Research Article Geometrical Applications of Split...

Research ArticleGeometrical Applications of Split Octonions

Merab Gogberashvili12 and Otari Sakhelashvili1

1Tbilisi Ivane Javakhishvili State University 3 Chavchavadze Avenue 0179 Tbilisi Georgia2Andronikashvili Institute of Physics 6 Tamarashvili Street 0177 Tbilisi Georgia

Correspondence should be addressed to Merab Gogberashvili gogbergmailcom

Received 16 August 2015 Accepted 28 September 2015

Academic Editor Yao-Zhong Zhang

Copyright copy 2015 M Gogberashvili and O Sakhelashvili This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

It is shown that physical signals and space-time intervals modeled on split-octonion geometry naturally exhibit properties fromconventional (3 + 1)-theory (eg number of dimensions existence of maximal velocities Heisenberg uncertainty and particlegenerations) This paper demonstrates these properties using an explicit representation of the automorphisms on split-octonionsthe noncompact form of the exceptional Lie group G

2 This group generates specific rotations of (3 + 4)-vector parts of split

octonions with three extra time-like coordinates and in infinitesimal limit imitates standard Poincare transformations In thispicture translations are represented by noncompact Lorentz-type rotations towards the extra time-like coordinates It is shownhow the G

2algebrarsquos chirality yields an intrinsic left-right asymmetry of a certain 3-vector (spin) as well as a parity violating effect

on light emitted by a moving quantum system Elementary particles are connected with the special elements of the algebra whichnullify octonionic intervals Then the zero-norm conditions lead to free particle Lagrangians which allow virtual trajectories alsoand exhibit the appearance of spatial horizons governing by mass parameters

1 Introduction

Many properties of physical systems can be revealed from theanalysis of proper mathematical structures used in descrip-tions of these systems The geometry of space-time one ofthemain physical characteristics of nature can be understoodas a reflection of symmetries of physical signals we receiveand of the algebra used in the measurement process Since allobservable quantities we extract from single measurementsare real in geometrical applications it is possible to restrictourselves to the field of real numbers To have a transitionfrom a manifold of the results of measurements to geometryone must be able to introduce a distance between someobjects and an etalon (unit element) for their comparison Inalgebraic language all these physical requirements mean thatto describe geometry we need a composition algebra with theunit element over the field of real numbers

Besides usual real numbers according to the Hurwitztheorem there are three unique normed division algebrasmdashcomplex numbers quaternions and octonions [1ndash3] Physicalapplications of the widest normed algebra of octonionsare relatively rare (see reviews [4ndash8]) One can point to

the possible impact of octonions on color symmetry [9ndash13]GUTs [14ndash17] representation of Clifford algebras [18ndash20]quantummechanics [21ndash29] space-time symmetries [30 31]field theory [32ndash35] quantum Hall effect [36] Kaluza-Kleinprogram without extra dimensions [37ndash42] Strings and M-theory [43ndash48] SUSY [49] and so forth

The essential feature of all normed composition algebrasis the existence of a real unit element and a different numberof hypercomplex units The square of the unit element isalways positive while the squares of the hypercomplex basisunits can be negative as well In physical applications onemainly uses division algebras with Euclidean norms whosehypercomplex basis elements have negative squares similarto the ordinary complex unit 119894 The introduction of vector-like basis elements (with positive squares) leads to the splitalgebras with pseudo-Euclidean norms

In this paper we propose parameterizing world-lines(paths) of physical objects by the elements of real splitoctonions [50ndash53]

119904 = 120596 + 120582119899119869119899+ 119909119899119895119899+ 119888119905119868 (119899 = 1 2 3) (1)

Hindawi Publishing CorporationAdvances in Mathematical PhysicsVolume 2015 Article ID 196708 14 pageshttpdxdoiorg1011552015196708

2 Advances in Mathematical Physics

A pair of repeated upper and lower indices by the standardconvention implies a summation that is 119909119899119895

119899= 120575119899119898119909119899119895119898

where 120575119899119898 is Kroneckerrsquos deltaIn geometric application four of the eight real parameters

in (1) 119905 and 119909119899 are space-time coordinates 120596 and 120582

119899 areinterpreted as the phase (classical action) and thewavelengthsassociated with the octonionic signals [50ndash53]

The eight basis units in (1) are represented by onescalar (denoted by 1) the three vector-like objects 119869

119899 the

three pseudovector-like elements 119895119899 and one pseudoscalar-

like unit 119868 The squares (inner products) of seven of thehypercomplex basis elements of split octonions give the unitelement 1 with the different signs

1198692

119899= 1

1198952

119899= minus1

1198682= 1

(2)

It is known that to generate a complete basis of split octonionsthe multiplication and distribution laws of only three vector-like elements 119869

119899are enough [1ndash3] The three pseudovector-

like basis units 119895119899 can be defined as the binary products

119895119899=1

2120576119899119898119896

119869119898119869119896

(119899119898 119896 = 1 2 3) (3)

(120576119899119898119896

is the totally antisymmetric unit tensor of rank three)and thus describes orthogonal planes spanned by two vector-like elements 119869

119899 The seventh basis unit 119868 (the oriented

volume) is defined as the triple product of all three vector-likeelements and has three equivalent representation in terms of119869119899 and 119895119899

119868 = 11986911198951= 11986921198952= 11986931198953 (4)

So the algebra of all noncommuting hypercomplex basis unitsof split octonions has the form

119869119899119869119898= minus119869119898119869119899= 120576119899119898119896

119895119896

119895119899119895119898= minus119895119898119895119899= 120576119899119898119896

119895119896

119895119898119869119899= minus119869119899119895119898= 120576119899119898119896

119869119896

119869119899119868 = minus119868119869

119899= 119895119899

119895119899119868 = minus119868119895

119899= 119869119899

(5)

Conjugations of octonionic basis units which can beunderstood as the reflection of vector-like elements

119869dagger

119899= minus119869119899 (6)

reverses the order of 119869119899in products that is

119895dagger

119899=1

2(120576119899119898119896

119869119898119869119896)dagger

=1

2120576119899119898119896

119869119896dagger119869119898dagger

= minus119895119899

119868dagger= (119869111986921198693)dagger

= 119869dagger

3119869dagger

2119869dagger

1= minus119868

(7)

So the conjugation of (1) gives

119904dagger= 120596 minus 120582

119899119869119899minus 119909119899119895119899minus 119888119905119868 (8)

Using (2) (5) and (8) one can find that the norm of (1) isgiven by

1198732= 119904119904dagger= 119904dagger119904 = 1205962minus 1205822+ 1199092minus 11988821199052 (9)

(where 1205822 = 120582119899120582119899 and 1199092 = 119909

119899119909119899) representing some kind of

8-dimensional space-timewith (4+4)-signature and reducingto the classical formula of Minkowski intervals if 1205962 minus1205822 = 0

As for the case of ordinary Minkowski space-time weassume that for physical events the corresponding ldquointervalsrdquogiven by (9) are nonnegative A second condition is that forphysical signals the vector part of split octonions (1) shouldbe time-like

11988821199052+ 120582119899120582119899gt 119909119899119909119899 (10)

The classification of split octonions by the values of theirnorms is presented in Appendix A

2 Transformations and Automorphisms

To find the geometry associated with the signals (1) weneed to represent rotations by split octonions due to thenonassociativity of the algebra the group of rotations ofoctonionic axes 1 119869

119899 119895119899 and 119868 119866119873119862

2is not equivalent to

the group of accompanied passive tensorial transformationsof coordinates 120596 120582

119899 119909119899 and 119905 (119878119874(4 4)) In Appendix B

it is shown that by rotations in four orthogonal planes ofoctonionic space (one of which always includes the axis ofreal numbers 120596) the octonion (1) always can be representedas the sum of four elements The left multiplication 119877119904 by anoctonion with unit norm (Appendix A)

119877 = 119890120598120579 (11)

where 120598 is the (3+4)-vector defined by seven basis units 119869119899 119895119899

and 119868 represents rotations by the angles 120579 in these four planesThe right product 119904119877minus1 will reverse the direction of rotationin the plane of 120596 So we can represent a rotation which onlyaffects the (3+4)-vector part of 119904 applying the unit half-angleoctonion twice and multiplying on both the left and the right(with its inverse)

1199041015840= 119877119904119877

minus1= 119877119904119877

dagger= 1198901205981205792

119904119890minus1205981205792

(12)

Thesemaps are well defined since the associator of the triplet(119877 119904 119877

minus1) vanishes The set of rotations (12) of the seven

octonionic coordinates119909119899120582119899 and 119905 in three planes which donot affect the scalar part 120596 of (1) contains 7times3 = 21 angles ofthe group 119878119874(3 4) of passive transformations of coordinates

To represent the active rotations in the space of 119904which preserve the norm (9) and multiplicative structure ofoctonions (5) as well we would need the transformationsto be automorphisms (Appendix C) An automorphism 119880 ofany two octonions 119874

1and 119874

2 gives the equation

(1198801198741119880minus1) (119880119874

2119880minus1) = 119880119874

11198742119880minus1 (13)

Advances in Mathematical Physics 3

which only holds in general if multiplication of 119880 1198741 and

1198742is associative Combinations of the rotations (12) around

different octonionic axes are not unique This means thatnot all transformations of 119878119874(3 4) form a group and canbe considered as real rotations Only the transformationsthat have a realization as associative multiplications shouldbe considered Also note that an automorphism of splitoctonions can be generated only by the octonions withpositive norms (Appendix A) since the set of split octonionswith negative norms do not form a group (it is not closedunder multiplication) So to model space-time symmetrieswe need to use the group of automorphisms of split octonionswith positive norms

It is known that associative transformations of split octo-nions can be done by the specific simultaneously rotations intwo (and not in three as for 119878119874(3 4)) orthogonal octonionicplanes (Appendix C) These rotations form a subgroup of119878119874(3 4) with 2 times 7 = 14 parameters known as the automor-phism group of split octonions 119866119873119862

2 Generators of this real

noncompact form of Cartanrsquos smallest exceptional Lie group1198662is presented in Appendix DInfinitesimal transformations of coordinates which

accompany the group of active transformations of octonionicbasis units 119866119873119862

2 and preserve the diagonal quadratic form

1199092minus 1205822minus 1198891199052ge 0 can be written in the form (Appendix C)

1199091015840

119899= 119909119899minus 120576119899119898119896

120572119898119909119896minus 120579119899119888119905

+1

2(1003816100381610038161003816120576119899119898119896

1003816100381610038161003816 120601119898+ 120576119899119898119896

120579119898) 120582119896

+ (120593119899minus1

3sum

119898

120593119898)120582119899

1198881199051015840= 119888119905 minus 120573

119899120582119899minus 120579119899119909119899

1205821015840

119899= 120582119899minus 120576119899119898119896

(120572119898minus 120573119898) 120582119896+ 120573119899119888119905

+1

2(1003816100381610038161003816120576119899119898119896

1003816100381610038161003816 120601119898minus 120576119899119898119896

120579119898) 119909119896

+ (120593119899minus1

3sum

119898

120593119898)119909119899

(14)

with no summing over 119899 in the last terms of 1199091015840119899and 1205821015840

119899 From

the five 3-angles in (14) 120572119898 120573119898 120601119898 120579119898 and 120593119898 only 14 areindependent because of the condition

sum

119899

(120593119899minus1

3sum

119898

120593119898) = 0 (15)

The Lorentz-type transformations (14) of the 7-dimen-sional space with four time-like coordinates should describespace-time symmetries if the split octonions form relevantalgebraic structures for microphysics The transformations(14) formally can be divided into three distinct classes(Appendix C) Euclidean rotations of the spatial 119909

119899 and

time-like (119905 and 120582119899) coordinates by the compact 3-angle

120572119899 and 120573119899 respectively boosts mixing of spatial and time-

like coordinates by the two hyperbolic 3-angle 120579119899 and 120601119899

and diagonal boosts of the spatial coordinates 119909119899 and

corresponding time-like parameters 120582119899 by the hyperbolic

angles 120593119899We notice that if we consider rotations by the angles 120572119899

120573119899 and 120579119899 that is assuming that

120601119898= 120593119898= 0 (16)

the passive1198661198731198622

-transformations (14) of only ordinary space-time coordinates 119909

119899and 119905 will imitate the ordinary infinites-

imal Poincare transformations of (3 + 1)-Minkowski space

1199091015840

119899= 119909119899minus 120576119899119898119896

120572119898119909119896minus 120579119899119888119905 + 119886119899

1198881199051015840= 119888119905 minus 120579

119899119909119899+ 1198860

(17)

Here the space-time translations

119886119899=1

2120576119899119898119896

120579119898120582119896

1198860= minus120573119899120582119899

(18)

are generated by the Lorentz-type rotations toward thetime-like directions 120582

119899 So in the language of octonionic

geometry any motion in ordinary space-time is generatedby 120582119899 sim 119901

1198991199012 Time translations 119886

0are smooth since 120573

119899

are compact angles However the angles 120579119898 are hyperbolicand for any active spatial translation 119886

119899there exists a horizon

(analogues to the Rindler horizon) which is equivalent to theintroduction of some mass scale or inertia (see Section 6)

For completeness note that there exists a second well-known representation of 119866119873119862

2as the symmetry group of

a ball rolling on a larger fixed ball without slipping ortwisting when the ratio of the balls radii is 13 [54ndash58]Understanding the exceptional Lie groups as the symmetrygroups of naturally occurring objects is a long-standingprogram in mathematics Symmetries of rolling balls arevisualized in (14) by the presence of two 3-vector 119909

119899and 120582

119899

one of which is time-like The fourth time-like coordinate(ordinary time 119905) which also is affected by (14) breaks thesymmetry between ldquoballsrdquo and the factor 13 of the ratio oftheir radii corresponds to the existence of the three extratime-like coordinates 120582

119899

3 Boosts Heisenberg Uncertaintyand Chirality

In the algebra of split octonions there exist only threespace-like parameters 119909

119899 and three independent compact

rotations around corresponding axes which explains whyphysical space described by octonionic signals has threespatial dimensions Let us analyze new features of the 119866119873119862

2-

transformations (14) in comparison with standard Lorentzrsquosformulas for (3 + 1)-Minkowski space


Euclidean rotations around one of the space-like axes 1199091

the automorphism (C1) correspond to the following passiveinfinitesimal transformations of coordinates

1199091015840

1= 1199091

1199091015840

2= 1199092+ 12057211199093

1199091015840

3= 1199093minus 12057211199092

1198881199051015840= 119888119905 minus 120573

11205821

1205821015840

1= 1205821+ 1205731119888119905

1205821015840

2= 1205822+ (1205721minus 1205731) 1205823

1205821015840

3= 1205823minus (1205721minus 1205731) 1205822

(19)

When 1205721

= 1205731this reduces to the standard rotation by

the Euler angle 1205721 which also causes translations of time

119905 due to mixing with the extra time-like parameter 1205821

In general any active Euclidean 3-rotation of the spatialcoordinates 119909119899 changes the time parameter by the amount ofthe corresponding 120582

119899 which can be understood as passing of

time in our worldAnalysis of boosts of 119866119873119862

2can help us in the physical

interpretation of the extra time-like parameters 120582119899 Consider

the automorphisms (C2) by the hyperbolic angles 1205791and 120601

1

1199091015840

1= 1199091minus 1205791119888119905

1199091015840

2= 1199092+1

2(1206011minus 1205791) 1205823

1199091015840

3= 1199093+1

2(1206011+ 1205791) 1205822

1198881199051015840= 119888119905 minus 120579

11199091

1205821015840

1= 1205821

1205821015840

2= 1205822+1

2(1206011+ 1205791) 1199093

1205821015840

3= 1205823+1

2(1206011minus 1205791) 1199092

(20)

The case 1206011= 1205791corresponds to ordinary boost in (119905 119909)-

planesmdashtransitions to the reference frame moving with thevelocity 120579

1along the axis 119909

1 Then if we consider the motion

of the origin of the moving system

1199091015840= 1199101015840= 1199111015840= 0 (21)

from the first line in (20) we find that

1205791=1199091

119888119905

1205823=1199092

1205791

1205822= minus

1199093

1205791

(22)

that is the quantities 1205822and 120582

3are inversely proportional

to the velocity 1199091119905 In the space of split octonions (1) there

are two classes of time-like parameters (119905 and 120582119899) and two

different light-cones So in addition to 119888 there must existthe second fundamental constant which can be extractedfrom 120582

119899 In quantum mechanics the quantity with the

dimension of length which is proportional to a fundamentalphysical constant and is inversely proportional to velocity (ormomentum 119901

119899) is called the wavelength So it is natural to

assume that

120582119899= ℏ

119901119899

1199012sim

ℏ

119901119899

(23)

where ℏ is the Planck constant and 119901119899 is the 3-momentum

associated with the octonionic signal So in our approachthe two fundamental physical constants 119888 and ℏ havegeometrical origin and correspond to two kinds of light-conesignals in the space of split octonions (9)

When 1206011= 0 from (20) we find that the ratios

Δ1199092

Δ1205822

= minus1205823

1199093

Δ1199093

Δ1205823

= minus1205822

1199092

(24)

(where Δ119909119899= 1199091015840

119899minus 119909119899and Δ120582

119899= 1205821015840

119899minus 120582119899) are unchanged

under infinitesimal transformations (20) Similar relationscan be obtained for the boosts along twoother space-like axesFrom the invariance of octonionic norms (B14) we know that

10038161003816100381610038161199091198991003816100381610038161003816 ≳

10038161003816100381610038161205821198991003816100381610038161003816 (25)

Then inserting (23) and (25) into (24) we can conclude thatthe uncertainty relations

Δ119909119899Δ119901119899ge ℏ (26)

in our model have the same geometrical meaning as theexistence of the maximal velocity 119888 [50ndash53 59]

From (20) we also notice that in the planes orthogonal toV1= 1199091119905 the pair 1199091015840

2 12058210158403increases while 1199091015840

3 12058210158402decreases

and for the case (21) we have

1199092

1205823

= minus1199093

1205822

= 1205791 (27)

Using similar relations for the boosts along other two spatialdirections we conclude that there exists some 3-vector (spin)

120590119899=1

ℏ120576119899119898119896

119909119898119901119896sim 120576119899119898119896

119909119898

120582119896

(28)

which characterizes relative rotations of 119909119899and 120582

119899in the

direction orthogonal to the velocity planes One can speculateon the connection of the left-right asymmetry in relativerotations of 119909

119899and 120582

119899in (20) with the right-handed neutrino

problem for massless case


4 Parity Violation

For the reference frame (21) for the finite angles1206011= 1205791 from

(20) one can obtain the standard relativistic expressions

tanh 1205791=V

119888

cosh 1205791=

1

radic1 minusV21198882

sinh 1205791=

V119888


(29)

whereV is the velocity ofmoving system along1199091Then from

(20) there follows the generalized rule of velocity addition

V10158401=

V1minus 119888 tanh 120579

1

1 minus tanh 1205791(V1119888)

V10158402=

V2minus tanh 120579

13

1 minus tanh 1205791(V1119888)

V10158403=

V3+ tanh 120579

12

1 minus tanh 1205791(V1119888)

(30)

We see that the standard expressions are altered only in theplanes orthogonal to V and that what is important are theterms with different sign One consequence of this fact is thatthe formula for the aberration of light will be modified

Consider photons moving in the (1199091 1199092)-plane

V1= 119888 cos 120574

12

V2= 119888 sin 120574

12

V10158401= 119888 cos 1205741015840

12

V10158402= 119888 sin 1205741015840

12

(31)

where 12057412and 120574101584012are angles between V

1and V10158401and the axes

1199091and 1199091015840

1 respectively Using (30) for the caseV119888 ≪ 1 we

have

sin 120574101584012=

sin 12057412minus (V119888

2) 3

1 minus (V119888) cos 12057412

cos 120574101584012=

cos 12057412minusV119888

1 minus (V119888) cos 12057412

(32)

where 3is the rate of Dopplerrsquos shift along an axis 119909

3

orthogonal to the photonrsquos trajectory Then angle of aberra-tion in the (119909

1 1199092)-plane give the value

Δ12057412= 1205741015840

12minus 12057412=V

119888sin 12057412minusV

11988823 (33)

Analogous to (33) the formula for the aberration angle in the(1199091 1199093)-plane has the form

Δ12057413=V

119888sin 12057413+V

11988822 (34)

where 2is the rate of the Doppler shift towards the

orthogonal to the (1199091 1199093)-plane directionThis spatial asym-

metry of aberration which distinguishes the left and rightcoordinate systems may be detectable by precise quantummeasurements

5 Spin and Hypercharge

Now consider the last class of automorphisms (Appendix C)

1199091015840

119899= 119909119899+ (120593119899minus1

3sum

119898

120593119898)120582119899

1199051015840= 119905

1205821015840

119899= 120582119899+ (120593119899minus1

3sum

119898

120593119898)119909119899

(35)

with no summing by 119899 These transformations representrotations of the three pairs of space-like and time-likecoordinates (119909

1 1205821) (1199092 1205822) and (119909

3 1205823) into each other

We have the three planes (1199091minus 1205821) (1199092minus 1205822) and (119909

3minus 1205823)

that undergo rotations through hyperbolic angles 1205931 1205932 and

1205933(of the three only two are independent) which are the only

hyper-planes in the space of split octonions that are affectedby one andnot two 3-angles of automorphisms So it is naturalto define the Abelian subalgebra of 119866119873119862

2by generators of two

independent rotations in these planes It is known that therank of the Cartan subalgebra of 119866119873119862

2is the same as of the

group 119878119880(3) [9ndash11 60] Indeed in terms of two parameters1198701and119870

2 which are related to the angles 120593

119899and (C4) as

1198701= 1198961=1

3(21205931minus 1205932minus 1205933)

1198702=radic3

2(1198961+ 1198961) = minus

1

2radic3(21205933minus 1205931minus 1205932)

(36)

the transformations (35) can be written more concisely

(

1205821015840

1+ 1198681199091015840

1

1205821015840

2+ 1198681199091015840

2

1205821015840

2+ 1198681199091015840

2

) = 119890(1198701Λ3+1198702Λ8)119868(

1205821+ 1198681199091

1205822+ 1198681199092

1205822+ 1198681199092

) (37)

where 119868 is the vector-like octonionic basis unit (1198682 = 1) andΛ3and Λ

8are the standard 3 times 3 Gell-Mann matrices

Λ3= (

1 0 0

0 minus1 0

0 0 0

)

Λ8=

1

radic3(

1 0 0

0 1 0

0 0 minus2

)

(38)

Then using analogies with 119878119880(3) one can classify irre-ducible representations of the space-time group of ourmodel119866119873119862

2 by the two fundamental simple roots (119870

1and 119870

2)


corresponding to spin and hypercharge It is known thatall quarks antiquarks and mesons can be imbedded in theadjoint representation of 119866119873119862

