On Properties of Generalized Quaternion Algebra

Ayoub B. Mamagani1* and Mehdi Jafari2

1- Department of mathematics, Islamic azad university, Salmas branch, Salmas, IRAN 2- Department of mathematics, University College of Science and Technology Elm o Fen, Urmia, IRAN

Corresponding author: Ayoub B. Mamagani

H , ABSTRACT: In this paper, we study the generalized quaternions,  and their algebraic properties. De Moivre's and Euler's formulas for these quaternions in different cases are investigated. The solutions n of equation q 1 is discussed mean while it has been shown that equation has uncountably many solutions for unit generalized quaternions. Finally, the relations between the powers of these quaternions are given.

Keywords: De Moivre’s formula, Generalized Quaternion, Lie group.

INTRODUCTION

Mathematically, quaternions represent the natural extension of complex numbers, forming an associative algebra under the addition and the multiplication. This algebra is an effective way for understanding many aspects of physics and kinematics. Nowadays, quaternions are used especially in the area of computer vision, computer graphics, animation, and to solve optimization problems involving the estimation of rigid body transformations as well. Obtaining the roots of a quaternion was given by Niven, (1942) and Brand, (1942). Brand proved De Moivre’s theorem and used it to find nth roots of a quaternion. Using De Moivre’s formula to find roots of a quaternion is more convenient. These formulas are also investigated in the cases of dual, split and complex quaternions (Kabadayi and Yayli, 2011; Ozdemir, 2009; Jafari, 2013). Whittlesey and Whittlesey, (1990) by the help of Euler's formula found the circles in the plane and the sphere in 3-space by means of the exponential expansions. In this paper, we briefly recall some fundamental properties of the generalized quaternions, and show that the set of all unit generalized quaternions with the group operation of quaternion multiplication is a Lie group of 3-dimension. Moreover, we obtain De-Moivre's and Euler's formulas for these quaternions in different cases. We use it to find n - th roots of a generalized quaternion. Finally, we give some example for the purpose of more clarification.

Preliminaries The Irish mathematician Rowan Hamilton struggled in vain to extend complex numbers to three dimensions. Eventually, he realized that it is necessary to go to four dimensions and he invented a new number system called the quaternions. Although, Hamilton did not use the ordered pair construction for quaternions, but he was the inventor of the pair construction for complex numbers. zz, A quaternion, is an ordered pair of complex numbers 12 i.e. z  , q( z12 , z ), z1 and 2 with addition and multiplication defined by (z1 , z 2 ) ( w 1 , w 2 )  ( z 1  w 1 , z 2  w 2 ), (zz , )( ww , ) ( zw  zwzw** ,  zw ), 12 12 11 2212 21 and azw(,)(,0)(,)(, a zw  azaw ), a  . qr, It turns out that multiplication is not commutative. That is, in general for quaternions we have

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rq qr.

This construction by pairs ties in nicely with the constructions of the rational, real, and complex numbers but is not the traditional approach. If we single out three special pairs and attach Hamilton’s notation to them as i( ,0), j  (0,1), k  (0, ) and identify (aa ,0) a , then we find (,)a aa  a  aaiajak    a  0 1 2 3 0 1 2 3 i which is the form Hamilton used to express quaternions. This form makes it quite clear that quaternions are a four- dimensional generalization of complex numbers.

The quaternions i,, j k satisfy the following relations: 2 2 2 i j  k  i j k  1. In the language of abstract algebra, the quaternions form a noncommutative, normed division algebra over . The eight-dimensional octonians O can be constructed from pairs of quaternions but there the chain ends. The only normed division algebras over are ,,H and O. (Heard, 2006).

