The fundamental algebraic properties of split quasi-octonions
Split kuazi-oktonyonların temel cebirsel özellikleri
Mehdi Jafari
A B S T R A C T
Ö Z E T A R T I C L E I N F O
Technical and Vocational University, Department of Mathematics, Urmia, IRAN
Received: 12 September 2015 Accepted: 10 April 2016
Available Online: 31 December 2016 Printing: 31 January 2017
Corresponding address: Mehdi Jafari E-mail: [email protected]
The fundamental properties of split quasi-octonion algebra, O’q, and definitions of fundamental operations such as scalar and vector parts, conjugate, norm, and polar form are presented. We explain the Cayley-Dickson construction of split quasi-octonion algebra, in particular we provide table of the octonion multiplication.
Keywords: Alternativity, Cayley-Dickson construction, Split quasi-octonion, Trigonometric form
Split kuaz-oktonyonların temel cebirsel özellikleri ve bazı temel operasyonlar tanımları, örneğin, skalar ve vektör parçaları, eşlenik, norm ve polar formu sunulmuştur. Split kuazi-oktonyonlar cebirin üzerinde Cayley-Dickson yapısını açıkladık, ve oktoniyon çarpım tablosunu temin ettik.
Anahtar sözcükler: Alternativite, Cayley-Dickson yapısı, Split kuazi-octonion, Trigonometrik form
Dicle University Institute of Natural and Applied Science Journal
journal homepage: http://www.dufed.org
Introduction
The Octonion, or the Cayley algebra O is an 8-dimensional non-associative algebra, which is defined by J.T. Graves and A. Cayley independently separated. Since octonions share with complex numbers and quaternions have many attractive mathematical properties, one might except that they would be equally useful. As a vector space, the octonions are
: , ,...
O
a
a e a a
i ia
R
i 0 1 7 0 1 7!
=
+
='
/
1
In our previous work, we investigated basic algebraic properties of real, split, complex, semi, and quasi octonions algebra. In following studies, here we study fundamental properties of split quasi-octonions, which is called split 1
4 -octonions in [9]. We review the generalized octonions algebra, and show that if put
,
1 0
a=- b=c= is obtained split quasi-octonions
algebra. Like real octonions, split semi-octonions form a non-associative algebra, but unlike real octonions, they are not division algebra. By Cayley-Dickson construction, e4 and H generates Oq’ as an algebra. We express any split quasi-octonions in trigonometric form similar to octonions and quaternions. In addition, we prove De Moivre’s theorem and Euler’s formula for these octonions.
1. Generalized Octonions Algebra
In this section, we give a brief summary of the generalized octonions. For detailed information about these octonions, we refer the reader to [1].
Definition 2.1. A generalized octonion
x
isdefined as
,
where
a
0-
a
7 are real numbers ande 0
i,(
≤ ≤
i
7
)
are octonionic units satisfying the equalities that are given in the following table;
.
e1 e2 e3 e4 e5 e6 e7 e1-
a
e3-
a
e
2 e5-
a
e
4 e7a
e
6 e2 e3-
b
b
e
1 e6 e7-
b
e
4-
b
e
5 e3a
e
2-
b
e
1-
ab
e7-
a
e
6b
e
5-
ab
e
4 e4 e5 e6 e7-
c
c
e
1c
e
2c
e
3 e5a
e
4 -e7a
e
6-
c
e
1-
ac
-
c
e
3ac
e
2 e6 e7b
e
4-
b
e
5-
c
e
2c
e
3-
bc
-
bc
e
1 e7-
a
e
6b
e
5ab
e
4-
c
e
3-
ac
e
2bc
e
1-
abc
Special Cases: 1. Ifa
=
b
=
c
=
1
, is considered, thenO a b c
( , , )
is the algebra of real octonions
O
[5].2. If
a
=
b
=
1
,
c
=-
1
, is considered, then is the algebra of split octonions (Psoudo-octonions)O
'
[4].