2[9ndash11] So the symmetry (35)

in addition to the uncertainty relations probably shows theexistence of three generations of objects which are necessaryto extract three spatial coordinates from any octonionicsignal 119904 To clarify this point note that (9) can be viewedas some kind of space-time interval with four time-likedimensions The ordinary time parameter 119905 corresponds tothe distinguished octonionic basis unit 119868 while the otherthree time-like parameters 120582

119899 have a natural interpreta-

tion as wavelengths that is they do not relate to timeas conventionally understood It is known that a uniquephysical system in multidimensional geometry generates alarge variety of ldquoshadowsrdquo in (3 + 1)-subspace as differentdynamical systems (in terms of different Hamiltonians) [61ndash70] The information of multidimensional structures whichis retained by these images of the initial system takes the formof hidden symmetries For the case of fundamental physicalsignals with three extra time-like dimensions in addition tomassless particles (which are not affected by extra times)one can observe three generations of particles with differentmass (corresponding to rotations of 119905 in one two or all threeextra time-like planes) see (E7) Note the factor 13 in frontof 120596 (action) in the first equation (E7) that appears dueto the existence of the three extra time-like parameters 120582

119899

Then three independent (120596 120582119899)-rotations in 8-dimensional

octonionic spacewill give the appearance of three generationsof particles in ordinary (3 + 1)-dimensions Such a possibilityis not ruled out by problems with ghosts and with unitaritysince split octonions with positive norms form the divisionalgebra

6 Free Particle Lagrangians

It is known that in split algebras there can be constructedspecial elements with zero norms which are called zerodivisors [1]These zero-normobjects are important structuresin physical applications [71] Zero divisors (light-cone opera-tors) of split octonions could serve as the unit signals and thusmay describe elementary particlesThen the number of prim-itive idempotents (eight) and nilpotents (twelve) numeratesthe types of fundamental particles (Appendix E) bosons andfermions respectively Indeed the properties that the productof two projection operators reduces to the same idempotent(E2) while the product of two Grassmann numbers is zero(E4) naturally explain the validity of Bose andFermi statisticsfor the corresponding elementary particles

The zero-norm condition 1198732 = 0 for the paths (9)of elementary particles be they massless or massive can bewritten in the form

1198891199042= 119889119904119889119904

dagger= 1198891205962minus 11988821198891199052(1 minus

V2

1198882+2

1198882) = 0 (39)

For photons |V| = 119888 the condition (39) is equivalent tothe Eikonal equation

119889120596

119889120582119899

119889120596

119889120582119899= 1 (40)

where the wavelengths 120582119899sim ℏ119901

119899 serve as the parameters of

trajectoryThe relation (40) justifies the interpretation of120596 asthe phase function (frequency) corresponding to the classicalaction of particles

For massive particles the time coordinate 119905 is goodparameter to describe motion and from (39) we find that thescalar part of a split octonionic signal 120596 which is unchangedunder automorphisms of 119866119873119862

2 is the conserved function

119889120596

119889119905= 0 (41)

Then (39) can be written as

120596 =119860

119898119888= minus119888int119889119905radic1 minus

V2

1198882+2

1198882 (42)

where119860 is the classical action of the particle with themass119898So using (23) the one-particle Lagrangian

119871 = minus1198981198882radic1 minus

V2

1198882+ ℏ2

2

11988821199014 (43)

contains an extra ldquoquantumrdquo term whichmay be relevant forthe relativistic velocities |V| sim 119888 or for the case of large forces

119899sim1198881199012

ℏ (44)

From (43) we see that on small distance scales simℏ|119901| aparticle can even exceed the speed of light (become virtual)since the new quantum term in (43) is positiveThe condition(39) and the invariance of the classical action (42) under119866119873119862

2

hold only for real trajectories However there exists the largergroup of invariances of the interval (9) 119878119874(4 4) the groupof passive tensorial transformations of all eight octonioniccoordinates which in general mixes space-like and time-likesubspaces and thus introduces virtual trajectories of particles

Applying the condition (10) (that for physical signals thevector part of a split octonion should be time-like) to (43) itsfollows that there should exist some maximal force

10038161003816100381610038161003816100381610038161003816 le |V|

1199012

ℏ= 1198982 1198883

ℏ (45)

where 119898 denotes the maximal possible value for the mass ofparticle associated with physical signals Using estimations of[72 73] for the maximum force 119865max = 119888

44119866 where 119866 is

the gravitational constant from (45) we find that themaximalmass of particles associated with any octonionic signal is thePlanck mass

1198982simℏ119888

119866 (46)

7 Conclusion

To conclude in this paper it was analyzed consequencesof describing physical signals in terms of split octonionsover the field of real numbers Eight real parameters of


split octonions were related to space-time coordinates thephase function (classical action) and wavelengths charac-terizing physical signals The (4 + 4)-norms of the realsplit octonions are invariant under the group 119878119874(4 4) oftensorial transformations of parameters (passive transforma-tions of coordinates) while the representation of rotationsby octonions themselves (ie the active transformations ofoctonionic basis units) corresponds to the real noncompactform of Cartanrsquos exceptional Lie group 119866

119873119862

2(the subgroup

of 119878119874(4 4)) which under certain conditions imitate thePoincare transformations in ordinary space-timeWithin thispicture in front of time-like coordinates in the expressionof pseudo-Euclidean octonionic intervals there naturallyappear two fundamental physical parameters the light speedand Planckrsquos constant From the requirement of positivedefiniteness of norms under 119866119873119862

2-transformations together

with the introduction of the maximal velocity 119888 there followconditions which are equivalent to uncertainty relations inquantum physics By examining of 119866119873119862

2-automorphisms it

is explicitly shown that there exists a 3-vector (spin) that isfully chiral which corresponds to the observed chirality ofneutrinos in nature We examined also the 119866119873119862

2-effects on

a quantum system moving at constant speed with respectto the observer which emits light in various planes relativeto the direction of motion and new parity-violating effecton aberration and Doppler shift was derived It was shownhow a particular class of 119866119873119862

2-automorphisms of space-time

coordinates can be expressed in concise form using twosimple roots of the 119878119880(3) Lie algebra Due to their conven-tional association with spin and hypercharge in physics weargued that our split-octonion geometry inherently modeledthree generations of fundamental particles In our approachelementary particles are interpreted as light-cone operatorsthat is they are connected with the special elements of thealgebra which nullify octonionic intervals Then the zero-normconditions in119866119873119862

2-geometry lead to corresponding free

particle Lagrangians which allow virtual particle trajectories(exceeding the speed of light in certain cases) and exhibit theexistence of spatial horizons governing by mass parameters

Appendices

A Classification of Split Octonions

The algebra of split octonions is a noncommutative nonasso-ciative nondivision ring Any element (1) of the ring can berepresented as

119904 = 120596 + 119881

119904dagger= 120596 minus 119881

(A1)

where the symbols 120596 and

119881 = 120582119899119869119899+ 119909119899119895119899+ 119888119905119868 (119899 = 1 2 3) (A2)

are called the scalar and vector parts of octonion respectivelySimilar to the case of split quaternions [59 74] there exist

three classes of split octonions having positive negative orzero norm

1198732= 119904119904dagger= 119904dagger119904 = 1205962minus 1198812 (A3)

where

1198812= minus1199092+ 11988821199052+ 1205822 (A4)

In addition one can distinguish split octonionswith time-likespace-like and light-like vector parts (A2)

1198812lt 0 (space minus like)

1198812gt 0 (time minus like)

1198812= 0 (light minus like)

(A5)

Split octonions with nonzero norms 1205962 = 1198812 have

multiplicative inverses

119904minus1=

119904dagger

1198732 (A6)

with the property

119904119904minus1= 119904minus1119904 = 1 (A7)

while octonions with zero norm 1205962 = 1198812 have no inverses

One can construct also the polar form of a split octonionwith nonzero norm

119904 = 119873119890120598120579 (A8)

where 120598 is a unit 7-vector (1205982 = plusmn1) For119873 = 0 the quantity

119877 =119904

119873(A9)

is called unit octonion which is useful to represent rotationsof (3 + 4)-vector spaces (A2)

Depending on the values of (A3) and (A4) split octo-nions have three different representations

(i) Every split octonion with negative norm 1205962 lt 1198812 can

be written in the form

119904 = 119873 (sinh 120579 + 120598 cosh 120579) (A10)

where

sinh 120579 = 120596

119873

cosh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A11)

and 120598 is a unit time-like 7-vector(ii) Every split octonionwith positive norm1205962 gt 119881

2 andwith time-like vector part1198812 gt 0 can be written in the form

119904 = 119873 (cosh 120579 + 120598 sinh 120579) (A12)


where

cosh 120579 = 120596

119873

sinh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A13)

so 120598 again is a unit time-like 7-vector(iii) Every split octonion with positive norm 1205962 gt 119881

2and with space-like vector part 1198812 lt 0 can be written in theform

119904 = 119873 (cos 120579 + 120598 sin 120579) (A14)

where

cos 120579 = 120596

119873

sin 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A15)

thus 120598 now is a unit space-like 7-vectorThe norm of a split octonion (A3) contains two types of

ldquolight-conesrdquo 119873 = 0 and 119881 = 0 and we need two constantsthat characterize the maximum spread angles of these conesThe first fundamental parameter the speed of light 119888 appearsin standard Lorentz boosts for the time-like signals 1198812 gt 0The second fundamental constant ℏ corresponds tomaximalrotations in (120596 119881)-planes when physical events have1198732 ge 0

and still can be described by the actions sim120596

B One-Side Products

The 8-dimensional octonionic space (1) always can be rep-resented as the sum of four elements by rotations in fourorthogonal planes (see (B5) (B10) and (B14) below) one ofwhich always includes the axis of real numbers120596 [75]Then itcan be shown that multiplication of one octonion by anotheroctonion of unit norm (A9) represents some kind of rotationin these four planes In the algebra of split octonions thereexist seven basis units 119869

119899 119895119899 and 119868 which define the 7-vector

120598 in the formulas for one-sided (eg the left) products

119877s = 1198901205981205792

119904 (B1)

So totally we have 4times7 = 28 parameters of the group 119878119874(4 4)which preserve the norm (9) [75] Octonionic hypercomplexbasis units have different algebraic properties and lead to thethree distinct classes of the rotation operator 119877

(i)The three pseudovector-like basis elements 119895119899 behave

as the ordinary complex unit (1198952119899= minus1) and rotations around

the corresponding spatial axes 119909119899 can be done by the unit

octonions with1198732 gt 0 having space-like vector part

119890119895119899120572119899 = cos120572

119899+ 119895119899sin120572119899 (B2)

where120572119899are three real compact angles To show this explicitly

let us consider the rotation operator 119877 in (B1) when 120598 =

1198951 In the octonionic algebra (5) the basis unit 119895

1has three

different representations by other basis elements that form anassociative triplets with 119895

1

1198951= 11986921198693= 11989521198953= 1198691119868 (B3)

or equivalently

11989511198693= 1198692

11989511198952= 1198953

1198951119868 = 1198691

(B4)

Then it is possible to ldquorotate outrdquo four octonionic axes andrewrite (1) in the equivalent form

119904 = radic1205962 + 1199092

11198901198951120572120596 + radic120582

2

2+ 1205822

311989011989511205721205821198693

+ radic1199092

2+ 1199092

311989011989511205721199091198952+ radic1199052 + 120582

2

11198901198951120572119905119868

(B5)

where four angles are given by

cos120572120596=

120596

radic1205962 + 1199092

1

cos120572120582=

1205823

radic1205822

2+ 1205822

3

cos120572119909=

1199092

radic1199092

2+ 1199092

3

cos120572119905=

119888119905

radic1199052 + 1205822

1

(B6)

Now it is obvious that when 119877 = 11989011989511205721 and 119904 is represented

by (B5) the product (B1) leads to simultaneous rotations inthe four planes (120596 119909

1) (1205822 1205823) (1199092 1199093) and (120582

1 119905) by the

same compact angle 1205721 Similar results can be obtained for

the rotations by the other two pseudovector-like units 1198952and

1198953(ii) For the three vector-like basis elements 119869

119899 with the

positive squares 1198692119899= 1 we need the unit octonions with

1198732gt 0 having time-like vector part

119890119869119899120579119899 = cosh 120579

119899+ 119869119899sinh 120579

119899 (B7)

where 120579119899are three real hyperbolic angles Let us consider the

example when 120598 = 1198691in (B1) The equivalent representations

of 1198691in the algebra (5) are

1198691= 11986921198953= minus11986931198952= 1198951119868 (B8)

or11986911198692= minus1198953

11986911198952= 1198693

1198691119868 = 1198951

(B9)


Then (1) can be rewritten as

119904 = radic1205962 minus 1205822

11198901198691120579120596 + radic119909

2

3minus 1205822

2119890minus11986911205791205821198692

+ radic1199092

2minus 1205822

311989011986911205791199091198952+ radic1199092

1minus 11990521198901198691120579119905119868

(B10)

where the angles are given by

cosh 120579120596=

120596

radic1205962 minus 1199092

1

cosh 120579120582=

1205822

radic1199092

3minus 1205822

2

cosh 120579119909=

1199092

radic1199092

2minus 1205822

3

cosh120572119905=

119905

radic1199092

1minus 1199052

(B11)

So when 119877 = 11989011986911205791 the product (B1) corresponds to the

simultaneous rotations in the planes (120596 1205821) (1205822 1199093) (1199092 1205823)

and (1199091 119905) by the hyperbolic angle 120579

1 We have similar results

for the rotations by the other two vector-like units 1198692and 1198693

(iii) The last vector-like basis element 119868 also has thepositive square 1198682 = 1 and to represent corresponding trans-formations we need the unit octonion with 119873

2gt 0 having

time-like vector part

119890119868120590= cosh120590 + 119868 sinh120590 (B12)

where 120590 is the real hyperbolic angle When in (B1) we put120598 = 119868 we can use the expressions (4) of the equivalentrepresentations of 119868 or

1198681198951= minus1198691

1198681198952= minus1198692

1198681198953= minus1198693

(B13)

Then the split octonion (1) can be rewritten as

119904 = radic1205962 minus 1199052119890119868120590120596 + radic119909

2

1minus 1205822

1119890minus11986812059011198951

+ radic1199092

2minus 1205822

2119890minus11986812059021198952+ radic1199092

3minus 1205822

3119890minus11986812059031198953

(B14)

where four hyperbolic angles are given by

cosh120590120596=

120596

radic1205962 minus 1199052

cosh1205901=

1199091

radic1199092

1minus 1205822

1

cosh1205902=

1199092

radic1199092

2minus 1205822

2

cosh1205903=

1199093

radic1199092

3minus 1205822

3

(B15)

So the left product (B1) when 119877 = 119890119868120590 corresponds to the



So the one-side products (B1) of an octonion 119904 byeach of the seven operators (B2) (B7) and (B12) yieldsimultaneous rotations in four mutually orthogonal planes ofthe octonionic (4 + 4)-space by the same angles [75] One ofthese four planes is formed by the unit element 1 and thehypercomplex basis unit that defines the axis of rotationTheremaining orthogonal planes are given by three pairs of otherbasis elementswhich formassociative triplets with the chosenbasis unit

The decomposition of a split octonion in the form (B5) isvalid only if its norm (9) is positive In contrast with uniformrotations around 119895

119899 we have limited rotations (B10) and

(B14) around 119869119899and 119868 For these cases we should require also

positiveness of the 2 norms of each of the four planes of rota-tions For example for (B14) the expressions of 2 norms offour orthogonal planes are radic1205962 minus 1199052 radic1199092

1minus 1205822

1 radic1199092

2minus 1205822

2

and radic1199092

3minus 1205822

3 These hyperbolic properties are the result of

the existence of zero divisors in split algebras (Appendix E)

C Automorphisms and 1198662

From the expressions of a hypercomplex unit of split octo-nions by two other basis elements which form associativetriplets with the selected unit one can find orthogonal to itplanes of rotations An automorphism (13) for each basis unitintroduces two angles of rotation in these planes So there areseven independent automorphisms each by two angles thatcorrespond to 2 times 7 = 14 generators of the algebra 119866119873119862

2 For

example the automorphism by the two real compact angles1205721and 120573

1 which leaves unchanged the basis unit 119895

1= 11986921198693=

11989521198953= 1198691119868 has the form

1198951015840

1= 1198951

1198951015840

2= 1198952cos1205721+ 1198953sin1205721

1198951015840

3= 1198951015840

11198951015840

2= 1198953cos1205721minus 1198952sin1205721

1198691015840

1= 1198691cos1205731+ 119868 sin120573

1

1198681015840= 1198691015840

11198951015840

1= 119868 cos120573

1minus 1198691sin1205731

1198691015840

2= 1198951015840

21198681015840= 1198692cos (120572

1minus 1205731) + 1198693sin (120572

1minus 1205731)

1198691015840

3= 1198951015840

31198681015840= 1198693cos (120572

1minus 1205731) minus 1198692sin (120572

1minus 1205731)

(C1)

It is known that automorphisms in the octonionic algebraare completely defined by the images of three basis elementsthat do not form associative subalgebras (119895

1 1198952 and 119869

1in

the case of (C1)) By the definition (13) an automorphismdoes not affect the scalar part of 119904 while the images of theother hypercomplex elements (119895

3 119868 1198692 and 119869

3in (C1)) are

determined using the algebra (5) Then it is obvious thatthe transformed basis 1198691015840

119899 1198951015840

119899 1198681015840 satisfy the samemultiplication

rules as 119869119899 119895119899 119868


Similar to (C1) there exist two automorphisms with fixed1198952= 11986931198691= 11989531198951= 1198692119868 and 119895

3= 11986911198692= 11989511198952= 1198693119868 axes each

generating the two compact angles 1205722 1205732and 120572

3 1205733 Three

different representations of a basis unit indicate the planes ofcorresponding automorphisms and the positive directions ofrotations

One can define also hyperbolic rotations (automor-phisms) around the three vector-like units 119869

119899by the angles

120579119899and 120601

119899 For example similar to the transformation (C1)

automorphism around the axis 1198691= 11989531198692= 11986931198952= 1198951119868 is

1198691015840

1= 1198691

1198691015840

2=1198692cosh (120601

1+ 1205791)

2+1198953sinh (120601

1+ 1205791)

2

1198951015840

3= 1198691015840

11198691015840

2=1198953cosh (120579

1+ 1206011)

2+1198692sinh (120579

1+ 1206011)

2

1198681015840= 119868 cosh 120579

1minus 1198951sinh 120579

1

1198951015840

1= 1198691015840

11198681015840= 1198951cosh 120579

1minus 119868 sinh 120579

1

1198951015840

2= 1198691015840

21198681015840=1198952cosh (120601

1minus 1205791)

2+1198693sinh (120601

1minus 1205791)

2

1198691015840

3= 1198951015840

31198681015840=1198693cosh (120601

1minus 1205791)

2+1198952sinh (120601

1minus 1205791)

2

(C2)

where we chose the representation with half-angle rotationsin order to write the transformations of 120582

119899and 119909

119899in

symmetric form The automorphisms with the fixed 1198692=

11989511198693= 11986911198953= 1198952119868 and 119869

3= 11989521198691= 11986921198951= 1198953119868 axes generate

hyperbolic angles 1205792 1206012and 1205793 1206013 respectively

Finally for the rotations around the time direction 119868 =

11986911198951= 11986921198952= 11986931198953 we have only two hyperbolic angles 119896

1

and 1198962

1198681015840= 119868

1198951015840

1= 1198951cosh 119896

1+ 1198691sinh 119896

1

1198951015840

2= 1198952cosh 119896

2+ 1198692sinh 119896

2

1198691015840

1= 1198951015840

11198681015840= 1198691cosh 119896

1+ 1198951sinh 119896

1

1198691015840

2= 1198951015840

21198681015840= 1198692cosh 119896

2+ 1198952sinh 119896

2

1198951015840

3= 1198951015840

11198951015840

2= 1198953cosh (119896

1+ 1198962) minus 1198693sinh (119896

1+ 1198962)

1198691015840

3= 1198951015840

31198681015840= 1198693cosh (119896

1+ 1198962) minus 1198953sinh (119896

1+ 1198962)

(C3)

In order to present the transformations of 120582119899and 119909

119899in (14)

in the symmetric form it is convenient to introduce insteadof 1198961and 1198962the three hyperbolic angles 120593

119899

1198961= 1205931minus1

3sum

119899

120593119899

1198962= 1205932minus1

3sum

119899

120593119899

minus (1198961+ 1198962) = 1205933minus1

3sum

119899

120593119899

(C4)

D Generators of 1198662

The group 1198661198731198622

first was studied by Cartan in his thesis [7677] He considered (3 + 4)-space with the seven coordinates(119910119899 119905 119911119899) and linear operators of their transformations

119883119899119899= minus119911119899

120597

120597119911119899

+ 119910119899

120597

120597119910119899

+1

3

3

sum

119898=1

(119911119898

120597

120597119911119898

minus 119910119898

120597

120597119910119898

)

(no summing over 119899)

1198831198990= minus2119905

120597

120597119911119899

+ 119910119899

120597

120597119905

+1

2120576119899119898119896

(119911119898 120597

120597119910119896

minus 119911119896 120597

120597119910119898

)

1198830119899= minus2119905

120597

120597119910119899

+ 119911119899

120597

120597119905

+1

2120576119899119898119896

(119910119898 120597

120597119911119896

minus 119910119896 120597

120597119911119898

)

119883119899119898

= minus119911119898

120597

120597119911119899

+ 119910119899

120597

120597119910119898

(119899 = 119898)

(D1)

which preserves the quadratic form 1198891199052+ 119911119899119910119899 Totally there

are 15 generators in (D1) since 119899119898 = 1 2 3 but the operators119883119899119899are linearly dependent

11988311+ 11988322+ 11988333= 0 (D2)

which explains why 1198661198731198622

is 14- and not 15-dimensionalBy transition to our coordinates 119909

119899and 120582

119899

119910119899= 120582119899+ 119909119899

120597

120597119910119899

=1

2(

120597

120597120582119899

+120597

120597119909119899

)

119911119899= 120582119899minus 119909119899

120597

120597119911119899

=1

2(

120597

120597120582119899

minus120597

120597119909119899

)

(D3)

from (D1) we can find the operators of the transformationof the vector part 119881 of (1) which preserves the diagonalquadratic form 1198812 = 120582

119899120582119899+ 1198891199052minus 119909119899119909119899

119883119899119899

= 119909119899

120597

120597119909119899

+ 120582119899

120597

120597120582119899

minus1

3

3

sum

119898=1

(119909119898

120597

120597119909119898

+ 120582119898

120597

120597120582119898

)


1198831198990

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

)

+ (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)+14120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

