Download Polar Representation of Complex Octonions Mehdi JAFARI

Polar Representation of Complex Octonions
Mehdi JAFARI1
Department of Mathematics, University College of Science and Technology Elm o Fan, Urmia, IRAN
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 [2,3]. 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.
AMS Subject Classification: 11R52
Keyword: De-Moiver’s formula, Euler’s formula, complex octonion.
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 [1]. A study of the
classical electromagnetism’s energy described by the complex octonions in sixteen dimensions is given by
Kansu et al [2]. 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.
COMPLEX-OCTONIONS ALGEBRA
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 ,
i 1
1
E-mail address: [email protected]
1
(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;
As a consequence of this definition, a complex octonion X can be written as
X  x  i x ',
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 ,
VX  A1 e1  A2 e2  A3 e3  A4 e4  A5 e5  A6 e6  A7 e7 .
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 ,
where e 2  1 and


Q, Q '  HC =
 Q  A0  A1e1  A2e2  A3e3 e12  e2 2  e32  1 , Ai  C ,
the complex quaternion division algebra (see [4]).
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 .
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
2
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 matrix-vector 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 
where X , Y  OC . Complex octonions multiplication is not associative, since
e1 ( e2 e4 )  e1e6  e7 ,
(e1e2 ) e4  e3e4  e7 .
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 .
i 1
The complex conjugate of X is
7
X   A0 e0   Ai ei  (a0  ia0' )e0  (a1  ia1' )e1  ...  (a7  ia7' )e7 .
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
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
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:
3
1) X  X , ( X * )*  X , ( X † )†  X
2) XY  Y X , ( XY )*  Y * X * , ( XY )†  Y † X †
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
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 
The norms of X 1 , X 2 , X 3 are
N X1  1, N X 2  0, N X 3  5  2i.
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 
The inverse of X1 , 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
X 11 
and X 2 not invertible.
Theorem 1. The set O1C of unit complex octonions is a subgroup of the group O C0 where OC0  OC  [0  0].
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,
the second subgroup requirement X 1  O1C .
4
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  ),
with
12
R
NX 
7
A
i 0
2
i
, cos  
A0
NX
 7 2
  Ai 
and sin    i 1 
NX
. The unit complex vector W  w  i w* is
given by
W  ( w1 , w2 ,..., w7 ) 
1
( A1 , A2 ,..., A7 ).
7
( A )
2 12
i
i 1
Example 2. The polar form of the complex octonions X 1  1  (i,1  i, 2i ,1  i , 2, 0,
2


X 1  cos  W1 sin
4
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 

2
3 is
)
2
 i ln(1  2),
where
3 and
1
)
W2 
(1  2i,  i  1, 2  i, 1, 2i, i  1, 5).
2
2
W1  2 (i,1  i, 2i ,1  i , 2, 0,
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
3!
4
4!
 ...  W ( 
2! 4!
 cos   W sin  ,
for any dual number

 ...
3
3!

5
5!
 ...)
.
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  N x (cos   W sin  ) be a complex octonions. Then for any
integer n;
X n  ( N x )n .(cos n   W sin n  )
5
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
Suppose that (cos   W sin  )n  cos n   W sin n  , we aim to show
(cos   W sin  )n1  cos(n  1)  W sin(n  1) .
Thus
(cos   W sin  ) n 1  (cos   W sin  ) n (cos   W sin  )
 (cos n  W sin n )(cos   W sin  )
 cos(n   )  W sin(n   )
 cos(n  1)  W sin(n  1) .
The formula holds for all integers n;
X 1  cos   W sin  ,
X  n  cos(n )  W sin(n )
 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
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.
X
ii)
N X (cos   W sin  ),
Complex octonions with real angles (ϕ=φ, φ∗=0); i.e.
X
N X (cos   W sin  ).
Theorem 3. De Moivre’s formula implies that there are uncountably many unit complex octonion
X  cos   W sin  satisfying X n  1 for n ≥3.
6
Proof: For every unit vector W , the unit complex octonion
X  cos
2
2
 W sin
,
n
n
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.
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 ,
thus, we find
cos n  cos ,
sin n  sin ,
So, the n roots of X are
Ak  cos(
Example 5. Let X 
  2 k
n
)  W sin(
  2 k
n
),
k  0,1, 2,..., n  1.
■
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.
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
1
18
1
X 1 3  cos
18
13
13
 W sin
 0.64  0.76 W and third one is
18
18
1
X 2 3  cos
25
25
 W sin
 0.34  0.93W .
18
18
7
1
1
1
Also, it is easy to see that X 03  X13  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  2  Z  {1},

then X  X
n
m
if and only if n  m (mod p).
Proof: Let n  m (mod p). Then we have n  ap  m, where 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)


 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 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 ).

■
Example 6. Let X  2  (1  2i,  3, 2i, i  1, 1 , i  1, 2) be a unit complex octonion. From Theorem 5,
2
2
m
2
 8, so we have
 /4
X  X 9  X 17  ...
X 2  X 10  X 18  ...
X 3  X 11  X 19  ...
X 4  X 12  X 20  ...  1
X 8  X 16  X 24  ...  1.
8
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 X  1 has uncountably many solutions
n
for any unit complex octonions (Theorem 3).
Futher work
We will give a complete investigation to real matrix representations of complex octonions, and give
any powers of these matrices.
References
[1] Baez J., The octonions, Bull. Amer. Math. Soc. 39 (2002), 145-205
[2] Kansu M.E., Emre, Tanışlı M., Süleyman D., Electromagnetic energy conservation with complex
octonions, Turk J Phys, 36 (2012), 438 – 445.
[3] James D., Edmonds J., Nine-vectors, complex octonion/quaternion hypercomplex numbers, Lie groups,
and the 'real' world, Foundations of physics, Vol. 8, ¾, 1978, 303-311.
[4] Jafari M., On the matrix algebra of complex quaternions, Accepted for publication in TWMS Journal
of Pure and Applied Mathematics, 2016.
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