3. If
a
=
b
=
1
,
c
=
0
, is considered, then( , , )
O a b c
is the algebra of semi-octonionsO
S[3].
4. If
a
=
b
=-
1
,
c
=
0
, is considered, then( , , )
O a b c
is the algebra of split semi-octonions'
O
S[5].5. If
a
=
1
,
b
=
c
=
0
, is considered, then( , , )
O a b c
is the algebra of quasi-octonionsO
q[6].
6. If
a
=-
1
,
b
=
c
=
0
, is considered, then( , , )
O a b c
is the algebra of split quasi-octonions'
O
q.7. If
a
=
b
=
c
=
0
, is considered, thenO a b c
( , , )
is the algebra of para-octonions
O
p[7].The generalized octonions algebra,
O a b c
( , , ),
is a non-commutative, non-associative, alternative, flexible and power-associative.
2. Split Quasi-Octonions Algebra
Definition 3.1. A split quasi-octonion
x
isexpressed as a set of eight real numbers
( , ,..., )
,
x
x x
x
x e
x e
i i i 0 1 7 0 0 1 7=
=
+
=/
where
x
0-
x
7 are real numbers. The multiplicationrules among the basis elements of octonions
(
≤ ≤
)
e
i0
i
7
can be expressed in the form:, , , , e e e e e e e e e e e e e e e e 1 1 2 1 2 3 2 1 1 1 1 1 4 5 4 6 7 6 = = =-= =-, ≤ ≤ , e k e e e e e e e e e e e e e e e 2 7 0 k 2 7 2 4 6 4 2 2 5 7 5 2 3 4 4 3 = = =-= =-= =-, e e e e e e e e e e e e e e e 5 5 3 3 1 3 2 3 1 1 4 1 5 6 5 = =-= =-=
=-The above multiplication rules are given in the following Table;
.
e1 e2 e3 e4 e5 e6 e7 e1 1 e3 e2 e5 e4 e7 -e6 e2 e3 0 0 e6 e7 0 0 e3 -e2 0 0 e7 e6 0 0 e4 e5 e6 e7 0 0 0 0 e5 -e4 -e7 -e6 0 0 0 0 e6 e7 0 0 0 0 0 0 e7 e6 0 0 0 0 0 0By using the Cayley-Dikson construction, a split quasi-octonion x can also be written as
(
)
(
)
,
x
a e
a e
a e
a e
a
a e
a e
a e e
q q l
' 0 0 1 1 2 2 3 3 1 2 3 4 4 5 6 7=
+
+
+
+
+
+
+
= +
where
l
2=
0
andq q
, '
are split semi-quaternions[2], i.e.
, '
,
,
q q
H
q
a
a e
a e e
1
e
e
0
a
R
SS O i 0 1 1 1 2 2 2 3 2 2 2!
!
=
=
+
+
=
=
=
(
2
This construction lets us view the split quasi-octonion as a two dimensional vector space over split semi-quaternions quaternions. Therefore,
'
.
O
q=
H
SSO5
H l
SSOA split quasi-octonion
x
can be decomposed interms of its scalar
( )
S
x and vector( )
V
ı x parts as
,
.
S
a V
a e
a e
a e
a e
a e
a e
a e
x 0 x 1 1 2 2 3 3 4 4 5 5 6 6 7 7 ı=
=
+
+
+
+
+
+
For two split semi-octonions
x
a e
i 1 1 0 7
=
=/
andw
b
e
i 1 1 0 7=
=/
the summation and substractionprocesses are given as
x
w
(
a
b
)
e
i i i i 0 7
!
=
!