+1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

1198830119899

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

) minus (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)

+1

4120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

minus1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

+1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

119883119899119898

= minus1

2(120582119899

120597

120597120582119898

minus 120582119898

120597

120597120582119899

)

minus1

2(119909119899

120597

120597119909119898

minus 119909119898

120597

120597119909119899

)

+1

2(119909119899

120597

120597120582119898

+ 120582119898

120597

120597119909119899

)

+1

2(120582119899

120597

120597119909119898

+ 119909119898

120597

120597120582119899

) (119899 = 119898)

(D4)

E Zero Divisors

In the algebra of split octonions two types of zero divisorsidempotent elements (projection operators) and nilpotentelements (Grassmann numbers) can be constructed [1 71]

There exist four noncommuting (totally eight) primitiveidempotents

119863plusmn(119869)

119899=1

2(1 plusmn 119869

119899)

119863plusmn(119868)

=1

2(1 plusmn 119868)

(119899 = 1 2 3)

(E1)

which obey the relations

119863plusmn(119869)

119899119863plusmn(119869)

119899= 119863plusmn(119869)

119899

119863+(119869)

119899119863minus(119869)

119899= 0

119863plusmn(119868)

119863plusmn(119868)

= 119863plusmn(119868)

119863+(119868)

119863minus(119868)

= 0

(E2)

We have also the four noncommuting classes (totallytwelve) of primitive nilpotents

119866plusmn(119869)

119899=1

2(119869119899plusmn 119895119899)

119866plusmn(119868)

119899=1

2(119868 plusmn 119895

119899)

(119899 = 1 2 3)

(E3)

with the properties

119866plusmn(119869)

119899119866plusmn(119869)

119899= 0

119866plusmn(119869)

119899119866∓(119869)

119899= 119863∓(119868)

119866plusmn(119868)

119899119866plusmn(119868)

119899= 0

119866plusmn(119868)

119899119866∓(119868)

119899= 119863plusmn(119869)

119899

(E4)

From these relations we see that separately nilpotents areGrassmann numbers but different 119866

119899-s do not commute

with each other in contrast to the projection operators (E2)Instead the quantities119866plusmn

119899are elements of the so-called algebra

of Fermi operators with the anticommutators

119866plusmn(119869)

119866∓(119869)

+ 119866∓(119869)

119866plusmn(119869)

= 1

119866plusmn(119868)

119866∓(119868)

+ 119866∓(119868)

119866plusmn(119868)

= 1

(E5)

The algebra of Fermi operators is some syntheses of theGrassmann and Clifford algebras

For the completeness we note that the idempotents andnilpotents obey the following algebra

119863plusmn(119869)

119899119866plusmn(119868)

119899= 119866plusmn(119868)

119899

119863plusmn(119869)

119899119866∓(119868)

119899= 0

119863plusmn(119868)

119866plusmn(119869)

119899= 0

119863plusmn(119868)

119866∓(119869)

119899= 119866∓(119869)

119899

(E6)

Using commuting zero divisors any octonion (1) can bewritten in the two equivalent forms

119904 = 119863+(119869)

119899(1

3120596 + 120582119899) + 119866

+(119868)

119899(119888119905 + 119909

119899)

+ 119863minus(119869)

119899(1

3120596 minus 120582119899) + 119866

minus(119868)

119899(119888119905 minus 119909

119899)


119904 = 119863+(119868)

(120596 + 119888119905) + 119866+(119869)

119899(120582119899+ 119909119899) + 119863minus(119868)

(120596 minus 119888119905)

+ 119866minus(119869)

119899(120582119899minus 119909119899)

(E7)

The norm of a split octonion (A4) contains two types ofldquolight-conesrdquo and we can introduce two types of critical sig-nals (elementary particles) visualized in the decompositions(E7)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported by the Shota Rustaveli NationalScience Foundation Grant ST09-798-4-100 Otari Sakhe-lashvili acknowledges also the scholarship of World Federa-tion of Scientists

References

[1] R D Schafer Introduction to Non-Associative Algebras DoverNew York NY USA 1995

[2] T A Springer and F D Veldkamp Octonions Jordan Algebrasand Exceptional Groups SpringerMonographs inMathematicsSpringer Berlin Germany 2000

[3] J C Baez ldquoTheOctonionsrdquo Bulletin of the AmericanMathemat-ical Society vol 39 pp 145ndash205 2002

[4] S Okubo Introduction to Octonion and Other Non-AssociativeAlgebras in Physics Cambridge University Press CambridgeUK 1995

[5] D R Finkelstein Quantum Relativity A Synthesis of the Ideasof Einstein and Heisenberg Springer-Verlag Berlin Germany1996

[6] F Gursey and C-H Tze On the Role of Division Jordan andRelated Algebras in Particle Physics World Scientific Singapore1996

[7] J Lohmus E Paal and L Sorgsepp Nonassociative Algebras inPhysics Hadronic Press Palm Harbor Fla USA 1994

[8] J Lohmus E Paal and L Sorgsepp ldquoAbout nonassociativity inmathematics and physicsrdquo Acta Applicandae Mathematica vol50 no 1-2 pp 3ndash31 1998

[9] M Gunaydin and F Gursey ldquoQuark structure and octonionsrdquoJournal of Mathematical Physics vol 14 no 11 article 1651 1973

[10] M Gunaydin and F Gursey ldquoAn octonionic representation ofthe Poincare grouprdquo Lettere al Nuovo Cimento vol 6 no 11 pp401ndash406 1973

[11] M Gunaydin and F Gursey ldquoQuark statistics and octonionsrdquoPhysical Review D vol 9 no 12 pp 3387ndash3391 1974

[12] S L Adler ldquoQuaternionic chromodynamics as a theory ofcomposite quarks and leptonsrdquo Physical Review D vol 21 no10 pp 2903ndash2915 1980

[13] KMorita ldquoOctonions quarks andQCDrdquoProgress ofTheoreticalPhysics vol 65 no 2 pp 787ndash790 1981

[14] T Kugo and P Townsend ldquoSupersymmetry and the divisionalgebrasrdquo Nuclear Physics B vol 221 no 2 pp 357ndash380 1983

[15] A Sudbery ldquoDivision algebras (pseudo)orthogonal groups andspinorsrdquo Journal of Physics AMathematical andGeneral vol 17no 5 pp 939ndash955 1984

[16] G Dixon ldquoDerivation of the standardmodelrdquo Il Nuovo CimentoB vol 105 no 3 pp 349ndash364 1990

[17] P Howe and PWest ldquoThe conformal group point particles andtwistorsrdquo International Journal of Modern Physics A vol 07 no26 pp 6639ndash6664 1992

[18] I R Porteous Clifford Algebras and the Classical GroupsCambridge University Press Cambridge UK 1995

[19] P Lounesto Clifford Algebras and Spinors Cambridge Univer-sity Press Cambridge UK 2001

[20] H L Carrion M Rojas and F Toppan Journal of High EnergyPhysics vol 2003 4 article 040 edition 2003

[21] D Finkelstein J M Jauch S Schiminovich and D SpeiserldquoFoundations of quaternion quantum mechanicsrdquo Journal ofMathematical Physics vol 3 pp 207ndash220 1962

[22] D Finkelstein J M Jauch S Schiminovich and D SpeiserldquoPrinciple of general Q covariancerdquo Journal of MathematicalPhysics vol 4 pp 788ndash796 1963

[23] G G Emch ldquoMechanique quantique quaternionienne et rel-ativitrsquoerestreinterdquo Helvetica Physica Acta vol 36 pp 739ndash7881963

[24] L P Horwitz and L C Biedenharn ldquoIntrinsic superselectionrules of algebraic Hilbert spacesrdquoHelvetica Physica Acta vol 38no 4 p 385 1965

[25] L P Horwitz and L C Biedenharn ldquoQuaternion quantummechanics second quantization and gauge fieldsrdquo Annals ofPhysics vol 157 no 2 pp 432ndash488 1984

[26] M Gunaydin C Piron and H Ruegg ldquoMoufang plane andoctonionic quantum mechanicsrdquo Communications in Mathe-matical Physics vol 61 no 1 pp 69ndash85 1978

[27] S De Leo and P Rotelli ldquoTranslations between quaternion andcomplex quantum mechanicsrdquo Progress of Theoretical Physicsvol 92 no 5 pp 917ndash926 1994

[28] V Dzhunushaliev ldquoNonperturbative operator quantization ofstrongly nonlinear fieldsrdquo Foundations of Physics Letters vol 16no 1 pp 57ndash70 2003

[29] V Dzhunushaliev ldquoA non-associative quantum mechanicsrdquoFoundations of Physics Letters vol 19 no 2 pp 157ndash167 2006

[30] F Gursey Symmetries in Physics (1600ndash1980) Proc of the 1stInternational Meeting on the History of Scientific Ideas SeminaridrsquoHistoria de les Ciencies Barcelona Spain 1987

[31] S De Leo ldquoQuaternions and special relativityrdquo Journal ofMathematical Physics vol 37 no 6 pp 2955ndash2968 1996

[32] K Morita ldquoQuaternionic weinberg-salam theoryrdquo Progress ofTheoretical Physics vol 67 no 6 pp 1860ndash1876 1982

[33] C Nash and G C Joshi ldquoSpontaneous symmetry breakingand the Higgs mechanism for quaternion fieldsrdquo Journal ofMathematical Physics vol 28 no 2 pp 463ndash467 1987

[34] C G Nash and G C Joshi ldquoPhase transformations in quater-nionic quantum field theoryrdquo International Journal of Theoreti-cal Physics vol 27 no 4 pp 409ndash416 1988

[35] S L Adler Quaternionic Quantum Mechanics and QuantumField Oxford University Press New York NY USA 1995


[36] B A Bernevig J-P Hu N Toumbas and S-C Zhang ldquoEight-dimensional quantum hall effect and lsquooctonionsrsquordquo PhysicalReview Letters vol 91 no 23 Article ID 236803 2003

[37] F D Smith Jr ldquoParticle masses force constants and Spin(8)rdquoInternational Journal of Theoretical Physics vol 24 no 2 pp155ndash174 1985

[38] F D Smith Jr ldquoSpin(8) gauge field theoryrdquo International Journalof Theoretical Physics vol 25 no 4 pp 355ndash403 1986

[39] M Pavsic ldquoKaluzandashKlein theory without extra dimensionscurved Clifford spacerdquo Physics Letters B vol 614 no 1-2 pp 85ndash95 2005

[40] M Pavsic ldquoSpin gauge theory of gravity in clifford space a real-ization of kaluza-klein theory in four-dimensional spacetimerdquoInternational Journal of Modern Physics A vol 21 pp 5905ndash5956 2006

[41] M Pavsic ldquoA novel view on the physical origin of E8rdquo Journal

of Physics A Mathematical and Theoretical vol 41 Article ID332001 2008

[42] C Castro ldquoOn the noncommutative and nonassociative geom-etry of octonionic space timemodified dispersion relations andgrand unificationrdquo Journal of Mathematical Physics vol 48 no7 Article ID 073517 2007

[43] D B Fairlie and C A Manogue ldquoLorentz invariance and thecomposite stringrdquo Physical Review D Particles and Fields vol34 no 6 pp 1832ndash1834 1986

[44] K W Chung and A Sudbery ldquoOctonions and the Lorentzand conformal groups of ten-dimensional space-timerdquo PhysicsLetters B vol 198 no 2 pp 161ndash164 1987

[45] J Lukierski and F Toppan ldquoGeneralized spacendashtime supersym-metries division algebras and octonionic M-theoryrdquo PhysicsLetters B vol 539 no 3-4 pp 266ndash276 2002

[46] J Lukierski and F Toppan ldquoOctonionic M-theory and 119863 = 11

generalized conformal and superconformal algebrasrdquo PhysicsLetters B vol 567 no 1-2 pp 125ndash132 2003

[47] L Boya ldquoOctonions andM-theoryrdquo inGROUP 24 Physical andMathematical Aspects of Symmetries CRC Press Paris France2002

[48] A Anastasiou L Borsten M J Duff L J Hughes and S NagyldquoAn octonionic formulation of theM-theory algebrardquo Journal ofHigh Energy Physics vol 2014 no 11 article 22 2014

[49] V Dzhunushaliev ldquoCosmological constant supersymmetrynonassociativity and big numbersrdquo The European PhysicalJournal C vol 75 no 2 article 86 2015

[50] M Gogberashvili ldquoObservable algebrardquo httparxivorgabshep-th0212251

[51] M Gogberashvili ldquoOctonionic geometryrdquo Advances in AppliedClifford Algebras vol 15 no 1 pp 55ndash66 2005

[52] M Gogberashvili ldquoOctonionic electrodynamicsrdquo Journal ofPhysics A vol 39 no 22 pp 7099ndash7104 2006

[53] M Gogberashvili ldquoOctonionic version of Dirac equationsrdquoInternational Journal of Modern Physics A vol 21 no 17 pp3513ndash3524 2006

[54] K Sagerschnig ldquoSplit octonions and generic rank two dis-tributions in dimension fiverdquo Archivum Mathematicum vol42 supplement pp 329ndash339 2006 httpwwwemisamsorgjournalsAM06-Ssagerpdf

[55] A A Agrachev ldquoRolling balls and octonionsrdquo Proceedings of theSteklov Institute of Mathematics vol 258 no 1 pp 13ndash22 2007

[56] I Agricola ldquoOld and newon the exceptional group 1198662rdquo Notices

of the American Mathematical Society vol 55 no 8 pp 922ndash929 2008

[57] R Bryant httpwwwmathdukeedusimbryantCartanpdf[58] J C Baez and J Huerta ldquoG

2and the rolling ballrdquo Transactions

of the American Mathematical Society vol 366 pp 5257ndash52932014

[59] M Gogberashvili ldquoSplit quaternions and particles in (2+1)-spacerdquo The European Physical Journal C vol 74 article 32002014

[60] J Beckers V Hussin and P Winternitz ldquoNonlinear equationswith superposition formulas and the exceptional group G

2 I

Complex and real forms of g2and their maximal subalgebrasrdquo

Journal of Mathematical Physics vol 27 p 2217 1986[61] R Mignani and E Recami ldquoDuration length symmetry in

complex three-space and interpreting superluminal Lorentztransformationsrdquo Lettere al Nuovo Cimento vol 16 no 15 pp449ndash452 1976

[62] E A B Cole ldquoSuperluminal transformations using eithercomplex space-time or real space-time symmetryrdquo Il NuovoCimento A vol 40 no 2 pp 171ndash180 1977

[63] E A B Cole ldquoParticle decay in six-dimensional relativityrdquoJournal of Physics A vol 13 no 1 article 109 1980

[64] M Pavsic ldquoUnified kinematics of bradyons and tachyons insix-dimensional space-timerdquo Journal of Physics AMathematicaland General vol 14 no 12 pp 3217ndash3228 1981

[65] M Gogberashvili ldquoBrane-universe in six dimensions with twotimesrdquo Physics Letters B vol 484 no 1-2 pp 124ndash128 2000

[66] M Gogberashvili and P Midodashvili ldquoBrane-universe in sixdimensionsrdquo Physics Letters B vol 515 no 3-4 pp 447ndash4502001

[67] M Gogberashvili and PMidodashvili ldquoLocalization of fields ona brane in six dimensionsrdquo Europhysics Letters vol 61 no 3 pp308ndash313 2003

[68] J Christian ldquoPassage of time in a planck scale rooted localinertial structurerdquo International Journal of Modern Physics Dvol 13 no 6 p 1037 2004

[69] J Christian ldquoAbsolute being versus relative becomingrdquo inRelativity and the Dimensionality of the World V PetkovEd vol 153 of Fundamental Theories of Physics pp 163ndash195Springer Berlin Germany 2007

[70] M V Velev ldquoRelativistic mechanics in multiple time dimen-sionsrdquo Physics Essays vol 25 no 3 pp 403ndash438 2012

[71] A Sommerfeld Atombau und Spektrallinien II Band ViewegBraunschweig Germany 1953

[72] G W Gibbons ldquoThe maximum tension principle in generalrelativityrdquo Foundations of Physics vol 32 no 12 pp 1891ndash19012002

[73] C Schiller ldquoGeneral relativity and cosmology derived fromprinciple of maximum power or forcerdquo International Journal ofTheoretical Physics vol 44 no 9 Article ID 0607090 pp 1629ndash1647 2005

[74] M Ozdemir and A A Ergin ldquoRotations with unit timelikequaternions in Minkowski 3-spacerdquo Journal of Geometry andPhysics vol 56 no 2 pp 322ndash336 2006

[75] C A Manogue and J Schray ldquoFinite Lorentz transformationsautomorphisms and division algebrasrdquo Journal ofMathematicalPhysics vol 34 no 8 pp 3746ndash3767 1993


[76] E Cartan Sur la structure des groupes de transformations finis etcontinus [These] Nony Paris France 1894

[77] E Cartan ldquoLes groupes reels simples finis et continusrdquo AnnalesScientifiques de lrsquoEcole Normale Superieure vol 31 pp 263ndash3551914 Reprinted in Oeuvres completes Gauthier-Yillars ParisFrance 1952

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


A pair of repeated upper and lower indices by the standardconvention implies a summation that is 119909119899119895

119899= 120575119899119898119909119899119895119898

where 120575119899119898 is Kroneckerrsquos deltaIn geometric application four of the eight real parameters

in (1) 119905 and 119909119899 are space-time coordinates 120596 and 120582

119899 areinterpreted as the phase (classical action) and thewavelengthsassociated with the octonionic signals [50ndash53]

The eight basis units in (1) are represented by onescalar (denoted by 1) the three vector-like objects 119869

119899 the

three pseudovector-like elements 119895119899 and one pseudoscalar-

like unit 119868 The squares (inner products) of seven of thehypercomplex basis elements of split octonions give the unitelement 1 with the different signs

1198692

119899= 1

1198952

119899= minus1

1198682= 1

(2)

It is known that to generate a complete basis of split octonionsthe multiplication and distribution laws of only three vector-like elements 119869

119899are enough [1ndash3] The three pseudovector-

like basis units 119895119899 can be defined as the binary products

119895119899=1

2120576119899119898119896

119869119898119869119896

(119899119898 119896 = 1 2 3) (3)

(120576119899119898119896

is the totally antisymmetric unit tensor of rank three)and thus describes orthogonal planes spanned by two vector-like elements 119869

119899 The seventh basis unit 119868 (the oriented

volume) is defined as the triple product of all three vector-likeelements and has three equivalent representation in terms of119869119899 and 119895119899

119868 = 11986911198951= 11986921198952= 11986931198953 (4)

So the algebra of all noncommuting hypercomplex basis unitsof split octonions has the form

119869119899119869119898= minus119869119898119869119899= 120576119899119898119896

119895119896

119895119899119895119898= minus119895119898119895119899= 120576119899119898119896

119895119896

119895119898119869119899= minus119869119899119895119898= 120576119899119898119896

119869119896

119869119899119868 = minus119868119869

119899= 119895119899

119895119899119868 = minus119868119895

119899= 119869119899

(5)

Conjugations of octonionic basis units which can beunderstood as the reflection of vector-like elements

119869dagger

119899= minus119869119899 (6)

reverses the order of 119869119899in products that is

119895dagger

119899=1

2(120576119899119898119896

119869119898119869119896)dagger

=1

2120576119899119898119896

119869119896dagger119869119898dagger

= minus119895119899

119868dagger= (119869111986921198693)dagger

= 119869dagger

3119869dagger

2119869dagger

1= minus119868

(7)

So the conjugation of (1) gives

119904dagger= 120596 minus 120582

119899119869119899minus 119909119899119895119899minus 119888119905119868 (8)

Using (2) (5) and (8) one can find that the norm of (1) isgiven by

1198732= 119904119904dagger= 119904dagger119904 = 1205962minus 1205822+ 1199092minus 11988821199052 (9)

(where 1205822 = 120582119899120582119899 and 1199092 = 119909

119899119909119899) representing some kind of

8-dimensional space-timewith (4+4)-signature and reducingto the classical formula of Minkowski intervals if 1205962 minus1205822 = 0

As for the case of ordinary Minkowski space-time weassume that for physical events the corresponding ldquointervalsrdquogiven by (9) are nonnegative A second condition is that forphysical signals the vector part of split octonions (1) shouldbe time-like

11988821199052+ 120582119899120582119899gt 119909119899119909119899 (10)

The classification of split octonions by the values of theirnorms is presented in Appendix A

2 Transformations and Automorphisms

To find the geometry associated with the signals (1) weneed to represent rotations by split octonions due to thenonassociativity of the algebra the group of rotations ofoctonionic axes 1 119869

119899 119895119899 and 119868 119866119873119862

2is not equivalent to

the group of accompanied passive tensorial transformationsof coordinates 120596 120582

119899 119909119899 and 119905 (119878119874(4 4)) In Appendix B

it is shown that by rotations in four orthogonal planes ofoctonionic space (one of which always includes the axis ofreal numbers 120596) the octonion (1) always can be representedas the sum of four elements The left multiplication 119877119904 by anoctonion with unit norm (Appendix A)

119877 = 119890120598120579 (11)

where 120598 is the (3+4)-vector defined by seven basis units 119869119899 119895119899

and 119868 represents rotations by the angles 120579 in these four planesThe right product 119904119877minus1 will reverse the direction of rotationin the plane of 120596 So we can represent a rotation which onlyaffects the (3+4)-vector part of 119904 applying the unit half-angleoctonion twice and multiplying on both the left and the right(with its inverse)

1199041015840= 119877119904119877

minus1= 119877119904119877

dagger= 1198901205981205792

119904119890minus1205981205792

(12)

Thesemaps are well defined since the associator of the triplet(119877 119904 119877

minus1) vanishes The set of rotations (12) of the seven

octonionic coordinates119909119899120582119899 and 119905 in three planes which donot affect the scalar part 120596 of (1) contains 7times3 = 21 angles ofthe group 119878119874(3 4) of passive transformations of coordinates

To represent the active rotations in the space of 119904which preserve the norm (9) and multiplicative structure ofoctonions (5) as well we would need the transformationsto be automorphisms (Appendix C) An automorphism 119880 ofany two octonions 119874

1and 119874

2 gives the equation

(1198801198741119880minus1) (119880119874

2119880minus1) = 119880119874

11198742119880minus1 (13)











1199091015840

119899= 119909119899minus 120576119899119898119896

120572119898119909119896minus 120579119899119888119905

+1

2(1003816100381610038161003816120576119899119898119896

1003816100381610038161003816 120601119898+ 120576119899119898119896

120579119898) 120582119896

+ (120593119899minus1

3sum

119898

120593119898)120582119899

1198881199051015840= 119888119905 minus 120573

119899120582119899minus 120579119899119909119899

1205821015840

119899= 120582119899minus 120576119899119898119896

(120572119898minus 120573119898) 120582119896+ 120573119899119888119905

+1

2(1003816100381610038161003816120576119899119898119896

1003816100381610038161003816 120601119898minus 120576119899119898119896

120579119898) 119909119896

+ (120593119899minus1

3sum

119898

120593119898)119909119899

(14)


