A Note on the Representation and Definition of Dual Split Semi-Quaternions Algebra

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Global Journal of Science Frontier Research: F Mathematics and Decision Sciences Volume 17 Issue 8 Version 1.0 Year 2017 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (USA) Online ISSN: 2249-4626 & Print ISSN: 0975-5896

A Note on the Representation and Definition of Dual Split Semi-Quaternions Algebra By Mehdi Jafari Afagh Higher Education Institute  Abstract- In this paper, dual split semi-quaternions algebra, Ηss, is defined for the first time, and some fundamental algebraic properties of its is studied. The set of all unit dual split semi-  quaternions is a subgroup of Ηss. Fortheremore, by De-Moivre’s formula, any powers of these quaternions are obtained.

Keywords: dual split semi-quaternion, de-moivre’s theorem, subgroup.

GJSFR-F Classification: MSC 2010: 11R52

ANoteontheRepresentationandDefinitionofDualSplitSemiQuaternionsAlgebra

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A Note on the Representation and Definition Notes of Dual Split Semi-Quaternions Algebra 201 r

Mehdi Jafari ea Y

 711 Abstract- In this paper, dual split semi-quaternions algebra, Ηss , is defined for the first time, and some

fundamental algebraic properties of its is studied. The set of all unit dual split semi-quaternions is a  subgroup of Ηss . Fortheremore, by De-Moivre’s formula, any powers of these quaternions are obtained.

Keywords: dual split semi-quaternion, de-moivre’s theorem, subgroup. V

I. Introduction VIII

ue ersion I s s

The quaternion number system was discovered by Hamilton, who was looking for I

an extension of the complex number system to use in various areas of mathematics. The different type of quaternions are suitable algebraic instructure for expressing important XVII space-time transformations as well as description of the classical and quantum filds. Dual numbers and dual quaternions were introduced in the 19th century by

W.K. Clifford, as a tool for his geometrical investigation. ) F In our previos work, we have studied the split semi-quaternions, and have (

presented some of their algebric properties. De Moivre’s and Euler’s formula for these quaternions are given (Jafari, 2015).We have shown that the set of all unit split semi- quaternions with the group operation of quaternion multiplication is a Lie group of 3- Research Volume dimension and find its Lie algebra and Killing bilinear form (Jafari, 2016). In this paper, we study the dual split semi-quaternions algebra and give some of their basic properties. We express De Moivre’s and Euler’s formulas for dual split semi- Frontier quaternions and find roots of a quaternion using these formulas. Finally, we give some

examples for more clarification.We hope that these results will contribute to the study Science of physical science. of

a) Split Semi-quaternions Algebra A split semi-quaternion has an expression of the form q Journal  q=+++ a01ai a2 j ak3

  Global where aaa012,, and a3 are real numbers and ijk,, ar e quaternionic units satisfying the equalities

 i2=1, jk 22 = = 0,       ij==−==− k ji,0 jk kj,

Author: Department of Mathematics, Afagh Higher Education Institute, Urmia, Iran. e-mail: [email protected]

and   ki= j = − ik.

The set of all split semi-quaternions is denoted by Ηss . For detailed information about this concept, we refer the reader to [3,4,6,9]. b) Dual Numbers Algebra Let and * be two real numbers, the combination a a Ref Aa= + εa*, 201

is called a dual number. Here ε is the dual unit. Dual numbers are considered as 3. r

ea polynomials inε subject to the rules , J Y afari M.,afari 721 ε≠=0, ε2 0, ε .rr== . εε, for all r .

The set of dual numbers, D, forms a commutative∈ℝ ring having the εa*( a* real) as

S divisors of zero, not field. Some properties of dual numbers are of on matrices the ome results V sin(aa+=εε**) sin aa+ cos a , VIII ** ue ersion I

s cos(aa+=εε ) cos aa − sin a , s

I a* a+=+εε a* a for a >0. XVII 2 a For detailed information about dual numbers algebra, we refer the reader to (Keler, 2000).

