Gttnaydin-Gureey theory 7 1.3. Subject of the present paper and some speculations 9 2. QUATERNIONS AND OCTONIONS 12 * 2.1. Definitions and some elementary properties . 13 2.2. General algebraic characterization of •quaternions and octonions. Their uniqueness. 16 2.3. The Cayley-Dickson procedure. Some modifications of quaternion and octonion algebras. 18 2.4. Iderapotents, split octonions and Zorn representation 23 2.5. Modifications of the Cayley multiplication table 27 3. REPRESENTATION PROBLEM 28 3.1. The octonion algebra as an alternative algebra 28 3.2. "Naive" approach to matrix representa:ion for an associative algebra 30 3.3. "Matrix"(bimodule) representation from alternativity '. 31 3.4. An example - the regular representation of th«s octonion algebra 34 3.5. Representation of an associator by different multiplication tables 37 APPENDICES (TABLES) .38 REFERENCES 46

О Academy of Sciences of Estonian SSR,1978 ABOUT NONASSOCIATXVITY IN PHYSICS AND CATHY-GRAVES OCTONIONS

Mathematical preliminaries

L.Sorgaepp and J.Lohmu»

1. INTRODOCTION

. Historical remark».

There have been several attempts to introduce new algebraic structures into physics other than Lie algebras. In the early days of quantum mechanics Jordan /1-3/ and Jordan, von Neman and Signer /4/ introduced commutative nonassociative algebras (today we call then commit- tativt Joieum algibiaA, further abbreviated as CJA; we have a good mathematical theory for them, see e.g. books /5-7/ by Schafer, Braun and Koecher, and Jacobson). In this approach the traditional formulation of quantum mechanics in terms of associative but nonccnmutative operators (leading to the Lie algebra type nonassoeiativity for commutator algebra) was replaced by an alternative one with commutative but nonassociative operators. (In quantum mechanics, using the ordinary product for hermitian operators, the product of two noncommutative operators is no longer hermitian. Since a physical meaning can only be given to hermitian operators, & new product АлВж^(АВ+ВА) which preserves the hermiticity of operators, was intro- - 4 -

duced. Obviously this multiplication is commutative but nonassociative). In recent papers /8,9/ Jordan suggested a possibility of relating nonassociativity with the elementary length. Hurt /10/ demonstrated that quantizable dynamical systems may be classified in terms of pure states on simple real Jordan algebras. The Lie-Jordan rings of classical and quantum mechanics are discussed in the book /11/ by Zaitsev. Santilli /12,13/ pointed out that in generalizations of physical theories Lie algebras should not be completely abandoned, but their structure must be generalized in some sense therewith preserving the content of the Lie algebra. For this purpose he proposed Lie.-admi&/>ible. а.1де.Ька& (Al- bert, /14/). (An algebra A with a product Xjf } *№A is called Lie-admissible if the commutator algebra 4~ with the product [Х,ч] * Жу-уАГ is a Lie algebra). Associative and Lie algebras are Lie-admissible. So are CJA, but their Lie admissibility is trivial, the algebra/)" being always zero algebra. If we formulate the quantum mechanics in terms of CJA, it may be generalized in terms of JoKdan- admiaiblt aLgtbiab /14/. (As for application and elabo- ration of these ideas, see /15-18/). Now, quite recently a nonassociative algebraic construction was introduced by Кtorides /19,20/, bearing a relation to a Lie algebra paralleling the relation between an associative enveloping algebra and Lie algebra. Some interesting possibilities of physical relevance are also considered. The ideas about the possible role of nonassociativity stimulated Kffiv and LoTunus to study the gtntfializtd difaK- ma.ti.onb of associative and Lie algebras, i.e. continuous transitions of these algebraic structures into more general nonassociative ones. In /21/ there are represented the ideology and some preliminary results (the finite - 5 -

first-order flexible transformations of all 2-djjnensional | power-associative algebras and the alternative and flexible first-order defecations of the Pauli algebra). The f concept of deformation in /21/ follows closely the same concept of Gerstenhaber /22/ and Nijenhuis and Richardson /23/. This is one possible line of thought. The other line originates from particle classification schemes. After the proposed l*oipin-4ttiangtm*4 scheme by Gell-Mann /24/ and Nakano and Nishijima /25/ many attempts have been made to generalize it by introducing higher symmetries. Yet, actual particle spectrum remained hidden and there was plenty of room for different speculative symmetry schemes and groups. Even for some period after the proposal of the Eightfold Way scheme by Gell-Mann /26,27/ and Ne'eman /28/ there were j rivals for SU(3)&% C(2 ,SO(7),SO(8)I and others. Perhaps I the most close-up reviews of that period are given in papers I by Behrends, Dreitlein, Fronsdal and Lee /29/, and by Spei- I ser and Tarski /30/. к Then and afterwards also possible octonion forroula- ;, tions of internal symmetries were discussed by Souriau and i Kastler /31, 32/, Pais /33,34/, Tiomno /35/, Gamba /36,37/ , (M|-algebra), Penney /38/, and Vollendorf /39/. In connec- ) tion with the particle equations see also Bourret /40/, Penney /41/, and Buoncristiani /140/* By now approximate SCf(3) symmetry of particle multiplets is well established. The inclusions SC/(5)/Z}C S0(?) and SU(3)c6eCSO(7)cSO(S) intuitively lead to the conclusion that octonions are intimately connected with the Slf(3)-aynmBtxy. The physical interest in octonions even strenghtens if we concern ourselves in the papers dealing : with the SO(t) symmetryt d'Espagnat /42/, Ne'eman /43,44/, •; Ne'eman and Ozwath /45/, Gursey /46/, Horn and Ne'eman /47/, Gourdin /48/, Cocho /49-51/, and Gueret, Vigier and Tait /52/. All these papers have their origin in weak inter- - 6 -

actions, some of them being connected with particular models later abandoned. Shekhter's baroinvariance idea /53/ also calls for some SO(i) framework. The mathematical theory of the S0(f) symmetry was developed and some in- • portant consequences for strong interactions were pointed ' out by Peaslee /54,55/. de Vos /56/, Esteve /57/, Ester* ?. and Tiemblo /58/, Todorov /59/, Fairbairn /60/, Kacfarlane /61/, and Liotta and Triolo /62/. A successful attempt to formulate quark models in rms of the split octonion algebra was recently accomplished by GQrsey and Gunaydin (next Sec. 1.2 with refe- '• rences therein will be devoted to their ideas). : In conclusion yet some separately standing papers about nonassociativity and its physical significance should ? be mentioned. , \ In /63/ diglane modified the concept of grot i repre- j sentation and obtained octonion units as a specia_ nonasso- • j ciative representation of the space reflection group. [ ' | Yaes /64/ pointed at nonassociatirity of the operators ! (propagators) in the crossing symmetric Bethe-Salpeter f• -; i equations. Whiston /65/ postulated that the space-time auto- morphisms generally need not form a group but a loop and he investigated some nonassociative loopextensions of the Galilei group. The potential significance of nonassociative algebras for the generalization of classical Hamiltonian dynamics was pointed out by Nambu /66/. Taking the Liouville theorem (about the conservation of the phase space volume) as a guiding principle, Maabu generalized the Hamiltonian dynamics to a three-dimensional phase space. The generalised Poisson brackets may be interpreted as associators, so, nonassociative algebras become involved (in this connection the paper contain* some remarks about Cayley- Dickson and Jordan algebras). Recent papers by Cohen, - 7 -

Garcia Sucre, Ruggeri and Kflnay on the Nambu mechanica are ausaiarized in a review article by Kalnay /67/. Kapuszik /68/ has deduced the most general form of the (nonassociative) composition law for quantum mechanical observablea. •ollendorf /41/ suggests that the five conservation laws of charge, hypercharge, baryon number and the .two lepton numbers lead to a commutative nonassociative 24- dimeasioaal linear algebra, each element of which being an ordered aet of three (split) octoniona. Further he applied octoniona to gravitation theory /69/ and outlined a bilocal field theory with the group SO@tj4) instead of the Lorentz group and with a ayatem of 24 coupled differential equations, /70/.

1.2. Gunaydin-Guraay theory.

A serious atep towarda an adequate and phyaically well- founded theory connecting the two linea of thought mentioned above was made in recent papers by Gflnaydin and Gttrsey /71-78, 141-143/, and others /79-81,144-153/ (see alao re- marka in the review /82/ by B.w.Lee). Their basic idea was a Hilbert space with octonion scalars. (Mathematical study of this concept has been accomplished by Goldstine, Horwitz and Biedenharn /83-85/). If we follow the theory of observable states developed by Birkhoff and von Neumann /86/ (the prepositional calculus of quantum mechanics) we can only have observable states in Hilbert spaces over associative composition algebras (see, e.g. /87/). According to the famous Hurwitz theorem 'f (Sec. 2.2.) there are only three such algebras over the field of real numbers - the real numbers themselves (R ), V complex numbers (€ ) and quaternions (Q ). The orthodox quantum mechanics explores the Hilbert space over complex rs. The quaternion formalism (with quaternion Hilbert spaces) of quantum mechanics was developed by Finkelstein, Jauch, Schlminoyitch and Speiser /88-91/ and Emch /92/. The general hope was to introduce additional internal degrees of freedom (isospin) by such an enlargening of the quantum- mechanical Hilbert space, but finally Jauch proved /93/ / that quaternionic representations of the Poincare" group did not generate new physical states. So, the abandoning of the associativity means the appearance of nonobservable states* It is quite an agree- le feature if the latest ideas and convictions about nonobservability of quarks (quark confinement or imprison- ment, /100, 101/) are to be taken seriously. Also in the algebraic formulation /4/ of quantum mechanics in terms of Jordan algebras octonion-valued ob- servables may be treated only in a nonobservable Hilbert < space (except the case of three degrees of freedc i). A - ; i field theory may be developed, where octonion fie..d , j equations would imply dynamical relations in the obser- ! vable Hilbert eubspace, /72,73,80/. " \ In /71/ the split-octonion algebra and the related -•' I groups S0(S) , $0(7), $2 and SU(5) are reformulated in a ' framework which explicitly exhibits the quark structure. ^ f In /72/ a representation of the Poincare group in split- : ootonionic Hilbert space is constructed, the SU(5) group ь appearing as the automorphism group of the octonionic representation leaving the complex (observable) subspace and the scalar product invariant. As the symmetry under this S(/(3) group cannot be broken without altering the underlying algebraic structure, the group was identified