=/
The product of two split semi-octonions
,
x
S
xV w
xS
wV
w ı ı=
+
=
+
is expressed as,
x
xw
=
S S
-
V V
C C+
S
V
C+
S V
C+
V V
C CThis product can be described by a matrix-vector product as
.
x w
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
a
b
b
b
b
b
b
b
b
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0 1 1 0 2 5 6 0 4 5 0 5 4 1 2 1 0 0 2 3 4 5 6 7 3 4 7 1 1 0 3 3 2 0 1 1 0 1 2 3 4 5 6 7=
--
-
-Split semi-octonions multiplication is not associative, since
(
)
,
(
)
.
e e e
e e
e
e e e
e e
e
1 1 1 2 4 6 7 2 4 3 4 7=
=-=
=
But it has the property of alternativity, that is, any two elements in it generate an associative subalgebra isomorphic to
R, ,C ,
C
0 1H H
S,
0.e
0 ande
i(
2
≤ ≤
i
7
)
generate a subalgebraisomorphic to
C
0 (dual numbers),e
0 ande
1 generate a subalgebra isomorphic toC
1(double (or split complex) numbers),
Subalgebra with bases
e e e e
0, , , (
1 i j2
≤
i j
,
≤
7
)
is isomorphic to split semi-quaternions algebera
H
S([2])
Subalgebra with bases
e e e e
0, , , (
1 j k2
≤
i j k
, ,
≤
7
)
is isomorphic to quasi-quaternions algebra
H
0.3. Some Properties of Split Quasi-Octonions
1) The conjugate of split quasi-octonionx
a e
iS
V
r
i i x 0 7=
=
+
=/
is.
x
a e
a e
i iS
V
r
x i x 0 0 1 7=
-
=
-=/
Conjugate of product of two split quasi-octonions and its own are described as
,
xy
=
yx x
- - --=
x
It is clear the scalar and vector parts of
x
is denoted byS
x=
x x
+
2
andV
r
x=
x x
-
2
.
-2) The norm ofx
is.
N
x=
xx
-=
xx
-=
x
2=
a
02-
a
12It satisfies the following property
N
xy=
N N
x y=
N N
y xThe modulus
x
of a split quasi-octonion x, like the modulus of a split quaternion, or split octonion, can be real or imaginary and can be equal to 0 for.
x
!
0
A split quasi-octonion x is called quasi-spacelike, quasi-timelike or quasi-lightlike(null), if
N
x<
0
,N
x>
0
orN
x=
0
, respectively.If
N
x=
1
, thenx
is called a unit splitquasi-octonion. We will use
O'
q1 to denote the set of unitsplit quasi-octonions.
3) The inverse of
x
withN
x]
0
, is.
x
N x
1
x
1
=
-
-4) The trace of element
x
is defined ast x
( )
= +
x x
-For every
x
!
O
'
q, we have(
x x x
+
-)
=
x
2+
xx
-=
x
2+
N 1
x.
, then,( )
x
2-
t x x N
+
x=
0
, therefore, the splitquasi-octonions algebra is quadratic.
The split quasi-octonions algebra is not division
algebra, because for every nonzero
x
!
O
'
q therelation
N
x=
0
, impliesx
]
0
.
Example 4.1. Consider the split quasi-octonions
( ,
, ,
, , , )
x
1= +
2
1
-
1 2
-
2 0 1 1
( ,
, ,
, , , )
x
2=
-
1
+
2
-
1 1
-
2 0 1 1
and( ,
,
,
, , ,
)
x
3= - +
2
1
2
1
-
1 2
-
2 2 1 1
;1. The vector parts of
x x
1,
2 are( , , , , , , ), ( , , , , , , ), Vıx 1 1 2 2 0 1 1 Vx 2 11 2 0 1 1 ı 2 1= - - = -
-2. The conjugates of
x x
1,
2 are( , , , , , ). ( , , , , , ),
x1= -2 1 -1 2 2 0 1 1- x2=-1- 2 -11-2 0 1 1
3. The norms are given by
,
,
.
