SUJEST, c.4, s.4, 2016 SUJEST, v.4, n.4, 2016 ISSN: 2147-9364 (Elektronik)
POLAR REPRESENTATION OF COMPLEX OCTONIONS
Mehdi JAFARI
1
Department of Mathematics, Technical and Vocational University, Urmia, IRAN
1
[email protected]
1
(Geliş/Received: 26.03.2016 ; Kabul/Accepted in Revised Form: 25.04.2016 ) ABSTRACT: The complex octonions are a non-associative extension of complex quaternions, are used in areas such as quantum physics, classical electrodynamics, the representations of robotic systems, kinematics etc. (Kansu et al., 2012, James et al., 1978). In this paper, we study the complex octonions and their basic properties. We generalize in a natural way De-Moivre’s and Euler’s formulae for division complex octonions algebra. Key Words: De-Moiver’s formula, Euler’s formula, Complex octonion
Kompleks Oktoniolarin Kutupsal Gösterimi ÖZ: Kompleks oktonyonlar, kompleks kuaterniyonların birleşimli olmayan ve kuantum fiziği, klasik elektrodinamik, robotik sistemlerin gösterimleri, kinematik (Kansu et al., 2012, James et al., 1978) gibi alanlarda kullanılan bir uzantısıdır. Bu makalede, kompleks oktonyonlar ve temel özelliklerini çalıştık. De-Moivre ve Euler formüllerini Kompleks oktonyonlar cebiri için tabii bir şekilde genelleştirdik. Anahtar Kelimeler: De Moivre’s formülü, Euler’s fromülü, Kompleks oktonyonlar. INTRODUCTION The octonions are the largest of the four normed division algebras. While somewhat neglected due to their non-associativity, they stand at the crossroads of many interesting fields of mathematics (Baez, 2002). A study of the classical electromagnetism’s energy described by the complex octonions in sixteen dimensions is given by Kansu et al. (Kansu et al., 2012). The complex exponential ei cos i sin generalizes to quaternions by replacing i by any unit quaternion µ since any unit pure quaternion is a root of -1. Hence, any quaternion may be represented in the polar form q q e where is a real angle. As with complex numbers and quaternions, any octonion can be written in polar form as
x r (cos w sin ) where r N x and w2 1. In this paper, we introduce the complex octonions algebra, OC, and study some fundamental algebraic properties of them. The polar representation of complex octonions are given, and then by means of the De-Moivre's theorem, any powers of these octonions are obtained. Finally, we give some examples for more clarification.
DOI: 10.15317/Scitech.2016.64
357
Polar Representation of Complex Octonions
MATERIAL AND METHOD A complex octonion X has an expression of the form 7
X A0 e0 A1 e1 A2 e2 A3 e3 A4 e4 A5 e5 A6 e6 A7 e7 A0 e0 Ai ei ,
(1)
i 1
where A0 A7 are complex numbers and ei , (0 i 7) are octonionic units satisfying the equalities that are given in the table below; Table 1. Octonionic units
.
e1
e2
e3
e4
e5
e6
e7
e1
-1
e3
-
e5
-
-
e6
e4
e2 e2
e3
-1
e1
e6
e7
e7 e4
e3
e2
-e1
-1
e7
e6
e5
e4
e5
e6
e7
-1
e1
e2
e3
e5
e4
-e7
e6
-e1
-1
-e3
e2
e6
e7
e4
- e5
-e2
e3
-1
-e1
e7
-e6
e5
e4
-e3
-e2
e1
-1
e5 e4
As a consequence of this definition, a complex octonion X can be written as
X x i x ',
(2)
where x and x ' , real and pure octonion components, respectively. The set of all complex octonions is denoted by OC. For defined octonion in equation (1), the scalar and vectorial parts can be given, respectively, as
S X A0 e0 ,
(3)
VX A1 e1 A2 e2 A3 e3 A4 e4 A5 e5 A6 e6 A7 e7 .
