SYMMETRY OF THE TRIPLE OCTONIONIC PRODUCT

Transcription

1 UDC 5 SYMMETRY OF THE TRIPLE OCTONIONIC PRODUCT г M V Kharinov (997 Rssia St Petersbrg line of VI 9 SPIIRS) khar@iiasspbs The Hermitian decomposition of a linear operator is generalized to the case of two or more operations n additive expansion of the prodct of three octonions into three parts is constrcted wherein each part either preserve or change the sign nder the action of the Hermitian conjgation and operation of inversion of the mltiplicative order of three hypercomplex nmbers as well as nder the composition of specified operations The prodct of three octonions in particlar qaternions with conjgate central factor is presented as the sm of mtally orthogonal triple anticommtator triple commtator and associator that vanishes in the case of associative qaternions The triple commtator is treated as a generalization of the cross prodct to the case of three argments both for qaternions and octonions generalized cross prodct is introdced as an antisymmetric component of the triple octonionic prodct that changes sign both for inversion of the mltiplicative order of three argments and for the Hermitian conjgation of the prodct considered respectively to the central argment The definition of the cross prodct of three hypercomplex nmbers dedced from symmetry considerations is compared with the soltion of S Okbo (99) and the modern soltion of T Dray and M Corinne (5) It is shown that the derived definition is eqivalent to the first soltion presented by SOkbo in an insfficiently perfect form Key words additive decomposition Hermitian conjgation symmetry antisymmetry qaternions octonions triple anticommtator triple cross prodct associator mtal orthogonality INTRODUCTION In this paper the generalization of a cross prodct to the case of three argments for a fordimensional and eight-dimensional vector spaces is considered t present one of the obstacles to solving this kind of problem is the commonly sed definition of a vector prodct by means of an intitively percepted "right-hand rle" In the toolkit of hypercomplex nmbers (non-commtative qaternions and non-associative octonions [ ]) the notion of a cross prodct of a pair of vectors is introdced withot referring to intition which simplifies the solving of the problem However nlike the cross prodct itself acqaintance of the reader with hypercomplex nmbers is sally limited since the hypercomplex nmbers are mentioned only in fewer corses on abstract algebra [] To qickly master the hypercomplex nmbers it is enogh to se the poplar book [] Many sefl rles for working with hypercomplex nmbers one may find in [] In the presented paper only formlas that most important for a context are listed Unnmbered formlas are mainly given either for memorization or for explaining the meaning of the notation In the paper we adhere to the notation of [] The aim of the paper is to generalize the additive Hermitian decomposition to sbstantiate the generalization of the cross prodct and to simplify the toolbox of hypercomplex nmbers which can prove to be especially sefl when working with nonassociative octonions ELEMENTRY INFORMTION By i we denote the mltiplicative nit that commtes with any hypercomplex nmber and when mltiplied leaves it nchanged

2 i i Note that i is a simple renaming of and say 5i 5 5 Let denotes the hypercomplex nmber which is conjgate to the nmber and is related to by the formla [] i i where i is the inner prodct of the vector and vector i In for algebras of hypercomplex nmbers (real nmbers complex nmbers qaternions and octonions) the sqare of length of the vector is introdced as the prodct of the vector and the conjgate vector and the inner prodct of the vectors and coincides with the half-sm of the vector and the conjgate vector i i From the last relations it is easy to establish the sefl identity () where the prodct is written withot brackets since the reslted vector does not depend on the order of mltiplication of hypercomplex factors In order to reliably relate to mltiplication of qaternions and octonions it is sefl to keep in the mind the following elementary formlas First formla expresses that conjgation changes the order of factors nd the following formla for the inner prodct i of vector and vector i expresses the transfer rle of the factor from one to another part of the inner prodct in accompany with simltaneos conjgation i i Finally the formla i i i states that the order of mltiplying of three hypercomplex factors and does not affect the inner prodct of and i So the prodct written withot brackets implies choosing either of the two alternative ways of their arrangement jst as in left part of () To sbstantiate the claimed additive decomposition in the toolkit of hypercomplex nmbers it is sefl to preliminary consider a system of three operations of the Hermitian conjgation type Let be a linear transformation of the vector of eight-dimensional space of octonions in particlar the for-dimensional space of qaternions Let s consider where and are fixed vectors Let the operation of the transformation of the operator be the operation of Hermitian conjgation v v for any pair of octonions and v Let operation be an operation of inversion of mltiplicative order which is carried ot by conjgation of together with both parameters and in combination with common conjgation Let operation be a doble conjgation operation consisting in replacing of the central argment by the conjgate one followed by the common conjgation The reslts of the composition of operations are smmarized in Tab