)

F c) Generalized Dual Quaternions Algebra ( A generalized dual quaternion Q has an expression of form

 Split Semi Q=++A01 Ai A2j + Ak3 Research Volume   where A,,AA and A are dual numbers and i,,jk are quaternionic units which 012 3

satisfy the equalities -quaternions, Frontier  i22=−=−=−α, jkβ,,2 αβ       

Science ij=k=−− ji , jk ==β i kj,

of and

 2016 ki=α j=−∈ ik ,αβ , R.

Journal , s The set of all generalized dual quaternions (abbreviated GDQ) are denoted by ubmitted.

Global  Hαβ . A generalized dual quaternion Q is a sum of a scalar and a vector, called scalar   part, SAQ = 0 , and vector part VQ =++Ai12 A j Ak3(Jafari,2015).

If SQ = 0, then Q is called pure generalized dual quaternion, we may be called its 3 generalized dual vector. The set of all generalized dual vectors denoted by Dαβ . Special cases:  1. αβ= =1, is considered, then Hαβ is the algebra of dual quaternions.

 2. αβ=1, = −1, is considered, then Hαβ is the algebra of split dual quaternions.  3. αβ=1, = 0, is considered, then Hαβ is the algebra of dual semi-quaternions.  4. αβ=−=1, 0, is considered, then Hαβ is the algebra of dual split semi-quaternions. 5. αβ= = is considered, then H is the algebra of dual quasi-quaternions (Jafari, 0, 0, αβ 2016). Notes Theorem 2. Every unit generalized dual quaternion is a screw operator. d) Dual split semi-quaternions Algebra A dual split semi-quaternion Qis defined as 201

 r

Q=A01+Ai +A2j+A 3k ea Y   731 where A01,,AA2and A3 are dual numbers and i,,jk are quaternionic units satisfying the equalities  2 22 i=1, jk = = 0,       V ij==−== k ji,0 jk kj VIII

and ue ersion I s  s ki = j = −ik. I XVII In other words, this may also be given as Qq= + ε q*, where qq, * are split semi- quaternions. The set of all dual split semi-quaternions(abbreviated dual SSQ) is denoted 

 by Ηss . We express the basic operations in terms of i, jk,. )  F ( Given Q=++A01 Ai A2j + Ak 3, A0 is called the scalar part of Q, denoted by

SAQ = 0 ,

 Research Volume and A12i++ A j A3k is called the vector part of Q, denoted by  

=++ Frontier V()Q Ai12 A j Ak3.

If SQ = 0, then Q is called pure dual SSQ. Science The addition becomes as of   QPAAiAjAkBBiBjBk+=++() + +++() + 01 2 3 012 3  Journal =(A0 +++ B0)( A11 Bi ) ++ ( A2 B 2 ) j ++() A 33 Bk

This rule preserves the associativity and commutativity properties of addition. The multiplication as Global   =++ + ++ + QP()AA01iA2jA 3k()BB 01iB 2jBk3  =(AB0 0 +++ AB 11)( AB 10 AB 01) i +   + −+ + + −++ ()AB20 AB 31 AB 0 2 AB 13 j() AB 30 AB 21 AB 12 AB 03 k Also, this can be written as

AA0100B0     AA00 B QP =  10 1  A− AAAB   2 301 2  A3−AAAB 210 3  Obviously, the quaternion multiplication is associative and distributive with respect to addition and subtraction, but the commutativity law does not hold in general.  Notes Corollary 1. Ηss with addition and multiplication has all the properties of a number field expect commutativity of the multiplication. It is therefore called the skew field of

201 quaternions. r

ea e) Some Properties of Dual Semi-Quaternions Y  1) The Ham ilton conjugate of =++ + is 741 Q A01Ai A2j Ak 3   Q=−−−A01Ai A2j Ak 3.  V The dual conjugate of Q=++A01 Ai A2j + Ak 3is

VIII  Q* = A01++ Ai A2 j+ Ak 3 ue ersion I

s  s ** * * =(a0−εε a0)+−( a 11 ai)+( a 2−ε a 2) j +−( a 33 ε ak ). I 

XVII The Hermitian conjugate of Q=++A01 Ai A2j + Ak3is  Q† = A− Ai− A j− Ak 01 2 3

** * * ) =(a0−εε a0)()(− a 11− aia− 2−ε aja 2)−− ( 33 ε ak ). F ( 2) The n orm of Q is

22 N Q =QQ = QQ = A01 − A Research Volume

The norm Q canbe dual number, real number, or zero. If NQ =1, then Q is called  1 Frontier a unit dual SSQ. We will use H ss to denote the set of all the unit dual SSQ.

