ANoteontheRepresentationandDefinitionofDualSplitSemiQuaternionsAlgebra

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1 Global Journal of Science Frontier Research: F Mathematics Decision Sciences Volume 7 Iue 8 Version.0 Year 07 Type : Double Blind Peer Reviewed International Research Journal Publisher: Global Journals Inc. (US) Online ISSN: & Print ISSN: Note on the Representation Definition of Dual Split Semi-uaternions lgebra By Mehdi Jafari fagh Higher Education Institute bstract- In this paper, dual split semi-quaternions algebra, Η, is defined for the first time, some fundamental algebraic properties of its is studied. The set of all unit dual split semiquaternions is a subgroup of Η. 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 Claification: MSC 00: R5 NoteontheRepresentationDefinitionofDualSplitSemiuaternionslgebra Strictly as per the compliance regulations of: 07. Mehdi Jafari. This is a research/review paper, distributed under the terms of the Creative Commons ttribution- Noncommercial.0 Unported License permitting all non commercial use, distribution, reproduction in any medium, provided the original work is properly cited.

2 Note on the Representation Definition of Dual Split Semi-uaternions lgebra Mehdi Jafari bstract- In this paper, dual split semi-quaternions algebra, Η, is defined for the first time, some fundamental algebraic properties of its is studied. The set of all unit dual split semi-quaternions is a subgroup of Η. Fortheremore, by De-Moivre s formula, any powers of these quaternions are obtained. Keywords: dual split semi-quaternion, de-moivre s theorem, subgroup. I. Introduction The quaternion number system was discovered by Hamilton, who was looking for an extension of the complex number system to use in various areas of mathematics. The different type of quaternions are suitable algebraic instructure for expreing important space-time transformations as well as description of the claical quantum filds. Dual numbers dual quaternions were introduced in the 9th century by W.K. Clifford, as a tool for his geometrical investigation. In our previos work, we have studied the split semi-quaternions, have presented some of their algebric properties. De Moivre s Euler s formula for these quaternions are given (Jafari, 05).We have shown that the set of all unit split semiquaternions with the group operation of quaternion multiplication is a Lie group of - dimension find its Lie algebra Killing bilinear form (Jafari, 06). In this paper, we study the dual split semi-quaternions algebra give some of their basic properties. We expre De Moivre s Euler s formulas for dual split semiquaternions 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 of physical science. a) Split Semi-quaternions lgebra split semi-quaternion q has an expreion of the form q= a + ai + a j+ ak 0 where a 0, a, a a are real numbers i, j, k are quaternionic units satisfying the equalities i =, j = k = 0, ij = k = ji, jk = 0 =kj, Global Journal of Science Frontier Research ( F ) Volume XVII Iue VIII V ersion I Year 07 7 uthor: Department of Mathematics, fagh Higher Education Institute, Urmia, Iran. mj_msc@yahoo.com 07 Global Journals Inc. (US)

3 Note on the Representation Definition of Dual Split Semi-uaternions lgebra Global Journal of Science Frontier Research ( F ) Volume XVII Iue VIII V ersion I Year 07 7 ki = j = ik. The set of all split semi-quaternions is denoted by Η. For detailed information about this concept, we refer the reader to [,4,6,9]. b) Dual Numbers lgebra Let a a be two real numbers, the combination = a+ a, is called a dual number. Here is the dual unit. Dual numbers are considered as polynomials in, subject to the rules 0, = 0,. r = r. =, for all 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 a+ a = a+ a a a+ a = aa a + = a + a > 0. sin( ) sin cos, cos( ) cos sin, a a a for a For detailed information about dual numbers algebra, we refer the reader to (Keler, 000). c) Generalized Dual uaternions lgebra generalized dual quaternion has an expreion of form = 0 + i + j + k where 0,, are dual numbers i, j, k are quaternionic units which satisfy the equalities i = α, j = β, k =αβ, ij = k =ji, jk = βi = kj, ki = αj = ik, αβ, R. The set of all generalized dual quaternions (abbreviated GD) are denoted by H αβ. generalized dual quaternion is a sum of a scalar a vector, called scalar part,, S = 0 vector part V = i + j + k (Jafari,05). If S = 0, then is called pure generalized dual quaternion, we may be called its generalized dual vector. The set of all generalized dual vectors denoted by D αβ. Special cases:. α = β =, is considered, then Hαβ is the algebra of dual quaternions. Ref. Jafari M., Some results on the matrices of Split Semi-quaternions, 06, submitted. 07 Global Journals Inc. (US)