3 Tab n example of three operations forming an belian grop of mtally inverse elements Tab describes the reslts of application of three operations and The table is symmetric and contains the same diagonal elements This indicates that the operations nder consideration form an belian grop of mtally inverse elements GENERLIZED DDITIVE DECOMPOSITION OF LINER OPERTOR For several operations of Hermitian conjgation type which constitte an belian grop of mtally inverse elements the linear operator is trivially decomposed into a sm of mixed symmetric skew-symmetric operators that nder the action of each operation either do not change or change the sign In a single operation this decomposition is well known This is the decomposition of the operator into a Hermitian and a skew-hermitian parts For two operations and an analogos decomposition into the sm of symmetric skewsymmetric operators is ermined by the relations and is written as () The above linear transformation of the operators into symmetric skew-symmetric operators and the inverse transformation are described by the normalized In the normalized Hadamard matrix the first row and the first colmn consist of only

4 symmetric orthogonal Hadamard matrix which coincides p to a factor with the inverse matrix E For a fixed first row in the matrix the permtation of the remaining three rows in general violates the symmetry of the matrix bt preserves the set of colmns For example by interchanging the last two lines we get a matrix in which the rows nmbered from top to bottom in order correspond to colmns alternating from left to right in the order of It is characteristic that the qartet of rows (colmns) of the matrix is an abelian grop with respect to the operation of termwise mltiplication In this case the rows (colmns) are alternated in the order of grop mltiplication which is not violated for any permtation of rows (colmns) For three operations and the expansion into a sm of symmetric skew-symmetric operators of the form whose action is ermined by the relations is presented in the form

5 In this case the matrix of size consisting of ± is symmetric and has the form of Fig e e Fig n example of a symmetric matrix with matching sets of rows and colmns Fig shows a matrix in which the alphabetic designations of the alternation of rows and colmns in a fixed order of grop mltiplication are written before the rows and above the colmns For clarity the cell fields containing are painted black symmetric matrix is a normalized orthogonal Hadamard matrix and p to a factor coincides with its inverse / E where E is the diagonal matrix of The eight rows (colmns) in Fig occpying places e constitte an belian grop with respect to termwise mltiplication and are ordered in the matrix accordingly to «dobling algorithm» [] (Fig ) It is noteworthy that the matrix posesses a permtation symmetry [5-9] sch that the set of colmns does not change nder of 6 permtations of rows preserving the order of the alternation of rows in accordance with the dobling algorithm These 6 permtations constitte a grop Of these permtations of rows preserve sets of colmns bt violate the diagonal symmetry of the matrix as for example in the matrix of Fig Fig shows a matrix of identical sets of rows and colmns For clarity the cell fields of the matrix containing are colored in black Before the rows and above the colmns of the matrix are written the alphabetical designations that indicate the location of rows and colmns placed in the order of grop term-by-term mltiplication On the left and speriorly the nderlined nmerical designations of rows and colmns are added which nmbered in the order of their alternation in the matrix in Fig The remaining of the 6 permtations of the rows of the matrix preserve not only the set of colmns bt also the diagonal symmetry of the matrix (Fig )

6 6 5 7 e e Fig Example of an asymmetric matrix with matching rows and colmns The alternation of rows (Fig ) in symmetric matrices is given in Tab Tab lternating of rows in symmetric matrices In the above table the leftmost colmn of nmbers alternating from top to bottom in a natral order describes the matrix of Fig Pairs of colmns from Tab define the permtations of the rows preserving the symmetry of matrix Fig For a fixed first colmn that coincides with a colmn of alternating nmbers the colmns of Tab describe permtations of rows preserving the diagonal symmetry of the matrix in Fig These permtations are not a grop bt represent the nion of an octet of cyclic sbgrops of and 7 order inclding two sbgrops of the seventh order three cyclic sbgrops of the third order and three cyclic sbgrops of the forth order Vertical lines in Tab separate colmns that describe permtations from one cyclic sbgrop It can be shown that the discssing normalized Hadamard matrices of size are ermined by their permtability so that the matrix itself is niqely reconstrcted from permtations that do not violate its diagonal symmetry and to recover the matrix it is sfficient to select in prescribed algorithm from twenty-eight colmns of Tab only three colmns that specify a pair of permtations [6-]; are characterized by the maximm permtation symmetry which is preserved with the maximm nmber of permtations of the rows different from the first row containing only [7] Owing to listed properties the discssed Hadamard matrices have fond practical application in a series of games with permtative symmetry [6-9]