/73,74,77/-with the exact "colour" group $l/cfi)<* *b* three-triplet quark models /96-99/; the SUc (5) nonsinglets being unobservable while all the physical (observable) particles are colour singlets. In /75/ GQrsey observed that the exceptional Jordan algebra /4/ (forming a representation of the exceptional - 9 -

froup Fif ) displays a lepton-quark structure and that it mifht be used te describe the intrinsic charge •pace ef particles (including also leptons). The exceptional f reups EQ , Ef , Ef have been investigated to provide a possibility for describing even richer intrinsic properties than celeur and charm for hadrons and perhaps also fer leptens (especially about the groups Eft tj , see /78,79,142, 144, 147, 153/). Gunaydin /76/ has constructed an exceptional reali- zation of the Lorenti group via the same exceptional Jordan algebra. The exceptional SL(2,C) multiplets generate a nonasseeiative algebra. Exceptional supersymmetry groups are introduced and investigated. In /80,81/ Casalbuoni, Domekos and Kovesi-Domokos have investigated some basic aspects of a quantum theory I based on the oetonionic fuark field. Physical assumptions I abeut the structure ef the quark field algebra were simi- | lar though not identical with those of Gursey and Gdnaydin. I The theory also leads to a phenomenon physically equivalent \] to quark confinement. '•) «гешя* /148/ pointed cut that celour gauging is not : possible in the ordinary paraquark model but is possible in the octonio» realisation of that siodel.

1«3«Subject of the Mcesent paper and some speculations.

In this paper we investigate a nonassociative algebra • (octonion algebra) and related mathematical structures, bearing in the mind possible physical applications. In its essence, it is a review and recollection of some facts "Д relevant to the further study but we hope that some aspects •Mi'. f| may be new. In Sec. 2 we give a review of the properties of ж Ш octonions and quaternions. Together with the real and

P: complex numbers they play a specific and essential role in - 10 -

mathematics and probably also in physical applications. He describe the Cayley-Dickson procedure which enables a unified construction of all modifications of quaternion and octonion algebras. In physics, of the most successful applications are split octonions. Finally we call attention to the simple reflection-type modifications of the Cayley multiplication table. In mathematical literature some of them are used but we hope that their assembled structure may be of some importance in mathematics (in the construction of the 16 -numbers in addition to real and complex numbers, quaternions and octonions) as well as in physics (description of the charge degeneration of states in an internal space). The modern particle physics is founded upon the quantum mechanical formalism which is essentially formulable I as "matrix mechanics", i.e. in terms of linear associative operators. If some nonassociative algebra is to b. of some importance in particle physics then we must have sane generalization of associative linear transformations (rep* resentations). Some mathematical apparatus is already a- waiting. We explore these problems in Sec. 3. He use a simple method to get the regular bimodule representation of an alternative algebra. The bimodule (R-L) representa- I tion for octonions generates a Clifford algebra with 64 basic elements, 6 of them being independent generic ele- С ments have signature -1, -1, -1, -1,-1, -1. It is remark- able that this Clifford algebra is related with the assembled structure of split octonions, the generic elements in this case being with signatures -1, -1, 1, 1, 1, 1 or 5

-lr -1, -1, 1, 1, 1. In the first case there occurs a re- ?-- lation with the space-time conformal group. There are also f- some projection operators related with bimodule represent- ^ ations. The associator for octonions is represented by the 16*assembly of modifications of the octonion multiplication table. There is a Sl/(3j invariance of associator the S(/(3) - 11 -

identifiable with the exact color symmetry. Another bridge between the classical and nonassociative formalisms is formed by some classical and exceptional groups. At firs4: there is the autcmorphisn group Ц» of octonion algebra with SCf(5) as a stationary subgroup. The group SO(8) may be regarded as generalized automxphisn group of octonion multiplication structure «here the elements of the group are represented threefold in octonion space (triality principle!) by vector representation and by two kinds of semispinors. The subgroup of S0($) containing Qj> is the group 50(7) » There are yet exceptional

groups Fifj EQ t £f} £j tightly related to octonions. The Clifford algebra connected with octonions, projection operators related with bimcdule representations, and classical and exceptional groups with interesting application possibilities will be treated more profoundly elsewhere. Also the Mal'tsev algebras (which are quite direct generalizations of Lie algebras) deserve special attention. Here we stop at bimoriile representations. Now we cannot resist the temptation to exhibit our own speculative ideas about nonassociativity in quantum and particle physics because these ideas have lead us to the mathematical structures explored in this paper. Our approach was the following. All of us know the applicability of real and complex numbers in physics and elsewhere. It is just the complex ; (imaginary) unit that seems to be responsible for the : space-time metric and the velocity of light may be possi- i bly related with it. • The next plausible hypercomplex number system is Щ the noncommutative associative quaternion algebra which й may be represented by Pauli matrices and may be intuiti- ш vely related (through spin concept ) with the fundamental We must clearly recognize that here relating noncossrata- tivity with the Planck constant through the epin concept we presume the origin of quantum effects exclusively from 1 the existence of spin. Prespin quantum theory (before \ Goudsrait, Uhlenbeck, Pauli) can and must be founded on spin concept only. - 12 -

constant of quantum mechanics (the Planck constant). According to the Hurvitz theorem (8ec. 2.2) there is yet only one hypercomplex composite division algebra, the Cayley algebra, and a "non-ccmpact" form (split octonion algebra) uniquely related to it. To be consistent there must be some fundamental constant related to nonasseciati- vity. For about half a century there exist difficulties in quantum field theory, for overcoming of which we need the concept of fundamental length. Can we not relate this fundamental length with the octonion nonassociativity? Are both these concepts somehow related to quark confinement? None of these questions cannot be answered here as yet. The present paper was aimed as a collection of preliminaries for approaching these problems. It is quite probable that cur way of applying mathematical concepts to physics is too straightforward and primitive to be right. Things may be more compl. cated. Still we believe that there is some close connection between exceptional (and fixed dimensional) mathematical structures and structures of fundamental microphysics.

2. QUATERNIONS AND OCTONIONS.

Quaternions were introduced by William Hamilton in 1843, /102-104/ and octonions by Arthur Cayley /105,106/ and John Graves in 1845. These number systems were the very first examples of hypercomplex numbers or algebras and they have had a significant impact on mathematics.

Graves published his results (Phil. Mag. (3), 26, 320 (1845)) only after Cayley's first paper /105/ Eut he had communicated these to Hamilton in a letter of Jan- \.i uary 4, 1844. Hamilton is his turn communicated this [Ц material to John Young working on similar problems. Щ Young published his results /107/, giving the credit to 0 Graves. /(About the history see /109,110,112,128,130/). ; - 13 -

Because of their beautiful and unique properties they have attracted many people to study and wonder at them, /108- 114,130/ with an aljsost Pythagorean belief into their fun- damentality. The uniqueness of quaternions and octonions finds a precise expression in the theorems of Frobenius and Hurwitz (Sec. 2.2).

2.1» Definitions and some elementary properties. QfiattAnZon {octonlon.) algebra may be defined as a linear algebra over the field of real numbers К with a general element of the form

where £. «1,2,3 for the qu.attH.nion algebra ФС , and С «1,2,...,7 for the oztonJLon algebra £) ; here and afterwards we assume a summation over repeating indices; ti are the quaternion (octonion) units satisfying (in a particular basis) multiplication rules

'о ТЬЫ

Cote - во, where O4Y is the usual Kronecker symbol and SljK ie the Levi-Civlta tensor for quaternions and a fully antisymmetric tensor with for octonions. We often call the triples in (2.3) cyclt*. The most profound characterisation of octonion units may be found in papers by Fr«s4enthal /154/ and van der Blij /155/. For convenience let us write the multiplication table explicitly. - 14 -

Table 1 Multiplication table of octonion units (Cayley's original variant)

ei e5 e^ ec

>,.• e3 eff e6 e? €i ез -ef, -ef €6 -e3 e/ ec 6? -e» -es- 4 e3 4 -ti -e. -^ es- -еч

-es -e6 -e7 -eo e^ 63 -€ et -ey -e* esr ен 7 -e3 ez -to -ey e7 -er «a «7 e7 e«r «^ -e3 -e2 ~e0

Remark. Further we shall deal mostly with octonions and values 1,2,.*.,7 are assumed for all indices i»j#k,... if not mentioned otherwise. In this section we do not specify indices and formulas may be adjusted for both quaternions and octonions. Comment. The definition given above relies upon the particular set (2.3) of structure constants in (2.2) used by Hamilton and Cayley in their original papers. These particular definitions are also used in modern algebra textbooks (about slight modifications of Cayley's multiplication table see Sec. 2.5.). Under linear transformations of the basis the structure constants transform according to the well-known tensor law and multiplication rules may be changed beyond recognition. Intrinsic (basis- independent) nature of quaternions and octonions may be understood in terms of their uniqueness. From (2.2) and (2.3) we get the commutation and anti- commutation rules - 2. £ (2.4)

(2.5) 4 - 15 -

and the aaoclaton. fa, e/j ъ}= (ei tj

^fret Ф^г'€^есу)ег (2<б) The quaternion units €ij€z,€$ are anticammutative and associative, the octonion units &?;&>•••; €7 anti- coimutative and antiassociative, the latter property mean-

ing that (е№)4кт-ес(фк) , or {е«.,$',е*7+= фу)ы>€С(фи)*0 for basic triples from (2.3). Because of the linearity we can now calculate products, commutators and associators also for arbitrary quaternions and octonions of form (2.1). It jLs easy to see from (2.2) and (2.3) that the quaternion algebra l£ is a subalgebra in the octonion algebra Ю . For en arbitrary quaternion_or octonion JC the con- j'ugatt quaternion or octonion J[ may be defined as

<2-7> the conjugation mapping JC~*X being an Involution, i.e.