If N = 0, then Q is called a null dual SSQ. A dual split semi-quaternion Q for Q  

Science which NQ = 0 has form Q=+== A2 j Ak3,( A 01 A 0) and it is a zero divisor.

of 3) The inverse of Q with NQ ≠ 0, is

−1 1

Journal QQ= . NQ  −1 −1 −11− Global Clearly QQ=1+++0 i 0j 0. k Note also that QP = PQ and (QP).= PQ

 1  0  0 Theorem 1. The set H ss of unit dualSSQ is a subgroup of the group Hss where Hss is the set of all non-zero dual split semi-quaternions.

 1  1 Proof: Let Q,.PH∈ ss We have NQP =1, i.e. QP ∈ H ss and thus the first subgroup requirement is satisfied. Also, by the property

= = = NQ NNQ Q−1 1,

−11 the second subgroup requirementQH∈ ss . Example 4. Consider the dual split semi-quaternions     Q1 =1 ++(1εε ) ij − + (ε − 1), kQi2 =2εε+ −(1 − ) j +− ( 1 +2ε ) k , and   =−εε+− +−ε+ = +ε+−ε Q34(1 ) (1 )i (1 3 ) j kQ, 2 2 i j k, Notes 1. The vector parts ofQQ12, are    V=(1 +εε)ijkVi − +−+ ( 1 2ε) ,= − (1 − ε )jk +−+ ( 1 2 ε ) . 201 QQ12 r ea 2. The Hamilton conjugates ofQQ23, are Y    751 Q23=2ε−+− i(1ε ) j − (−+ 1 2 ε ) kQ , =− (1 εε ) −− (1 )i −− (1 3 ε ) jk − ,

3. The dual conjugates ofQQ23, are

   V ** Q23=−2ε +++ i(1ε ) j −−− ( 1 2 ε ) kQ , =+ (1 εε ) ++ (1 )i ++ (1 3 ε ) jk + , VIII

ue ersion I 4. The Hermitian conjugate of are s QQ14, s

    I † † Q1 =1 −−(1εε ) ij − ++ (1ε ) k , Q4 =22+εε ij −− k, XVII 5. The norms are given by

N=−2ε,N =−== 1, NN 0, 2 Q1 Q2 QQ 34 ) F

( 6. The inverses of QQ12, are

1    −−11= − +εε + −ε− =−ε − + − ε −−+ ε

Q [1 (1 )i j ( 1) kQ ] , [2 i (1 ) j ( 1 2 ) k ] , Research Volume 122ε

and Q not invertible. 3 7. One can realize the following operations Frontier   QQ12+ =(1 +2εε ) + (2 + )i− (1 + 2ε ) j +−+ ( 2 3 ε ) k  Science

QQ23− =−+(1 3)εε +i +−+ (2 2) ε j +−+ (2 2) ε k of  QQ14=(2 + 2)[2εε + ++ (2 2)]i +   Journal ++−[1 (1 2 )εε ]jk − [(1 − 2 ) + 2 ] .

f) Trigonometric Form and De Moivre s Theorem Global ’ In this section, we express De-Moivre’s formula for dual SSQ. For this, we can

cosider two different cases:

Case 1. Let the norm of dual SSQ be positive.

The trigonometric (polar) form of a non-null dual SSQ  Q=++A01 Ai A2j + Ak 3

is  QR= (coshφφ+ W sinh )

where RN= Q , and

A A2 A coshφ = 0 , sinhφ = 1 = 1 . R R AA22− 12 ef  R φ = ϕ+ εϕ* is a dual angle and the unit dual vector W is given by

201   11 7. r Wwww=( 1,,)[23 = AiAjAk1 ++2 3] = ( AAA12 , ,3 ). ea 2 A (1997). R Y A1 1 osenfe 761 This is similar to polar coordinate expression of asplit quaternion [7], split semi- quaternion [3]. ld B.,ld Example 1.5. The trigonometric forms of the dual split semi-quaternions

