**Representation:**
Since octonions
do not form an associative algebra they can not be represented by matrices.
However this does not preclude us from constructing matrices with octonion
entrices or representing adjoint algebra of octonions in terms of matrices.
The following describes two methods of representing octonions.
**(1) Quaternionic representation:**
An octonion, A, is written as an ordered pair of two
4-dimensional quaternions, Q_{1} and Q_{2}
as
A := ( Q_{1};
Q_{2} ) .
For the algebra of octonions, O
, to satisfy the basic properties outlined in Introduction one needs to
define the multiplication rules of two octonions, A and B.
Writing B in terms of two quaternions, Q_{3}
and Q_{4}, we define the multiplication of
A and B, via Cayley-Dickson process
by
AB := ( Q_{1};
Q_{2} )( Q_{3};
Q_{4} )
:= ( Q_{1}Q_{3}
+ bQ_{4}*Q_{2};
Q_{2}Q_{3}* +
Q_{4}Q_{1} )
,
where Q* is a conjugate of Q, and b
is a field parameter.
**(2) Coordinate representation:**
An octonion, A, may be written componentwise
as
A := _{m=0}**S**^{7}
a_{m} i_{m}
= a_{0 }i_{0}
+ _{j=1}**S**^{7 }
a_{j} i_{j} ,
where im
are the basal elements of O. The basal
element, i_{0}, is chosen as the basis
of reals, R. For the algebra of
octonions to satisfy the basic properties outlined in Introduction one
needs to define the multiplication rules for the basis. There are many
choices for this task but we will describe just two of them below.
(**I**) via congruence modulo
7:
Define i_{j} i_{j}
= -1 ; i_{j} i_{j+1}
:= i_{j+3} := i_{j-4}
, (mod 7)
with i_{2k}
i_{2m} = i_{2n},
whenever i_{k} i_{m}
= i_{n} .
Then the multiplication rule for the seven basal elements,
i_{j }( j = 1,...7)
is summed up in the following seven cycles:
(1,2,4) ,
(1,3,7) , (1,5,6) , (2,3,5) , (2,6,7) , (3,4,6)
, (4,5,7) .
The cycle (j,k,m) is to mean
i_{j}i_{k}=
i_{m} , i_{k}
i_{m} = i_{j}
, i_{k} i_{j}
= - i_{m }.
(**II**) via Cayley-Dickson process
from the quaternion basis:
Write octonion basal element as ordered pair of quaternion
basal elements
i_{m}
:= ( i_{m} ; 0
) , and
i_{m+4} :=
( 0 ; i_{m} ),
for m = 0,1,2,3.
e.g., i_{3}
= ( i_{3 }; 0 ) ,
and i_{6} = ( 0 ; i_{2}
) .
Then by Cayley-Dickson
process the multiplication rule can be found,
for example, i_{3 }i_{6}=
( 0 ; i_{2 }i_{3})
= ( 0 ; i_{1} ) = i_{5},
and summarized in the following seven cycles:
(1,2,3) , (1,4,5) , (1,7,6) , (2,4,6) , (2,5,7) , (3,4,7)
, (3,6,5) .
..... details |