(4)
A complex octonion X can also be written as
X ( A0 e0 A1 e1 A2 e2 A3 e3 ) ( A4 A5 e1 A6 e2 A7 e3 )e4 Q Q ' e ,
(5)
where e 2 1 and
Q, Q ' HC Q A0 A1e1 A2e2 A3e3 e12 e22 e32 1 , Ai C , the complex quaternion division algebra (Jafari, 2016).
(6)
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M. JAFARI
7
7
i 0
i 0
For two complex octonions X Ai ei and Y Bi ei , the summation and substraction processes are given as 7
X Y ( Ai Bi ) ei .
(7)
i 0
Addition and subtraction of complex octonions is done by adding and subtracting corresponding terms and hence their coefficients, like quaternions. The product of two complex octonions X S X VX , Y SY VY is expressed as
X Y S X SY VX ,VY S X VY SY VX VX VY
(8)
Multiplication is distributive over addition, so the product of two octonions can be calculated by summing the product of all the terms, again like quaternions. This product can be described by a matrixvector product as
A0 A 1 A2 A X .Y 3 A4 A5 A 6 A7
A1 A0 A3 A2 A5 A4 A7 A6
A2 A3 A0 A1 A6 A7 A4 A5
A3 A2 A1 A0 A7 A6 A5 A4
A4 A5 A6 A7 A0 A1 A2 A3
A5 A4 A7 A6 A1 A0 A3 A2
A6 A7 A4 A5 A2 A3 A0 A1
A7 B0 A6 B1 A5 B2 A4 B3 , A3 B4 A2 B5 A1 B6 A0 B7
(9)
where X , Y OC . Complex octonions multiplication is not associative, since
e1 ( e2 e4 ) e1e6 e7 , (e1e2 ) e4 e3e4 e7 .
(10)
It is clear that subalgebra with bases e0 , e1 , ei , e j (2 i, j 7) is isomorphic to complex quaternions algebera HC .
SOME PROPERTIES OF COMPLEX OCTONIONS 7
1) The Hamilton conjugate of X Ai ei S X VX is i 0
7
X A0 e0 Ai ei S X VX .
(11)
i 1
The complex conjugate of X is 7
X A0 e0 Ai ei (a0 ia0' )e0 (a1 ia1' )e1 ... (a7 ia7' )e7 .
(12)
i 1
The Hermitian conjugate of X is 7
X † ( X )* A0 e0 Ai ei (a0 ia0' )e0 (a1 ia1' )e1 ... (a7 ia7' )e7 . i 1
(13)
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Polar Representation of Complex Octonions
It is clear the scalar and vector parts of X is denoted by S X X X and VX X X . 2 2 2)
The norm of X is
N X XX XX X
2
7
Ai2 C
(14)
i 0
If N X 1, then X is called a unit complex octonion. We will use O1C to denote the set of unit complex octonions. If N X 0, then X is called a null complex octonion. Lemma 1. Let X , Y OC . The conjugate and norm of complex octonions satisfy the following properties:
1) X X , ( X * )* X , ( X † )† X 2) XY Y X , ( XY )* Y * X * , ( XY )† Y † X †
(15)
3) X Y X Y , ( X Y ) X Y , ( X Y ) X Y 4) N X N X , N XY N X NY *
*
*
†
†
†
3) The inverse of X with N X 0, is
X 1
1 X. NX
(16)
Example 1. Consider the complex octonions
3 1 2 e1 (1 i)e2 2i e3 (1 i) e4 i e5 e6 e7 , 2 2 1 1 X2 e1 (1 i )e2 2i e3 (1 i ) e4 2i e5 e6 2 e7 , 2 2 X 3 1 (2 i ) e1 ie2 e3 (1 i) e4 i e5 e6 e7 , X1
(17)
The norms of X 1 , X 2 , X 3 are
N X1 1, N X 2 0, N X3 5 2i.