Generalized Quaternion Algebra q This section summarizes the essentials of the algebra of generalized quaternions. A generalized quaternion is an expression of the form q a  a i  a j  a k 0 1 2 3 where a0,, a 1 a 2 and a3 are real numbers and i,, j k are quaternionic units satisfying the equalities i2-,-,-, j 2   k 2   ij k -,-, ji jk  i  kj and ki j-,,. ik   Î R  The set of all generalized quaternions is denoted by  (Jafari, 2012). We express the basic operations in the

form. The addition becomes as

()()a0 ai 1  ajak 2  3  b 0  bi 1  bjbk 2  3 ()()()()ab   abiabjabk      0 0 1 1 2 2 3 3 and the multiplication as

(a0 ai 1  aj 2  akb 3 )( 0  bi 1  bj 2  bk 3 )

()a0 b 0  a 1 b 1   a 2 b 2   a 3 b 3

()a1 b 0  a 0 b 1  a 3 b 2  a 2 b 3 i

()a2 b 0  a 3 b 1  a 0 b 2  a 1 b 3 j ().a b  a b  a b  a b k 3 0 2 1 1 2 0 3 q a  a i  a j  a k a q, Given 0 1 2 3 , 0 is called the scalar part of denoted by S q  a , 0

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a i a j a k q, and 1 2 3 is called the vector part of denoted by V(). q a i  a j  a k 1 2 3 q The conjugate of is q a  a i  a j  a k. 0 1 2 3 The norm of is N qq  qq  a2  a 2   a 2   a 2. q 0 1 2 3 N  0, The inverse of with q is 1 qq1  . N q 1 1  1  1 Clearly qq1  0 i  0 j  0 k . Note also that qp pq and ().qp p q

De Moivre’s Formula for Generalized Quaternions We investigate the properties of the generalized quaternions in two different cases.

Case 1: , are positive numbers. S3 S2 Let G be the set of all unit dual generalized quaternions and G be the set of all unit generalized vectors, that is, S3  {q  H : N  1}  H , G q  S2 {VqaiajakN ( )    :  a 2   a 2   a 2  1}. G 1 2 3Vq() 1 2 3 S3 Theorem 1. Under quaternionic multiplication, G is a Lie group of dimension 3. f :H  Proof: To show that with the multiplication is a group, let  be a differentiable function given as f(). q a2  a 2   a 2   a 2 0 1 2 3 S31 f  (1) H, G is a submanifold of  since 1 is a regular value of function f . Also, the following maps 3 3 3 33 1  :SSSGGG (,)qp qp  :SSGG q q sending to and sending to are both differentiable. q a  a i  a j  a k Every nonzero generalized quaternion 0 1 2 3 can be written in the polar form q N(cos u sin ) q 2 2 2 a0 a1  a 2  a 3 u cos  sin  . The unit generalized vector is given by where N and q N q 1 u( u1 , u 2 , u 3 ) ( a 1 , a 2 , a 3 ), a2  a 2  a 2 1 2 3 a2  a 2   a 2  0. with 1 2 3 u S,2 2 For any G since u 1 we have a natural generalization of Euler's formula for generalized quaternions 2  3  4 eu 1  u   u   ... 2! 3! 4! 2  4  3  5 1    ... u (    ...) 2! 4! 3! 5! cosu sin , for any real number  . For detailed information about Euler's formula, see (Whittlesey, 1990).

Lemma 1. For any we have

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cosu sin  cos   u sin   cos     u sin     . 1 1 2 2 1 2 1 2 u Theorem 2. (De Moivre's formula) Let q e cos  u sin be a unit generalized quaternion. Then for any n; integer n nu q e cos n  u sin n . Proof: The proof will be by induction on nonnegative integers n. For n  2 and on using the validity of theorem as lemma 1, one can show 2 (cosuu sin  )  cos2   sin2  n Suppose that (cosu sin  )  cos n   u sin n  , we aim to show n1 (cosu sin  )  cos( n  1)   u sin( n  1)  . Thus (cosu sin  )nn1  (cos   u sin  ) (cos   u sin  ) (cosn  u sin n  )(cos   u sin  ) cos(n   )  u sin( n    ) cos(n  1)  u sin( n  1) .

The formula holds for all integers n; 1 qucos sin , qn cos(  n )  u sin(  n )

cosn u sin n . Special case: If 1, then Theorem 2 holds for real quaternions, see (Cho,1998).