7 THE DECOMPOSITION OF THE TRIPLE OCTONIONIC PRODUCT INTO THE SUM OF THE NTICOMMUTTOR COMMUTTOR ND SSOCITOR Let's obtain an additive decomposition of the prodct of three octonions and for two commting operations namely for the operation of Hermitian conjgation and the operation of inversion of the mltiplicative order Using Tab we rewrite the trivial relations from the right-hand side of () as () () (5) (6) where the term vanishes and the remaining terms of the additive decomposition receive the names that reflect their expression by commtating the argments and changing by the opposite the mltiplicative order of and So matching to the traditional terminology is called a commtator is called an associator and is called an anticommtator ccording to () () (6) they coincide with the symmetric skew-symmetric (antisymmetric) components of the additive decomposition of the triple prodct of octonions аnd with the conjgate central argment NOTE The validity of (5) and the eqalities of the half-sms in () () (6) follows from the anticommtativity of the associator which in trn trivially follows from associator zeroing if any two of the three argments coincide [-] Concerning formlas (6) it shold be noted that they can be considered as a definition of a triple commtator It is noteworthy that similar definitions are fond in the literatre for example in [] on the page By means of the inverse expressions from the left-hand side of () the terms and as (see Tab ) will be written via (7) Formla (7) gives the decomposition of the triple octonionic prodct commtator into the sm of and asso- three mtally orthogonal terms namely anticommtator ciator () (9) () and () () give the expressions и mtally orthogonal terms as a linear combinations of the listed

8 It is easy to verify the mtal orthogonality of и sing the explicit expression of via and () and also by the expression of via of the mltiplicative nit i and the cross prodcts and i i i i () The formla () for is derived from identity () by replacing of with The formla () for is derived from its expression (6) as one or another of the half-sm and the following expression of the prodct of two hypercomplex nmbers via [] i i i () The cross prodct of two argments is obtained by replacing in the central argment by the mltiplicative nit i that according to (6) coincides with the commtator of two vectors и i () where the vector is the cross prodct of a pair of hypercomplex vectors and [ ] The cross prodct of the hypercomplex nmbers and is orthogonal to the mltiplicative nity i and vanishes when i is taken as one or another argment ( i ) i i In the rest the cross prodct preserves the properties of the conventional cross prodct which is introdced in three-dimensional space sing the intitively perceived "right-hand rle" Under the preservation of properties it is meant that if we introdce the operation of annlling the real component of a hypercomplex nmber denoting by a prime i i then the cross prodct of the vectors and coincides with the conventional cross prodct of vectors and of the three-dimensional sbspace that is orthogonal to a mltiplicative nit i Using () it is easy to establish the orthogonality of the to each of three argments and and to verify that and are mtally orthogonal sing () The orthogonality of the associator to each of the three argments and is derived from its expression by means of half-sms () The same property of the orthogonality of the associator to each of its argments and is dedced from its expression sing the half-sm () The orthogonality of the triple commtator and the associator to its argments implies the anticommtativity of mixed prodcts for example

9 From () it is easy to establish that the triple vector prodct vanishes when a pair of its argments coincide with one another which implies its anticommtativity for example The associator has the same properties of vanishing to zero and anticommtativity for example Moreover it is known [-] that the associator vanishes in the case of three argments from one qaternionic sbalgebra Then it follows from the anticommtativity of the mixed prodct that the associator is orthogonal to the vectors of qaternionic sbalgebras generated by any pair of three vectors and Therefore in addition to the vectors and the associator is also orthogonal to the for following vectors i In particlar the associator is orthogonal to the triple anticommtator and commtator which are linear combinations () () of the specified seven vectors For brevity frther formlas are given withot proof since they are obtained by applying the previosly written ones NOTE In calclations it is recommended to avoid the rt tilizing of the orts i j k etc [ ] withot a special need for a spatial basis since this leads to cmbersome expressions especially when working with octonions 5 INTERPRETTION Interpretation of terms и in the additive decomposition (7) - () of prodcts of three octonions and with conjgate central argment is as follows ccording to the defining relations () () is a triple anticommtator which is expressed as a linear combination of the argments and does not change when the first and last argments are interchanged When the central argment is replaced by the mltiplicative nit i the anticommtator is converted into an anticommtator for the prodct of two hypercomplex nmbers [] The triple commtator introdced in (5) and represented in () as a linear combination of the mltiplicative nit i and pairwise cross prodcts of the argments is treated as a generalized cross prodct of three hypercomplex nmbers (qaternions or octonions) When one of the argments is replaced by the mltiplicative nit i the triple cross prodct p to a sign coincides with the conventional cross prodct of two other argments When the central argment is sbstitted by the mltiplicative nit i the triple commtator is converted into an ordinary cross prodct of two argments and [] The associator is well known in the scientific papers [-] Here it is introdced by formlas () where nlike the conventional definitions conjgation of the central argment is sed and the associator is calclated with the coefficient / These differences do not affect the basic properties of the associator 6 LENGTHS OF DDITIVE DECOMPOSITION TERMS The length sqares of the triple anticommtator the commtator (triple cross prodct) and the associator are expressed as follows