(18)
The conjugates of X 1 , X 2 , X 3 are
3 1 2 e1 (1 i)e2 2i e3 (1 i) e4 i e5 e6 e7 , 2 2 1 1 X 2* e1 (1 i )e2 2i e3 (1 i) e4 2i e5 e6 2 e7 , 2 2 X 3† 1 (2 i) e1 ie2 e3 (1 i) e4 i e5 e6 e7 , X1
(19)
The inverse of X 1 , X 3 are
3 1 2 e1 (1 i)e2 2i e3 (1 i) e4 i e5 e6 e7 , 2 2 1 X3 [1 (2 i) e1 ie2 e3 (1 i) e4 i e5 e6 e7 ], 5 2i X11
(20)
and X 2 not invertible. Theorem 1. The set O1C of unit complex octonions is a subgroup of the group OC0
OC0 OC [0 0].
where
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M. JAFARI
Proof: Let X , Y O1C . We have N XY 1, i.e. XY O1C and thus the first subgroup requirement is satisfied. Also, by the property
N X N X N X 1 1,
(21)
the second subgroup requirement X 1 O1C .
RESULT AND DISCUSSION Trigonometric Form and De Moivre’s Theorem 7
Every non-null complex octonion X
Ai ei can be written in the trigonometric (polar) form
i 0
X R(cos W sin ),
(22)
with 12
R
NX
7
A i 0
2 i
7 2 Ai and sin i 1 NX
A0
, cos
NX
. The unit complex vector W w i w* is
given by
W ( w1 , w2 ,..., w7 )
1 ( A ) i 1
2 12 i
Example 2. The polar form of the complex octonions X1 1 (i,1 i, 2i ,1 i , 2, 0,
2
X 1 cos
4
(23)
( A1 , A2 ,..., A7 ).
7
W1 sin
3 is ) 2
(24)
4
and X 2 i (1 2i, i 1, 2 i, 1, 2i, i 1, 5) is
X 2 cos W2 sin , cos 1 i
i ln(1 2), 2
(25)
where
W1 2 (i,1 i, 2i ,1 i , 2, 0,
3 and 1 ) W2 (1 2i, i 1, 2 i, 1, 2i, i 1, 5). 2 2
(26)
It is clear that NW NW 1 and WW 1 1 W2W2 1. 1 2 Since W
2
1 we have a natural generalization of Euler's formula for generalized quaternions eW 1 W 1
2
2 2!
4
W
.
3!
4 4!
... W (
2! 4! cos W sin , for any dual number
3
...
3 3!
5 5!
...)
(27)
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Polar Representation of Complex Octonions
Lemma 2. For every unit vector W , we have
cos W sin cos 1
1
2
W sin 2 cos 1 2 W sin 1 2 .
Theorem 2. (De-Moivre's formula) Let X
(28)
N x (cos W sin ) be a complex octonions. Then for any
integer n;
X n ( N x ) n .(cos n W sin n )
(29)
Proof: The proof will be by induction on nonnegative integers n and let N X 1 . 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
(30)
Suppose that (cos W sin )n cos n W sin n , we aim to show
(cos W sin )n1 cos(n 1) W sin(n 1).
(31)
Thus
(cos W sin ) n 1 (cos W sin ) n (cos W sin ) (cos n W sin n )(cos W sin )
(32)
cos(n ) W sin(n ) cos(n 1) W sin(n 1) . The formula holds for all integers n;
X 1 cos W sin ,
(33)
X n cos( n ) W sin( n )
(34)
cos n W sin n . ■
Example 3. Let X 3 (1 i, i, 2i ,1 i ,1, 1, 2) be a complex octonion. Every power of this octonion is found with the aid of Theorem 1. For example, 20-th and 83-th powers are
5 5 W sin 20 ) 6 6 1 3 220 ( W ) 219 [1 3(1 i, i, 2i ,1 i ,1, 1, 2)], 2 2
X 20 220 (cos 20
(35)
and
5 5 W sin 83 ) 6 6 82 2 ( 3 W ).
x83 283 (cos83
We investigate some properties of the complex octonions by separating them in two cases: i)
Complex octonions with complex angles (ϕ=φ+iφ∗); i.e.
(36)
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M. JAFARI
N X (cos W sin ),
X
ii)
(37)
Complex octonions with real angles (ϕ=φ, φ∗=0); i.e.
N X (cos W sin ).