4 Note on the Representation Definition of Dual Split Semi-uaternions lgebra. α =, β =, is considered, then H αβ is the algebra of split dual quaternions.. α =, β = 0, is considered, then H αβ is the algebra of dual semi-quaternions. 4. α =, β = 0, is considered, then H αβ is the algebra of dual split semi-quaternions. 5. α = 0, β = 0, is considered, then H αβ is the algebra of dual quasi-quaternions (Jafari, 06). Theorem. Every unit generalized dual quaternion is a screw operator. d) Dual split semi-quaternions lgebra dual split semi-quaternion is defined as = 0 + i + j + k where 0,, are dual numbers i, j, k are quaternionic units satisfying the equalities i =, j = k = 0, ij = k = ji, jk = 0= kj ki = j = ik. In other words, this may also be given as = q+ q, where qq, are split semiquaternions. The set of all dual split semi-quaternions(abbreviated dual SS) is denoted by Η. We expre the basic operations in terms of i, jk,. Given = 0 + i + j + k, 0 is called the scalar part of, denoted by i j k + + S = is called the vector part of, denoted by If S = 0, then is called pure dual SS. The addition becomes as 0, V( ) = i + j+ k. + P= ( 0 + i + j + k ) + ( B0 + Bi + Bj + Bk) = ( + B ) + ( + B ) i + ( + B ) j + ( + B ) k 0 0 This rule preserves the aociativity commutativity properties of addition. The multiplication as P= ( 0 + i + j + k )( B0 + Bi + Bj + Bk) = ( B B ) + ( B 0 + B 0 ) i + + ( B B + B + B ) j+ ( B B + B + B ) k Global Journal of Science Frontier Research ( F ) Volume XVII Iue VIII V ersion I Year 07 7 lso, this can be written as 07 Global Journals Inc. (US)

5 Note on the Representation Definition of Dual Split Semi-uaternions lgebra Global Journal of Science Frontier Research ( F ) Volume XVII Iue VIII V ersion I Year B B 0 B 0 B P = Obviously, the quaternion multiplication is aociative distributive with respect to addition subtraction, but the commutativity law does not hold in general. Corollary. Η with addition multiplication has all the properties of a number field expect commutativity of the multiplication. It is therefore called the skew field of quaternions. e) Some Properties of Dual Semi-uaternions ) The Hamilton conjugate of = + i + j+ kis The dual conjugate of 0 = i j k 0. = + i + j+ kis 0 = + i+ j+ k = ( a a ) + ( a a ) i + ( a a ) j + ( a a ) k. The Hermitian conjugate of = + i + j+ kis ) The norm of is = ( a a ) ( a a ) i ( a a ) j ( a a ) k. 0 i j k = 0 0 N = = = 0 The norm canbe dual number, real number, or zero. If N =, then is called a unit dual SS. We will use H to denote the set of all the unit dual SS. If N = 0, then is called a null dual SS. dual split semi-quaternion for which N = 0 has form = j+ k,( 0 = = 0) it is a zero divisor. ) The inverse of with N 0, is = Clearly = + 0i + 0 j + 0 k. Note also that P = P ( P) = P. Theorem. The set H 0 of unit dualss is a subgroup of the group H 0 where H is the set of all non-zero dual split semi-quaternions. Proof: Let P, H. We have N P =, i.e. P H thus the first subgroup requirement is satisfied. lso, by the property N. N = N = N =, 07 Global Journals Inc. (US)

6 Note on the Representation Definition of Dual Split Semi-uaternions lgebra the second subgroup requirement H. Example 4. Consider the dual split semi-quaternions = + ( + ) i j + ( ) k, ( ) i j ( ) = k, = ( ) + ( ) i + ( ) j + k, = + i + j k, 4. The vector parts of, are V = ( + ) i j + ( + ) k, V = i ( ) j + ( + ) k.. The Hamilton conjugates of, are = i + ( ) j ( + ) k, = ( ) ( ) i ( ) j k,. The dual conjugates of, are = + i + ( + ) j ( ) k, = ( + ) + ( + ) i + (+ ) j + k, 4. The Hermitian conjugate of, are 4 = ( ) i j + ( + ) k, 4 = + i j k, 5. The norms are given by 6. The inverses of, are N =, N =, N = 0, N = 4 [ ( ) ( ) ], = + i + j k = [ i + ( ) j ( + ) k ], not invertible. 7. One can realize the following operations + = (+ ) + ( + ) i (+ ) j + ( + ) k = ( + ) + i + ( + ) j + ( + ) k 4 = ( + ) + [ + (+ ) ] i+ + [ + ( ) ] j[( ) + ] k. f) Trigonometric Form De Moivre s Theorem In this section, we expre De-Moivre s formula for dual SS. For this, we can cosider two different cases: Case. Let the norm of dual SS be positive. The trigonometric (polar) form of a non-null dual SS = + i + j+ k 0 Global Journal of Science Frontier Research ( F ) Volume XVII Iue VIII V ersion I Year Global Journals Inc. (US)