Then the покт ЩХ)=XX* XX=(xi +XiXi)tb (2.8) may be defined for an arbitrary quaternion or octonion JC , and now the tnvtKiz element Х~*6(& (orО) for а Х^й X

The norm form (2.8) is nondegenerate and positively definite for both quaternions and octonions (over defined by (2.1-3), and therefore every element ) has the unique inverse element ЛГ~'б£ - 16 -

The algebras & and 0 are dlvUlon algtbtieu (i.e. algebras with unique division, without zero divisors), but they have nonisomorphic companions with degenerate and nondefinite norm forms (Sec* 2.3.). Speaking about division in a noncomnutative algebra left and right quotients must be considered. Let us have two quaternions or octonions A*3 ; ЗФО . By the definition, the lt£t quotient B\4 of A divided by 3 "from the left" is the solution of the equation

(2Л0а) and the tight quotient A/3 of /} divided by./? "from the right" is the solution of the equation

It is easy to see and verify that

In quaternion and octonion algebras the norm (2.8) satisfies the decomposition property

The algebras with such property are called composition algtbKa*. Only if N(X) is nondegenerate and definite, (XjY)—N(XY) obeys the axioms of the scalar product. Al- gebra is then called the noftmecf

2.2. General algebraic characteritation of quaternions and octonions. Their uniqueness,. The quaternion algebra щ, is a 6impli aaociatlve. but noncommutative division algebra with involution. - 17 -

The octonion algebra 0 is a lisaplz alttAnativt and nonaesociative division algebra with involution. Both algebras are the nornted composition ones. All alternative algebras are poweA-aiAociative. and so is the octonion algebra (an algebra /7 is power-associative if for any XeA , XnXin s Хл+Л! , see /14/). These facts may easily be inferred from the previous section (Sec. 2.1.). For the notion of alternativity see Sec. 3.1., as regards other general notions, terms and facts about algebras, references to some textbooks for example to some of /5,115-]22/ are made . We restrict ourselves to real numbers H\, as the basic underlying field of scalare, i.e. we shall deal only with the algebras over JR. . Such restriction greatly simplifies the whole picture, because, as said Hermann Weyl /123/, really bewildering things occur not in algebras themselves but in their underlying fields. The quaternion algebra is associative and therefore representable by matrices. We get the simplest nontrivial representation if we choose the Pauli matrices accompanied by the unit matrix as quaternion units* their multiplica-

tion rules exactly coincide with (2.2) for €o,€if^2)^5 • Also a real 4-dimensional matrix representation may be constructed. The nonassociativity of octonions does not permit matrix representation in usual sense. In the next section (Sec. 3) we shall construct a "twofold" matrix representation by the regular alternative blmodulz. And now let us give a brief sketch of uniqueness properties. Among the infinite set of all algebras there are some extraordinary and unique ones - the algebra of real numbers, the algebra of complex numbers, the quaternion algebra and the octonion algebra (the latter will also be referred to as the Cayley algebra). Their uniqueness lies - 18 -

in some important properties shared by then and lacking at the other algebras and it nay be expressed by the following theorems. Probenius' theorem (/124/, see also /119/)t Bach associative division algebra is isomorphic el-? ther to the algebra of real numbers? to the algebra of complex numbers or to the quaternion algebra. Generalised Frobenius' theorem (Kurosh /119/)t Each alternative division algebra is isomorphic ei- her to the algebra of real numbers, to the algebra of complex numbers, to the quaternion algebra or to the oe- tonion algebra. •

Dimensions 1,2,4,8f are obligatory also for nonalter- native finite-dimensional division algebras (Bott, Milnor /125/). Hurwitz' theorem, /126/t Bach normed composition algebra with a unit element is isomorphic to one of the following algebras: to the algebra of real numbers,, to the algebra of complex numbers, to the quaternion algebra or to the octonion algebra. • For elementary proofs of these theorems we refer to the book by Kantor and Solodovnikoff, /114/. j Here the modern formulations of the Frobenius' and I Hurwitz* theorems ахв given. In the original papers /124, 126/ the problems were solved in terms of quadratic forms. In 1938, Linnik /127/ proved a theorem (in terms of algebras) from which both the Frobenius'and Hurwitz'theorems followed. In terms of algebras the Hurwitz'theorem was also proved by Albert /128/ and Dubish /129/ (see also references in /5/, p. 73 and /112/).

2.3. The Cayley-Dickson procedure. Some modifications of ..? quaternion and octonion algebras.- j| Let A be an n-dimensional linear algebra over Л ft with an identity element €o and involution a-*Cf , - 19 - , ЛЯеА * а+а , "йГ t &&tt€-fl . By the Caylty-Vickson pKoctduttt /109,5/ we can construct a 2n-dlmensional linear algebra В with the identity element and involution and Ad3 as i a aubalgebra. • The construction is the following» Let us form all

possible ordered pairs ( CLt}CLi ), CLi,ui€A and define addition and multiplication by scalars componentwise, and multiplication by the formula

The set of all pairs (0H,0Li)-= i ; GLi>Qi€A becomes a 2n- dimensional algebra Jb over the field IK with the

identity element (to} 0 )} €Q being an identity element of the initial algebra A (in the following the identity element (io,O) will be denoted also by 4Q ). The elements (&}0)9 CL6A form a subalgebra /)'<=£ isomorphic to A • The element 4 ^(Pfto) is a special one: &*а-НСо t and the elements 4-(Q.i,Cll) of the constructed algebra 3 may be represented in the form e , (2.14) with the multiplication determined by (2.13)t

я 3" and with an involution ©-»• i i - a# -Ck€ . • (2.16) Proceeding from real numbers M\ we have at the first step the algebras C(f4) with general elements - 20 -

l\f(c) = CC"= б"С = Xo f Taking values M>i *±ijO, we have three number algebras: la) complex numb ел.A £s С(4) ; lb) double, numb гл.6 or hplit complex питЬелл £(~f) t lc) dual питЬгы complex number algebra is the normed division algebra. For double and dual numbers and other higher modifications, such as semiquaternions, etc., we refer to an earlier exposition of hypercomplex numbers by Cartan and Study in Encyclopedic. dtb Science* Uathimatlquu /130/ and to the books /131,132/ by Rosenfeld. Starting with the complex numbers С we h ve at the second step algebras (ft (У>Мг) with general elements

There are three nonitomorphic algebras \bl(yjHlJ s 2a) Hamilton's quattKnion* Й - fi(w * 2b) hpliX quatifinlont> or antiquatiKnioni vL\1y~ij r~ (the latter term is from /131,132/)* {

2c) AtmiquateAnioni @{ft0) (term from /131,132/). At the third step, starting with quaternions, we are lead to the algebras Ю (4) f)M}y with general elements - 21 -

(2.19)

We here 3 noniacmarphic algebras 0(1,1}Ji%) •- 3a) Cayl«y«s octonlonb

3b) 4p££C ОС^ОИ-СОПА ОГ Эс) itmioctonionb (0(4,1,0) . At «ach step l)-3) we have a division algebra a split algebra ( уи = -/ ) and an algebra with nilideal ( ft*0 ). la general, after three steps we have for the Cayley- Dickson algebras 0(/Ч/О,/*з) the followinf wultiplica- tion table, /5/j see p. 22. Pro» this table we can easily specify the multiplication rules for the above mentioned algebras la) - 3c). The norm forms for general algebras C(MI) and О (fli,/*Z,f*i) are the following:

(2.20a)

xi +j4 x} -hfn MI tfLi/ii x}+ (2.20c)

'• I - 22 -

Table 2

m IS « 5 •A 1: i at. t (A (б «M V ro 1 **> 1; 3 ' ^* * Л| g I» CD 5t

^ X. 1 -Dic k >i ft? ^» m * Cay l 1 p 1

о • •o re a f •4iH 1 I f

$ лм % th e u i

r-l • (л XI >> 1 а i ! 4 1 •o IS. licat ]

•H v> (s. ^> 1 V I - 23 -

The problem of isomorphism may be tackled by the following theorem (quoted by Jacobson in /133/): Two Cayley-Dickeon algebras OCpht-*-)]1*) and are isomorphic if their bilinear forms Q(}l) - £[,N -N(X)-N(?)] are equivalent. • In particular, the algebras ^(f*-O)j*-1 - i^O are the only three nonisomorphic algebras among the algebras С (fit) , eo are the algebras ^.("ff/i$)j/il=l/,0 among the algebras (Q. (4,Цг) and the algebras O^^M^ -1,0 among the algebras Ofyj^MiJ At each step in the Cayley-Dickson process we had only one normed division algebra. The normed division algebras allowed by Frobenius* and Hurwitz' theorems are exhausted after the first three steps. It is also easy to verify that the Cayley-Dickson procedure gives alternative algebras only if the initial algebra is an associative one.