119899 From


sum

119899

(120593119899minus1

3sum

119898

120593119898) = 0 (15)


119899 and








120601119898= 120593119898= 0 (16)

the passive1198661198731198622




1199091015840

119899= 119909119899minus 120576119899119898119896

120572119898119909119896minus 120579119899119888119905 + 119886119899

1198881199051015840= 119888119905 minus 120579

119899119909119899+ 1198860

(17)


119886119899=1

2120576119899119898119896

120579119898120582119896

1198860= minus120573119899120582119899

(18)






119899







119899and 120582

119899


119899





2-





1199091015840

1= 1199091

1199091015840

2= 1199092+ 12057211199093

1199091015840

3= 1199093minus 12057211199092

1198881199051015840= 119888119905 minus 120573

11205821

1205821015840

1= 1205821+ 1205731119888119905

1205821015840

2= 1205822+ (1205721minus 1205731) 1205823

1205821015840

3= 1205823minus (1205721minus 1205731) 1205822

(19)

When 1205721










1

1199091015840

1= 1199091minus 1205791119888119905

1199091015840

2= 1199092+1

2(1206011minus 1205791) 1205823

1199091015840

3= 1199093+1

2(1206011+ 1205791) 1205822

1198881199051015840= 119888119905 minus 120579

11199091

1205821015840

1= 1205821

1205821015840

2= 1205822+1

2(1206011+ 1205791) 1199093

1205821015840

3= 1205823+1

2(1206011minus 1205791) 1199092

(20)






1199091015840= 1199101015840= 1199111015840= 0 (21)


1205791=1199091

119888119905

1205823=1199092

1205791

1205822= minus

1199093

1205791

(22)









assume that

120582119899= ℏ

119901119899

1199012sim

ℏ

119901119899

(23)




Δ1199092

Δ1205822

= minus1205823

1199093

Δ1199093

Δ1205823

= minus1205822

1199092

(24)

(where Δ119909119899= 1199091015840

119899minus 119909119899and Δ120582

119899= 1205821015840



10038161003816100381610038161199091198991003816100381610038161003816 ≳

10038161003816100381610038161205821198991003816100381610038161003816 (25)


Δ119909119899Δ119901119899ge ℏ (26)



2 12058210158403increases while 1199091015840

3 12058210158402decreases


1199092

1205823

= minus1199093

1205822

= 1205791 (27)


120590119899=1

ℏ120576119899119898119896

119909119898119901119896sim 120576119899119898119896

119909119898

120582119896

(28)


119899in the


119899and 120582




4 Parity Violation



tanh 1205791=V

119888

cosh 1205791=

1


sinh 1205791=

V119888


(29)



V10158401=

V1minus 119888 tanh 120579

1

1 minus tanh 1205791(V1119888)

V10158402=

V2minus tanh 120579

13

1 minus tanh 1205791(V1119888)

V10158403=

V3+ tanh 120579

12

1 minus tanh 1205791(V1119888)

(30)



V1= 119888 cos 120574

12

V2= 119888 sin 120574

12

V10158401= 119888 cos 1205741015840

12

V10158402= 119888 sin 1205741015840

12

(31)



1199091and 1199091015840


have

sin 120574101584012=

sin 12057412minus (V119888

2) 3

1 minus (V119888) cos 12057412

cos 120574101584012=

cos 12057412minusV119888

1 minus (V119888) cos 12057412

(32)


3



Δ12057412= 1205741015840

12minus 12057412=V

119888sin 12057412minusV

11988823 (33)


Δ12057413=V

119888sin 12057413+V

11988822 (34)






1199091015840

119899= 119909119899+ (120593119899minus1

3sum

119898

120593119898)120582119899

1199051015840= 119905

1205821015840

119899= 120582119899+ (120593119899minus1

3sum

119898

120593119898)119909119899

(35)


1 1205821) (1199092 1205822) and (119909



3minus 1205823)









119899and (C4) as

1198701= 1198961=1

3(21205931minus 1205932minus 1205933)

1198702=radic3

2(1198961+ 1198961) = minus

1


(36)


(

1205821015840

1+ 1198681199091015840

1

1205821015840

2+ 1198681199091015840

2

1205821015840

2+ 1198681199091015840

2

) = 119890(1198701Λ3+1198702Λ8)119868(

1205821+ 1198681199091

1205822+ 1198681199092

1205822+ 1198681199092

) (37)



Λ3= (

1 0 0

0 minus1 0

0 0 0

)

Λ8=

1

radic3(

1 0 0

0 1 0

0 0 minus2

)

(38)



1and 119870

2)







119899






1198891199042= 119889119904119889119904


V2

1198882+2

1198882) = 0 (39)


119889120596

119889120582119899

119889120596

119889120582119899= 1 (40)






119889120596

119889119905= 0 (41)


120596 =119860


V2

1198882+2

1198882 (42)



V2

1198882+ ℏ2

2

11988821199014 (43)


119899sim1198881199012

ℏ (44)


2



10038161003816100381610038161003816100381610038161003816 le |V|

1199012

ℏ= 1198982 1198883

ℏ (45)


44119866 where 119866 is


1198982simℏ119888

119866 (46)

7 Conclusion




119873119862

2(the subgroup




2-automorphisms it


2-effects on






Appendices



119904 = 120596 + 119881

119904dagger= 120596 minus 119881

(A1)


119881 = 120582119899119869119899+ 119909119899119895119899+ 119888119905119868 (119899 = 1 2 3) (A2)




where

1198812= minus1199092+ 11988821199052+ 1205822 (A4)





(A5)



119904minus1=

119904dagger

1198732 (A6)

with the property

119904119904minus1= 119904minus1119904 = 1 (A7)



119904 = 119873119890120598120579 (A8)


119877 =119904

119873(A9)





119904 = 119873 (sinh 120579 + 120598 cosh 120579) (A10)

where

sinh 120579 = 120596

119873

cosh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A11)



119904 = 119873 (cosh 120579 + 120598 sinh 120579) (A12)


where

cosh 120579 = 120596

119873

sinh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A13)



119904 = 119873 (cos 120579 + 120598 sin 120579) (A14)

where

cos 120579 = 120596

119873

sin 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A15)




B One-Side Products




119877s = 1198901205981205792

119904 (B1)






119890119895119899120572119899 = cos120572

119899+ 119895119899sin120572119899 (B2)




1has three


1

1198951= 11986921198693= 11989521198953= 1198691119868 (B3)

or equivalently

11989511198693= 1198692

11989511198952= 1198953

1198951119868 = 1198691

(B4)


119904 = radic1205962 + 1199092

11198901198951120572120596 + radic120582

2

2+ 1205822

311989011989511205721205821198693

+ radic1199092

2+ 1199092

311989011989511205721199091198952+ radic1199052 + 120582

2

11198901198951120572119905119868

(B5)


cos120572120596=

120596

radic1205962 + 1199092

1

cos120572120582=

1205823

radic1205822

2+ 1205822

3

cos120572119909=

1199092

radic1199092

2+ 1199092

3

cos120572119905=

119888119905

radic1199052 + 1205822

1

(B6)



1) (1205822 1205823) (1199092 1199093) and (120582

1 119905) by the




119899 with the



119890119869119899120579119899 = cosh 120579

119899+ 119869119899sinh 120579

119899 (B7)




1198691= 11986921198953= minus11986931198952= 1198951119868 (B8)

or11986911198692= minus1198953

11986911198952= 1198693

1198691119868 = 1198951

(B9)



119904 = radic1205962 minus 1205822

11198901198691120579120596 + radic119909

2

3minus 1205822

2119890minus11986911205791205821198692

+ radic1199092

2minus 1205822

311989011986911205791199091198952+ radic1199092

1minus 11990521198901198691120579119905119868

(B10)


cosh 120579120596=

120596

radic1205962 minus 1199092

1

cosh 120579120582=

1205822

radic1199092

3minus 1205822

2

cosh 120579119909=

1199092

radic1199092

2minus 1205822

3

cosh120572119905=

119905

radic1199092

1minus 1199052

(B11)







2gt 0 having


119890119868120590= cosh120590 + 119868 sinh120590 (B12)


1198681198951= minus1198691

1198681198952= minus1198692

1198681198953= minus1198693

(B13)



2

1minus 1205822

1119890minus11986812059011198951

+ radic1199092

2minus 1205822

2119890minus11986812059021198952+ radic1199092

3minus 1205822

3119890minus11986812059031198953

(B14)


cosh120590120596=

120596

radic1205962 minus 1199052

cosh1205901=

1199091

radic1199092

1minus 1205822

1

cosh1205902=

1199092

radic1199092

2minus 1205822

2

cosh1205903=

1199093

radic1199092

3minus 1205822

3

(B15)









1minus 1205822

1 radic1199092

2minus 1205822

2

and radic1199092

3minus 1205822





2 For



1= 11986921198693=

11989521198953= 1198691119868 has the form

1198951015840

1= 1198951

1198951015840

2= 1198952cos1205721+ 1198953sin1205721

1198951015840

3= 1198951015840

11198951015840

2= 1198953cos1205721minus 1198952sin1205721

1198691015840

1= 1198691cos1205731+ 119868 sin120573

1

1198681015840= 1198691015840

11198951015840

1= 119868 cos120573

1minus 1198691sin1205731

1198691015840

2= 1198951015840

21198681015840= 1198692cos (120572

1minus 1205731) + 1198693sin (120572

1minus 1205731)

1198691015840

3= 1198951015840

31198681015840= 1198693cos (120572

1minus 1205731) minus 1198692sin (120572

1minus 1205731)

(C1)


1 1198952 and 119869

1in


3 119868 1198692 and 119869

3in (C1)) are


119899 1198951015840


rules as 119869119899 119895119899 119868



3= 11986911198692= 11989511198952= 1198693119868 axes each


3 1205733 Three



119899by the angles

120579119899and 120601



1198691015840

1= 1198691

1198691015840

2=1198692cosh (120601

1+ 1205791)

2+1198953sinh (120601

1+ 1205791)

2

1198951015840

3= 1198691015840

11198691015840

2=1198953cosh (120579

1+ 1206011)

2+1198692sinh (120579

1+ 1206011)

2

1198681015840= 119868 cosh 120579

1minus 1198951sinh 120579

1

1198951015840

1= 1198691015840

11198681015840= 1198951cosh 120579

1minus 119868 sinh 120579

1

1198951015840

2= 1198691015840

21198681015840=1198952cosh (120601

1minus 1205791)

2+1198693sinh (120601

1minus 1205791)

2

1198691015840

3= 1198951015840

31198681015840=1198693cosh (120601

1minus 1205791)

2+1198952sinh (120601

1minus 1205791)

2

(C2)


119899and 119909

119899in


11989511198693= 11986911198953= 1198952119868 and 119869

3= 11989521198691= 11986921198951= 1198953119868 axes generate




1

and 1198962

1198681015840= 119868

1198951015840

1= 1198951cosh 119896

1+ 1198691sinh 119896

1

1198951015840

2= 1198952cosh 119896

2+ 1198692sinh 119896

2

1198691015840

1= 1198951015840

11198681015840= 1198691cosh 119896

1+ 1198951sinh 119896

1

1198691015840

2= 1198951015840

21198681015840= 1198692cosh 119896

2+ 1198952sinh 119896

2

1198951015840

3= 1198951015840

11198951015840

2= 1198953cosh (119896

1+ 1198962) minus 1198693sinh (119896

1+ 1198962)

1198691015840

3= 1198951015840

31198681015840= 1198693cosh (119896

1+ 1198962) minus 1198953sinh (119896

1+ 1198962)

(C3)


119899in (14)


119899

1198961= 1205931minus1

3sum

119899

120593119899

1198962= 1205932minus1

3sum

119899

120593119899

minus (1198961+ 1198962) = 1205933minus1

3sum

119899

120593119899

(C4)


The group 1198661198731198622


119883119899119899= minus119911119899

120597

120597119911119899

+ 119910119899

120597

120597119910119899

+1

3

3

sum

119898=1

(119911119898

120597

120597119911119898

minus 119910119898

120597

120597119910119898

)


1198831198990= minus2119905

120597

120597119911119899

+ 119910119899

120597

120597119905

+1

2120576119899119898119896

(119911119898 120597

120597119910119896

minus 119911119896 120597

120597119910119898

)

1198830119899= minus2119905

120597

120597119910119899

+ 119911119899

120597

120597119905

+1

2120576119899119898119896

(119910119898 120597

120597119911119896

minus 119910119896 120597

120597119911119898

)

119883119899119898

= minus119911119898

120597

120597119911119899

+ 119910119899

120597

120597119910119898

(119899 = 119898)

(D1)



11988311+ 11988322+ 11988333= 0 (D2)



119899and 120582

119899

119910119899= 120582119899+ 119909119899

120597

120597119910119899

=1

2(

120597

120597120582119899

+120597

120597119909119899

)

119911119899= 120582119899minus 119909119899

120597

120597119911119899

=1

2(

120597

120597120582119899

minus120597

120597119909119899

)

(D3)


119899120582119899+ 1198891199052minus 119909119899119909119899

119883119899119899

= 119909119899

120597

120597119909119899

+ 120582119899

120597

120597120582119899

minus1

3

3

sum

119898=1

(119909119898

120597

120597119909119898

+ 120582119898

120597

120597120582119898

)


1198831198990

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

)

+ (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)+14120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

+1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

1198830119899

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

) minus (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)

+1

4120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

minus1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

+1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

119883119899119898

= minus1

2(120582119899

120597

120597120582119898

minus 120582119898

120597

120597120582119899

)

minus1

2(119909119899

120597

120597119909119898

minus 119909119898

120597

120597119909119899

)

+1

2(119909119899

120597

120597120582119898

+ 120582119898

120597

120597119909119899

)

+1

2(120582119899

120597

120597119909119898

+ 119909119898

120597

120597120582119899

) (119899 = 119898)

(D4)

E Zero Divisors



119863plusmn(119869)

119899=1

2(1 plusmn 119869

119899)

119863plusmn(119868)

=1

2(1 plusmn 119868)

(119899 = 1 2 3)

(E1)


119863plusmn(119869)

119899119863plusmn(119869)

119899= 119863plusmn(119869)

119899

119863+(119869)

119899119863minus(119869)

119899= 0

119863plusmn(119868)

119863plusmn(119868)

= 119863plusmn(119868)

119863+(119868)

119863minus(119868)

= 0

(E2)


119866plusmn(119869)

119899=1

2(119869119899plusmn 119895119899)

119866plusmn(119868)

119899=1

2(119868 plusmn 119895

119899)

(119899 = 1 2 3)

(E3)

with the properties

119866plusmn(119869)

119899119866plusmn(119869)

119899= 0

119866plusmn(119869)

119899119866∓(119869)

119899= 119863∓(119868)

119866plusmn(119868)

119899119866plusmn(119868)

119899= 0

119866plusmn(119868)

119899119866∓(119868)

119899= 119863plusmn(119869)

119899

(E4)






119866plusmn(119869)

119866∓(119869)

+ 119866∓(119869)

119866plusmn(119869)

= 1

119866plusmn(119868)

119866∓(119868)

+ 119866∓(119868)

119866plusmn(119868)

= 1

(E5)



119863plusmn(119869)

119899119866plusmn(119868)

119899= 119866plusmn(119868)

119899

119863plusmn(119869)

119899119866∓(119868)

119899= 0

119863plusmn(119868)

119866plusmn(119869)

119899= 0

119863plusmn(119868)

119866∓(119869)

119899= 119866∓(119869)

119899

(E6)


119904 = 119863+(119869)

119899(1

3120596 + 120582119899) + 119866

+(119868)

119899(119888119905 + 119909

119899)

+ 119863minus(119869)

119899(1

3120596 minus 120582119899) + 119866

minus(119868)

119899(119888119905 minus 119909

119899)


119904 = 119863+(119868)

(120596 + 119888119905) + 119866+(119869)

119899(120582119899+ 119909119899) + 119863minus(119868)

(120596 minus 119888119905)

+ 119866minus(119869)

119899(120582119899minus 119909119899)

(E7)




Acknowledgments


References


































































2 I




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















1199091015840

119899= 119909119899minus 120576119899119898119896

120572119898119909119896minus 120579119899119888119905

+1

2(1003816100381610038161003816120576119899119898119896

1003816100381610038161003816 120601119898+ 120576119899119898119896

120579119898) 120582119896

+ (120593119899minus1

3sum

119898

120593119898)120582119899

1198881199051015840= 119888119905 minus 120573

119899120582119899minus 120579119899119909119899

1205821015840

119899= 120582119899minus 120576119899119898119896

(120572119898minus 120573119898) 120582119896+ 120573119899119888119905

+1

2(1003816100381610038161003816120576119899119898119896

1003816100381610038161003816 120601119898minus 120576119899119898119896

120579119898) 119909119896

+ (120593119899minus1

3sum

119898

120593119898)119909119899

(14)


119899 From


sum

119899

(120593119899minus1

3sum

119898

120593119898) = 0 (15)


119899 and








120601119898= 120593119898= 0 (16)

the passive1198661198731198622




1199091015840

119899= 119909119899minus 120576119899119898119896

120572119898119909119896minus 120579119899119888119905 + 119886119899

1198881199051015840= 119888119905 minus 120579

119899119909119899+ 1198860

(17)


119886119899=1

2120576119899119898119896

120579119898120582119896

1198860= minus120573119899120582119899

(18)






119899







119899and 120582

119899


119899





2-





1199091015840

1= 1199091

1199091015840

2= 1199092+ 12057211199093

1199091015840

3= 1199093minus 12057211199092

1198881199051015840= 119888119905 minus 120573

11205821

1205821015840

1= 1205821+ 1205731119888119905

1205821015840

2= 1205822+ (1205721minus 1205731) 1205823

1205821015840

3= 1205823minus (1205721minus 1205731) 1205822

(19)

When 1205721










1

1199091015840

1= 1199091minus 1205791119888119905

1199091015840

2= 1199092+1

2(1206011minus 1205791) 1205823

1199091015840

3= 1199093+1

2(1206011+ 1205791) 1205822

1198881199051015840= 119888119905 minus 120579

11199091

1205821015840

1= 1205821

1205821015840

2= 1205822+1

2(1206011+ 1205791) 1199093

1205821015840

3= 1205823+1

2(1206011minus 1205791) 1199092

(20)






1199091015840= 1199101015840= 1199111015840= 0 (21)


1205791=1199091

119888119905

1205823=1199092

1205791

1205822= minus

1199093

1205791

(22)









assume that

120582119899= ℏ

119901119899

1199012sim

ℏ

119901119899

(23)




Δ1199092

Δ1205822

= minus1205823

1199093

Δ1199093

Δ1205823

= minus1205822

1199092

(24)

(where Δ119909119899= 1199091015840

119899minus 119909119899and Δ120582

119899= 1205821015840



10038161003816100381610038161199091198991003816100381610038161003816 ≳

10038161003816100381610038161205821198991003816100381610038161003816 (25)


Δ119909119899Δ119901119899ge ℏ (26)



2 12058210158403increases while 1199091015840

3 12058210158402decreases


1199092

1205823

= minus1199093

1205822

= 1205791 (27)


120590119899=1

ℏ120576119899119898119896

119909119898119901119896sim 120576119899119898119896

119909119898

120582119896

(28)


119899in the


119899and 120582




4 Parity Violation



tanh 1205791=V

119888

cosh 1205791=

1


sinh 1205791=

V119888


(29)



V10158401=

V1minus 119888 tanh 120579

1

1 minus tanh 1205791(V1119888)

V10158402=

V2minus tanh 120579

13

1 minus tanh 1205791(V1119888)

V10158403=

V3+ tanh 120579

12

1 minus tanh 1205791(V1119888)

(30)



V1= 119888 cos 120574

12

V2= 119888 sin 120574

12

V10158401= 119888 cos 1205741015840

12

V10158402= 119888 sin 1205741015840

12

(31)



1199091and 1199091015840


have

sin 120574101584012=

sin 12057412minus (V119888

2) 3

1 minus (V119888) cos 12057412

cos 120574101584012=

cos 12057412minusV119888

1 minus (V119888) cos 12057412

(32)


3



Δ12057412= 1205741015840

12minus 12057412=V

119888sin 12057412minusV

11988823 (33)


Δ12057413=V

119888sin 12057413+V

11988822 (34)






1199091015840

119899= 119909119899+ (120593119899minus1

3sum

119898

120593119898)120582119899

1199051015840= 119905

1205821015840

119899= 120582119899+ (120593119899minus1

3sum

119898

120593119898)119909119899

(35)


1 1205821) (1199092 1205822) and (119909



3minus 1205823)









119899and (C4) as

1198701= 1198961=1

3(21205931minus 1205932minus 1205933)

1198702=radic3

2(1198961+ 1198961) = minus

1


(36)


(

1205821015840

1+ 1198681199091015840

1

1205821015840

2+ 1198681199091015840

2

1205821015840

2+ 1198681199091015840

2

) = 119890(1198701Λ3+1198702Λ8)119868(

1205821+ 1198681199091

1205822+ 1198681199092

1205822+ 1198681199092

) (37)



Λ3= (

1 0 0

0 minus1 0

0 0 0

)

Λ8=

1

radic3(

1 0 0

0 1 0

0 0 minus2

)

(38)



1and 119870

2)







119899






1198891199042= 119889119904119889119904


V2

1198882+2

1198882) = 0 (39)


119889120596

119889120582119899

119889120596

119889120582119899= 1 (40)






119889120596

119889119905= 0 (41)


120596 =119860


V2

1198882+2

1198882 (42)



V2

1198882+ ℏ2

2

11988821199014 (43)


119899sim1198881199012

ℏ (44)


2



10038161003816100381610038161003816100381610038161003816 le |V|

1199012

ℏ= 1198982 1198883

ℏ (45)


44119866 where 119866 is


1198982simℏ119888

119866 (46)

7 Conclusion




119873119862

2(the subgroup




2-automorphisms it


2-effects on






Appendices



119904 = 120596 + 119881

119904dagger= 120596 minus 119881

(A1)


119881 = 120582119899119869119899+ 119909119899119895119899+ 119888119905119868 (119899 = 1 2 3) (A2)




where

1198812= minus1199092+ 11988821199052+ 1205822 (A4)





(A5)



119904minus1=

119904dagger

1198732 (A6)

with the property

119904119904minus1= 119904minus1119904 = 1 (A7)



119904 = 119873119890120598120579 (A8)


119877 =119904

119873(A9)





