The algebraic structure of dual semi-quaternions

Journal of Selçuk University Natural and Applied Science

The Algebraic Structure of Dual

Department of Mathematics, Technical and Vocational University, Urmia, IRAN

Abstract

We introduce the dual semi-quaternions show that the set of unit dual semi

quaternions are given, and then bymeans of the

Keywords: De-Moiver’s theorem; dual semi

Introduction

The quaternion algebra was first introduced by William Rowan Hamilton as a successor to the complex numbers. Most recently, quaternions have enjoyed prominence in computer science, because they are t

rotations in three and four dimensions. Dual numbers and dual quaternions were introduced in the 19th century by W.K. Clifford as a tool for his geometrical investigation. A brief introduction to the semi

1997). Dyachkova(Dyachkova, 2007

of semi-quaternions with the quaternion product is a Lie group. In our previous work, we studied the algebraic properties of semi

Euler's formulas for these quaternions. The De Moivre's formula implies that there are uncountable many unit semi

2013).Then the matrix associated with a semi Moivre’s formula the n -th

By the Hamilton operators, we derived the kinematic mapping of Bla ———

1

Corresponding author. Tel.:

+0-Journal of Selçuk University Natural and Applied Science

Online ISSN: 2147-3781

w w w . j o s u n a s . o r g 5 ( 3 ) : 1 5 - 2 4 , 2 0 1 6

The Algebraic Structure of Dual Semi-quaternions Mehdi JAFARI1

Department of Mathematics, Technical and Vocational University, Urmia, IRAN

quaternions algebra, Hds, and study some fundamental algebraic properties of them.

show that the set of unit dual semi-quaternions is a subgroup of Hds. The polar representation of dual semi means of the De-Moivre's theorem, any powers of these quaternions are obtained.

semi-quaternion; Trigonometric Form

The quaternion algebra was first introduced by William Rowan Hamilton as a successor to the complex numbers. Most recently, quaternions have enjoyed prominence in computer science, because they are the simplest algebraic tools for describing rotations in three and four dimensions. Dual numbers and dual quaternions were century by W.K. Clifford as a tool for his geometrical investigation. A brief introduction to the semi-quaternions is provided in (Rosenfeld, Dyachkova, 2007) has showed that the set of all invertible elements quaternions with the quaternion product is a Lie group. In our previous work, we studied the algebraic properties of semi-quaternions and gave the De

Euler's formulas for these quaternions. The De Moivre's formula implies that there are uncountable many unit semi-quaternions satisfying _{q }n 1 _{for n 2(Mortazaasl, Jafari}

Then the matrix associated with a semi-quaternion was studied and using the De

n power of such a matrix was obtained (Jafari By the Hamilton operators, we derived the kinematic mapping of Bla

-000-000-0000 ; fax: +0-000-000-0000 ; e-mail: mj_msc@yahoo.com

Journal of Selçuk University Natural and Applied Science

Department of Mathematics, Technical and Vocational University, Urmia, IRAN

and study some fundamental algebraic properties of them. We The polar representation of dual semi-, any powers of these quaternions are obtained.

The quaternion algebra was first introduced by William Rowan Hamilton as a successor to the complex numbers. Most recently, quaternions have enjoyed prominence he simplest algebraic tools for describing rotations in three and four dimensions. Dual numbers and dual quaternions were century by W.K. Clifford as a tool for his geometrical ons is provided in (Rosenfeld, has showed that the set of all invertible elements quaternions with the quaternion product is a Lie group. In our previous work, s and gave the De-Moivre's and Euler's formulas for these quaternions. The De Moivre's formula implies that there are 2 Mortazaasl, Jafari, quaternion was studied and using the

De-Jafari, Molaei, 2015). By the Hamilton operators, we derived the kinematic mapping of Blaschke and

Grünwald. It was argued that the corresponding geometry is a quasi-elliptic geometry (Jafari, 2015).

In this paper, we introduce the dual semi-quaternions algebra, Hds, and study some

fundamental algebraic properties of them, for the first time. We show that the set of unit dual semi-quaternions is a subgroup of Hds. The polar representations of dual

semi-quaternions are given, and then by means of the De-Moivre's theorem, any powers of these quaternions are obtained.Finally, we give some examples for more clarification.