2.4. Idempotents, split octonions and Zorn's representation. The division and split octonion algebras have some differences which may characteristically be expressed in terms of idvnpottntl (by definition, an element Ц.ФО is idempotent, if U2»U ?. At first let us recall briefly some general results. Any finite-dimensional power-associative algebra which is not a nilalgebra contains an idempotent (JL^-0 (/5/, p. 32).

The idempotents Ui)UiJ...tU.m€fl (arbitrary algebra) are called раЛкшЛлг orthogonal in case UClL =0 for *V fy-4*'->n • *a idempotent ueft is called if there do not exist orthogonal idempotents 1WsO) so that U-»tr-t/W . In a finite- dimensional algebra A , any idempotent may be written as a sum of pairwise orthogonal primitive idempotents. By the technique of idempotents and Pierce decompo- - 24 -

sitions lorn /134,135/ investigated the general structure ©£ simple alternative algebra*. Bis results may be summa- rised as follows. Finite-dimensional simple alternative algebra with a dtcompotltton o( Mtmtltf 4• Uf ^Uzt^Mim Кы,шИ£ ;

илШ*О , tty'j *#/ • 4j2/—)0l , 1Т)&Ъ is associative. -y (lorn /134/i /5/, p. 51). / Finite-dimensional simple alternative nonassoclative

algebra with 4- U1+U& ; U4iUt being orthogonal primi- • '-?• idempotents, is the split octonion algebra. (/134/; /5/, p. 52). In the case of /77*/ the identity element is a primitive idempotent and the algebra under consideration (finite dimensional simple alternative algebra) is the Cayley

division algebra. (/134/i /S/, p. 5€,c These three theorems exhaust the problem. Let us illustrate the situation by a direct Lsxvesti- gation of the idempotents of the Cayley division and split octonion algebras. Applying the condition U**U to general octonion (2.1) we get the most general form of idem-

potent, it is an octonion U*Uc€e+Ui4f + ,.t +Uf€?- with tte*-£- , UiUi -~tj » l*4,lr..,1- г summation I It is impossible to satisfy the latter condition for the division algebra and therefore the only idempotent is the identity 4& €Q (in accordance with the results reviewed). In the case of split octonion algebra the second r> condition is af+Ui+u} -14%—Itf-ltf- which may be satisfied choosing, e.gÛÛ^, The identity element may be decomposed as i — Ef+Ez. "itii a particular choice £f~ f"'• "^ T ^ ^ •••* T^*" » — % &4 -,., — •£ €? . The elements E/Êz «ê «его divisors t E1E2.—O . Split octonions may be obtained from the Cayley division algebra, multiplying the units €y, €?, €(t €+ by an imaginary unit С commuting with all octonion units. - 25 -

e The new units to=€o, ty=€iy •••} <4~lt4 } 5-t-*f} satisfy the multiplication rules for split octonions, ob- S tained from table 2 setting /*fW t fiz~^ t /*5 "^ . It may be said the Cayley division algebra and split octonion algebra are the two nea,l io*mt> or ^ complex octonion algebfia (Q&d? . Zorn /132/ proved that the split octonion algebra may be represented by a special vector-matrix algebra [ViktoKmatAizen&ybtem) over j\ . (About this representation-eee also /71,133,135,156/). We here give a brief presentation of the construction. • Let us consider a set elements ("vector-matrices") of the following form

(2.21) where oCjfi € R and CLj Ь 6 \§ , the latter being a 3- dimensional vector space over Щ, • This set of "vector- matrices" becomes an algebra if addition and multiplication by scalars from Bi are defined componentwise and the multiplication operation by the following formula «a'-tb'a-ext') ' a/ /(2#22 ) where d*i and dXv denote the scalar, and vector products of vectors from r§ . • The split octonion units with multiplication table 2 (where /k-y*^4i=s-yk3=

(2.23) - 26 -

•'•«•*»'•

where the connection with the Gflnaydin-GSreey /71-74/ split units UojtlfjU^Ui (i=4,2,3) is also indicated. These latter units have the multiplication table where zero divisors are explicitly presented: Table 3 Multiplication table of split octonion units (Gursey-Gfinaydin /71-74/). i u. Uo 0 0

Ho 0 u* 0 UQ "I Ui 0 4 £ij< ut Ш 0 -§ut грик - 27 -

Here Suft is the usual antieynunetric tensor vith From papers /71-74/ we have an impressive example ex- hibiting the physical fitness of zero divisors and idempotents (it is specially noted also by Sommerfeld in his book /157/). Following the octonion quark field construction (for example, in /73/) everyone experiences the advan- tages of the split algebra. Zero divisors enable to project or split out some parts from many-component entities.

2.5. Modifications of the Cayley multiplication table. As we have already mentioned, the structure constants of a given linear algebra may be specified up to nonsingu- lar linear transformation of the basic units. However, there are some particular simple modifications of the Cayley multiplication table (Table 1, p. 14 ) that appear very fruitful for several purposes (especially for writ- ing out generators for some classical groups. Sec. 4). These simple modifications confine themselves to the changes of signs (reflections) of some octonion units. This is connected with nonunigueness of the definition of the antisymmetric tensor Sun, in (2.2-3). The signs for CijK. may be arbitrarily chosen only for 4 cycles (c/K), then the other 3 are uniquely determined. So there are 16 such modifications of octonion multiplication tables. On the other hand, each combination of reflections (^,'->-&! for some С ) is equivalent to some choice of signs for

In Appendix I we present all the 16 modifications of ;A the Cayley multiplication table, the corresponding posi- f| tive cycles are separately tabulated. The Cayley original i| table (Table 1 on p. 14 ) is 0, the invtKtt table (all cycles inversed., i.e. with inversed signs) is Si0 *, the other

pairs of mutually inversed tables are «trl| 2,2; ...; 7,7* In the literature about octonions we have net the If following multiplication tables: - 28 -

Cayley /105,106/, Kantor-Solodovnikofff /114/, Kurosh /119/, Linnik /111/, Penney /38,41/, Siglane /63/; it ia the original Cayley table; de Voa /56/ (modification 7);

3) till = -^3 ; tit4*Z?} €lt4 *-&; £ Cartan /158/ , Freudenthal /154/ (modification 2); 4) Ticmno /35/ (modification 4);

5) е/вг гг3/ е^^ег, titH'tQ titHt^ Buaemann /159/ (modification T).

6) «f€i*€3; €/

3. REPRESENTATIOtT PROBLPi.

3.1, The octonion algebra aa an alternative algebra. The Cayley algebra О is a simple alternative algebra. By definition, an algebra A ia eULttKn&tbit if the following Lt.it and Klqkt alttKnatlvlty condition» are satisfiedt

or in terms of aaaociatora (2.6)t

An alternative algebra also may be defined aa an algebra in which all aubalgebras generated by two elements - 29 -

are associative. Then the alternativity conditions (3.1) or (Э.11) follow as consequences (Artin'a theorem, see e.g. /5,119,0. Eqs. (3.1•) may be "linearised*t

we then get the "true" alternativity of the associatest

where rXt njy nt denotes the permutation of the elements Л/У/3" i £*±У , depending on the parity of the permutation. If the characteristics of the basic underlying field is not two then (3.2) define an alternative algebra, /274/. So here we are allowed to use freely this form of alternativity condition. Remark. It must be mentioned that in defining relations (3.2) the elements Х,Ч,Ь occur "linearly", i.e. each element once. Then it ^suffices to superimpose conditions (3.2) only for the basic units €£ , for the general elements X*Xc€£ , 4**/]€/ , £«&*£« it follows automatically. It is not *во in the case of "non- linear" defining relations (3.1) or (3.11), where one and the same element occurs twice. For an exact notion of simplicity we refer to /S/, for a general notion of the representation of nonassociative algebras, to /136-139)5/. In this section, we de- rive a "twofold" (bimodule) representation directly from the alternativity condition, as it is at times done in the case of the associative algebras, deducing the matrix representation directly from the associativity condition. For the sake of completeness and clarity we at first discuss breafly also the associative case. 4] u - 30 -

3.2. "Naive* approach to matrix representation for an associative algebra. For an associative algebra A the associativity condition for the basic units €ij£z) »»>€h also implies the associativity for the whole algebra, i.e. for the gen-

eral elements X~Xlti , i^if2,»>} П . Then the associativity condition means simply /4*,...,л, (з.з)

where /7 is the dimension of the algebra under consider-

ation. Using the AtAactatit constant* Cty defined by Zitj = Ctj *K , (3.4) б о we can rewrite (3.3) in the form and consequently ^ f - S .S - Г (3.5) Here in Sees. 3.2. and 3.3. all the indices have values 4jZj,,.fn and summation over repeating indices is as sumed. Now the right hand side of (3.5) may be interpreted as a product of two matrices C& and C/fc , where Г is the column index and S the row index for the first matrix» and /* is the row index and К the column index for the second matrix (i щ. for these matrices tht mpptti Indtx лл the кот indtx and tht Atcond towvi Jindtx Ы tkt column Indtx): and the indices с and J specify different matrices. щ The left hand side of (3.5) may be interpreted as the § matrices [Cf)% multiplied by the structure constants ;' - 31 - I • 11 So from (3.3) we have got the Ktgulafi ma.tA.Lx tlon toe the associative basic units €i t

(3.6)

and also for the general elements C of the algebra A .