10 ) )( )( ( (5) ) ( (6) ) ( (7) where for any hypercomplex vector the designation denotes its length and is sed as an eqivalent of to shorten the record So in (5)-(7) It is sefl to note that the length sqare of the anticommtative component of the triple octonionic prodct is expressed by the erminant of the symmetric positive definite matrix of scalar prodcts of the argments and NOTE In applications it may be sefl to represent (6) (7) in another form sing an expression of in terms of pairwise scalar prodcts and scalar prodcts of one argment to the conjgate second argment It is pertinent to note that in the special theory of relativity of Einstein the qantity is called «the space-time interval» and is treated as an analog of the spatial distance where is the fordimensional space-time vector identified with the hypercomplex nmber by WR Hamilton himself and by his contemporary followers [5] From the decomposition (7) of the prodct into orthogonal terms by means of the smmation (5)-(7) we obtain the eqality expressing the axiomatic property of normed algebras which consists in the fact that the sqare of the prodct length of two hypercomplex vectors is eqal to the prodct of the sqares of the lengths of these vectors 7 SOURCE SOLUTIONS The generalizing of cross prodct of two vectors to the case of more than three dimensions was investigated by Zrab K Silagadze in [6] The soltion is constrcted by postlating axioms which have a physical meaning and generalize the characteristic properties of the traditional three-

11 dimensional cross prodct It is shown that the only possible dimensionality for a jstified generalization of a vector prodct is dimension 7 The generalization of a cross prodct as the octonionic commtator of two argments is given as an example It is noteworthy that in addition to the prodct of three vectors in [6] the prodct of three argments is also considered In this connection a nmber of interesting relations are derived Namely the known formla "'BC' mins 'CB'" (rssian) is treated B C B C C B B C where in the notation sed here B and C For comparison of formlas the difference in notations is presented in Tab Tab Matching of notations [6] Notion Notation In [6] In present paper Mltiplicative nit e i Inner prodct ( Eclidean scalar prodct ) Cross prodct of two argments ( Vector prodct ) ssociator ( Ternary prodct ) In [] Ssm Okbo considered the prodct of three octonions extracting ot the anticommtative part Ocbo in the additive decomposition The soltion obtained by a fairly long chain of calclations is given on the page in the form i Ocbo wherein is disclosed in or notations by the following expression Okbo i i i i Okbo To simplify the comparison of the decomposition of the triple octonionic prodct present paper with the decomposition of the triple octonionic prodct of co-assignments of notations in the in [] we give the Tab Matching of notations [] Notation Notion In [] In present paper Mltiplicative nit e i Inner prodct Cross prodct of two argments ssociator Taking into accont the expressions () for expressed by the formla i and () for Tab the soltion [] is

12 which is eqivalent to the formla (7) for the decomposition of the triple prodct of triple anticommtator triple commtator and associator Ths the formlas (7)-() improve the reslts originally obtained in [] for to trivially express the idea of decomposition of and into the sm and allow into the sm of mtally orthogonal by commtating of factors and inversing of the mltiplicative order of the argments and t the same time the identities ()-(6) develop the interpretation of and as symmetric antisymmetric or avoiding ambigos terminology symmetric skew-symmetric parts of the prodct In contrast to [] in [7] the athors Tevian Dray and Corinne Manoge consider the prodct of octonions and with the conjgate central element bt the triple octonion prodct denoted they defined as where the generalized cross prodct and the associator are combined in one difference expression that obtained from (9) () Sch the prodct is anticommtative bt when the prodct is decomposed into a sm of three orthogonal terms it does not lead to a transparent expression (7) jstified by trivial commtation rles and rles for inversing of the mltiplicative order of the argments ()-(6) SYMMETRY CONSIDERTIONS The paper [6] is well known in abstract algebra (section R Rings and lgebras) In this paper we draw the attention of the reader to the remarkable reslts in this field of the well-known physics Ssm Okbo who in 99 in a triple octonionic prodct has extracted ot two additive anticommtative parts giding by symmetry considerations In the contination of S Okbo's research one of the parts is proposed to be interpreted as a generalization of the cross prodct for for-dimensional and eight-dimensional Eclidean space and the case of three argments basing on symmetry considerations It is important that the notion of symmetry is associated with the alternative notion of antisymmetry and is interpreted in the following senses as commtativity and anticommtativity of the ternary operation with vector argments in which the permtation of the argments either does not affect the reslt of the operation or changes the sign of the resltant vector to the opposite; as the properties of several commting operations on linear operators in a vector space which are introdced as a generalization of the Hermitian decomposition of the operator into a symmetric and antisymmetric (skew-symmetric) parts So in this paper an additive decomposition of the prodct of three octonions with conjgate central argment is obtained The main reslts are as follows The additive decomposition consists of three parts namely the triple anticommtator the commtator (triple vector prodct) and the associator ermined by changing the seqence order and the order of mltiplication of the three argments ; all three terms of the additive decomposition are mtally orthogonal; two terms and are anticommtative; all three terms (with zero forth term ) are symmetric skew-symmetric components of the representations of the operator of mltiplication of two constant octonions by a central conjgate octonion considered