7 Note on the Representation Definition of Dual Split Semi-uaternions lgebra is = R(coshφ+ W sinh φ) where R= N, Global Journal of Science Frontier Research ( F ) Volume XVII Iue VIII V ersion I Year cosh R 0 φ =, sinh φ = =. R φ = ϕ + ϕ is a dual angle the unit dual vector W is given by W = ( w, w, w ) = [ i + j+ k ] = (,, ). This is similar to polar coordinate expreion of asplit quaternion [7], split semiquaternion []. Example.5. The trigonometric forms of the dual split semi-quaternions where N = N =. W W = + i + ( + ) j + k, is = coshφ + W sinh φ, = ( + ) + i + ( ) j k, is = [coshφ + W sinh φ ] cos h φ =, sin φ =, W = (, +, ) + cosh φ =, sinh φ =, W = (,, ), Theorem.5. (De Moivre's Theorem) If = R(coshφ+ W sinh φ) be a dual SS n is any positive integer, then n n = R (cosh nφ+ W sinh nφ) Proof: The proof is easily followed by induction on n. The Theorem holds for all integers n, since Example.5. Let = + i + ( + ) j k. Solution: First write in trigonometric form. whereφ = ln, W = (, +, ) pplying de Moivre s Theorem gives: = R (coshφw sinh φ), n n = R [cosh( nφ) + Wsinh( nφ)] n = R [cosh nφ Wsinh nφ] 0 Find 45. = (coshφ+ W sinh φ), Ref 7. Rosenfeld B., Geometry of Lie groups, Kluwer cademic Publishers, Netherls, (997). 07 Global Journals Inc. (US)

8 Note on the Representation Definition of Dual Split Semi-uaternions lgebra + = (cosh0φ+ sinh0 φ) = ( + ) W W = ( ) (cosh 45φ W sinh 45 φ) n Corollary.5. The equation =, does not have solution for a unit dual split semiquaternion. Example.5. Let = + i+ ( + ) j+ k, be a dual split semi-quaternion. There is no n n (n> 0) such that =. Case. Let the norm of dual SS be negative, i.e. N = < The polar form of a non-null dual SS is where R= N, sinh = + i + j+ k 0 = R(sinhψ + W cosh ψ) R 0 ψ =, 0 0. cosh φ = =. R φ = ϕ + ϕ is a dual angle the unit dual vector W is given by W = ( w, w, w ) = [ i + j+ k ] = (,, ). Futher Work By the Hamilton operators, dual split semi-quaternions have been expreed in terms of 4 4 matrices. With the aid of the De-Moivre's formula, we will obtain any power of these matrices. References Références Referencias. Jafari M., Split semi-quaternions algebra in semi-euclidean 4-space,Cumhuriyet Science Journal, Vol 6() (05) Jafari M.,Matrices of generalized dual quaternions,konuralp journal of mathematics, Vol. (), (05)0-.. Jafari M., Some results on the matrices of Split Semi-quaternions, 06, submitted. 4. Jafari M., Introduction to Dual uasi-quaternions: lgebra Geometry, researchgate.net/publication/ Jafari M., The lgebraic Structure of Dual Semi-quaternions, accepted for publication in Journal of Selçuk University Natural pplied Science. 6. Keler Max L., On the theory of screws the dual method, Proceeding of a symposium commemorating the Legacy, works, Life of Sir Robert Stawell Ball Upon the 00 th nniversary of a Treatise on the theory of Screws, University of Cambridge, Trinity College, July 9-, 000. Global Journal of Science Frontier Research ( F ) Volume XVII Iue VIII V ersion I Year Global Journals Inc. (US)

9 Note on the Representation Definition of Dual Split Semi-uaternions lgebra Rosenfeld B., Geometry of Lie groups, Kluwer cademic Publishers, Netherls, (997). Whittlesey J.,Whittlesey K.,Some Geometrical Generalizations of Euler's Formula, International journal of mathematical education in science & technology, () (990) Global Journal of Science Frontier Research ( F ) Volume XVII Iue VIII V ersion I Year Global Journals Inc. (US)