3.3. 'Matrix* (binodule) representatiob from alternativitv. Now let us proceed analogously to the case of an alternative algebra. Here, for the basic units we cannot start from (3.1) because these equations involve only two different elements and are too poor for determining a representation. Let us start with a fragment of (3.2):

, ej} и} + Ucj ек, ej] =o. 0.7) Then

'$1 and introducing structure constants 1 с* cfK + (ckC$ -Сл с} Interpreting here also the structure constants Си with fixed С as matrices, we get

where ** \Г (3.9)

Щ Щ - 32 -

etc. We call K^ljr *h* iwmt representation, obviously

cJ/C = CKi = (С*^ . (3.10)

Formula (3.8) is a fundamental one. Representing €i —* (C X€-n (light and Lt{t multiplication*) are defined respectively by U-*4X and «-»XV , Уу£# . Our tC-*(cc)'f is the Itit xtgulal лерле-

депзд&соп and ^t'-* ^ )f the night Ktgulafi KtpKthtn- tation, Eqe. (3.8) may be written (tUnohhq ( [] From

and

(3.12) we get respectively (3.13b,c) which together with (3.8•) form the btuic tquation* (3.13) for the regular left (L) and right (R) representations* К Ц Lj + [Ц Rj] = e& L RcRj +№, Lj] - ЩЯл - 33 -

Prom these Egs. we can deduce all the other equations following from (3.7), (3.11) and (3.12) by index permutations. Among them we find some quite interesting relations:

LlLj +LjU •((& +CJC)LK , (3.14a) RiRi +/?/& -(c$ + c.?)RK, (з.мы

LiU -hfijki = CS(LIC+8K)4 (3.14C)

LjLi + RiQi = tjc (U +R*). (3.i4d) In the case of associative algebras we have a) from \l(,lj,tK}*0 U b) fnm f*,y,e;}«0 о / fAy We can see that [L)U,]-[LpftJ^O and we therefore have quite a normal matrix representation. Eqs. (3.13) may be laid into foundation of the general representation theory /136-139/. Let V be a vector space over some field F . A biKtpKtitntation (S,T) or bimodule. H.tpie.6e.nta£Lon of an alternative algebra is a pair of linear mappings X -* S* r X —> 7д of/) into the algebra L(V/ of all linear transformations on У satisfying

The representation space V , in which linear transforma tions QT act, is turned into an alt ел native. bJUnodu.lt. - 34 -

by the following definitions

$A,rxeL(v). Every alternative algebra A over г hae a repre sentation, particularly the regular representation constructed above. Then there is a correspondence /jj *"* Z_x / "* *~* where LX/Rx are te.it and light multiplication respectively:

3.4. An example - regular representation of the octonion algebra. It is easy to write regular birepresentation matrices explicitly using (3.9) within fixed multiplication table. But it is still easier to write these matrices immediately from the division table.* (Tables 4a,b; p. 35).

Por example, *

Finally e.Ao = f»e* « Си ее»(k)c ec. Now we see that the division table ^к/ti (K,I - -0,/,...,?- ) is the 8x8 table (d)j €i (in fact there is no summation over i because the structure constants С if for fixed t^j are different from zero only for one value of i ), so we can immediately write the left representation matrix Li = (Ci) from the (right) division table €K/€J taking from it the coefficients at - 35 -

Division tables for octonion units e/c/g/ Table 4a

e0 ft e3 e* ft e7 -ei ec -e* -ft -ev -er -ft -er ti во -ез -er ^ e7 -ft

ft ei €з e0 -ey -ft -e7 ev *r ej e$ -*z e/ -ft e« -er ev *V ev e? €e ^7 -e* -ft -ез

^7 -ft ^© €з -e2 e< ft -ft -ev -ез €o e? ^t -e^ ft и; в С u~ •

Table 4b ey €з e0 e2 ev ft e7

to e0 -e/ -ег -e* -e? "ft -e7 -fiy €/ ti ec e* -ег e» "ft ec ft tt -e3 e* е-/ ft e7 -ev -ft

e^ ez -ef e0 e7 -ft er ev e« -^ -ft e ei ft e ш e 3 es es- -e7 ft -e4 -ft ft I ft e« ev -ft e* -ey e? e7 -& ^v -e$ e/

•a ?•г r • - 36 -

the units 6(" and preserving the row-column numeration. Analogously from the division table ^'\€& we have

*A€/c — (Ct- Jj €i t so we can immediately write the right representation matrix Ri -(Сч) from the (left) division table €j\ ZK , taking from it the coefficients at the units li . For illustration let us write out the

matrices LifKi , p. 35; full tables in Appendix IIA,B. Finally, let us remark that for a mutually inversed pair of the multiplication tables 0 and 0 (as also for the other pairs) the right and left division tables are reciprocal: the table €;/бк for 0 coincides with V^f I * foT

3.5. Representation of an associator by different multiplication tables. Let us investigate the associator j £tj6/4fcj> <^a^...f" of the octonion units. Let H be fixed for a moment, we

denote it by ф , /C=//2/.../^' . Then the terms (t№)4& and {(Tl/Cftd) т&У be regarded as some peculiar "products" of ti and €#' . In terms of structure constants these "products" may be expressed as follows:

there is summation over ft S and /^J * Qt The "multiplication tables" for the "products" and И.{%1%ф) differ exactly in the same way as does the Cayley multiplication table for octonion units from the k-th modification (see Table I in Appendix I; all С if in (3.18) and (3.19) are from the Cayley table, i.e. from the modification 0). - 37 -

So we can «rite that С/ф * CCj Сгф , (3.20) where the subscript8 in parenthesihi s belobl w the structure constants indicate their belonging to the multiplication table ( Us. 4,1,t..,^ ). Now we may write

an< for arbitrary octonions У * 4j tij x z/=

Every term AZ~XX belongs to some 6-subspace (fro* where the oc ton ion units €cj l/C are left out; this property may easily be traced comparing the multiplication tables 0 and К ). This 6-subspace is the action arena for the group $Щ$) , the automorphism group of octonions when one of the oc ton ion units €if £ZJ*»J fy is fixed (in our case the fixed unit is IK ). SU&) is stationary subgroup in - 38 -

APPENDICES

In APPENDICES we present 16 modifications of the original octonion multiplication tables (Appendix I) and L-, R-matrices of the regular (adjoint) bimodule representation for all these modifications (Appendix II). The origin of these modifications was discussed in Sec. 2.5 (p. 27). Now at first let us write positive cycles for all modifications (for 0-modification they are already given by (2.3), p. 13).

Positive cycles for modifications 0 123 145 176 246 257 347 365 1 123 145 176 26£ 275_ 37£ 356_ 2 123 15£ 167_ 246 257 374 356 3 123 15£ 167 26£ 275_ 347 365 4 132. 145 167_ 246 275_ 3*7 356_ 5 132 145 16_7 26± 257 374_ 365 6 132_ 15£ 17* 246 275_ 37_£ 365 7 132 15£ 176 26£ 257 347 356 0 13_2 15£ 167_ 26£ 275_ 37£ 356_ 1 132_ 15£ 167_ 246 257 347 365 2 132. 1Л5 176 £6£ 275_ 347 365 3 13_i I45 176 246 257 37£ 356_ % 123 15£ 176 ?64 257 374_ 365 5 123 154_ 176 246 275_ 347 3516 ? 123 145 167 26± 257 347 356 7 123 145 167 2*6 275 374 365 - 39 -

In tables of Appendix I there are given changes of signs in the "basic" multiplication rules £*£<: = €.4 , ti*.n •» «5- , е*е* •• ее , егвц - e? i these changes are in 1-1 correspondence with those of positive cycles/ and may also be obtained by suitable reflections of basic units ^ij^Zy/^7* For example, we can take basic multiplication rules 6/в.г г -6з * €I€H =• -€$" t eze*, = - вб , eje* ж €7 (modification 7), then positive cycles are 132, 154, 176, 26±, 257, 347, 356 (changes compared witK~0-modification cycles are under- lined). This modification may also be obtained by the reflections £/ -* -e/ , e* -9 -вб / or by €2 ~* -^2 ' 65- -»-3r # or by e$ -» -ез , ец -* - e« In tables of Appendix I we write out only the multiplication table for 0-modification. In the rest of the tables we only hatch the "boxes" where signs at product units are reversed compared with 0-modification (Cayley table). In Appendix II we write out L-matrices (Table A) and R-matrices (Table B) of the regular bimodule representation of octonions (Sec. 3.4) for the modifications 0, 1, ...,7. For 0-modification (Cayley table) they may be obtained directly from Tables 4a,b (p. 35). As mentioned on p. 36, for mutually reversed pair of multiplication tables the right and left division tables are reciprocal: the table ec /€i j for k-th modification ( К «0,1,...,7) coincides wit(Thhe modificationa€j\t,i fors thhave emodificatio their origin n к.in anthde v-ccchangee s of po- sitiv(Th e dificycles.i Reversinh g thalil cycleiis foir K-th h hmodification we obtain the modification К . From the right or left division tables for k-th modification we have L- or li- ma trices. The transition L--> R means the change of order of factors and due to the anticommutativity it is equivalent to the reversion of all cycles in k-th modification, i.e. to К -9 к ). Therefore we dj> not write out L- or R- i matrices for the modifications К ( \ =0,1,,..,7). For example, the matrix L. for modification 7 is the matrix R. for modification 2. л л л

We write the symbols LitLzr-> *-/ and Ki,RZ)-,f<7 only to the left of the left column of tables II A, B. The numbers CJ on the left in other columns indicate that the correspondig matrix may be obtained as a product Li Li (or RiRj ), where Lc,Lj (or /?,',#/ ) are i- (or^-) matrices* of 0-modif ication. (j - 40 -

APPENDIX I. Octonion multiplication tables (Sec. 2.5)