13 as a variable argment where the pper and lower signs inside the brackets denote the property of mixed symmetry either to preserve or change the sign nder the operation of Hermitian conjgation and either preserve or change the sign nder the operation of inversion of the mltiplicative order respectively The additional symmetry consideration obtained and patented as a by-prodct [7 ] is that the symmetric normalized Hadamard matrices arising in the generalization of the additive Hermitian decomposition of an operator into symmetric skew-symmetric parts have the property of preserving the mirror diagonal symmetry nder the maximm nmber of row or colmn permtations Symmetry considerations provide a proper choice of the definition of a generalized triple vector prodct from the nmber of possible generalizations Ths the term most likely is the appropriate variant of the triple cross prodct generalized to the case of for-dimensional and eightdimensional Eclidean space 9 CONCLUSION In addition to the cross prodct definition qaternions and octonions are famos for the fact that they graceflly describe the rotation of space orthogonal to the mltiplicative nit i In a later paper we are going to show that this also applies to the Lorentz transformations which in this case are considered in Eclidean space [-] pparently the refinement of the representation of Lorentz transformations in terms of qaternions and octonions is promising for application in Einstein's theory of relativity [5] Dirac qantm mechanics [5] and other fields of physics [] However basing on the common experience of or preceding work the development of applications in the field of digital image processing is closer to s Therefore it is planned first of all to contine research in this context relying on or own experience in image processing as well as on long-term stdies of the Yoshkar-Ola's school of scientists who sccessflly apply qaternions for image processing [-] s is known in image processing the pixel of the color image is mapped to the point of the threedimensional Eclidean color space presented in one or another coordinate system To describe the color transformation of an image the rotation of the color space is sally sed Using the Lorentz transformation instead of simple rotation we will constrct programmatically implement and explore the generalized color transformation of the image to advance in the practical soltion of one of the many problems of modern image processing REFERENCES Hamilton WR «Lectres on qaternions» Hodges and Smith 5 76 pp Kantor IL Solodovnikov S «Hypercomplex nmbers an elementary introdction to algebras» Springer pp Vavilov N «Specific grop theory» Sankt-Petersbrg 6 75 pp (in Rssian) Okbo S «Triple prodcts and Yang Baxter eqation I Octonionic and qaternionic triple systems» Jornal of mathematical physics 99 Vol No 7 pp 7-9 https//arxivorg/pdf/hep-th/95pdf 5 Kharinov MV «Permtational and hidden symmetry on the example of isomorphic Hadamard matrices pplications in the field of artificial intelligence» Proceedings of Second Int Conf «Means of mathematical modeling» Sankt-Petersbrg SPbSTU Pblishing Hose 999 Vol 5 pp 7-5 (in Rssian) 6 Kharinov MV «Materials on the series of games "DaltoRainbow"» https//drivegooglecom/drive/folders/b_pnb5xi5vknrmxfae5cm (in Rssian) electronic game https//drivegooglecom/drive/folders/b_pnb5xi5sbptkirwhrrt 7 Kharinov MV «Color logical pzzle game "DaltoRainbow"» Deposited in VINITI No 57- B9 M 99 7 pp (in Rssian) Kharinov MV «Color logical pzzle game DaltoRainbow» USSR patent No 7997 from Oct 9 pp Blletin of Inventions 99 No pp 56 (in Rssian) 9 vilov NI «Kharinov s symmetrical pictograms» Mathematics No pp6 (in Rssian)

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