X
(38)
Theorem 3. De Moivre’s formula implies that there are uncountably many unit complex octonion X cos W sin satisfying X 1 for n ≥3. n
Proof: For every unit vector W , the unit complex octonion
2 2 W sin , n n
X cos
(39)
is of order n. For n 1 or n 2, the complex octonion X is independent of W .
■
3 1 3 is of order 8 and X (1 i, i, 2i ,1 i , 1, , 2) is of ) 2 2 2
Example 4. X 1 (i,1 i, 2i ,1 i , 2, 0, 2 order 12.
Theorem 4. Let X cos W sin be a unit complex octonion. The equation An X has n roots, and they are
Ak cos(
2 k n
) W sin(
2 k n
k 0,1, 2,..., n 1.
),
(40)
Proof: We assume that A cos W sin is a root of the equation An X , since the vector parts of X and A are the same. From Theorem 2, we have
An cos n W sin n ,
(41)
thus, we find
cos n cos ,
sin n sin ,
So, the n roots of X are
Ak cos(
2 k n
) W sin(
2 k n
),
k 0,1, 2,..., n 1.
(42)
■ Example 5. Let X
3 1 (1 2i, 3, 2i, i 1, , i 1, 2) cos W sin be a unit complex octonion. 2 2 6 6
The cube roots of the octonion X are 1
X k 3 cos(
6 2 k 3
) w sin(
6 2 k 3
),
k 0,1, 2.
(43)
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Polar Representation of Complex Octonions
For k 0 , the first root is X 0 3 cos W sin 0.98 0.17 W , and the second one for k 1 is 18 18 1
1
X 1 3 cos
13 13 W sin 0.64 0.76 W and third one is 18 18 1
X 2 3 cos 1
1
25 25 W sin 0.34 0.93W . 18 18
(44)
1
Also, it is easy to see that X 03 X 13 X 23 0. The relation between the powers of complex octonions can be found in the following Theorem. Theorem 5. Let X be a unit complex octonion with the polar form X cos W sin . If m
n m 2 Z {1}, then X X if and only if n m (mod p).
Proof: Let n m (mod p). Then we have n a. p m, where
(45)
a Z.
X n cos n W sin n cos(ap m) W sin(ap m) 2 2 cos(a m) W sin(a m)
(46)
cos(m a 2 ) W sin( m a 2 ) cos m W sin m X m. Now suppose
X n cos n W sin n and X m cos m W sin m. If
(47)
X n X m then we get cos n cos m and sin n sin m , which means n m 2 a, a Z.
Thus n m
2 a or n m(mod p ).
(48) (49) ■
Example 6. Let X
m
2 1 (1 2i, 3, 2i, i 1, , i 1, 2) be a unit complex octonion. From Theorem 5, 2 2
2 8, so we have /4
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M. JAFARI
X X 9 X 17 ... X 2 X 10 X 18 ... X 3 X 11 X 19 ... X X 4
12
X
20
(50)
... 1
X 8 X 16 X 24 ... 1.
CONCLUSION In this paper, we defined and gave some of algebraic properties of complex octonions and investigated the De Moivre’s formulas for these octonions. The relation between the powers of complex octonions is given in Theorem 5. We also showed that the equation solutions for any unit complex octonions (Theorem 3).
X n 1
has uncountably many
FUTHER WORK We will give a complete investigation to real matrix representations of complex octonions, and give any powers of these matrices. REFERENCES Baez, J., 2002, ‚The Octonions‛, Bulletin (New Series) Of The American Mathematical Society (Bull. Amer. Math. Soc.) Vol. 39, pp. 145-205. Kansu, M.E., Tanışlı M., Süleyman D., 2012, ‘’Electromagnetic Energy Conservation with Complex Octonions’’, Turk Journal of Physics, Vol. 36, pp. 438 – 445. James, D., Edmonds, J., 1978, ‚Nine-vectors, Complex Octonion/quaternion Hypercomplex Numbers, Lie groups, and The 'Real' World‛, Foundations of Physics, Vol. 8, pp. 303-311. Jafari, M., 2016, ‚On The Matrix Algebra of Complex Quaternions‛, Accepted for publication in TWMS Journal of Pure and Applied Mathematics. DOI: 10.13140/RG.2.1.3565.2321