I: к - 41 -

ш - 42 -

L-matrices of the regular bimodule rep- APPENDIX IIA. / 2 "3 о - 43 -

resentation of octonions (Sec. 3.4 ). Ц 5 6 7 -f -f, . -1. , 1 I • 1 . , . /

• ; - m 1 У 7 f • •7 .. . -1

! i 115 У * _.. -1 - ' ' j i 4 : : : нЛ ! -/: ' '- .. - - f - j- -- r L •r -- ? ; : Г 1 , l ! * 4 1 i ' • i ...... 1 ' -1 ! : ; • 1 1 ' «i^1 -i ; I f -f 1 ...... i ! I ' Г ! • * i-

...... --- -t __LL -i • ' "! f i f I f

• ; t !/ •f •• I i к ™ -У r r j j --?,* • fi ... •. _

_ 7_ ... r—-••• -- A.. - • - •• j' . -•—••

; , ' ; 4 .:. ,....;•_ •• i i i i "If* • i i 4. ; • 'ft ^ ! j . ...4L f 1 : i i—i ... •* — • :.....-/ —1 — ..

t '. . . -. .. i ; jT-:- -i -f

•**•• :.:— 4 "/* w 4 - . f. _ :1 . -/ ' • ! 3L. d:. ; r- - 44 -

APPENDIX IIB. R-matrices of the regular bimodule rep- 0 7 23

.1 I

s 1 I i •Ha! L&I 1-rt "П LL i-4 t I " +-r

тт

«-/I

н—j—<- . I ; m: i tt -U : 3D •4-+- F i 21 3l

1 i lL -p I ! 1 reaentation of octonions (S«c.3.4.)> 5 6

4 - 46 -

REFERENCES

Jordan algzbfia.6 and quantum me.chanA.ci 1. Jordan P., Über eine Klasse nichtassoziativer hyper- komplexer Algebren. Göttingen Nachr. 1932, s. 569-575. 2. Jordan P., Über Verallgemeinerungs-moglichkeiten des Formalismus der Quantenmechanik. Göttingen Nachr. 1933, s. 209-217. 3. Jordan P., Ober die Multiplikation quanteranechanischen Grossen. Z. Phys. 8£, N5-6, 285-291 (1933). 4. Jordan P., von Neumann J., Wigner E., On an algebraic generalization of the quantum mechanical formalism. Ann. of Math. (USA) 35, Hi, 29-64 (1934). 5. Schafer R.D., An introduction to non-associative algebras. Academic Press, New York-London, 1966. 6. Braun H., Koecher M., Jordan-Algebren. Springer-Verlag, Berlin-Heidelberg-New York 1966. 7. Jacobson N., Structure and representations of Jordan algebras. Amer. Math. Soc. Coll. Pub. 39_ (Providence 1968) „ 8. Jordan P., Über das Verhältnis der Theorie der Elemen- , tarlange zur Quantentheorie, I, II. Communs Math. Phys. 9, N4, 279-292 (1968); 11, N4, 293-296 (1969). " 9. Jordan P., Zur Frage einer physikalischen Verwendbarkeit nichtassoziativer Algebren. Z. Phys. 229_, N3-5, 193-198 (1969). 10. Hurt N.E., Jordan algebras and quantizable dynamical ,j" systems. : Lett. Nuovo Cimento 1, Nil, 473-474 (1971). r 11. 3a«ueB r.A., AjireGpamcecKRe npoOneMti MaTeManraecKoA ;..,. H reopeTHvecKoft QHSKKH. - Hsa. "Hayxa", $.-M., M. 1974. ß Llt-admiAiibte. alge.bx.cu |v fc.- 12. Santilli R.M., Imbedding of Lie algebras in non-asso- " ciative structures. Nuovo Cimento A51» N2, 570-576 (1967). 13. Santilli R.M., An introduction to Lie-ad»is«ible algebras. Nuovo Ctaento, Suppl. (1), £, 1225-1249 (1968). 14. Albert A.A., Power-associative rings. - 47 -

Trans. Alter. Math. Soc. 6±, N4, 552-593 (1948) 15. Santilli R.M., Soliani G., Statistics and parastatistics formal unification. Univ. of Torino, internal note (1967). 16. Santilli R.M., Dissipativity and Lie-admissible algebras. Meccanica 1, N1, 3-11 (1969). 17. Roman P., Santilli R.M., A Lie-admissible model for dissipative plasmas. Lett. Nuovo Cimento 2_, N9, 449-455 (1969). 18. Santilli R.M., Haag theorem and Lie-admissible algebras. In "Analytical Methods in Mathematical Physics (ed. by R.P.Gilbert and R.G.Newton), Gordon & Breach, New York 1970; paper 11-18. 19. Ktorides C.N., Possibilities for a novel algebraic approach to symmetry breaking. Preprint IC/74/52 (Int. Report), Miramare- Trieste, June 1974. 20. Ktorides C.N., A non-Lie algebraic framework and its possible merits for symmetry descriptions. J. Math. Phys. 1£, N10, 2130-2141 (1975). Ve.4oKmat4.on o& nonanoclatlvz alge.bH.ai 21. Koiv M., Lobmus J., Generalized deformations of nonassociative algebras. Definition and some simple examples. Preprint FAI-18, Institute of Physics & Astro- nomy, Tartu 1972. 22. Gerstenhaber M., On the deformation of rings and algebras, I-III. Ann. of Math. (USA) 79, N1, 59-103 (1964); 84, HI, 1-19 (1966); 88,TTl, 1-34 (1968). 23. Nijenhuis A., Richardson R.W., Cohomology and deformations in graded Lie algebras. Bull. Am. Math. Soc. 7_2, N1, 1-29 (1966). and SU{3] - рарелл 24. Gell-Mann M., Isotopic spin and new unstable particles. Phys. Rev. 9£, N3, 833-834 (1953). 25. Nakano Т., Nishijima K., Charge independence for V- par tides. Progr. Theor. Phys. 10_, N5, 581-582 (1953). 26. Gell-Mann M., The Eightfold Way: a theory of strong - 48 -

interaction symmetry. Report CTSL-20, CALTECH, Pasadena 1961. 27. Gell-Mann M., Symmetries of baryons and mesons. Phys. Rev. 125_, N3, 1067-1084 (1962). 28. Ne'eman Y., Derivation of strong interactions from a gauge invariance. ; NUcl. Phys. 26,, N2, 222-229 (1961). - лоте, laity fte.vlzw& 29. Behrends R.E., Dreitlein J., Fronsdal C, Lee B.W., Simple groups and strong interaction symmetries. Rev. Mod. Phys. 34_, N1, 1-40 (1962). \r 30. Speiser D.R., Tarski J., Possible schemes for global У symmetry. J. Math. Phys. 4, N5, 588-612 (1963). Application o{ octonloni to paAtlcle cla.iililca.tlon 31. Souriau J.-M., Theorie algebrique dee mesons et baryons. C.R.Acad. Sci. Paris 25£, N16, 2807 (1960). 32. Souriau J.-M., Kastler D., Cayley octonions and strong interactions. "Conf. Internet. Aix-en-Provence sur particules llementaires, 1961", vol. 1, Gap 1962; pp. 169-170. 33. Pais A., Remark on the algebra of interactions. Phys. Rev. Lett. 7, N7, 291-293 (1961). 34. Pais A., On spinors in n dimensions. J. Math. Phys. 3_, N6, 1135-1139 (1962). 35. Tiomno J., Octonions and super-global symmetry. In "Theoretical Physics" (Trieste Seminar, 1962), ;• Vienna 1963; pp. 251-264. 8 '<) 36. Gamba A., On Jordan algebra M,. 'J "High-Energy Phys. and Elementary Particles" ;• (Trieste Seminar, 1964)% Vienna 1965; pp. 641- 646. ; 37. Gamba A., Peculiarities of the eight dimensional space. J. Math. Phys. £, N4, 775-781 (1967). о 38. Penney R., Octonions and isospin. Ы Nuovo Cimento B3_, N1, 95-113 (1971). Щ 39. Vollendorf P., Cayley-Zahlen und Erhaltangsgesetze §1 fflr Elementarteilchen. g* Z. Naturforechr. 30A, N4, 431-433 (1975). | 40. Bourret R., Particle equations from non-associative Щ algebra. Canad. J. Phys. 37_, N2, 183-188 (1959) ft* - 49 -

41. Penney R., Octonions and the Dirac equation. Amer. J. Phys. 36, N10, 871-873 (1968). Tht **«uj» 0(1) In clmilicAtion iukemt* And It* KtlAtlon to SU(3) Ub9tit OH) * ее AUO 42. d'Espagnat В., Greup-theoretical methods. Free. 1962 Int. Cenf. on High-Energy Physics, CERN, Geneva, 1962; pp. 917-923. 43. Ne'enan Y., Restricted AQ/AS * -1 weak currents, CP and the R(t) extension of SU(3). Phys. Rev. Lett. 13_, N25, 769-771 (1964). 44. Ne'eman У., Higher symmetries and current conservation. Phya. Lett. 4, N2, 81-83 (1963); Erratum: ibid. 4, N5, 312 (Г963). 45. Ne'enan Y., Ozsvath I., Unitary symmetry viewed as a broken rotational invariance. Phys. Rev. B138, N6, 1474-1487 (1965). 46. Gursey P., On the structure and parity of weak inter— action currents. Ann. of Phys. (USA) 12,, N1, 91-117 (1961). 47. Horn D., Ne'eman Y., Unitary symmetry and the weak currents, I, II. Nuevo-Cimente 29, N3, 760-770 (1963); 31, N4, 879-883 (1964)7~ 48. Gourdin M., Eightfold way and weak interactions. Nuovo CiMento3£, N2, 587-602 (1963). 49. Cocho G., El mesen В у la simetria global R-. Rev. Mexic. fie. 13_, HI, 7-13 (1964). 50. Cocho G., Weak interactions and higher symmetries. Rev. mexic. fi«. 1£, N1, 17-28 (1965). 51. Cocho G., Unitary or octal synmetry? Phys. Rev. B137, N5, 1255-1258 (1965). 52. Gueret P., Vigier J.P., Tait W., A symmetry scheme for hadrons, leptons and intermediate vector bosons. Nuovo Cimento АГ7/ N4, 663-680 (1973). 53. Шехтер В.М., Бароинвариантност» ешкиых взаимодействий. ЖЭТФ 48, »5, 1459-1478 (1965).