N
x1=
3
N
x2=-
3
N
x3=
0
4. The inverses are
( ,
, ,
, , , ),
x
N x
1
3
2
3
1 1 1 2 2 0 1 1
x 11 1 1=
=
-
-
---( ,
, ,
, , , ),
x
13
1
3
1
2
1
1
2 0 1 1
2=
+
-
-- andx
3 not invertible.5. One can realize the following operations
( ,
, ,
, , , )
( , , , , , )
( ,
,
,
, , ,
)
( , ,
, , , , )
.
x
x
x
x
x x
x x
N
N N
N
0
2
2 3
4 0 2 2
0
0 2
2
1
1 1
1 1
2
0
2
0 0 1
0 0
1
1
1
1
1
1
x x x x x x 1 2 1 2 1 2 1 2 1 2 1 2 2 1+
= +
-
-= +
=
+
-
-
-=- +
-
-=
=
=
--
-Theorem 4.1. The set
O
1S1 of unit splitsemi-octonions is a subgroup of the group
O
S10 where.
O
OS=
O
S-
6
0
-
0
ı@
Proof: Let
x y
,
!
O
'
S1. We haveN
xy=
1
i.e.'
xy
!
O
S and thus the first subgroup requirement issatisfied. Also, by the property
,
N
x=
N
x=
N
x-1=
1
the second subgroup requirement
x
1!
O
' .
S1
-4. Trigonometric Form and De Moivre’s
Theorem
Trigonometric(polar) form of the nonzero split
quasi-octonion
x
a e
i i i 0 7=
=/
is as follows:i) Every quasi-spacelike octonion x can be written in the form
x
=
N
x(
sinh
m
+
w
r
cosh
m
)
where
,
sinh
cosh
N
a
N
a
N
a
x x x 0 12 1m
=
m
=
=
the unit octonion vector
w
ı is given by( , ,..., )
( , ,..., ).
w
w w
w
a
a a
a
1
r
1 2 7 1 2 1 2 7=
=
Since
w
r
2=
1
; we have a natural generalization ofEuler’s formula for unit split quasi-octonion
( ) ! ( ) ! ( ) ! ( ) ... ( ! ! ...) ( ( !) ( !) ...) ! cosh sinh e w w w w w w w w w 1 3 1 2 3 2 4 5 4 5 r r r r r r r r r w 2 2 3 3 4 5 4 5 r m m m m m m m m m m m m = + + + + = + + + + + + + = + + + m
Example 5.1. The trigonometric forms of the split quasi-octonions 1
( ,
, , , , ,
)
x
1= +
1
2
-
1 0 1 1 1
-
1
is,
cosh ln
sinh ln
x
1=
3
7
3
+
w
r
13
A
1 The inverse hyperbolic sine and cosine are defined sinh x-1 =Ln x( + x2+1)
(
,
, , ,
, , )
x
2= +
1
2
-
1 0 1
-
1
2
1
is(
)
(
)
sinh ln
cosh ln
x
2=
1
+
2
+
w
r
21
+
2
where ( , , , , , , ) ( , , , , , , ) . w w and N N 2 1 2 1 0 1 1 1 1 2 1 2 1 0 1 1 2 1 1 r r w w 1 2 r1 r2 = - -= - - ==-ii) Every quasi-timelike octonion x can be written in the form
(
cosh
sin
)
x
=
N
xi
+
u
r
i
where,
cosh
sinh
N
a
N
a
N
a
x x x 0 12 1i
=
i
=
=
,the unit octonion vector
u N
ı
(
ur=
1
)
is given by( , ,..., )
( , ,..., ).
u
u u
u
a
a a
a
1
r
1 2 7 1 2 1 2 7=
=
Example 5.2. The polar forms of the split quasi-octonions
(
, ,
, , , ,
)
x
2
1
2
1 1 1 1 2 1 2
1=
+ -
-
-
is(
)
(
) ,
cosh ln
sinh ln
x
1=
2
1
7
1
+
2
+
u
r
11
+
2
A
(
, ,
,
, ,
, )
x
2=
3
+ -
2 0 2
-
1 1
-
1 1
is(
)
(
),
cosh ln
sinh ln
x
2=
2
+
3
+
u
r
22
+
3
where ( , , , , ) ( , , , , , , ), , , u u 2 1 2 2 1 1 1 1 2 21 1 1 1 2 1 2 0 r r 2 1= - - - = - --iii) Every null octonion x can be written in the form
x
= +
1
f
ıwhere
f
ı is a null vector (N
fr=-
1
).Example 5.3. The polar form of the split quasi-octonions
x
= +
1
^
1 0
, ,
-
1 1 1
, , ,
-
1
,
-
2
h
isx
= +
1
f
ıwhere
f
ı=
( , ,
1 0
-
1 1 1
, , ,
-
1
,
-
2
).