11111 21111 2 Example 1. Let q   ( , , )cos   ( , , )sin be a unit generalized quaternion. Every 2 2   33    3 power of this quaternion is found with the aid of theorem 3. For example, 9-th and 53-th powers are 2 1 1 1 1 2 q9 cos9 ( , , )sin9 333    1, and 2 1 1 1 1 2 q53 cos53 ( , , )sin 53 333    1 1 1 1 1    ( , , ). 22   

Theorem 3. De Moivre’s formula implies that there are uncountably many unit generalized quaternions q satisfying n q 1 for n≥3. u S,2 Proof: For every G the unit generalized quaternion 22 qucos sin nn q is of order n. For n 1or n 2, the generalized quaternion is independent of u.

1 1 1 1  1 1 1 1 1  Example 2. qu1  ( ,0, )  cos  sin is of order 8 and q2 (, , ) cosu sin is of order 6. 2 2  4 4 22  33

n n Theorem 4. Let qucos sin be a unit generalized quaternion. The equation xq has roots, and they are

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22kk    xk cos( )  u sin( ), k  0,1,2,..., n  1. nn xucos sin Proof: We assume that is a root of the equation xqn  , since the vector parts of x and q are the same. From Theorem 3, we have n xcos n u sin n , thus, we find cosnn cos  , sin  sin  , 22kk    So, the n roots of are xk cos( )  u sin( ), k  0,1,2,..., n  1. nn The relation between the powers of generalized quaternion can be found in the following theorem. 2 Theorem 5. Let q be a unit generalized quaternion with the polar form qucos sin . If m   {1},  np then qq if and only if n p(mod m ). n p(mod m ). Proof: Let Then we have n am p, where a . qn cos n u sin n cos(am  p )  u sin( am  p ) 22 cos(a  p )  u sin( a  p )  cos(p  a 2  )  u sin( p   a 2  ) cosp u sin p p  q . qqnp Now suppose qn cos n u sin n and qp cos p u sin p . If then we get cosn 

2 cos p and sinnp sin , which means n p  2  a , a  . Thus n p a or n p(mod m ).  1 1 1 1 2 Example 3. Let q1 ( ,0, ) be a unit generalized quaternion. From theorem 4, m 8, 2 2    /4 so we have q q9  q 17  ... q2 q 10  q 18  ... q3 q 11  q 19  ... q4 q 12  q 20 ...   1 ... 8 16 24 q q  q ...  1. Case 2: Let  be a positive number and  be a negative number. q a  a i  a j  a k , In this case, for a generalized quaternion 0 1 2 3 we can consider three different subcases. Subcase (i): Let norm of generalized quaternion be positive and the norm of its vector part negative, i.e. N a2  a 2   a 2   a 2 0, V   a 2   a 2   a 2  0. qq0 1 2 3 1 2 3 q In this case, the polar form of is defined as q r(cosh w sinh ), with r N  a2  a 2   a 2   a 2 , q 0 1 2 3

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a 2 2 2 0 a1   a 2   a 3 w cosh  and sinh  . The unit generalized vector (axis of quarternion) is defined as Nq N q 1 w( w1 , w 2 , w 3 ) ( a 1 , a 2 , a 3 ). a2   a 2   a 2 1 2 3 2 2 2 2 222 q Special case: If Nq  a0  a 1  a 2  a 3  0 and V a  a  a  0. 1,  1, q 1 2 3 Then the is a timelike quaternion with spacelike vector part and its polar form is qN(cosh  sinh  ), q 1 3 where   (,,)a1 a 2 a 3 is a spacelike unit vector in E and (Ozdemir, 2006) aaa222   1 1. 1 2 3

NV0, 0. Theorem 6. (De Moivre’s formula) Let q r(cosh w sinh ) be a generalized quaternion with qq n, Then for any integer we have nn q r(cosh n w sinh n ). nn Proof: We use induction on positive integers n.Assume that q r(cosh n w sinh n ) holds. Then, qnn1  r(cosh n  w sinh n  ) r (cosh   w sinh  ) rn1(cosh n  w sinh n  )(cosh   w sinh  ) rn1[(cosh n cosh   sinh n  sinh  )  w (cosh n  sinh   sinh n  cosh  ) rn1[cosh( n  1)  w sinh( n  1) ].