54. Peaslee D.C., Particle charge syinaetriee as Ra sub- groups. J. Math. Phys. 4, N7, 910-914 (1963). 55. Peaslee D.C., Uniqueness of 4- and 8-di»ensional spaces. J. Math. Phys., 5_, N7, 897-899 (1964). 56. de Ves J.A., The internal symmetries of elestentary particles. Nucl. Phys. 59_, N1, 65-80 (1964). - 50 -

57. Esteve A., Strong interaction symmetries. Nuovo Cimento 3£, N3, 788-790 (1964). 58. Esteve A., Tiemblo A., Vector invariants of SU,/Z,. Nuovo Cimento 39> N3, 880-885 (1965). 3 * 59. Тодоров И., Формулировка октетной модели, включающей лишь целочисленные эгдряды. Докл. Болт. АН 18_, №3, 199-202 (1965). 60. Fairbairn W.M., The eight-dimensional orthogonal group 0(8) and internal symmetry. Nucl. Phys. Bl, N3, 127-138 (1967). 61. Macfarlane A.J., Description of the symmetry group SU(3)/Z_ of the octet model. Communs Math. Phys. 11, N2, 91-98 (1968). 62. Liotta R.S., Triolo L., Regular representation of the group SU(3): eigenfunctions and eigenvalues of the Beltrami-Laplace operator defined on the 4-dimen-

sional manifold in the euclidean space Rfi. Nuovo Cimento A59_, N2, 303-307 (1969)° bU.&ce.llane.ou& on 63. Ыйтлане X., О некоторых возможных обобщениях понятия представления группы. Труды Института физики и астрономии АН ЭССР 16, 90-105 (1961). . 64. Yaes R.J., Nonassociativity of the operators in the crossing-symmetric Bethe-Salpeter equation. Phys. Rev. £2, N10, 2457-2463 (1970). 65. Whiston G.S., Associative and non-associative auto- morphisms of Newtonian space-time. Int. J. Theor. Phys. £, N1-2, 79-103 (1972). 66. Nambu Y., Generalized Hamiltonian dynamics. Phys. Rev. D7, N8, 2405-2412 (1973). 67. Kalnay A.J., On the new Nambu mechanics/ its classical partners and its quantization. Talk given at the "Colloque Int. de Geometrie Symplectique et Phys. Mathematique", Aix-en- Provence, France, 24-28 June 1974; Internal Report IC/74/89, Trieste-Miramare, August 1974. 68. Kapuscik E., The multiplication law in quantum physics. I. The non-relativistic quantum mechanics; II. The relativistic quantum field theory. Acta phys. polon. B5, N4, 497-505; N5, 663-675 (1974). -~ 69. Vollendorf F., Gravitation und bilokale Feldtheorie. Z. Naturforsch. 30A, N5, 642-644 (1975). - 51 -

70. Vollendorf F., Bilokale Feldtheorie der Elementar- teilchen. Z. Naturforsch. 30A, N6, 891-895 (1975). GU.na.yd in-Gümty theory Uee al&o /141-153/) and tome. Kttatzd papzKi /83-93/ 71. Günaydin M., Gürsey F., Quark structure and octonions. J. Math. Phys. 1£, Nil, 1651-1667 (1973). 72. Gfinaydin M., GOrsey F., An octonionic representation of the Poincaré group. Lett. Nuovo Cimento 6_, Nil, 401-406 (1973). 73. Gûnaydin M., Gürsey F., Quark statistics and octonions. Phys. Rev. D9, N12, 3387-3391 (1974). 74. Gürsey F., Color quarks and octonions. Proceedings of the "Baltimore 1974 Johns Hopkins Workshop on current problems in high energy particle theory", Baltimore 1974; pp. 15-42. 75. Gürsey F., Algebraic methods and quark structure. Proc. of the Kyoto Conf. on Math. Problems in Theoretical Physics; Kyoto Jan 23-29, 1975. Preprint, Yale university 1975. 76. Günaydin M., Exceptional realizations of Lorentz group: supersymmetries and leptons. Preprint S.N.S. 5/1975, Scuola Normale Superiore, Pisa, April 1975. 77. Günaydin M., Octonionic Hubert spaces, the Poincaré group and SU(3). Preprint, Bonn University, 1975. 78. Gürsey F., Ramond P., Sikivie P., An universal gauge theory model based on E(6). Phys. Lett. 60B, N2, 177-180 (1976). 79. Ramond P., Unified theory of strong, electromagnetic, and weak interactions based on the vector-like group E(7). Preprint, CALT-68-540, CALTECH, Pasadena 1976. 80. Casalbuoni R., Domokos G., Kövesi-Domokos S., Algebraic Approach to the quark problem. Preprint COO-3285-19, John Hopkins University, Baltimore 1975. 81. Casalbuoni R., Domokos G., Kövesi-Domokos S., Coloured supersymmetry. Preprint TH.2150-CERN, Geneva, 31 March 1976. 82. Lee B.W., My Perspectives on Particle Physics: Summary of Orbis Scientiae, 1976. FERMILAB-Conf-76/20-THY/EXP, Feb 1976. 83. Gold s tine H.H., Horwitz L.P., On a Hubert space with - 52 -

nonassociative scalars. Prec. Nat. Acad. Sci. (USA), 4S, N7, 1134-1142 (1962). ~~ 84. C#14«tine K.H., Eorwitz L.P., Hilbert apace with nonassociative scalars I, II. Math. Ann. 154, N1, 1-27 (1964); 164, N4, 291- 316 (1966). r. 85. Hexwitz L.P., Biedenharn L.C., Intrinsic Superselection rules of algebraic Hilbert space. Helv. Phys. Acta 3S_, N4, 385-401 (1965). 86. Birkhoff G., von Neumann J., The logic of quantum mechanics. Ann. of Math. (USA) 3J7, N4, 823-843 (1936). : 87. Jauch J.M., Foundations of Quantum Mechanics. I" Addisen-Wesley, Reading, 1968. 88. Finkelstein D., Jauch J.M., Schlminevitch S., Speiser D., Foundations of quaternion quantum mechanics. J. Math. Phys. 3_, N2, 207-220 (1962). 89. Finkelstein D., Jauch J.M., Schiminovitch S., Speiser D., Seme physical consequences of general Q-covari- ance. Helv. Phys. Acta 35, N4-5, 328-329 (1962).

90. Finkelstein D., Jauch J.M., Speiser D.r Quaternionic < Representations of Compact Groups. J. Math. Phys. £, N1, 136-140 (1963).

91. Finkelstein D., Jauch J.M., Schiminovitch S.f Speiser D., Principle of general Q-cevariance. J. Math. Phys. £, N6, 788-796 (1963). 92. Emch G.G., Mecanique quantieue quaternionienne et i

... i' - о*.А.д<<.пл1 piptiu i 94. Gell-Mann M., A schematic model of baryons and mesens. • Phys. Lett. S_, N3, 214-215 (1964). f 95. Zweig G., An SU- medal for strong interaction symmetry ['•• and its breaking. ;-y CEKN preprints TH.401, 412; Geneva 1964. Щ - 53 -

- colouK Iv&iy Incomplète. Hit oi paptKi) 96. Gell-Mann M., Quarks. Lectures given at XI Internationale Universltäts- wochen fur Kernphysik, Schladming, Feb 21 - March 4, 1972; CERN preprint TH.1543; Aug 1, 1972. Acta phys. austriaca. Suppl. 9_, 733 (1972). 97. Nambu Y., Ran M.-Y., Three-triplets, paraquarks and "colored" quarks. Phys. Rev. Dl£, N2, 674-683 (1974). 98. Glashow Sh.L., Quarks with color and flavor. Scientific American 233, N4 , 38-50 (Oct 1975). 99. Chanowitz M.S., Color and experimental physics. Preprint LBL-4237, Berkeley 1975. - conilntmtnt Uome ptptn.*, whtAt numtoKouA KtltKtncti on tkt mbjt&t may be {êund) 100. Ramond P., Of strings, shells and bags ... Preprint, Yale university. New Haven 1974. 101. Böse S.K., Jabs A., Muller-Kirsten H.J.W., Vahedi N., Aspects of quark confinement. Preprint, university of.Kaiserslautern, 1975. Qu.euttKn.ioni> and octonlêni - original piptK* *.nd *omt Ktvltw* lite. aXho /154/) 102. Hamilton W.R., (The first report about quaternions, no heading). Proc. Roy. Irish. Acad. 2_. (1843), pp. 423-434. 103. Hamilton W.R., Researches respecting Quaternions. First Series. Trans. Roy. Irish. Acad. 21 (1848), pp. 199-296. 104. Hamilton W.R., Lectures on quaternions. Dublin 1853. 105. Cayley A., On Jacob!'s elliptic functions, in reply to the Hev. Brice Bronwin; and on quaternions* Phil. Mag. (London) 26 (1845), pp. 210-213; The Collected Math. PTpers I, Cambridge, 1889; p. 127. 106. Cayley A., Note on a system of imaginäries. Phil. Mag. (London) 30 (1847), pp. 257-258; The Collected MathT Papers I, Cambridge, 1889; p. 301. 107. Young J.R., On an extension of a theorem of Kuler, with a determination of the limit beyond which it fails. - 54 -