Theorem 5.1. (De Moivre’s formula)Let
(
sinh
cosh
)
x
=
N
xm
+
w
r
n
m
be aquasi-spacelike octonion. We have
(
) sinh
(
cosh
)
x
nN
w
r
n
x nm
m
=
+
forn
odd and(
) (
cos
h
sin
h
)
x
nN
n
w
r
n
x nm
m
=
+
forn
even.Proof: The proof is easily followed by induction on n.
Example 5.4. Let x= + -1 ( 2,-1 0 1 2 2, , , , ,-1). Find
x
26 andx
43Solution: First write x in trigonometry form:
(
)
(
)
sinh ln
cosh ln
x
=
1
+
2
+
w
r
1
+
2
( ) ( ) . cosh ln sinh ln x25= 266 2+1 @+wr 266 2+1 @ ( ) ( ) . h ln ln sin cosh x43= 436 2+1 @+wr 436 2+1 @Theorem 5.2. (De Moivre’s formula) Let
(
cosh
sinh
)
x
=
N
x{
+
v
r
{
be a quasi-timelikeoctonion. Then for any integer
(
) cosh
(
sinh
)
x
nN
n
u
r
n
x n
{
{
=
+
Proof: The proof is easily followed by induction on n.
Theorem 5.3. (De Moivre’s formula)If
x
a e
i1
r
i i 0 7f
=
= +
=/
be a null octonion. Then forany integer
.
x
n= +
1
n
f
r
5. The roots of a Split Quasi-Octonion
Theorem 6.1. Let
x
=
N
x(
sinh
m
+
w
r
cosh
m
)
be a quasi-spacelike octonion. The equation
a
n=
x
has only one root and this is
(
sinh
cosh
)
a
=
2nN
xn
m
+
w
r
n
m
Theorem 6.2. Let
x
=
N
x(
cosh
m
+
v
r
sinh
m
)
be a quasi-timelike octonion. The equation
a
n=
x
has only one root and this is
(
cosh
sinh
)
a
=
2nN
xn
m
+
v
r
n
m
Proof: We assume that
a
=
M
(
cosh
m
+
vı
sinh
m
)
is a root of the equation
a
n=
x
, since the vector partsof
x
anda
are the same. From Theorem 5.2, we have(
cosh
sinh
)
a
n=
M
nn
m
+
v
r
n
m
Now, we find
, cosh cosh , sinh sinh .
M= Nx {= nm {= nm
So,
a
nN
x(
cosh
n
v
r
sinh
n
)
{
{
=
+
is a root ofequation
a
n=
x
.
If we suppose that there are tworoots satisfying the equality, we obtain that these roots must be equal to each other.
Example 6.1. Let x= 3+( 2 1 2 1 1, , - , ,-1 1, ). Find
x
41Solution: First we write x in polar form:
(
)
(
),
cosh ln
sinh ln
x
=
2
+
3
+
u
r
22
+
3
Then,(
)
(
)
,
cosh
ln
sinh
ln
x
2
4
3
u
r
22
4
3
1 4=
+
+
+
Conclusion
In this paper, we defined and gave some of algebraic properties of split quasi-octonions and investigated the De Moivre’s formulas for these octonions. We gave some examples for more clarification.
We hope that this work will contribute to the study of physics and other sciences.
Futher Work
We will give a complete investigation to real matrix representations of split quasi-octonions, and consider a relation between the powers of these matrices.
References
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