V  

Q1 =2 +i+(1 +εε) jk+2 , is QW1= coshφφ11+ sinh 1, Geo

VIII   metry

ue ersion I = +εε ++ − − is = εφ+ φ s Q2 (1 ) i (1 ),jk QW2 2 [cosh22 sinh2 ] s

I where of Lie groups Lie of

XVII  cos hφ1= 2 , sinφ11= 1,W = (1,1 + εε , 2 )

11+ ε  φ= φ= = −− ε cosh 2, sinh22 ,W (1,1 , 1), ) 22εε F (

 , and NN= = −1. Kluw WW12  Theorem 1.5. (De Moivre's Theorem) If QR= (coshφφ+ W sinh ) be a dual SSQ and n is Research Volume any positive integer, then Netherlands, Publishers, Academic er  Qnn= R(cosh nWφφ+ sinhn ) Frontier

Proof: The proof is easily followed by induction on n.■ The Theorem holds for all integers n, since Science  of QR−−11= (coshφφ− Wsinh ),  Q−−nn=R[cosh(−+ nWφφ ) sinh( − n )] Journal  = R−n[cosh nWφφ− sinh n ]

Global  Example 2.5. LetQ=2 ++i(1 +εε ) jk − 2 . Find Q10 and Q−45.

Solution: First write Q in trigonometric form.  QW= 3(coshφφ + sinh ),  whereφ=ln 3 ,W = (1,1 +−εε , 2 ) Applying de Moivre’s Theorem gives:

35+−3−5 33 55− QW10=3 5 (cosh10φφ+=+ sinh10 ) 35 ( W) 22  QW−−45 = ( 3)45 (cosh 45φφ− sinh 45 )

Corollary 1.5. The equation Qn =1, does not have solution for a unit dual split semi- quaternion. Example 3.5. Let Q=1+3εi ++ (2εε ) jk + , be a dual split semi-quaternion. There is no n otes N (n> 0) such that Qn =1. Case 2. Let the norm of dual SSQ be negative, i.e. N=−

Q 01 201

The polar form of a non-null dual SSQ r ea Y  Q=++A01 Ai A2j + Ak 3 771 is  = ψψ+ QR(sinh W cosh ) V

VIII

where RN= Q , and ue ersion I s s

A A2 A I sinhψ = 0 , coshφ = 1 = 1 . R R 22 XVII AA1− 2  φ = ϕ+ εϕ* is a dual angle and the unit dual vector W is given by

)

 11 ( Wwww=( ,,)[ = AiAjAk ++] =( AAA , , ). 123 2 1 2 3 123 A A1 1

Futher Work Research Volume By the Hamilton operators, dual split semi-quaternions have been expressed in terms of 4×4 matrices. With the aid of the De-Moivre's formula, we will obtain any power of these matrices. Frontier

References Références Referencias Science 1. Jafari M., Split semi-quaternions algebra in semi-Euclidean 4-space,Cumhuriyet of Science Journal, Vol 36(1) (2015) 70-77. 2. Jafari M.,Matrices of generalized dual quaternions,Konuralp journal of mathematics,

Vol. 3(2), (2015)110-121. Journal 3. Jafari M., Some results on the matrices of Split Semi-quaternions, 2016, submitted. 4. Jafari M., Introduction to Dual Quasi -quaternions: Algebra and Geometry, Global researchgate.net/publication/282332207. 5. Jafari M., The Algebraic Structure of Dual Semi-quaternions, accepted for publication in “Journal of Selçuk University Natural and Applied Science ”. 6. Keler Max L., On the theory of screws and the dual method, Proceeding of a symposium commemorating the Legacy, works, and Life of Sir Robert Stawell Ball Upon the 100th Anniversary of a Treatise on the theory of Screws, University of Cambridge, Trinity College, July 9-11, 2000.

7. Rosenfe ld B., Geometry of Lie groups, Kluwer Academic Publishers, Netherlands, (1997). 8. Whittlesey J.,Whittlesey K.,Some Geometrical Generalizations of Euler's Formula, International journal of mathematical education in science & technology, 21(3) (1990) 461-468.

Notes 201 r ea Y

781 V VIII ue ersion I s s I XVII

) F ( Research Volume Frontier Science of Journal Global