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On Properties of Generalized Quaternion Algebra Journal of Novel Applied Sciences Available online at www.jnasci.org ©2013 JNAS Journal-2013-2-12/683-689 ISSN 2322-5149 ©2013 JNAS 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 J Nov . Appl Sci., 2 (12): 683-689, 2013 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 , 0 684 J Nov . Appl Sci., 2 (12): 683-689, 2013 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 qq1 . Nq 1 1 1 1 Clearly qq1 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, 3 SG {q H : Nq 1} H , 2 2 2 2 SG {VqaiajakN ( ) 1 2 3 : a 1 a 2 a 3 1}. Vq() 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. Every nonzero generalized quaternion can be written in the polar form q Nq (cos u sin ) 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! cosu sin , for any real number . For detailed information about Euler's formula, see (Whittlesey, 1990). q a0 a 1 i a 2 j a 3 k Lemma 1. For any we have 685 J Nov . Appl Sci., 2 (12): 683-689, 2013 cos1u sin 1 cos 2 u sin 2 cos 1 2 u sin 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 (cosuu sin ) cos2 sin2 Suppose that (cosu sin )n cos n u sin n , we aim to show n1 (cosu sin ) cos( n 1) u sin( n 1) . Thus (cosu sin )nn1 (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 qucos sin , qn 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 gen- eralized 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 qn 1 for n ≥3. 2 Proof: Foru everyS,G the unit generalized quaternion 22 qucos sin nn q is of order n.
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