Semi-quaternions Algebra

A semi-quaternion is defined as

01 1 2 3

qa a ia ja k

wherea a a₀, ₁, ₂ and a₃are real numbers and 1, , ,i j k  of q may be interpreted as the four basic vectors of Cartesian set of coordinates; and they satisfy the non-commutative multiplication rules 2 2 2 1, 0 , 0 , i j k ij k ji jk kj                     and . ki j ik

The set of all semi-quaternions are denoted by Hs.A semi-quaternion may be defined as

a pair (Sq,Vq), where Sq=a0 ∈ℝ is scalar part and Vq=a i1 a j2 a k3



 

is the vector part of q. The quaternion product of two quaternions q and p is defined as

, q p q p q p p q p q qp  S S  V V   S V S V V V where 0 0 1 1 3 1 1 3 1 2 2 1 , , , , 0 + ( ) + ( ) . q p q p q p S a S b V V a b V V i a b a b j a b a b k               It could be written 0 1 0 1 0 1 2 3 0 1 2 3 2 1 0 3 0 0 0 0 . a a b a a b qp a a a a b a a a a b                            

Obviously, quaternion multiplication is associative and distributive with respect to addition and subtraction, but the commutative law does not hold in general.

The conjugate of the semi-quaternionqS_qV_qis denoted by q, and defined as .

q q

q S V The norm of a semi-quaternionqa₀1a i₁a j₂a k₃: ( , , a a a a_{0 1 2}, ₃)is defined by

2 2 0 1, q

N q qq qa a

and say that q₀ q N_qis a unit semi-quaternion where N _q 0,which means that

0 0 1

a  a and a2,a3 are any real-numbers.The set S32 containing of all the unit

semi-quaternions is the 2-fold cover of the special orthogonal group SO(3). It is an analogue of the Hopf bundle.The inverse of q isdefined as

1 , q 0. q q q N N   

For detailed information about these concepts, we refer the reader to (Mortezaasl and Jafari, 2013).

Dual numbers Algebra

Letaanda be two real numbers, the combination*

*

,

A a a

is called a dual number. Here



is the dual unit. Dual numbers are considered as polynomials in , subject to the rules

2

0, 0 , .r r. ,

        forall r∈ℝ. The set of dual numbers, ,D forms a commutative ring having the *

a

 (a*real) as divisors of zero, not field. Some properties of dual numbers are

* *

* *

* *

sin( ) sin cos ,

cos( ) cos sin ,

0. 2 a a a a a a a a a a a a a a for a a                

For detailed information about dual numbers algebra, we refer the reader to (Keler, 2000).

Dual Semi-quaternions Algebra

A dual semi-quaternion Q has an expression of form

01 1 2 3

QA A iA jA k

whereA A A and A₀, ₁, ₂ ₃ are dual numbers and i j k , , are quaternionic units which satisfy the equalities

2 _1, 2 2 ₀ , 0 , i j k ij k ji jk kj                     and . ki j ik

* *

, , _s

Qqq q q H

whereq and _{q are real and pure dual semi-quaternion components, respectively.} *

A quaternion QA₀1A i₁A j₂A k₃ is pieced into two parts with scalar pieceSQ A0

and vectorial piece V_Q A i₁A j₂A k₃. We also write Q S _QV_Q. The set of all dual semi-quaternions (abbreviated DSQ) are denoted by Hds.

The addition rule for DSQ is component-wise addition:

0 1 2 3 0 1 2 3 0 0 1 1 2 2 3 3 ( ) ( ) ( ) ( ) ( ) ( ) . Q P A A i A j A k B B i B j B k A B A B i A B j A B k                          

This rule preserves the associativity and commutativity properties of addition. The product of scalar and a dual semi-quaternion is defined in a straightforward manner. If c is a scalar andQH_ds,

0 ( 1) ( 2) ( 3)

cQcA  cA i cA j cA k

Dual semi-quaternionicmultiplication of two quaternionQ S _QV_Qand PS_PV_Pis defined 0 0 1 1 0 1 1 0 2 0 3 1 0 2 1 3 0 3 1 2 2 1 3 0 * * , ( ) ( ) ( ) ( ) ( ). Q P Q P Q P P Q Q P QP S S V V S V S V V V A B A B A B A B i A B A B A B A B j A B A B A B A B k qp  qp q p                              

The generalized quaternion product can be described by a matrix-vector product as

0 1 0 1 0 1 2 3 0 1 2 3 2 1 0 3 0 0 0 0 . A A B A A B QP A A A A B A A A A B                            

Obviously, the quaternion multiplication is associative and distributive with respect to addition and subtraction, but the commutativity law does not hold in general.