119904 = 119873 (sinh 120579 + 120598 cosh 120579) (A10)

where

sinh 120579 = 120596

119873

cosh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A11)



119904 = 119873 (cosh 120579 + 120598 sinh 120579) (A12)


where

cosh 120579 = 120596

119873

sinh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A13)



119904 = 119873 (cos 120579 + 120598 sin 120579) (A14)

where

cos 120579 = 120596

119873

sin 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A15)




B One-Side Products




119877s = 1198901205981205792

119904 (B1)






119890119895119899120572119899 = cos120572

119899+ 119895119899sin120572119899 (B2)




1has three


1

1198951= 11986921198693= 11989521198953= 1198691119868 (B3)

or equivalently

11989511198693= 1198692

11989511198952= 1198953

1198951119868 = 1198691

(B4)


119904 = radic1205962 + 1199092

11198901198951120572120596 + radic120582

2

2+ 1205822

311989011989511205721205821198693

+ radic1199092

2+ 1199092

311989011989511205721199091198952+ radic1199052 + 120582

2

11198901198951120572119905119868

(B5)


cos120572120596=

120596

radic1205962 + 1199092

1

cos120572120582=

1205823

radic1205822

2+ 1205822

3

cos120572119909=

1199092

radic1199092

2+ 1199092

3

cos120572119905=

119888119905

radic1199052 + 1205822

1

(B6)



1) (1205822 1205823) (1199092 1199093) and (120582

1 119905) by the




119899 with the



119890119869119899120579119899 = cosh 120579

119899+ 119869119899sinh 120579

119899 (B7)




1198691= 11986921198953= minus11986931198952= 1198951119868 (B8)

or11986911198692= minus1198953

11986911198952= 1198693

1198691119868 = 1198951

(B9)



119904 = radic1205962 minus 1205822

11198901198691120579120596 + radic119909

2

3minus 1205822

2119890minus11986911205791205821198692

+ radic1199092

2minus 1205822

311989011986911205791199091198952+ radic1199092

1minus 11990521198901198691120579119905119868

(B10)


cosh 120579120596=

120596

radic1205962 minus 1199092

1

cosh 120579120582=

1205822

radic1199092

3minus 1205822

2

cosh 120579119909=

1199092

radic1199092

2minus 1205822

3

cosh120572119905=

119905

radic1199092

1minus 1199052

(B11)







2gt 0 having


119890119868120590= cosh120590 + 119868 sinh120590 (B12)


1198681198951= minus1198691

1198681198952= minus1198692

1198681198953= minus1198693

(B13)



2

1minus 1205822

1119890minus11986812059011198951

+ radic1199092

2minus 1205822

2119890minus11986812059021198952+ radic1199092

3minus 1205822

3119890minus11986812059031198953

(B14)


cosh120590120596=

120596

radic1205962 minus 1199052

cosh1205901=

1199091

radic1199092

1minus 1205822

1

cosh1205902=

1199092

radic1199092

2minus 1205822

2

cosh1205903=

1199093

radic1199092

3minus 1205822

3

(B15)









1minus 1205822

1 radic1199092

2minus 1205822

2

and radic1199092

3minus 1205822





2 For



1= 11986921198693=

11989521198953= 1198691119868 has the form

1198951015840

1= 1198951

1198951015840

2= 1198952cos1205721+ 1198953sin1205721

1198951015840

3= 1198951015840

11198951015840

2= 1198953cos1205721minus 1198952sin1205721

1198691015840

1= 1198691cos1205731+ 119868 sin120573

1

1198681015840= 1198691015840

11198951015840

1= 119868 cos120573

1minus 1198691sin1205731

1198691015840

2= 1198951015840

21198681015840= 1198692cos (120572

1minus 1205731) + 1198693sin (120572

1minus 1205731)

1198691015840

3= 1198951015840

31198681015840= 1198693cos (120572

1minus 1205731) minus 1198692sin (120572

1minus 1205731)

(C1)


1 1198952 and 119869

1in


3 119868 1198692 and 119869

3in (C1)) are


119899 1198951015840


rules as 119869119899 119895119899 119868



3= 11986911198692= 11989511198952= 1198693119868 axes each


3 1205733 Three



119899by the angles

120579119899and 120601



1198691015840

1= 1198691

1198691015840

2=1198692cosh (120601

1+ 1205791)

2+1198953sinh (120601

1+ 1205791)

2

1198951015840

3= 1198691015840

11198691015840

2=1198953cosh (120579

1+ 1206011)

2+1198692sinh (120579

1+ 1206011)

2

1198681015840= 119868 cosh 120579

1minus 1198951sinh 120579

1

1198951015840

1= 1198691015840

11198681015840= 1198951cosh 120579

1minus 119868 sinh 120579

1

1198951015840

2= 1198691015840

21198681015840=1198952cosh (120601

1minus 1205791)

2+1198693sinh (120601

1minus 1205791)

2

1198691015840

3= 1198951015840

31198681015840=1198693cosh (120601

1minus 1205791)

2+1198952sinh (120601

1minus 1205791)

2

(C2)


119899and 119909

119899in


11989511198693= 11986911198953= 1198952119868 and 119869

3= 11989521198691= 11986921198951= 1198953119868 axes generate




1

and 1198962

1198681015840= 119868

1198951015840

1= 1198951cosh 119896

1+ 1198691sinh 119896

1

1198951015840

2= 1198952cosh 119896

2+ 1198692sinh 119896

2

1198691015840

1= 1198951015840

11198681015840= 1198691cosh 119896

1+ 1198951sinh 119896

1

1198691015840

2= 1198951015840

21198681015840= 1198692cosh 119896

2+ 1198952sinh 119896

2

1198951015840

3= 1198951015840

11198951015840

2= 1198953cosh (119896

1+ 1198962) minus 1198693sinh (119896

1+ 1198962)

1198691015840

3= 1198951015840

31198681015840= 1198693cosh (119896

1+ 1198962) minus 1198953sinh (119896

1+ 1198962)

(C3)


119899in (14)


119899

1198961= 1205931minus1

3sum

119899

120593119899

1198962= 1205932minus1

3sum

119899

120593119899

minus (1198961+ 1198962) = 1205933minus1

3sum

119899

120593119899

(C4)


The group 1198661198731198622


119883119899119899= minus119911119899

120597

120597119911119899

+ 119910119899

120597

120597119910119899

+1

3

3

sum

119898=1

(119911119898

120597

120597119911119898

minus 119910119898

120597

120597119910119898

)


1198831198990= minus2119905

120597

120597119911119899

+ 119910119899

120597

120597119905

+1

2120576119899119898119896

(119911119898 120597

120597119910119896

minus 119911119896 120597

120597119910119898

)

1198830119899= minus2119905

120597

120597119910119899

+ 119911119899

120597

120597119905

+1

2120576119899119898119896

(119910119898 120597

120597119911119896

minus 119910119896 120597

120597119911119898

)

119883119899119898

= minus119911119898

120597

120597119911119899

+ 119910119899

120597

120597119910119898

(119899 = 119898)

(D1)



11988311+ 11988322+ 11988333= 0 (D2)



119899and 120582

119899

119910119899= 120582119899+ 119909119899

120597

120597119910119899

=1

2(

120597

120597120582119899

+120597

120597119909119899

)

119911119899= 120582119899minus 119909119899

120597

120597119911119899

=1

2(

120597

120597120582119899

minus120597

120597119909119899

)

(D3)


119899120582119899+ 1198891199052minus 119909119899119909119899

119883119899119899

= 119909119899

120597

120597119909119899

+ 120582119899

120597

120597120582119899

minus1

3

3

sum

119898=1

(119909119898

120597

120597119909119898

+ 120582119898

120597

120597120582119898

)


1198831198990

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

)

+ (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)+14120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

+1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

1198830119899

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

) minus (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)

+1

4120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

minus1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

+1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

119883119899119898

= minus1

2(120582119899

120597

120597120582119898

minus 120582119898

120597

120597120582119899

)

minus1

2(119909119899

120597

120597119909119898

minus 119909119898

120597

120597119909119899

)

+1

2(119909119899

120597

120597120582119898

+ 120582119898

120597

120597119909119899

)

+1

2(120582119899

120597

120597119909119898

+ 119909119898

120597

120597120582119899

) (119899 = 119898)

(D4)

E Zero Divisors



119863plusmn(119869)

119899=1

2(1 plusmn 119869

119899)

119863plusmn(119868)

=1

2(1 plusmn 119868)

(119899 = 1 2 3)

(E1)


119863plusmn(119869)

119899119863plusmn(119869)

119899= 119863plusmn(119869)

119899

119863+(119869)

119899119863minus(119869)

119899= 0

119863plusmn(119868)

119863plusmn(119868)

= 119863plusmn(119868)

119863+(119868)

119863minus(119868)

= 0

(E2)


119866plusmn(119869)

119899=1

2(119869119899plusmn 119895119899)

119866plusmn(119868)

119899=1

2(119868 plusmn 119895

119899)

(119899 = 1 2 3)

(E3)

with the properties

119866plusmn(119869)

119899119866plusmn(119869)

119899= 0

119866plusmn(119869)

119899119866∓(119869)

119899= 119863∓(119868)

119866plusmn(119868)

119899119866plusmn(119868)

119899= 0

119866plusmn(119868)

119899119866∓(119868)

119899= 119863plusmn(119869)

119899

(E4)






119866plusmn(119869)

119866∓(119869)

+ 119866∓(119869)

119866plusmn(119869)

= 1

119866plusmn(119868)

119866∓(119868)

+ 119866∓(119868)

119866plusmn(119868)

= 1

(E5)



119863plusmn(119869)

119899119866plusmn(119868)

119899= 119866plusmn(119868)

119899

119863plusmn(119869)

119899119866∓(119868)

119899= 0

119863plusmn(119868)

119866plusmn(119869)

119899= 0

119863plusmn(119868)

119866∓(119869)

119899= 119866∓(119869)

119899

(E6)


119904 = 119863+(119869)

119899(1

3120596 + 120582119899) + 119866

+(119868)

119899(119888119905 + 119909

119899)

+ 119863minus(119869)

119899(1

3120596 minus 120582119899) + 119866

minus(119868)

119899(119888119905 minus 119909

119899)


119904 = 119863+(119868)

(120596 + 119888119905) + 119866+(119869)

119899(120582119899+ 119909119899) + 119863minus(119868)

(120596 minus 119888119905)

+ 119866minus(119869)

119899(120582119899minus 119909119899)

(E7)




Acknowledgments


References


































































2 I




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1199091015840

1= 1199091

1199091015840

2= 1199092+ 12057211199093

1199091015840

3= 1199093minus 12057211199092

1198881199051015840= 119888119905 minus 120573

11205821

1205821015840

1= 1205821+ 1205731119888119905

1205821015840

2= 1205822+ (1205721minus 1205731) 1205823

1205821015840

3= 1205823minus (1205721minus 1205731) 1205822

(19)

When 1205721










1

1199091015840

1= 1199091minus 1205791119888119905

1199091015840

2= 1199092+1

2(1206011minus 1205791) 1205823

1199091015840

3= 1199093+1

2(1206011+ 1205791) 1205822

1198881199051015840= 119888119905 minus 120579

11199091

1205821015840

1= 1205821

1205821015840

2= 1205822+1

2(1206011+ 1205791) 1199093

1205821015840

3= 1205823+1

2(1206011minus 1205791) 1199092

(20)






1199091015840= 1199101015840= 1199111015840= 0 (21)


1205791=1199091

119888119905

1205823=1199092

1205791

1205822= minus

1199093

1205791

(22)









assume that

120582119899= ℏ

119901119899

1199012sim

ℏ

119901119899

(23)




Δ1199092

Δ1205822

= minus1205823

1199093

Δ1199093

Δ1205823

= minus1205822

1199092

(24)

(where Δ119909119899= 1199091015840

119899minus 119909119899and Δ120582

119899= 1205821015840



10038161003816100381610038161199091198991003816100381610038161003816 ≳

10038161003816100381610038161205821198991003816100381610038161003816 (25)


Δ119909119899Δ119901119899ge ℏ (26)



2 12058210158403increases while 1199091015840

3 12058210158402decreases


1199092

1205823

= minus1199093

1205822

= 1205791 (27)


120590119899=1

ℏ120576119899119898119896

119909119898119901119896sim 120576119899119898119896

119909119898

120582119896

(28)


119899in the


119899and 120582




4 Parity Violation



tanh 1205791=V

119888

cosh 1205791=

1


sinh 1205791=

V119888


(29)



V10158401=

V1minus 119888 tanh 120579

1

1 minus tanh 1205791(V1119888)

V10158402=

V2minus tanh 120579

13

1 minus tanh 1205791(V1119888)

V10158403=

V3+ tanh 120579

12

1 minus tanh 1205791(V1119888)

(30)



V1= 119888 cos 120574

12

V2= 119888 sin 120574

12

V10158401= 119888 cos 1205741015840

12

V10158402= 119888 sin 1205741015840

12

(31)



1199091and 1199091015840


have

sin 120574101584012=

sin 12057412minus (V119888

2) 3

1 minus (V119888) cos 12057412

cos 120574101584012=

cos 12057412minusV119888

1 minus (V119888) cos 12057412

(32)


3



Δ12057412= 1205741015840

12minus 12057412=V

119888sin 12057412minusV

11988823 (33)


Δ12057413=V

119888sin 12057413+V

11988822 (34)






1199091015840

119899= 119909119899+ (120593119899minus1

3sum

119898

120593119898)120582119899

1199051015840= 119905

1205821015840

119899= 120582119899+ (120593119899minus1

3sum

119898

120593119898)119909119899

(35)


1 1205821) (1199092 1205822) and (119909



3minus 1205823)









119899and (C4) as

1198701= 1198961=1

3(21205931minus 1205932minus 1205933)

1198702=radic3

2(1198961+ 1198961) = minus

1


(36)


(

1205821015840

1+ 1198681199091015840

1

1205821015840

2+ 1198681199091015840

2

1205821015840

2+ 1198681199091015840

2

) = 119890(1198701Λ3+1198702Λ8)119868(

1205821+ 1198681199091

1205822+ 1198681199092

1205822+ 1198681199092

) (37)



Λ3= (

1 0 0

0 minus1 0

0 0 0

)

Λ8=

1

radic3(

1 0 0

0 1 0

0 0 minus2

)

(38)



1and 119870

2)







119899






1198891199042= 119889119904119889119904


V2

1198882+2

1198882) = 0 (39)


119889120596

119889120582119899

119889120596

119889120582119899= 1 (40)






119889120596

119889119905= 0 (41)


120596 =119860


V2

1198882+2

1198882 (42)



V2

1198882+ ℏ2

2

11988821199014 (43)


119899sim1198881199012

ℏ (44)


2



10038161003816100381610038161003816100381610038161003816 le |V|

1199012

ℏ= 1198982 1198883

ℏ (45)


44119866 where 119866 is


1198982simℏ119888

119866 (46)

7 Conclusion




119873119862

2(the subgroup




2-automorphisms it


2-effects on






Appendices



119904 = 120596 + 119881

119904dagger= 120596 minus 119881

(A1)


119881 = 120582119899119869119899+ 119909119899119895119899+ 119888119905119868 (119899 = 1 2 3) (A2)




where

1198812= minus1199092+ 11988821199052+ 1205822 (A4)





(A5)



119904minus1=

119904dagger

1198732 (A6)

with the property

119904119904minus1= 119904minus1119904 = 1 (A7)



119904 = 119873119890120598120579 (A8)


119877 =119904

119873(A9)





119904 = 119873 (sinh 120579 + 120598 cosh 120579) (A10)

where

sinh 120579 = 120596

119873

cosh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A11)



119904 = 119873 (cosh 120579 + 120598 sinh 120579) (A12)


where

cosh 120579 = 120596

119873

sinh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A13)



119904 = 119873 (cos 120579 + 120598 sin 120579) (A14)

where

cos 120579 = 120596

119873

sin 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A15)




B One-Side Products




119877s = 1198901205981205792

119904 (B1)






119890119895119899120572119899 = cos120572

119899+ 119895119899sin120572119899 (B2)




1has three


1

1198951= 11986921198693= 11989521198953= 1198691119868 (B3)

or equivalently

11989511198693= 1198692

11989511198952= 1198953

1198951119868 = 1198691

(B4)


119904 = radic1205962 + 1199092

11198901198951120572120596 + radic120582

2

2+ 1205822

311989011989511205721205821198693

+ radic1199092

2+ 1199092

311989011989511205721199091198952+ radic1199052 + 120582

2

11198901198951120572119905119868

(B5)


cos120572120596=

120596

radic1205962 + 1199092

1

cos120572120582=

1205823

radic1205822

2+ 1205822

3

cos120572119909=

1199092

radic1199092

2+ 1199092

3

cos120572119905=

119888119905

radic1199052 + 1205822

1

(B6)



1) (1205822 1205823) (1199092 1199093) and (120582

1 119905) by the




119899 with the



119890119869119899120579119899 = cosh 120579

119899+ 119869119899sinh 120579

119899 (B7)




1198691= 11986921198953= minus11986931198952= 1198951119868 (B8)

or11986911198692= minus1198953

11986911198952= 1198693

1198691119868 = 1198951

(B9)



119904 = radic1205962 minus 1205822

11198901198691120579120596 + radic119909

2

3minus 1205822

2119890minus11986911205791205821198692

+ radic1199092

2minus 1205822

311989011986911205791199091198952+ radic1199092

1minus 11990521198901198691120579119905119868

(B10)


cosh 120579120596=

120596

radic1205962 minus 1199092

1

cosh 120579120582=

1205822

radic1199092

3minus 1205822

2

cosh 120579119909=

1199092

radic1199092

2minus 1205822

3

cosh120572119905=

119905

radic1199092

1minus 1199052

(B11)







2gt 0 having


119890119868120590= cosh120590 + 119868 sinh120590 (B12)


1198681198951= minus1198691

1198681198952= minus1198692

1198681198953= minus1198693

(B13)



2

1minus 1205822

1119890minus11986812059011198951

+ radic1199092

2minus 1205822

2119890minus11986812059021198952+ radic1199092

3minus 1205822

3119890minus11986812059031198953

(B14)


cosh120590120596=

120596

radic1205962 minus 1199052

cosh1205901=

1199091

radic1199092

1minus 1205822

1

cosh1205902=

1199092

radic1199092

2minus 1205822

2

cosh1205903=

1199093

radic1199092

3minus 1205822

3

(B15)









1minus 1205822

1 radic1199092

2minus 1205822

2

and radic1199092

3minus 1205822





2 For



1= 11986921198693=

11989521198953= 1198691119868 has the form

1198951015840

1= 1198951

1198951015840

2= 1198952cos1205721+ 1198953sin1205721

1198951015840

3= 1198951015840

11198951015840

2= 1198953cos1205721minus 1198952sin1205721

1198691015840

1= 1198691cos1205731+ 119868 sin120573

1

1198681015840= 1198691015840

11198951015840

1= 119868 cos120573

1minus 1198691sin1205731

1198691015840

2= 1198951015840

21198681015840= 1198692cos (120572

1minus 1205731) + 1198693sin (120572

1minus 1205731)

1198691015840

3= 1198951015840

31198681015840= 1198693cos (120572

1minus 1205731) minus 1198692sin (120572

1minus 1205731)

(C1)


1 1198952 and 119869

1in


3 119868 1198692 and 119869

3in (C1)) are


119899 1198951015840


rules as 119869119899 119895119899 119868



3= 11986911198692= 11989511198952= 1198693119868 axes each


3 1205733 Three



119899by the angles

120579119899and 120601



1198691015840

1= 1198691

1198691015840

2=1198692cosh (120601

1+ 1205791)

2+1198953sinh (120601

1+ 1205791)

2

1198951015840

3= 1198691015840

11198691015840

2=1198953cosh (120579

1+ 1206011)

2+1198692sinh (120579

1+ 1206011)

2

1198681015840= 119868 cosh 120579

1minus 1198951sinh 120579

1

1198951015840

1= 1198691015840

11198681015840= 1198951cosh 120579

1minus 119868 sinh 120579

1

1198951015840

2= 1198691015840

21198681015840=1198952cosh (120601

1minus 1205791)

2+1198693sinh (120601

1minus 1205791)

2

1198691015840

3= 1198951015840

31198681015840=1198693cosh (120601

1minus 1205791)

2+1198952sinh (120601

1minus 1205791)

2

(C2)


119899and 119909

119899in


11989511198693= 11986911198953= 1198952119868 and 119869

3= 11989521198691= 11986921198951= 1198953119868 axes generate




1

and 1198962

1198681015840= 119868

1198951015840

1= 1198951cosh 119896

1+ 1198691sinh 119896

1

1198951015840

2= 1198952cosh 119896

2+ 1198692sinh 119896

2

1198691015840

1= 1198951015840

11198681015840= 1198691cosh 119896

1+ 1198951sinh 119896

1

1198691015840

2= 1198951015840

21198681015840= 1198692cosh 119896

2+ 1198952sinh 119896

2

1198951015840

3= 1198951015840

11198951015840

2= 1198953cosh (119896

1+ 1198962) minus 1198693sinh (119896

1+ 1198962)

1198691015840

3= 1198951015840

31198681015840= 1198693cosh (119896

1+ 1198962) minus 1198953sinh (119896

1+ 1198962)

(C3)


119899in (14)


119899

1198961= 1205931minus1

3sum

119899

120593119899

1198962= 1205932minus1

3sum

119899

120593119899

minus (1198961+ 1198962) = 1205933minus1

3sum

119899

120593119899

(C4)


The group 1198661198731198622


119883119899119899= minus119911119899

120597

120597119911119899

+ 119910119899

120597

120597119910119899

+1

3

3

sum

119898=1

(119911119898

120597

120597119911119898

minus 119910119898

120597

120597119910119898

)


1198831198990= minus2119905

120597

120597119911119899

+ 119910119899

120597

120597119905

+1

2120576119899119898119896

(119911119898 120597

120597119910119896

minus 119911119896 120597

120597119910119898

)

1198830119899= minus2119905

120597

120597119910119899

+ 119911119899

120597

120597119905

+1

2120576119899119898119896

(119910119898 120597

120597119911119896

minus 119910119896 120597

120597119911119898

)

119883119899119898

= minus119911119898

120597

120597119911119899

+ 119910119899

120597

120597119910119898

(119899 = 119898)

(D1)



11988311+ 11988322+ 11988333= 0 (D2)



119899and 120582

119899

119910119899= 120582119899+ 119909119899

120597

120597119910119899

=1

2(

120597

120597120582119899

+120597

120597119909119899

)

119911119899= 120582119899minus 119909119899

120597

120597119911119899

=1

2(

120597

120597120582119899

minus120597

120597119909119899

)

(D3)