Trans. Roy. Irish, Acad. 21, Science Section, pp. 311-341 (1848). 108. bacfarlane A., Bibliography of quaternions and allied systems of mathematics. Dublin, 1904. 109. Dickson L.E., On quaternions and their generalization and the history of the eight square theorem. Ann. of Math. (USA) 20, N1, 155-171 (1919). 110. Quaternion Centenary Celebration. Proc. Roy. Irish Acad. 50A, N6, 69-121 (1945). 111. Линник Ю.В., Кватернионы и числа Кали; некоторые при- ложения арифметики кватернионов. УМН 4, №5, 49-98 (1949). 112. van der Blij P., Histery of the octaves. Simon Stevin 34_, N3, 106-127 (19C0/61). 113. Rastall P., Quaternions in Relativity. Rev. Mod. Phys. 3£, N3, 820-832 (1964). 114. Кантор И.Л., Солодовников А.С, Гиперкомилекстю числа. Изд. "Наука", ф.-м., М. 1973. 4#me medtAn algebra ttxtbook* 115. van der Waerden B.L., Moderne algebra I, II. Berlin 1930/31; further eds. 1937/40, 1950,1955. Ван дер Варден Б.Л., Современная алгебра I, II. ОНТИ, М.-Л., 1934, 1937» Гостехиздат, М.-Л.1947. 116. Albert A.A., Structure of algebras. Amer. Math. Soc. Coll. Publ., vol. 24; New York, 1939. 117. Birkhoff 6., MacLane S., A survey of modern algebra. New York, 1941; 2nd ed. 1953. 118. Jacobson N., Lectures in abstract algebra, I-III. Van Nostrand, Princeton 1951, 1953, 1964. 119. Курош А.Г., Лекции по общей алгебре. Физматгиз, М. 1962| Kurosh A.6., Lectures on general algebra. Chelsea, New York, 1963. 120. Lang A., Algebra. Addison-Wesley Publ. Co., Reading, Mass., 1965; Лент С, Алгебра. Изд. "Мир", М. 1968. 121. Мальцев А.И., Алгебраические системы. Изд. "Наука", ф.-м., М. 1970. 122. Калужнин А.А., Введение в овцу» алгебры. Изд. "Наука", ф.-м., М. 1973. - 55 -

123. Weyl H., A half-century of mathematics. Amer. Math. Monthly 58., N3, 523-553 (1951). - unlqutntii the.0Ke.mi Uee AIAO /109,111,112,114/ and : ttxtbooki) 124. Frobenius 6., Ober Lineare Substitutionen und bi- lineare Formen. Journal fur d. reine u. angewandte Math. 84, N1, 1-63 (1878). "~ : 125. Bott R., Milnor J., On the parallelizability of the spheres. .Bull. Amer. Math. Soc. 64, N1 (Part 1), 87-89 (1958). ~~ 126. Hurwitz A., Uber die Komposition der quadratischen

:; Formen von beliebig vielen variabeln. J Nachr. Ges. Wise. Gottingen (1898), s. 309-316. I 127. Линник Ю.В., Обобщение теоремы Frobenius'а к установ- ление связи с теоремой Гурвица о композиции квад- ратичных форм. > Изв. АН СССР, сер. матам. (1938), »1, стр.41-52; 128. Albert A., Quadratic forms permitting composition. Ann. of Math. (USA) 43_, N1, 161-177 (1942). ' 129. Dubish R., Composition of quadratic forms. i; Ann. of Math. (USA) 47_, N3., 510-527 (1946). • > About vaKiotiA hyp&Kcompltx ly&ttmi, ? 130. Study E., Cartan Б., Nombres complexes. • Bncyclopedie Sci. Math. Pures et Appl. I, vol. 1, fasc. 1, Paris-Leipzig 1904; pp. 329-468. : 131. Розенфельд Б.А., Неевклидовы геометрии. | Гостехиэдат, М. 1955. :- 132. Розенфельд Б.А., Многомерные пространства. Ш: Изд. "Наука", ф.-м., М. 1966.

133. Jacobson N., Lie algebras. Interscience, New York 1962. Джекобсон Н., Алгебры Ли. Изд. "Мир", М. 1964. 2окп'л vtttoK-matKix KtpKtbtnt&tion o£ iplit octonioni, 134. Zorn M., Theorie der alternativen Rings. Abh. Math. Sex. Hamburg 8_, 123-147 (1931). 131. Zorn M., AlternativkSrper und quadratische Systeme. • Abb. Math. Sen. Hamburg 9_, 395-402 (1933). - 56 -

Gtnfi&l JLtpn.tAtnte.tlon tktoKy o{. eJL§tbKai 136. Eilenberg S., Extensions of general algebras. Ann. Soc. Polon. Math. 21, N1, 125-134 (1948). 137. Schafer R.D., Representations of alternative algebras. Trans. Amer. Math. Soc. 72., N1, 1-17 (1952). 138. Schafer R.D., The Casinir operation for alternative algebras. Proc. Amer. Math. Soc. £, N3, 444-451 (1953). 139. Jacobson N., Structure of alternative and Jordan bi- modules. Osaka Math. J. 6_, N1, 1-71 (1954).

140. Buoncristiani A.M., The Cayley algebra and linear field equations. Util. Math. (5, 23-44 (1974). 141. Gursey F., Ramond P., Sikivie P., Six-quark model for the supression of Дб-l neutral currents. Phys. Rev. D12_, N7, 2166-2168 (1975). 142. Gursey F., Sikivie P., E- as a universal gauge group. Phys. Rev. Lett.,3£, N14, 775-778 (1976). 143. Gursey F., Exceptional groups and elementary particles. Lect. Notes Phys. 5£, 225-233 (1976). 144. Ramond P., Is there an exceptional group in your future? E- and the travails of the symmetry breaking. Preprint CALT-68-579, Pasadena, 1976. 145. Ramond P., Introduction to exceptional Lie groups and algebras. Preprint CALT-68-577, Pasadena, 1976. г 146. wybourne B.G., Bowick M.J., Basic properties of ex- f ceptional Lie groups. [)' Preprint, Canterbury Univ., New Zealand, 1976. Г. 147. Wybourne B.G., SU xSU| scalars in E_ irreps. [v Preprint, Univfi . of Canterbury, New Zealand; to be published in J. Math. Phys. 148. Freund P.G.O., Quark parastatistics and color gauging. Phys. Rev. D13, N8, 2322-2324 (1976). 149. Casalbuoni R., Colored quarks and octonions. Preprint INFN, Sezione Firenze, Dec 1976. 150. Buccella F., uuarks, color and octonions. Preprint, Istituto di Fisica "G.Marconi", Roma. 151. Buccella P., Falcioni M., Pugliese A., Octonions and unified theories with orthogonal groups. Preprint, Istituto di Fisica dell'Universita Roma. - 57 -

152. Fritzsch H., Universality of quarks and leptons. Preprint CERN TH. 2309, 14 April 1977. 153..Cung V.K., Kim C.W., Lepton assignment in E- model and multilepton production by neutrinos. Preprint JHU-HET-773, Baltimore 1977. 154. Freudenthal H., Oktaven, Auenahmegruppen und Oktavengeometrie. Math. Inst, der Rijksuniversiteit, Utrecht 1951. 155. Blij, van der F., Units of octaves. Proc. Koninkl. nederl. akad. wetensch. A69, 2£, N2, 127-130 (1966); (Indag. Math. 28J7 156. Sagle A.A., Malcev algebras. Trans. Amer. Math. Soc. 101, N3, 426-458 (1961). 157. Sommerfeld A., Atombau und Spektrallinien, II„ Braunschweig 1951. Зоммерфельд А., Строение атома и спектры, 11. ГИТТЛ, М. 1956. 158. Cartan E., Leçons sur la théorie des spineurs. Paris, 1938. Карган Э., Теория спиноров. ГИИЛ, м.,1947. 159. Busentann H., The geometry of geodesies. Academic Press, New York 1955. Буэеман г., Геометрия геодезических. «извтгиз, M. 1962.

Received October 31, 1977 - 58 -

О НЕАССОЦИАТИВНОСТИ В ФИЗИКЕ И ОКТОНИОНАХ КЭЛИ-ГРБЙВСА.

Математическое введение

Л.Свртсепп, Я.Лыхмус

Резюме

Статья представляет собой обзор о некоторых неассо- циативных закономерностях в физик* и о применении теории неассоциативных алгебр (в частности, алгебры октонионов) в квантовой физике частиц. Дано математическое описание гиперкомплексной алгебры октонионов, рассмотрена проблема представлений. Подробнее изучено регулярное бимодульное представление октонионов. Библ. 159 назв., 8 табл.

I АКАДЕМИЯ НАУК ЭСТОНСКОЙ ССР. Отделение физико-математических и технических наук. Лео Карлович Соргсепг , Яак Херманн-Карлович Л ы х - мус. О неассоциативности в физике и октонионах Кэли-Гревса. 1. Математическое введение. Препринт F-7. На английском языке. Редакционно-иэдательскяй совет АН ЭССР, Таллин. Toimetaja L. Palgi. Trükkida antud 01.09.78. Paber 30X40/8. Trükipoognaid 3,75. Tingtrükipoognaid 3,48. Arvestuspoognaid 2,35. Trükiarv 300. MB-04444. ENSV TA Toimetue- ja Kirjastus- nCukocru, Tallinn, Sakala 3. ENSV TA rotaprint, Tallinn, Sakala 3. Telllnuse nr. 23S. | Hind 35 kop.