Corollary 1.Hds 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 quaternions.

Some Properties of Dual Semi-Quaternions

1) The Hamiltonconjugate of dual semi-quaternion

01 1 2 3 Q Q

Q A A iA jA k S V

01 1 2 3 Q Q

Q A A iA jA kS V

Conjugate of product of two dual semi-quaternion and its own are described as ,

PQ Q P Q Q  It is clear the scalar and vector parts of Qis denoted by

2 Q Q Q S   and . 2 Q Q Q V   2) The norm of Qis 2 2 0 1. Q N Q QQQA A

It satisfies the following property

PQ P Q Q P

N N N N N

If N _Q 1, then Qis called a unit dual semi-quaternion. We will use 1 ds

H to denote the set of unit DSQ. If N _Q 0,thenQis called a null dual semi-quaternion.

3) The inverse of QwithN _Q 0, is 1 1

,

Q

Q Q

N



 with the following properties;

i) (QP)⁻¹ _{= P}⁻¹_Q⁻¹_{, ii) (λQ)}⁻¹ _{= (1/λ) Q}⁻¹_{, iii) N}

Q⁻¹ = 1/NQ.

Theorem 1. The set 1

ds

H of unit DSQ is a subgroup of the group 0 ds

H where 0

ds

H is the set of all non-zero dual semi-quaternions.

Proof: Let 1

, _ds.

Q PH We have N_{Q P}1,i.e. 1

ds

Q PH and thus the first subgroup requirement is satisfied. Also, by the property

1 1,

Q Q Q

N N N  

the second subgroup requirement 1 1

. ds

Q H

Trigonometric Form and De Moivre’s Theorem

Every non-null dual semi-quaternion

01 1 2 3 ,

QA A iA jA k

withcan be written in the trigonometric (polar) form (cos sin ), QR W  where 2 2 0 1 Q R N  A A and

0 cos A R  andsin A1 . R  Here *

    is a dual angle and the unit dual vector Wis given by

1 2 3 1 1 , 0. A i A j A k W A A        

This is similar to polar coordinate expression of a semi-quaternion. Example 2. The trigonometric forms of the DSQ

1 1 3 3 1 ( ) ( ) 2 2 2 2 Q      i jk isQ₁cosW₁sin , and 2 1 1 (1 ) 2 2 Q   i j  k is ₂ 1 (cos ₂sin ), 4 4 2 Q   W  where 1 2 1 , , 2 2(1 ) . 3 3 1 3 1 2 2 2 2 W i j k W i j k                            It is clear that 1 2 1. W W N N 

Lemma 1. For every unit vector 2

, (since 1)

W W   we have



cos1Wsin1



cos2Wsin2



cos



12



Wsin



12



.

  

Theorem 2. (De-Moivre's formula) Let Q cosWsin , be a unit dual semi-quaternion. Then for any integer n;

cos sin ,

n

Q  nW n

Proof: The proof will be by induction on nonnegative integern.

For n 2 and on using the validity of theorem as lemma 1, one can show



cosWsin



2 cos 2Wsin 2 . Suppose that



cosWsin



n cosnWsinn,we aim to show





1

cosWsin n cos(n1)Wsin(n1) .

Thus



cos sin



1 (cos sin ) (cos sin )

(cos sin )(cos sin )

cos( ) sin( ) cos( 1) sin( 1) . n _n W W W n W n W n W n n W n                                       

The formula holds for all integersn; 1 _{cos sin ,} Q _ _W  cos( ) sin( ) cos sin . n Q n W n n W n      _{   }     Example 3. Let (1 3) ( 3 1 ) 2 2 2 2

Q     i jk be a unit dual semi-quaternion.