119899120582119899+ 1198891199052minus 119909119899119909119899

119883119899119899

= 119909119899

120597

120597119909119899

+ 120582119899

120597

120597120582119899

minus1

3

3

sum

119898=1

(119909119898

120597

120597119909119898

+ 120582119898

120597

120597120582119898

)


1198831198990

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

)

+ (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)+14120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

+1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

1198830119899

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

) minus (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)

+1

4120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

minus1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

+1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

119883119899119898

= minus1

2(120582119899

120597

120597120582119898

minus 120582119898

120597

120597120582119899

)

minus1

2(119909119899

120597

120597119909119898

minus 119909119898

120597

120597119909119899

)

+1

2(119909119899

120597

120597120582119898

+ 120582119898

120597

120597119909119899

)

+1

2(120582119899

120597

120597119909119898

+ 119909119898

120597

120597120582119899

) (119899 = 119898)

(D4)

E Zero Divisors



119863plusmn(119869)

119899=1

2(1 plusmn 119869

119899)

119863plusmn(119868)

=1

2(1 plusmn 119868)

(119899 = 1 2 3)

(E1)


119863plusmn(119869)

119899119863plusmn(119869)

119899= 119863plusmn(119869)

119899

119863+(119869)

119899119863minus(119869)

119899= 0

119863plusmn(119868)

119863plusmn(119868)

= 119863plusmn(119868)

119863+(119868)

119863minus(119868)

= 0

(E2)


119866plusmn(119869)

119899=1

2(119869119899plusmn 119895119899)

119866plusmn(119868)

119899=1

2(119868 plusmn 119895

119899)

(119899 = 1 2 3)

(E3)

with the properties

119866plusmn(119869)

119899119866plusmn(119869)

119899= 0

119866plusmn(119869)

119899119866∓(119869)

119899= 119863∓(119868)

119866plusmn(119868)

119899119866plusmn(119868)

119899= 0

119866plusmn(119868)

119899119866∓(119868)

119899= 119863plusmn(119869)

119899

(E4)






119866plusmn(119869)

119866∓(119869)

+ 119866∓(119869)

119866plusmn(119869)

= 1

119866plusmn(119868)

119866∓(119868)

+ 119866∓(119868)

119866plusmn(119868)

= 1

(E5)



119863plusmn(119869)

119899119866plusmn(119868)

119899= 119866plusmn(119868)

119899

119863plusmn(119869)

119899119866∓(119868)

119899= 0

119863plusmn(119868)

119866plusmn(119869)

119899= 0

119863plusmn(119868)

119866∓(119869)

119899= 119866∓(119869)

119899

(E6)


119904 = 119863+(119869)

119899(1

3120596 + 120582119899) + 119866

+(119868)

119899(119888119905 + 119909

119899)

+ 119863minus(119869)

119899(1

3120596 minus 120582119899) + 119866

minus(119868)

119899(119888119905 minus 119909

119899)


119904 = 119863+(119868)

(120596 + 119888119905) + 119866+(119869)

119899(120582119899+ 119909119899) + 119863minus(119868)

(120596 minus 119888119905)

+ 119866minus(119869)

119899(120582119899minus 119909119899)

(E7)




Acknowledgments


References


































































2 I




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










4 Parity Violation



tanh 1205791=V

119888

cosh 1205791=

1


sinh 1205791=

V119888


(29)



V10158401=

V1minus 119888 tanh 120579

1

1 minus tanh 1205791(V1119888)

V10158402=

V2minus tanh 120579

13

1 minus tanh 1205791(V1119888)

V10158403=

V3+ tanh 120579

12

1 minus tanh 1205791(V1119888)

(30)



V1= 119888 cos 120574

12

V2= 119888 sin 120574

12

V10158401= 119888 cos 1205741015840

12

V10158402= 119888 sin 1205741015840

12

(31)



1199091and 1199091015840


have

sin 120574101584012=

sin 12057412minus (V119888

2) 3

1 minus (V119888) cos 12057412

cos 120574101584012=

cos 12057412minusV119888

1 minus (V119888) cos 12057412

(32)


3



Δ12057412= 1205741015840

12minus 12057412=V

119888sin 12057412minusV

11988823 (33)


Δ12057413=V

119888sin 12057413+V

11988822 (34)






1199091015840

119899= 119909119899+ (120593119899minus1

3sum

119898

120593119898)120582119899

1199051015840= 119905

1205821015840

119899= 120582119899+ (120593119899minus1

3sum

119898

120593119898)119909119899

(35)


1 1205821) (1199092 1205822) and (119909



3minus 1205823)









119899and (C4) as

1198701= 1198961=1

3(21205931minus 1205932minus 1205933)

1198702=radic3

2(1198961+ 1198961) = minus

1


(36)


(

1205821015840

1+ 1198681199091015840

1

1205821015840

2+ 1198681199091015840

2

1205821015840

2+ 1198681199091015840

2

) = 119890(1198701Λ3+1198702Λ8)119868(

1205821+ 1198681199091

1205822+ 1198681199092

1205822+ 1198681199092

) (37)



Λ3= (

1 0 0

0 minus1 0

0 0 0

)

Λ8=

1

radic3(

1 0 0

0 1 0

0 0 minus2

)

(38)



1and 119870

2)







119899






1198891199042= 119889119904119889119904


V2

1198882+2

1198882) = 0 (39)


119889120596

119889120582119899

119889120596

119889120582119899= 1 (40)






119889120596

119889119905= 0 (41)


120596 =119860


V2

1198882+2

1198882 (42)



V2

1198882+ ℏ2

2

11988821199014 (43)


119899sim1198881199012

ℏ (44)


2



10038161003816100381610038161003816100381610038161003816 le |V|

1199012

ℏ= 1198982 1198883

ℏ (45)


44119866 where 119866 is


1198982simℏ119888

119866 (46)

7 Conclusion




119873119862

2(the subgroup




2-automorphisms it


2-effects on






Appendices



119904 = 120596 + 119881

119904dagger= 120596 minus 119881

(A1)


119881 = 120582119899119869119899+ 119909119899119895119899+ 119888119905119868 (119899 = 1 2 3) (A2)




where

1198812= minus1199092+ 11988821199052+ 1205822 (A4)





(A5)



119904minus1=

119904dagger

1198732 (A6)

with the property

119904119904minus1= 119904minus1119904 = 1 (A7)



119904 = 119873119890120598120579 (A8)


119877 =119904

119873(A9)





119904 = 119873 (sinh 120579 + 120598 cosh 120579) (A10)

where

sinh 120579 = 120596

119873

cosh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A11)



119904 = 119873 (cosh 120579 + 120598 sinh 120579) (A12)


where

cosh 120579 = 120596

119873

sinh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A13)



119904 = 119873 (cos 120579 + 120598 sin 120579) (A14)

where

cos 120579 = 120596

119873

sin 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A15)




B One-Side Products




119877s = 1198901205981205792

119904 (B1)






119890119895119899120572119899 = cos120572

119899+ 119895119899sin120572119899 (B2)




1has three


1

1198951= 11986921198693= 11989521198953= 1198691119868 (B3)

or equivalently

11989511198693= 1198692

11989511198952= 1198953

1198951119868 = 1198691

(B4)


119904 = radic1205962 + 1199092

11198901198951120572120596 + radic120582

2

2+ 1205822

311989011989511205721205821198693

+ radic1199092

2+ 1199092

311989011989511205721199091198952+ radic1199052 + 120582

2

11198901198951120572119905119868

(B5)


cos120572120596=

120596

radic1205962 + 1199092

1

cos120572120582=

1205823

radic1205822

2+ 1205822

3

cos120572119909=

1199092

radic1199092

2+ 1199092

3

cos120572119905=

119888119905

radic1199052 + 1205822

1

(B6)



1) (1205822 1205823) (1199092 1199093) and (120582

1 119905) by the




119899 with the



119890119869119899120579119899 = cosh 120579

119899+ 119869119899sinh 120579

119899 (B7)




1198691= 11986921198953= minus11986931198952= 1198951119868 (B8)

or11986911198692= minus1198953

11986911198952= 1198693

1198691119868 = 1198951

(B9)



119904 = radic1205962 minus 1205822

11198901198691120579120596 + radic119909

2

3minus 1205822

2119890minus11986911205791205821198692

+ radic1199092

2minus 1205822

311989011986911205791199091198952+ radic1199092

1minus 11990521198901198691120579119905119868

(B10)


cosh 120579120596=

120596

radic1205962 minus 1199092

1

cosh 120579120582=

1205822

radic1199092

3minus 1205822

2

cosh 120579119909=

1199092

radic1199092

2minus 1205822

3

cosh120572119905=

119905

radic1199092

1minus 1199052

(B11)







2gt 0 having


119890119868120590= cosh120590 + 119868 sinh120590 (B12)


1198681198951= minus1198691

1198681198952= minus1198692

1198681198953= minus1198693

(B13)



2

1minus 1205822

1119890minus11986812059011198951

+ radic1199092

2minus 1205822

2119890minus11986812059021198952+ radic1199092

3minus 1205822

3119890minus11986812059031198953

(B14)


cosh120590120596=

120596

radic1205962 minus 1199052

cosh1205901=

1199091

radic1199092

1minus 1205822

1

cosh1205902=

1199092

radic1199092

2minus 1205822

2

cosh1205903=

1199093

radic1199092

3minus 1205822

3

(B15)









1minus 1205822

1 radic1199092

2minus 1205822

2

and radic1199092

3minus 1205822





2 For



1= 11986921198693=

11989521198953= 1198691119868 has the form

1198951015840

1= 1198951

1198951015840

2= 1198952cos1205721+ 1198953sin1205721

1198951015840

3= 1198951015840

11198951015840

2= 1198953cos1205721minus 1198952sin1205721

1198691015840

1= 1198691cos1205731+ 119868 sin120573

1

1198681015840= 1198691015840

11198951015840

1= 119868 cos120573

1minus 1198691sin1205731

1198691015840

2= 1198951015840

21198681015840= 1198692cos (120572

1minus 1205731) + 1198693sin (120572

1minus 1205731)

1198691015840

3= 1198951015840

31198681015840= 1198693cos (120572

1minus 1205731) minus 1198692sin (120572

1minus 1205731)

(C1)


1 1198952 and 119869

1in


3 119868 1198692 and 119869

3in (C1)) are


119899 1198951015840


rules as 119869119899 119895119899 119868



3= 11986911198692= 11989511198952= 1198693119868 axes each


3 1205733 Three



119899by the angles

120579119899and 120601



1198691015840

1= 1198691

1198691015840

2=1198692cosh (120601

1+ 1205791)

2+1198953sinh (120601

1+ 1205791)

2

1198951015840

3= 1198691015840

11198691015840

2=1198953cosh (120579

1+ 1206011)

2+1198692sinh (120579

1+ 1206011)

2

1198681015840= 119868 cosh 120579

1minus 1198951sinh 120579

1

1198951015840

1= 1198691015840

11198681015840= 1198951cosh 120579

1minus 119868 sinh 120579

1

1198951015840

2= 1198691015840

21198681015840=1198952cosh (120601

1minus 1205791)

2+1198693sinh (120601

1minus 1205791)

2

1198691015840

3= 1198951015840

31198681015840=1198693cosh (120601

1minus 1205791)

2+1198952sinh (120601

1minus 1205791)

2

(C2)


119899and 119909

119899in


11989511198693= 11986911198953= 1198952119868 and 119869

3= 11989521198691= 11986921198951= 1198953119868 axes generate




1

and 1198962

1198681015840= 119868

1198951015840

1= 1198951cosh 119896

1+ 1198691sinh 119896

1

1198951015840

2= 1198952cosh 119896

2+ 1198692sinh 119896

2

1198691015840

1= 1198951015840

11198681015840= 1198691cosh 119896

1+ 1198951sinh 119896

1

1198691015840

2= 1198951015840

21198681015840= 1198692cosh 119896

2+ 1198952sinh 119896

2

1198951015840

3= 1198951015840

11198951015840

2= 1198953cosh (119896

1+ 1198962) minus 1198693sinh (119896

1+ 1198962)

1198691015840

3= 1198951015840

31198681015840= 1198693cosh (119896

1+ 1198962) minus 1198953sinh (119896

1+ 1198962)

(C3)


119899in (14)


119899

1198961= 1205931minus1

3sum

119899

120593119899

1198962= 1205932minus1

3sum

119899

120593119899

minus (1198961+ 1198962) = 1205933minus1

3sum

119899

120593119899

(C4)


The group 1198661198731198622


119883119899119899= minus119911119899

120597

120597119911119899

+ 119910119899

120597

120597119910119899

+1

3

3

sum

119898=1

(119911119898

120597

120597119911119898

minus 119910119898

120597

120597119910119898

)


1198831198990= minus2119905

120597

120597119911119899

+ 119910119899

120597

120597119905

+1

2120576119899119898119896

(119911119898 120597

120597119910119896

minus 119911119896 120597

120597119910119898

)

1198830119899= minus2119905

120597

120597119910119899

+ 119911119899

120597

120597119905

+1

2120576119899119898119896

(119910119898 120597

120597119911119896

minus 119910119896 120597

120597119911119898

)

119883119899119898

= minus119911119898

120597

120597119911119899

+ 119910119899

120597

120597119910119898

(119899 = 119898)

(D1)



11988311+ 11988322+ 11988333= 0 (D2)



119899and 120582

119899

119910119899= 120582119899+ 119909119899

120597

120597119910119899

=1

2(

120597

120597120582119899

+120597

120597119909119899

)

119911119899= 120582119899minus 119909119899

120597

120597119911119899

=1

2(

120597

120597120582119899

minus120597

120597119909119899

)

(D3)


119899120582119899+ 1198891199052minus 119909119899119909119899

119883119899119899

= 119909119899

120597

120597119909119899

+ 120582119899

120597

120597120582119899

minus1

3

3

sum

119898=1

(119909119898

120597

120597119909119898

+ 120582119898

120597

120597120582119898

)


1198831198990

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

)

+ (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)+14120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

+1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

1198830119899

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

) minus (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)

+1

4120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

minus1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

+1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

119883119899119898

= minus1

2(120582119899

120597

120597120582119898

minus 120582119898

120597

120597120582119899

)

minus1

2(119909119899

120597

120597119909119898

minus 119909119898

120597

120597119909119899

)

+1

2(119909119899

120597

120597120582119898

+ 120582119898

120597

120597119909119899

)

+1

2(120582119899

120597

120597119909119898

+ 119909119898

120597

120597120582119899

) (119899 = 119898)

(D4)

E Zero Divisors



119863plusmn(119869)

119899=1

2(1 plusmn 119869

119899)

119863plusmn(119868)

=1

2(1 plusmn 119868)

(119899 = 1 2 3)

(E1)


119863plusmn(119869)

119899119863plusmn(119869)

119899= 119863plusmn(119869)

119899

119863+(119869)

119899119863minus(119869)

119899= 0

119863plusmn(119868)

119863plusmn(119868)

= 119863plusmn(119868)

119863+(119868)

119863minus(119868)

= 0

(E2)


119866plusmn(119869)

119899=1

2(119869119899plusmn 119895119899)

119866plusmn(119868)

119899=1

2(119868 plusmn 119895

119899)

(119899 = 1 2 3)

(E3)

with the properties

119866plusmn(119869)

119899119866plusmn(119869)

119899= 0

119866plusmn(119869)

119899119866∓(119869)

119899= 119863∓(119868)

119866plusmn(119868)

119899119866plusmn(119868)

119899= 0

119866plusmn(119868)

119899119866∓(119868)

119899= 119863plusmn(119869)

119899

(E4)






119866plusmn(119869)

119866∓(119869)

+ 119866∓(119869)

119866plusmn(119869)

= 1

119866plusmn(119868)

119866∓(119868)

+ 119866∓(119868)

119866plusmn(119868)

= 1

(E5)



119863plusmn(119869)

119899119866plusmn(119868)

119899= 119866plusmn(119868)

119899

119863plusmn(119869)

119899119866∓(119868)

119899= 0

119863plusmn(119868)

119866plusmn(119869)

119899= 0

119863plusmn(119868)

119866∓(119869)

119899= 119866∓(119869)

119899

(E6)


119904 = 119863+(119869)

119899(1

3120596 + 120582119899) + 119866

+(119868)

119899(119888119905 + 119909

119899)

+ 119863minus(119869)

119899(1

3120596 minus 120582119899) + 119866

minus(119868)

119899(119888119905 minus 119909

119899)


119904 = 119863+(119868)

(120596 + 119888119905) + 119866+(119869)

119899(120582119899+ 119909119899) + 119863minus(119868)

(120596 minus 119888119905)

+ 119866minus(119869)

119899(120582119899minus 119909119899)

(E7)




Acknowledgments


References


































































2 I




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















119899






1198891199042= 119889119904119889119904


V2

1198882+2

1198882) = 0 (39)


119889120596

119889120582119899

119889120596

119889120582119899= 1 (40)






119889120596

119889119905= 0 (41)


120596 =119860


V2

1198882+2

1198882 (42)



V2

1198882+ ℏ2

2

11988821199014 (43)


119899sim1198881199012

ℏ (44)


2



10038161003816100381610038161003816100381610038161003816 le |V|

1199012

ℏ= 1198982 1198883

ℏ (45)


44119866 where 119866 is


1198982simℏ119888

119866 (46)

7 Conclusion




119873119862

2(the subgroup




2-automorphisms it


2-effects on






Appendices



119904 = 120596 + 119881

119904dagger= 120596 minus 119881

(A1)


119881 = 120582119899119869119899+ 119909119899119895119899+ 119888119905119868 (119899 = 1 2 3) (A2)




where

1198812= minus1199092+ 11988821199052+ 1205822 (A4)





(A5)



119904minus1=

119904dagger

1198732 (A6)

with the property

119904119904minus1= 119904minus1119904 = 1 (A7)



119904 = 119873119890120598120579 (A8)


119877 =119904

119873(A9)





119904 = 119873 (sinh 120579 + 120598 cosh 120579) (A10)

where

sinh 120579 = 120596

119873

cosh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A11)



119904 = 119873 (cosh 120579 + 120598 sinh 120579) (A12)


where

cosh 120579 = 120596

119873

sinh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A13)



119904 = 119873 (cos 120579 + 120598 sin 120579) (A14)

where

cos 120579 = 120596

119873

sin 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A15)




B One-Side Products




119877s = 1198901205981205792

119904 (B1)






119890119895119899120572119899 = cos120572

119899+ 119895119899sin120572119899 (B2)




1has three


1

1198951= 11986921198693= 11989521198953= 1198691119868 (B3)

or equivalently

11989511198693= 1198692

11989511198952= 1198953

1198951119868 = 1198691

(B4)


119904 = radic1205962 + 1199092

11198901198951120572120596 + radic120582

2

2+ 1205822

311989011989511205721205821198693

+ radic1199092

2+ 1199092

311989011989511205721199091198952+ radic1199052 + 120582

2

11198901198951120572119905119868

(B5)


cos120572120596=

120596

radic1205962 + 1199092

1

cos120572120582=

1205823

radic1205822

2+ 1205822

3

cos120572119909=

1199092

radic1199092

2+ 1199092

3

cos120572119905=

119888119905

radic1199052 + 1205822

1

(B6)



1) (1205822 1205823) (1199092 1199093) and (120582

1 119905) by the




119899 with the



119890119869119899120579119899 = cosh 120579

119899+ 119869119899sinh 120579

119899 (B7)




1198691= 11986921198953= minus11986931198952= 1198951119868 (B8)

or11986911198692= minus1198953

11986911198952= 1198693

1198691119868 = 1198951

(B9)



119904 = radic1205962 minus 1205822

11198901198691120579120596 + radic119909

2

3minus 1205822

2119890minus11986911205791205821198692

+ radic1199092

2minus 1205822

311989011986911205791199091198952+ radic1199092

1minus 11990521198901198691120579119905119868

(B10)


cosh 120579120596=

120596

radic1205962 minus 1199092

1

cosh 120579120582=

1205822

radic1199092

3minus 1205822

2

cosh 120579119909=

1199092

radic1199092

2minus 1205822

3

cosh120572119905=

119905

radic1199092

1minus 1199052

(B11)







2gt 0 having


119890119868120590= cosh120590 + 119868 sinh120590 (B12)


1198681198951= minus1198691

1198681198952= minus1198692

1198681198953= minus1198693

(B13)



2

1minus 1205822

1119890minus11986812059011198951

+ radic1199092

2minus 1205822

2119890minus11986812059021198952+ radic1199092

3minus 1205822

3119890minus11986812059031198953

(B14)


cosh120590120596=

120596

radic1205962 minus 1199052

cosh1205901=

1199091

radic1199092

1minus 1205822

1

cosh1205902=

1199092

radic1199092

2minus 1205822

2

cosh1205903=

1199093

radic1199092

3minus 1205822

3

(B15)









1minus 1205822

1 radic1199092

2minus 1205822

2

and radic1199092

3minus 1205822





2 For



1= 11986921198693=

11989521198953= 1198691119868 has the form

1198951015840

1= 1198951

1198951015840

2= 1198952cos1205721+ 1198953sin1205721

1198951015840

3= 1198951015840

11198951015840

2= 1198953cos1205721minus 1198952sin1205721

1198691015840

1= 1198691cos1205731+ 119868 sin120573

1

1198681015840= 1198691015840

11198951015840

1= 119868 cos120573

1minus 1198691sin1205731

1198691015840

2= 1198951015840

21198681015840= 1198692cos (120572

1minus 1205731) + 1198693sin (120572

1minus 1205731)

1198691015840

3= 1198951015840

31198681015840= 1198693cos (120572

1minus 1205731) minus 1198692sin (120572

1minus 1205731)

(C1)


1 1198952 and 119869

1in


3 119868 1198692 and 119869

3in (C1)) are


119899 1198951015840


rules as 119869119899 119895119899 119868



3= 11986911198692= 11989511198952= 1198693119868 axes each


3 1205733 Three



119899by the angles

120579119899and 120601



1198691015840

1= 1198691

1198691015840

2=1198692cosh (120601

1+ 1205791)

2+1198953sinh (120601

1+ 1205791)

2

1198951015840

3= 1198691015840

11198691015840

2=1198953cosh (120579

1+ 1206011)

2+1198692sinh (120579

1+ 1206011)

2

1198681015840= 119868 cosh 120579

1minus 1198951sinh 120579

1

1198951015840

1= 1198691015840

11198681015840= 1198951cosh 120579

1minus 119868 sinh 120579

1

1198951015840

2= 1198691015840

21198681015840=1198952cosh (120601

1minus 1205791)

2+1198693sinh (120601

1minus 1205791)

2

1198691015840

3= 1198951015840

31198681015840=1198693cosh (120601

1minus 1205791)

2+1198952sinh (120601

1minus 1205791)

2

(C2)