Hence, the formula is ture.

Subcase (ii): Let the norm of generalized quaternion be positive and the norm of its vector part positive, i.e. N a2  a 2   a 2   a 2 0, V   a 2   a 2   a 2  0. qq0 1 2 3 1 2 3 In the case, the polar form of is defined as q r(cos u sin ), with r N  a2  a 2   a 2   a 2 , q 0 1 2 3 a 2 2 2 0 a1  a 2  a 3 u cos  and sin  . The unit generalized vector (axis of quarternion) is defined as N N q q 1 u( u1 , u 2 , u 3 ) ( a 1 , a 2 , a 3 ). a2  a 2  a 2 1 2 3 Special case: If and V a222  a  a  0. q 1 2 3 Then the is a timelike quaternion with timelike vector part and its polar form is q N(cos u sin ), q E3 where u is a timelike unit vector in 1 and uu 1. (Ozdemir, 2006) NV0, 0. Theorem 7. (De Moivre’s formula) Let q r(cos u sin ) be a generalized quaternion with qqThen nn q r(cos n u sin n ), for any integer Proof: The proof of this theorem can be done using induction, similarly, to the proof of the Theorem 2.

Subcase (iii): The norm of generalized quaternion is negative, i.e.

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N a2  a 2   a 2   a 2  0. q 0 1 2 3 0 a2   a 2   a 2   a 2 a2  a 2   a 2  0. q Since 0 1 2 3 then 1 2 3 In the case, the polar form of is defined as q r(sinh u cosh ), with r N  a2  a 2   a 2   a 2 , q 0 1 2 3 2 2 2 a0 a   a   a sinh  cosh  1 2 3 . The unit generalized vector u (axis of quarternion) is defined as N and q N q 1 u( u1 , u 2 , u 3 ) ( a 1 , a 2 , a 3 ). a2   a 2   a 2 1 2 3

N a2  a 2  a 2  a 2  0. Special case: If 1,  1and norm is negative number, i.e. q 0 1 2 3 Then the is a spacelike quaternion and its polar form is q N(sinh u cosh ). q E.3 where u is a spacelike unit vector in 1 The product of two spacelike quaternions is timelike. (Ozdemir, 2006)

N  0. Theorem 8. (De Moivre’s formula) Let q r(sinh u cosh ) be a generalized quaternion with q Then for n, any integer we have n n r(sinh n u cosh n ), n is odd . q   rn (cosh n u sinh n ), n is even .  Proof: The proof follows immediately from the induction.

CONCULSION

In this paper, we defined and gave some of algebraic properties of generalized quaternion and investigated the Euler’s and De Moivre’s formulas for these quaternions in several cases. The relation between the powers of n generalized quaternions is given in theorem 5. We also showed that the equation q 1 has uncountably many solutions for any general unit generalized quaternion (Theorem 4).

REFERENCES

Brand L.1942. The roots of a Quaternion, American Mathematical Monthly 49 (8) 519-520. Cho E. 1998. De-Moivre’s formula for quaternions, Applied Mathematics Letters, 11(6) 33-35. Niven I. 1942. The roots of a quaternion, American Mathematical Monthly 49 (6) 386-388. Jafari M. 2013. On the matrix algebra of complex quaternions, Accepted for publication in journal Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica. Jafari M. 2012. Generalized Hamilton operators and their Lie groups, PhD thesis, Ankara university, Ankara, Turkey. Kabadayi H and Yayli Y. 2011. De Moivre's formula for dual quaternions, Kuwait journal of science, Vol. 38(1) 15-23. Heard William B. 2006. Rigid body mechanics, WILEY-VCH Verlag GmbH & Co., Weinheim, Germany. Ozdemir M. 2009. The roots of a split quaternion, Applied Mathematics Letters 22, 258-263. Whittlesey J and Whittlesey K. 1990. Some geometrical generalizations of Euler’s formula, International journal of mathematics education Sci. & Tech., 21(3) 461-468.

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