Every power of this quaternion is found with the aid of Theorem 2. For example, 30-th poweris 30 cos 30( ) sin 30( ), 3 3 1 30 . Q W W              Since 2 1

W   , we have a natural generalization of Euler's formula for dual quaternions

2 3 4 2 4 3 5 1 ... 2! 3! 4! (1 ...) ( ...) 2! 4! 3! 5! cos sin , W e W W W W  _                               

for any dual number.For detailed information about Euler's formula, see (Whittlesey, 1990).

We investigate some properties of the dual semi-quaternions by separating them to two cases:

1) Dual semi-quaternions with dual angles(  )

  ; i.e.

(cos sin ).

Q

Q N W 

2) Dual semi-quaternions with real angles(  ,  0)

  ; i.e.

(cos sin ).

Q

Q N W 

Theorem 3. De Moivre’s formula implies that there are uncountably many unit DSQ

cos sin

Q W satisfying Q n 1forn≥3.

Proof: For every unit vector W,the unit semi-quaternion

2 2 cos sin , Q W n n     

Example 4. 2 ( 2,1 , 2 1) 2 2 Q    is of order 8 and 3 ( , 11 , 2 ) 2 2 Q      is of order 12.

Remark 1.The equation _{Q }n 1_{does not have any solution for a general unit DSQ.}

Example5.Let (1 3) ( 3 1 )

2 2 2 2

Q     ijk be a unit DSQ. There is non (n 0)

such that Q n 1.

Theorem 4. Let QcosWsinbe a unit DSQ. The equation n

X Qhasnroots, and they are 2 2 cos( ) sin( ) , 0,1, 2,..., 1. k k k X W k n n n        

Proof: We assume thatX cosWsin is a root of the equation_Xn_{Q, since the} vector parts ofQand Xare the same. From Theorem 2, we have

cos sin ,

n

X  nW n

thus, we find

cosncos , sinnsin ,

So, the nroots of _Xn_Qare 2 2

cos( ) sin( ) , 0,1, 2,..., 1. k k k X W k n n n         

Example 6. Let 3 ( , 21 1,1 ) cos sin

2 2 6 6

Q      W  be a unit DSQ.The cube roots

of the quaternionQare 1 3 _cos( 6 2 _{) sin(} 6 2 _{) , 0,1, 2.} 3 3 k k k Q     W    k

For k 0, the first root is 13

0 cos sin

18 18

Q   W  ≈0.98 0.17W , and the second one for

1 k  is 13 1 13 13 cos sin 18 18

Q   W  ≈0.64 0.76W and third one is

1 3 2 25 25 cos sin 18 18 Q   W  ≈0.34 0.93 W. 

Also, it is easy to see that

1 1 1

3 3 3

0 1 2 0.

Q Q Q 

Theorem 5. Let Qbe a unit DSQ with the polar formQcosWsin.If 2 Z {1} p   

   then QnQm if and only if nm(mod ).p Proof: Let nm(mod ).p Then we havenap m ,where a Z.

cos sin cos( ) sin( ) 2 2 cos( ) sin( ) n Q n W n ap m W ap m a m W a m                        cos( 2 ) sin( 2 ) cos sin . m m a W m a m W m Q                Now suppose cos sin n

Q  nW nand Qm cosmWsinm. If Qn Qm then we get cosncosmandsinnsinm, which means

2 , Z. nm a a Thus n m 2a    ornm(mod ).p Example 7. Let 2 ( 2,1 , 2 ) 2 2

x     be a unit DSQ. From Theorem 5, 2 8,

/ 4 p     so we have 9 17 2 10 18 3 11 19 4 12 20 8 16 24 ... ... ... ... 1 ... 1. Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q                    Conclusion

In this paper, we defined and gave some of algebraic properties of dual semi-quaternions and investigated the De Moivre’s formulas for these semi-quaternions. The relation between the powers of DSQ is given in Theorem 2. We hope that these results will contribute to the study of physical science.

Futherwork

We will give a complete investigation to real matrix representations of DSQ, and give any powers of these matrices.

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