119899and 119909

119899in


11989511198693= 11986911198953= 1198952119868 and 119869

3= 11989521198691= 11986921198951= 1198953119868 axes generate




1

and 1198962

1198681015840= 119868

1198951015840

1= 1198951cosh 119896

1+ 1198691sinh 119896

1

1198951015840

2= 1198952cosh 119896

2+ 1198692sinh 119896

2

1198691015840

1= 1198951015840

11198681015840= 1198691cosh 119896

1+ 1198951sinh 119896

1

1198691015840

2= 1198951015840

21198681015840= 1198692cosh 119896

2+ 1198952sinh 119896

2

1198951015840

3= 1198951015840

11198951015840

2= 1198953cosh (119896

1+ 1198962) minus 1198693sinh (119896

1+ 1198962)

1198691015840

3= 1198951015840

31198681015840= 1198693cosh (119896

1+ 1198962) minus 1198953sinh (119896

1+ 1198962)

(C3)


119899in (14)


119899

1198961= 1205931minus1

3sum

119899

120593119899

1198962= 1205932minus1

3sum

119899

120593119899

minus (1198961+ 1198962) = 1205933minus1

3sum

119899

120593119899

(C4)


The group 1198661198731198622


119883119899119899= minus119911119899

120597

120597119911119899

+ 119910119899

120597

120597119910119899

+1

3

3

sum

119898=1

(119911119898

120597

120597119911119898

minus 119910119898

120597

120597119910119898

)


1198831198990= minus2119905

120597

120597119911119899

+ 119910119899

120597

120597119905

+1

2120576119899119898119896

(119911119898 120597

120597119910119896

minus 119911119896 120597

120597119910119898

)

1198830119899= minus2119905

120597

120597119910119899

+ 119911119899

120597

120597119905

+1

2120576119899119898119896

(119910119898 120597

120597119911119896

minus 119910119896 120597

120597119911119898

)

119883119899119898

= minus119911119898

120597

120597119911119899

+ 119910119899

120597

120597119910119898

(119899 = 119898)

(D1)



11988311+ 11988322+ 11988333= 0 (D2)



119899and 120582

119899

119910119899= 120582119899+ 119909119899

120597

120597119910119899

=1

2(

120597

120597120582119899

+120597

120597119909119899

)

119911119899= 120582119899minus 119909119899

120597

120597119911119899

=1

2(

120597

120597120582119899

minus120597

120597119909119899

)

(D3)


119899120582119899+ 1198891199052minus 119909119899119909119899

119883119899119899

= 119909119899

120597

120597119909119899

+ 120582119899

120597

120597120582119899

minus1

3

3

sum

119898=1

(119909119898

120597

120597119909119898

+ 120582119898

120597

120597120582119898

)


1198831198990

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

)

+ (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)+14120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

+1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

1198830119899

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

) minus (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)

+1

4120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

minus1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

+1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

119883119899119898

= minus1

2(120582119899

120597

120597120582119898

minus 120582119898

120597

120597120582119899

)

minus1

2(119909119899

120597

120597119909119898

minus 119909119898

120597

120597119909119899

)

+1

2(119909119899

120597

120597120582119898

+ 120582119898

120597

120597119909119899

)

+1

2(120582119899

120597

120597119909119898

+ 119909119898

120597

120597120582119899

) (119899 = 119898)

(D4)

E Zero Divisors



119863plusmn(119869)

119899=1

2(1 plusmn 119869

119899)

119863plusmn(119868)

=1

2(1 plusmn 119868)

(119899 = 1 2 3)

(E1)


119863plusmn(119869)

119899119863plusmn(119869)

119899= 119863plusmn(119869)

119899

119863+(119869)

119899119863minus(119869)

119899= 0

119863plusmn(119868)

119863plusmn(119868)

= 119863plusmn(119868)

119863+(119868)

119863minus(119868)

= 0

(E2)


119866plusmn(119869)

119899=1

2(119869119899plusmn 119895119899)

119866plusmn(119868)

119899=1

2(119868 plusmn 119895

119899)

(119899 = 1 2 3)

(E3)

with the properties

119866plusmn(119869)

119899119866plusmn(119869)

119899= 0

119866plusmn(119869)

119899119866∓(119869)

119899= 119863∓(119868)

119866plusmn(119868)

119899119866plusmn(119868)

119899= 0

119866plusmn(119868)

119899119866∓(119868)

119899= 119863plusmn(119869)

119899

(E4)






119866plusmn(119869)

119866∓(119869)

+ 119866∓(119869)

119866plusmn(119869)

= 1

119866plusmn(119868)

119866∓(119868)

+ 119866∓(119868)

119866plusmn(119868)

= 1

(E5)



119863plusmn(119869)

119899119866plusmn(119868)

119899= 119866plusmn(119868)

119899

119863plusmn(119869)

119899119866∓(119868)

119899= 0

119863plusmn(119868)

119866plusmn(119869)

119899= 0

119863plusmn(119868)

119866∓(119869)

119899= 119866∓(119869)

119899

(E6)


119904 = 119863+(119869)

119899(1

3120596 + 120582119899) + 119866

+(119868)

119899(119888119905 + 119909

119899)

+ 119863minus(119869)

119899(1

3120596 minus 120582119899) + 119866

minus(119868)

119899(119888119905 minus 119909

119899)


119904 = 119863+(119868)

(120596 + 119888119905) + 119866+(119869)

119899(120582119899+ 119909119899) + 119863minus(119868)

(120596 minus 119888119905)

+ 119866minus(119869)

119899(120582119899minus 119909119899)

(E7)




Acknowledgments


References


































































2 I




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119873119862

2(the subgroup




2-automorphisms it


2-effects on






Appendices



119904 = 120596 + 119881

119904dagger= 120596 minus 119881

(A1)


119881 = 120582119899119869119899+ 119909119899119895119899+ 119888119905119868 (119899 = 1 2 3) (A2)




where

1198812= minus1199092+ 11988821199052+ 1205822 (A4)





(A5)



119904minus1=

119904dagger

1198732 (A6)

with the property

119904119904minus1= 119904minus1119904 = 1 (A7)



119904 = 119873119890120598120579 (A8)


119877 =119904

119873(A9)





119904 = 119873 (sinh 120579 + 120598 cosh 120579) (A10)

where

sinh 120579 = 120596

119873

cosh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A11)



119904 = 119873 (cosh 120579 + 120598 sinh 120579) (A12)


where

cosh 120579 = 120596

119873

sinh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A13)



119904 = 119873 (cos 120579 + 120598 sin 120579) (A14)

where

cos 120579 = 120596

119873

sin 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A15)




B One-Side Products




119877s = 1198901205981205792

119904 (B1)






119890119895119899120572119899 = cos120572

119899+ 119895119899sin120572119899 (B2)




1has three


1

1198951= 11986921198693= 11989521198953= 1198691119868 (B3)

or equivalently

11989511198693= 1198692

11989511198952= 1198953

1198951119868 = 1198691

(B4)


119904 = radic1205962 + 1199092

11198901198951120572120596 + radic120582

2

2+ 1205822

311989011989511205721205821198693

+ radic1199092

2+ 1199092

311989011989511205721199091198952+ radic1199052 + 120582

2

11198901198951120572119905119868

(B5)


cos120572120596=

120596

radic1205962 + 1199092

1

cos120572120582=

1205823

radic1205822

2+ 1205822

3

cos120572119909=

1199092

radic1199092

2+ 1199092

3

cos120572119905=

119888119905

radic1199052 + 1205822

1

(B6)



1) (1205822 1205823) (1199092 1199093) and (120582

1 119905) by the




119899 with the



119890119869119899120579119899 = cosh 120579

119899+ 119869119899sinh 120579

119899 (B7)




1198691= 11986921198953= minus11986931198952= 1198951119868 (B8)

or11986911198692= minus1198953

11986911198952= 1198693

1198691119868 = 1198951

(B9)



119904 = radic1205962 minus 1205822

11198901198691120579120596 + radic119909

2

3minus 1205822

2119890minus11986911205791205821198692

+ radic1199092

2minus 1205822

311989011986911205791199091198952+ radic1199092

1minus 11990521198901198691120579119905119868

(B10)


cosh 120579120596=

120596

radic1205962 minus 1199092

1

cosh 120579120582=

1205822

radic1199092

3minus 1205822

2

cosh 120579119909=

1199092

radic1199092

2minus 1205822

3

cosh120572119905=

119905

radic1199092

1minus 1199052

(B11)







2gt 0 having


119890119868120590= cosh120590 + 119868 sinh120590 (B12)


1198681198951= minus1198691

1198681198952= minus1198692

1198681198953= minus1198693

(B13)



2

1minus 1205822

1119890minus11986812059011198951

+ radic1199092

2minus 1205822

2119890minus11986812059021198952+ radic1199092

3minus 1205822

3119890minus11986812059031198953

(B14)


cosh120590120596=

120596

radic1205962 minus 1199052

cosh1205901=

1199091

radic1199092

1minus 1205822

1

cosh1205902=

1199092

radic1199092

2minus 1205822

2

cosh1205903=

1199093

radic1199092

3minus 1205822

3

(B15)









1minus 1205822

1 radic1199092

2minus 1205822

2

and radic1199092

3minus 1205822





2 For



1= 11986921198693=

11989521198953= 1198691119868 has the form

1198951015840

1= 1198951

1198951015840

2= 1198952cos1205721+ 1198953sin1205721

1198951015840

3= 1198951015840

11198951015840

2= 1198953cos1205721minus 1198952sin1205721

1198691015840

1= 1198691cos1205731+ 119868 sin120573

1

1198681015840= 1198691015840

11198951015840

1= 119868 cos120573

1minus 1198691sin1205731

1198691015840

2= 1198951015840

21198681015840= 1198692cos (120572

1minus 1205731) + 1198693sin (120572

1minus 1205731)

1198691015840

3= 1198951015840

31198681015840= 1198693cos (120572

1minus 1205731) minus 1198692sin (120572

1minus 1205731)

(C1)


1 1198952 and 119869

1in


3 119868 1198692 and 119869

3in (C1)) are


119899 1198951015840


rules as 119869119899 119895119899 119868



3= 11986911198692= 11989511198952= 1198693119868 axes each


3 1205733 Three



119899by the angles

120579119899and 120601



1198691015840

1= 1198691

1198691015840

2=1198692cosh (120601

1+ 1205791)

2+1198953sinh (120601

1+ 1205791)

2

1198951015840

3= 1198691015840

11198691015840

2=1198953cosh (120579

1+ 1206011)

2+1198692sinh (120579

1+ 1206011)

2

1198681015840= 119868 cosh 120579

1minus 1198951sinh 120579

1

1198951015840

1= 1198691015840

11198681015840= 1198951cosh 120579

1minus 119868 sinh 120579

1

1198951015840

2= 1198691015840

21198681015840=1198952cosh (120601

1minus 1205791)

2+1198693sinh (120601

1minus 1205791)

2

1198691015840

3= 1198951015840

31198681015840=1198693cosh (120601

1minus 1205791)

2+1198952sinh (120601

1minus 1205791)

2

(C2)


119899and 119909

119899in


11989511198693= 11986911198953= 1198952119868 and 119869

3= 11989521198691= 11986921198951= 1198953119868 axes generate




1

and 1198962

1198681015840= 119868

1198951015840

1= 1198951cosh 119896

1+ 1198691sinh 119896

1

1198951015840

2= 1198952cosh 119896

2+ 1198692sinh 119896

2

1198691015840

1= 1198951015840

11198681015840= 1198691cosh 119896

1+ 1198951sinh 119896

1

1198691015840

2= 1198951015840

21198681015840= 1198692cosh 119896

2+ 1198952sinh 119896

2

1198951015840

3= 1198951015840

11198951015840

2= 1198953cosh (119896

1+ 1198962) minus 1198693sinh (119896

1+ 1198962)

1198691015840

3= 1198951015840

31198681015840= 1198693cosh (119896

1+ 1198962) minus 1198953sinh (119896

1+ 1198962)

(C3)


119899in (14)


119899

1198961= 1205931minus1

3sum

119899

120593119899

1198962= 1205932minus1

3sum

119899

120593119899

minus (1198961+ 1198962) = 1205933minus1

3sum

119899

120593119899

(C4)


The group 1198661198731198622


119883119899119899= minus119911119899

120597

120597119911119899

+ 119910119899

120597

120597119910119899

+1

3

3

sum

119898=1

(119911119898

120597

120597119911119898

minus 119910119898

120597

120597119910119898

)


1198831198990= minus2119905

120597

120597119911119899

+ 119910119899

120597

120597119905

+1

2120576119899119898119896

(119911119898 120597

120597119910119896

minus 119911119896 120597

120597119910119898

)

1198830119899= minus2119905

120597

120597119910119899

+ 119911119899

120597

120597119905

+1

2120576119899119898119896

(119910119898 120597

120597119911119896

minus 119910119896 120597

120597119911119898

)

119883119899119898

= minus119911119898

120597

120597119911119899

+ 119910119899

120597

120597119910119898

(119899 = 119898)

(D1)



11988311+ 11988322+ 11988333= 0 (D2)



119899and 120582

119899

119910119899= 120582119899+ 119909119899

120597

120597119910119899

=1

2(

120597

120597120582119899

+120597

120597119909119899

)

119911119899= 120582119899minus 119909119899

120597

120597119911119899

=1

2(

120597

120597120582119899

minus120597

120597119909119899

)

(D3)


119899120582119899+ 1198891199052minus 119909119899119909119899

119883119899119899

= 119909119899

120597

120597119909119899

+ 120582119899

120597

120597120582119899

minus1

3

3

sum

119898=1

(119909119898

120597

120597119909119898

+ 120582119898

120597

120597120582119898

)


1198831198990

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

)

+ (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)+14120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

+1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

1198830119899

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

) minus (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)

+1

4120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

minus1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

+1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

119883119899119898

= minus1

2(120582119899

120597

120597120582119898

minus 120582119898

120597

120597120582119899

)

minus1

2(119909119899

120597

120597119909119898

minus 119909119898

120597

120597119909119899

)

+1

2(119909119899

120597

120597120582119898

+ 120582119898

120597

120597119909119899

)

+1

2(120582119899

120597

120597119909119898

+ 119909119898

120597

120597120582119899

) (119899 = 119898)

(D4)

E Zero Divisors



119863plusmn(119869)

119899=1

2(1 plusmn 119869

119899)

119863plusmn(119868)

=1

2(1 plusmn 119868)

(119899 = 1 2 3)

(E1)


119863plusmn(119869)

119899119863plusmn(119869)

119899= 119863plusmn(119869)

119899

119863+(119869)

119899119863minus(119869)

119899= 0

119863plusmn(119868)

119863plusmn(119868)

= 119863plusmn(119868)

119863+(119868)

119863minus(119868)

= 0

(E2)


119866plusmn(119869)

119899=1

2(119869119899plusmn 119895119899)

119866plusmn(119868)

119899=1

2(119868 plusmn 119895

119899)

(119899 = 1 2 3)

(E3)

with the properties

119866plusmn(119869)

119899119866plusmn(119869)

119899= 0

119866plusmn(119869)

119899119866∓(119869)

119899= 119863∓(119868)

119866plusmn(119868)

119899119866plusmn(119868)

119899= 0

119866plusmn(119868)

119899119866∓(119868)

119899= 119863plusmn(119869)

119899

(E4)






119866plusmn(119869)

119866∓(119869)

+ 119866∓(119869)

119866plusmn(119869)

= 1

119866plusmn(119868)

119866∓(119868)

+ 119866∓(119868)

119866plusmn(119868)

= 1

(E5)



119863plusmn(119869)

119899119866plusmn(119868)

119899= 119866plusmn(119868)

119899

119863plusmn(119869)

119899119866∓(119868)

119899= 0

119863plusmn(119868)

119866plusmn(119869)

119899= 0

119863plusmn(119868)

119866∓(119869)

119899= 119866∓(119869)

119899

(E6)


119904 = 119863+(119869)

119899(1

3120596 + 120582119899) + 119866

+(119868)

119899(119888119905 + 119909

119899)

+ 119863minus(119869)

119899(1

3120596 minus 120582119899) + 119866

minus(119868)

119899(119888119905 minus 119909

119899)


119904 = 119863+(119868)

(120596 + 119888119905) + 119866+(119869)

119899(120582119899+ 119909119899) + 119863minus(119868)

(120596 minus 119888119905)

+ 119866minus(119869)

119899(120582119899minus 119909119899)

(E7)




Acknowledgments


References


































































2 I




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where

cosh 120579 = 120596

119873

sinh 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A13)



119904 = 119873 (cos 120579 + 120598 sin 120579) (A14)

where

cos 120579 = 120596

119873

sin 120579 = |119881|

119873

120598 =120582119899119869119899+ 119909119899119895119899+ 119888119905119868

|119881|

(A15)




B One-Side Products




119877s = 1198901205981205792

119904 (B1)






119890119895119899120572119899 = cos120572

119899+ 119895119899sin120572119899 (B2)




1has three


1

1198951= 11986921198693= 11989521198953= 1198691119868 (B3)

or equivalently

11989511198693= 1198692

11989511198952= 1198953

1198951119868 = 1198691

(B4)


119904 = radic1205962 + 1199092

11198901198951120572120596 + radic120582

2

2+ 1205822

311989011989511205721205821198693

+ radic1199092

2+ 1199092

311989011989511205721199091198952+ radic1199052 + 120582

2

11198901198951120572119905119868

(B5)


cos120572120596=

120596

radic1205962 + 1199092

1

cos120572120582=

1205823

radic1205822

2+ 1205822

3

cos120572119909=

1199092

radic1199092

2+ 1199092

3

cos120572119905=

119888119905

radic1199052 + 1205822

1

(B6)



1) (1205822 1205823) (1199092 1199093) and (120582

1 119905) by the




119899 with the



119890119869119899120579119899 = cosh 120579

119899+ 119869119899sinh 120579

119899 (B7)




1198691= 11986921198953= minus11986931198952= 1198951119868 (B8)

or11986911198692= minus1198953

11986911198952= 1198693

1198691119868 = 1198951

(B9)



119904 = radic1205962 minus 1205822

11198901198691120579120596 + radic119909

2

3minus 1205822

2119890minus11986911205791205821198692

+ radic1199092

2minus 1205822

311989011986911205791199091198952+ radic1199092

1minus 11990521198901198691120579119905119868

(B10)


cosh 120579120596=

120596

radic1205962 minus 1199092

1

cosh 120579120582=

1205822

radic1199092

3minus 1205822

2

cosh 120579119909=

1199092

radic1199092

2minus 1205822

3

cosh120572119905=

119905

radic1199092

1minus 1199052

(B11)







2gt 0 having


119890119868120590= cosh120590 + 119868 sinh120590 (B12)


1198681198951= minus1198691

1198681198952= minus1198692

1198681198953= minus1198693

(B13)



2

1minus 1205822

1119890minus11986812059011198951

+ radic1199092

2minus 1205822

2119890minus11986812059021198952+ radic1199092

3minus 1205822

3119890minus11986812059031198953

(B14)


cosh120590120596=

120596

radic1205962 minus 1199052

cosh1205901=

1199091

radic1199092

1minus 1205822

1

cosh1205902=

1199092

radic1199092

2minus 1205822

2

cosh1205903=

1199093

radic1199092

3minus 1205822

3

(B15)









1minus 1205822

1 radic1199092

2minus 1205822

2

and radic1199092

3minus 1205822





2 For



1= 11986921198693=

11989521198953= 1198691119868 has the form

1198951015840

1= 1198951

1198951015840

2= 1198952cos1205721+ 1198953sin1205721

1198951015840

3= 1198951015840

11198951015840

2= 1198953cos1205721minus 1198952sin1205721

1198691015840

1= 1198691cos1205731+ 119868 sin120573

1

1198681015840= 1198691015840

11198951015840

1= 119868 cos120573

1minus 1198691sin1205731

1198691015840

2= 1198951015840

21198681015840= 1198692cos (120572

1minus 1205731) + 1198693sin (120572

1minus 1205731)

1198691015840

3= 1198951015840

31198681015840= 1198693cos (120572

1minus 1205731) minus 1198692sin (120572

1minus 1205731)

(C1)


1 1198952 and 119869

1in


3 119868 1198692 and 119869

3in (C1)) are


119899 1198951015840


rules as 119869119899 119895119899 119868



3= 11986911198692= 11989511198952= 1198693119868 axes each


3 1205733 Three



119899by the angles

120579119899and 120601



1198691015840

1= 1198691

1198691015840

2=1198692cosh (120601

1+ 1205791)

2+1198953sinh (120601

1+ 1205791)

2

1198951015840

3= 1198691015840

11198691015840

2=1198953cosh (120579

1+ 1206011)

2+1198692sinh (120579

1+ 1206011)

2

1198681015840= 119868 cosh 120579

1minus 1198951sinh 120579

1

1198951015840

1= 1198691015840

11198681015840= 1198951cosh 120579

1minus 119868 sinh 120579

1

1198951015840

2= 1198691015840

21198681015840=1198952cosh (120601

1minus 1205791)

2+1198693sinh (120601

1minus 1205791)

2

1198691015840

3= 1198951015840

31198681015840=1198693cosh (120601

1minus 1205791)

2+1198952sinh (120601

1minus 1205791)

2

(C2)


119899and 119909

119899in


11989511198693= 11986911198953= 1198952119868 and 119869

3= 11989521198691= 11986921198951= 1198953119868 axes generate




1

and 1198962

1198681015840= 119868

1198951015840

1= 1198951cosh 119896

1+ 1198691sinh 119896

1

1198951015840

2= 1198952cosh 119896

2+ 1198692sinh 119896

2

1198691015840

1= 1198951015840

11198681015840= 1198691cosh 119896

1+ 1198951sinh 119896

1

1198691015840

2= 1198951015840

21198681015840= 1198692cosh 119896

2+ 1198952sinh 119896

2

1198951015840

3= 1198951015840

11198951015840

2= 1198953cosh (119896

1+ 1198962) minus 1198693sinh (119896

1+ 1198962)

1198691015840

3= 1198951015840

31198681015840= 1198693cosh (119896

1+ 1198962) minus 1198953sinh (119896

1+ 1198962)

(C3)


119899in (14)


119899

1198961= 1205931minus1

3sum

119899

120593119899

1198962= 1205932minus1

3sum

119899

120593119899

minus (1198961+ 1198962) = 1205933minus1

3sum

119899

120593119899

(C4)


The group 1198661198731198622


119883119899119899= minus119911119899

120597

120597119911119899

+ 119910119899

120597

120597119910119899

+1

3

3

sum

119898=1

(119911119898

120597

120597119911119898

minus 119910119898

120597

120597119910119898

)


1198831198990= minus2119905

120597

120597119911119899

+ 119910119899

120597

120597119905

+1

2120576119899119898119896

(119911119898 120597

120597119910119896

minus 119911119896 120597

120597119910119898

)

1198830119899= minus2119905

120597

120597119910119899

+ 119911119899

120597

120597119905

+1

2120576119899119898119896

(119910119898 120597

120597119911119896

minus 119910119896 120597

120597119911119898

)

119883119899119898

= minus119911119898

120597

120597119911119899

+ 119910119899

120597

120597119910119898

(119899 = 119898)

(D1)



11988311+ 11988322+ 11988333= 0 (D2)



119899and 120582

119899

119910119899= 120582119899+ 119909119899

120597

120597119910119899

=1

2(

120597

120597120582119899

+120597

120597119909119899

)

119911119899= 120582119899minus 119909119899

120597

120597119911119899

=1

2(

120597

120597120582119899

minus120597

120597119909119899

)

(D3)


119899120582119899+ 1198891199052minus 119909119899119909119899

119883119899119899

= 119909119899

120597

120597119909119899

+ 120582119899

120597

120597120582119899

minus1

3

3

sum

119898=1

(119909119898

120597

120597119909119898

+ 120582119898

120597

120597120582119898

)


1198831198990

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

)

+ (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)+14120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

+1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

1198830119899

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

) minus (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)

+1

4120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

minus1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

+1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

119883119899119898

= minus1

2(120582119899

120597

120597120582119898

minus 120582119898

120597

120597120582119899

)

minus1

2(119909119899

120597

120597119909119898

minus 119909119898

120597

120597119909119899

)

+1

2(119909119899

120597

120597120582119898

+ 120582119898

120597

120597119909119899

)

+1

2(120582119899

120597

120597119909119898

+ 119909119898

120597

120597120582119899

) (119899 = 119898)

(D4)

E Zero Divisors



119863plusmn(119869)

119899=1

2(1 plusmn 119869

119899)

119863plusmn(119868)

=1

2(1 plusmn 119868)

(119899 = 1 2 3)

(E1)


119863plusmn(119869)

119899119863plusmn(119869)

119899= 119863plusmn(119869)

119899

119863+(119869)

119899119863minus(119869)

119899= 0

119863plusmn(119868)

119863plusmn(119868)

= 119863plusmn(119868)

119863+(119868)

119863minus(119868)

= 0

(E2)


119866plusmn(119869)

119899=1

2(119869119899plusmn 119895119899)

119866plusmn(119868)

119899=1

2(119868 plusmn 119895

119899)

(119899 = 1 2 3)

(E3)

with the properties

119866plusmn(119869)

119899119866plusmn(119869)

119899= 0

119866plusmn(119869)

119899119866∓(119869)

119899= 119863∓(119868)

119866plusmn(119868)

119899119866plusmn(119868)

119899= 0

119866plusmn(119868)

119899119866∓(119868)

119899= 119863plusmn(119869)

119899

(E4)






119866plusmn(119869)

119866∓(119869)

+ 119866∓(119869)

119866plusmn(119869)

= 1

119866plusmn(119868)

119866∓(119868)

+ 119866∓(119868)

119866plusmn(119868)

= 1

(E5)



119863plusmn(119869)

119899119866plusmn(119868)

119899= 119866plusmn(119868)

119899

119863plusmn(119869)

119899119866∓(119868)

119899= 0

119863plusmn(119868)

119866plusmn(119869)

119899= 0

119863plusmn(119868)

119866∓(119869)

119899= 119866∓(119869)

119899

(E6)


119904 = 119863+(119869)

119899(1

3120596 + 120582119899) + 119866

+(119868)

119899(119888119905 + 119909

119899)

+ 119863minus(119869)

119899(1

3120596 minus 120582119899) + 119866

minus(119868)

119899(119888119905 minus 119909

119899)


119904 = 119863+(119868)

(120596 + 119888119905) + 119866+(119869)

119899(120582119899+ 119909119899) + 119863minus(119868)

(120596 minus 119888119905)

+ 119866minus(119869)

119899(120582119899minus 119909119899)

(E7)




Acknowledgments


References


































































2 I




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119904 = radic1205962 minus 1205822

11198901198691120579120596 + radic119909

2

3minus 1205822

2119890minus11986911205791205821198692

+ radic1199092

2minus 1205822

311989011986911205791199091198952+ radic1199092

1minus 11990521198901198691120579119905119868

(B10)


cosh 120579120596=

120596

radic1205962 minus 1199092

1

cosh 120579120582=

1205822

radic1199092

3minus 1205822

2

cosh 120579119909=

1199092

radic1199092

2minus 1205822

3

cosh120572119905=

119905

radic1199092

1minus 1199052

(B11)







2gt 0 having


119890119868120590= cosh120590 + 119868 sinh120590 (B12)


1198681198951= minus1198691

1198681198952= minus1198692

1198681198953= minus1198693

(B13)



2

1minus 1205822

1119890minus11986812059011198951

+ radic1199092

2minus 1205822

2119890minus11986812059021198952+ radic1199092

3minus 1205822

3119890minus11986812059031198953

(B14)


cosh120590120596=

120596

radic1205962 minus 1199052

cosh1205901=

1199091

radic1199092

1minus 1205822

1

cosh1205902=

1199092

radic1199092

2minus 1205822

2

cosh1205903=

1199093

radic1199092

3minus 1205822

3

(B15)









1minus 1205822

1 radic1199092

2minus 1205822

2

and radic1199092

3minus 1205822





2 For



1= 11986921198693=

11989521198953= 1198691119868 has the form

1198951015840

1= 1198951

1198951015840

2= 1198952cos1205721+ 1198953sin1205721

1198951015840

3= 1198951015840

11198951015840

2= 1198953cos1205721minus 1198952sin1205721

1198691015840

1= 1198691cos1205731+ 119868 sin120573

1

1198681015840= 1198691015840

11198951015840

1= 119868 cos120573

1minus 1198691sin1205731

1198691015840

2= 1198951015840

21198681015840= 1198692cos (120572

1minus 1205731) + 1198693sin (120572

1minus 1205731)

1198691015840

3= 1198951015840

31198681015840= 1198693cos (120572

1minus 1205731) minus 1198692sin (120572

1minus 1205731)

(C1)


1 1198952 and 119869

1in


3 119868 1198692 and 119869

3in (C1)) are


119899 1198951015840


rules as 119869119899 119895119899 119868



3= 11986911198692= 11989511198952= 1198693119868 axes each


3 1205733 Three



119899by the angles

120579119899and 120601



1198691015840

1= 1198691

1198691015840

2=1198692cosh (120601

1+ 1205791)

2+1198953sinh (120601

1+ 1205791)

2

1198951015840

3= 1198691015840

11198691015840

2=1198953cosh (120579

1+ 1206011)

2+1198692sinh (120579

1+ 1206011)

2

1198681015840= 119868 cosh 120579

1minus 1198951sinh 120579

1

1198951015840

1= 1198691015840

11198681015840= 1198951cosh 120579

1minus 119868 sinh 120579

1

1198951015840

2= 1198691015840

21198681015840=1198952cosh (120601

1minus 1205791)

2+1198693sinh (120601

1minus 1205791)

2

1198691015840

3= 1198951015840

31198681015840=1198693cosh (120601

1minus 1205791)

2+1198952sinh (120601

1minus 1205791)

2

(C2)


119899and 119909

119899in


11989511198693= 11986911198953= 1198952119868 and 119869

3= 11989521198691= 11986921198951= 1198953119868 axes generate




1

and 1198962

1198681015840= 119868

1198951015840

1= 1198951cosh 119896

1+ 1198691sinh 119896

1

1198951015840

2= 1198952cosh 119896

2+ 1198692sinh 119896

2

1198691015840

1= 1198951015840

11198681015840= 1198691cosh 119896

1+ 1198951sinh 119896

1

1198691015840

2= 1198951015840

21198681015840= 1198692cosh 119896

2+ 1198952sinh 119896

2

1198951015840

3= 1198951015840

11198951015840

2= 1198953cosh (119896

1+ 1198962) minus 1198693sinh (119896

1+ 1198962)

1198691015840

3= 1198951015840

31198681015840= 1198693cosh (119896

1+ 1198962) minus 1198953sinh (119896

1+ 1198962)

(C3)


119899in (14)


119899

1198961= 1205931minus1

3sum

119899

120593119899

1198962= 1205932minus1

3sum

119899

120593119899

minus (1198961+ 1198962) = 1205933minus1

3sum

119899

120593119899

(C4)


The group 1198661198731198622


119883119899119899= minus119911119899

120597

120597119911119899

+ 119910119899

120597

120597119910119899

+1

3

3

sum

119898=1

(119911119898

120597

120597119911119898

minus 119910119898

120597

120597119910119898

)


1198831198990= minus2119905

120597

120597119911119899

+ 119910119899

120597

120597119905

+1

2120576119899119898119896

(119911119898 120597

120597119910119896

minus 119911119896 120597

120597119910119898

)

1198830119899= minus2119905

120597

120597119910119899

+ 119911119899

120597

120597119905

+1

2120576119899119898119896

(119910119898 120597

120597119911119896

minus 119910119896 120597

120597119911119898

)

119883119899119898

= minus119911119898

120597

120597119911119899

+ 119910119899

120597

120597119910119898

(119899 = 119898)

(D1)



11988311+ 11988322+ 11988333= 0 (D2)



119899and 120582

119899

119910119899= 120582119899+ 119909119899

120597

120597119910119899

=1

2(

120597

120597120582119899

+120597

120597119909119899

)

119911119899= 120582119899minus 119909119899

120597

120597119911119899

=1

2(

120597

120597120582119899

minus120597

120597119909119899

)

(D3)


119899120582119899+ 1198891199052minus 119909119899119909119899

119883119899119899

= 119909119899

120597

120597119909119899

+ 120582119899

120597

120597120582119899

minus1

3

3

sum

119898=1

(119909119898

120597

120597119909119898

+ 120582119898

120597

120597120582119898

)


1198831198990

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

)

+ (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)+14120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

+1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

1198830119899

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

) minus (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)

+1

4120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

minus1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

+1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

119883119899119898

= minus1

2(120582119899

120597

120597120582119898

minus 120582119898

120597

120597120582119899

)

minus1

2(119909119899

120597

120597119909119898

minus 119909119898

120597

120597119909119899

)

+1

2(119909119899

120597

120597120582119898

+ 120582119898

120597

120597119909119899

)

+1

2(120582119899

120597

120597119909119898

+ 119909119898

120597

120597120582119899

) (119899 = 119898)

(D4)

E Zero Divisors



119863plusmn(119869)

119899=1

2(1 plusmn 119869

119899)

119863plusmn(119868)

=1

2(1 plusmn 119868)

(119899 = 1 2 3)

(E1)


119863plusmn(119869)

119899119863plusmn(119869)

119899= 119863plusmn(119869)

119899

119863+(119869)

119899119863minus(119869)

119899= 0

119863plusmn(119868)

119863plusmn(119868)

= 119863plusmn(119868)

119863+(119868)

119863minus(119868)

= 0

(E2)


119866plusmn(119869)

119899=1

2(119869119899plusmn 119895119899)

119866plusmn(119868)

119899=1

2(119868 plusmn 119895

119899)

(119899 = 1 2 3)

(E3)

with the properties

119866plusmn(119869)

119899119866plusmn(119869)

119899= 0

119866plusmn(119869)

119899119866∓(119869)

119899= 119863∓(119868)

119866plusmn(119868)

119899119866plusmn(119868)

119899= 0

119866plusmn(119868)

119899119866∓(119868)

119899= 119863plusmn(119869)

119899

(E4)






119866plusmn(119869)

119866∓(119869)

+ 119866∓(119869)

119866plusmn(119869)

= 1

119866plusmn(119868)

119866∓(119868)

+ 119866∓(119868)

119866plusmn(119868)

= 1

(E5)



119863plusmn(119869)

119899119866plusmn(119868)

119899= 119866plusmn(119868)

119899

119863plusmn(119869)

119899119866∓(119868)

119899= 0

119863plusmn(119868)

119866plusmn(119869)

119899= 0

119863plusmn(119868)

119866∓(119869)

119899= 119866∓(119869)

119899

(E6)


119904 = 119863+(119869)

119899(1

3120596 + 120582119899) + 119866

+(119868)

119899(119888119905 + 119909

119899)

+ 119863minus(119869)

119899(1

3120596 minus 120582119899) + 119866

minus(119868)

119899(119888119905 minus 119909

119899)


119904 = 119863+(119868)

(120596 + 119888119905) + 119866+(119869)

119899(120582119899+ 119909119899) + 119863minus(119868)

(120596 minus 119888119905)

+ 119866minus(119869)

119899(120582119899minus 119909119899)

(E7)




Acknowledgments


References


































































2 I




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











3= 11986911198692= 11989511198952= 1198693119868 axes each


3 1205733 Three



119899by the angles

120579119899and 120601



1198691015840

1= 1198691

1198691015840

2=1198692cosh (120601

1+ 1205791)

2+1198953sinh (120601

1+ 1205791)

2

1198951015840

3= 1198691015840

11198691015840

2=1198953cosh (120579

1+ 1206011)

2+1198692sinh (120579

1+ 1206011)

2

1198681015840= 119868 cosh 120579

1minus 1198951sinh 120579

1

1198951015840

1= 1198691015840

11198681015840= 1198951cosh 120579

1minus 119868 sinh 120579

1

1198951015840

2= 1198691015840

21198681015840=1198952cosh (120601

1minus 1205791)

2+1198693sinh (120601

1minus 1205791)

2

1198691015840

3= 1198951015840

31198681015840=1198693cosh (120601

1minus 1205791)

2+1198952sinh (120601

1minus 1205791)

2

(C2)


119899and 119909

119899in


11989511198693= 11986911198953= 1198952119868 and 119869

3= 11989521198691= 11986921198951= 1198953119868 axes generate




1

and 1198962

1198681015840= 119868

1198951015840

1= 1198951cosh 119896

1+ 1198691sinh 119896

1

1198951015840

2= 1198952cosh 119896

2+ 1198692sinh 119896

2

1198691015840

1= 1198951015840

11198681015840= 1198691cosh 119896

1+ 1198951sinh 119896

1

1198691015840

2= 1198951015840

21198681015840= 1198692cosh 119896

2+ 1198952sinh 119896

2

1198951015840

3= 1198951015840

11198951015840

2= 1198953cosh (119896

1+ 1198962) minus 1198693sinh (119896

1+ 1198962)

1198691015840

3= 1198951015840

31198681015840= 1198693cosh (119896

1+ 1198962) minus 1198953sinh (119896

1+ 1198962)

(C3)


119899in (14)


119899

1198961= 1205931minus1

3sum

119899

120593119899

1198962= 1205932minus1

3sum

119899

120593119899

minus (1198961+ 1198962) = 1205933minus1

3sum

119899

120593119899

(C4)


The group 1198661198731198622


119883119899119899= minus119911119899

120597

120597119911119899

+ 119910119899

120597

120597119910119899

+1

3

3

sum

119898=1

(119911119898

120597

120597119911119898

minus 119910119898

120597

120597119910119898

)


1198831198990= minus2119905

120597

120597119911119899

+ 119910119899

120597

120597119905

+1

2120576119899119898119896

(119911119898 120597

120597119910119896

minus 119911119896 120597

120597119910119898

)

1198830119899= minus2119905

120597

120597119910119899

+ 119911119899

120597

120597119905

+1

2120576119899119898119896

(119910119898 120597

120597119911119896

minus 119910119896 120597

120597119911119898

)

119883119899119898

= minus119911119898

120597

120597119911119899

+ 119910119899

120597

120597119910119898

(119899 = 119898)

(D1)



11988311+ 11988322+ 11988333= 0 (D2)



119899and 120582

119899

119910119899= 120582119899+ 119909119899

120597

120597119910119899

=1

2(

120597

120597120582119899

+120597

120597119909119899

)

119911119899= 120582119899minus 119909119899

120597

120597119911119899

=1

2(

120597

120597120582119899

minus120597

120597119909119899

)

(D3)


119899120582119899+ 1198891199052minus 119909119899119909119899

119883119899119899

= 119909119899

120597

120597119909119899

+ 120582119899

120597

120597120582119899

minus1

3

3

sum

119898=1

(119909119898

120597

120597119909119898

+ 120582119898

120597

120597120582119898

)


1198831198990

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

)

+ (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)+14120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

+1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

1198830119899

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

) minus (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)

+1

4120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

minus1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

+1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

119883119899119898

= minus1

2(120582119899

120597

120597120582119898

minus 120582119898

120597

120597120582119899

)

minus1

2(119909119899

120597

120597119909119898

minus 119909119898

120597

120597119909119899

)

+1

2(119909119899

120597

120597120582119898

+ 120582119898

120597

120597119909119899

)

+1

2(120582119899

120597

120597119909119898

+ 119909119898

120597

120597120582119899

) (119899 = 119898)

(D4)

E Zero Divisors



119863plusmn(119869)

119899=1

2(1 plusmn 119869

119899)

119863plusmn(119868)

=1

2(1 plusmn 119868)

(119899 = 1 2 3)

(E1)


119863plusmn(119869)

119899119863plusmn(119869)

119899= 119863plusmn(119869)

119899

119863+(119869)

119899119863minus(119869)

119899= 0

119863plusmn(119868)

119863plusmn(119868)

= 119863plusmn(119868)

119863+(119868)

119863minus(119868)

= 0

(E2)


119866plusmn(119869)

119899=1

2(119869119899plusmn 119895119899)

119866plusmn(119868)

119899=1

2(119868 plusmn 119895

119899)

(119899 = 1 2 3)

(E3)

with the properties

119866plusmn(119869)

119899119866plusmn(119869)

119899= 0

119866plusmn(119869)

119899119866∓(119869)

119899= 119863∓(119868)

119866plusmn(119868)

119899119866plusmn(119868)

119899= 0

119866plusmn(119868)

119899119866∓(119868)

119899= 119863plusmn(119869)

119899

(E4)






119866plusmn(119869)

119866∓(119869)

+ 119866∓(119869)

119866plusmn(119869)

= 1

119866plusmn(119868)

119866∓(119868)

+ 119866∓(119868)

119866plusmn(119868)

= 1

(E5)



119863plusmn(119869)

119899119866plusmn(119868)

119899= 119866plusmn(119868)

119899

119863plusmn(119869)

119899119866∓(119868)

119899= 0

119863plusmn(119868)

119866plusmn(119869)

119899= 0

119863plusmn(119868)

119866∓(119869)

119899= 119866∓(119869)

119899

(E6)


119904 = 119863+(119869)

119899(1

3120596 + 120582119899) + 119866

+(119868)

119899(119888119905 + 119909

119899)

+ 119863minus(119869)

119899(1

3120596 minus 120582119899) + 119866

minus(119868)

119899(119888119905 minus 119909

119899)


119904 = 119863+(119868)

(120596 + 119888119905) + 119866+(119869)

119899(120582119899+ 119909119899) + 119863minus(119868)

(120596 minus 119888119905)

+ 119866minus(119869)

119899(120582119899minus 119909119899)

(E7)




Acknowledgments


References


































































2 I




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1198831198990

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

)

+ (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)+14120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

+1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

1198830119899

= (120582119899

120597

120597119905minus 119905

120597

120597120582119899

) minus (119909119899

120597

120597119905+ 119905

120597

120597119909119899

)

+1

4120576119899119898119896

(120582119898 120597

120597120582119896

minus 120582119896 120597

120597120582119898

)

minus1

4120576119899119898119896

(119909119898 120597

120597119909119896

minus 119909119896 120597

120597119909119898

)

minus1

4120576119899119898119896

(120582119898 120597

120597119909119896

+ 119909119896 120597

120597120582119898

)

+1

4120576119899119898119896

(119909119898 120597

120597120582119896

+ 120582119896 120597

120597119909119898

)

119883119899119898

= minus1

2(120582119899

120597

120597120582119898

minus 120582119898

120597

120597120582119899

)

minus1

2(119909119899

120597

120597119909119898

minus 119909119898

120597

120597119909119899

)

+1

2(119909119899

120597

120597120582119898

+ 120582119898

120597

120597119909119899

)

+1

2(120582119899

120597

120597119909119898

+ 119909119898

120597

120597120582119899

) (119899 = 119898)

(D4)

E Zero Divisors



119863plusmn(119869)

119899=1

2(1 plusmn 119869

119899)

119863plusmn(119868)

=1

2(1 plusmn 119868)

(119899 = 1 2 3)

(E1)


119863plusmn(119869)

119899119863plusmn(119869)

119899= 119863plusmn(119869)

119899

119863+(119869)

119899119863minus(119869)

119899= 0

119863plusmn(119868)

119863plusmn(119868)

= 119863plusmn(119868)

119863+(119868)

119863minus(119868)

= 0

(E2)


119866plusmn(119869)

119899=1

2(119869119899plusmn 119895119899)

119866plusmn(119868)

119899=1

2(119868 plusmn 119895

119899)

(119899 = 1 2 3)

(E3)

with the properties

119866plusmn(119869)

119899119866plusmn(119869)

119899= 0

119866plusmn(119869)

119899119866∓(119869)

119899= 119863∓(119868)

119866plusmn(119868)

119899119866plusmn(119868)

119899= 0

119866plusmn(119868)

119899119866∓(119868)

119899= 119863plusmn(119869)

119899

(E4)






119866plusmn(119869)

119866∓(119869)

+ 119866∓(119869)

119866plusmn(119869)

= 1

119866plusmn(119868)

119866∓(119868)

+ 119866∓(119868)

119866plusmn(119868)

= 1

(E5)



119863plusmn(119869)

119899119866plusmn(119868)

119899= 119866plusmn(119868)

119899

119863plusmn(119869)

119899119866∓(119868)

119899= 0

119863plusmn(119868)

119866plusmn(119869)

119899= 0

119863plusmn(119868)

119866∓(119869)

119899= 119866∓(119869)

119899

(E6)


119904 = 119863+(119869)

119899(1

3120596 + 120582119899) + 119866

+(119868)

119899(119888119905 + 119909

119899)

+ 119863minus(119869)

119899(1

3120596 minus 120582119899) + 119866

minus(119868)

119899(119888119905 minus 119909

119899)


119904 = 119863+(119868)

(120596 + 119888119905) + 119866+(119869)

119899(120582119899+ 119909119899) + 119863minus(119868)

(120596 minus 119888119905)

+ 119866minus(119869)

119899(120582119899minus 119909119899)

(E7)




Acknowledgments


References


































































2 I




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119904 = 119863+(119868)

(120596 + 119888119905) + 119866+(119869)

119899(120582119899+ 119909119899) + 119863minus(119868)

(120596 minus 119888119905)

+ 119866minus(119869)

119899(120582119899minus 119909119899)

(E7)




Acknowledgments


References


































































2 I




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































2 I




























Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Geometrical Applications of Split...

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