|Peter M Jack
science forum beginner
Joined: 30 Jan 2006
|Posted: Sat Jul 01, 2006 4:22 am Post subject:
Quatro-Quaternions and the matrix representations of octonions.
"Quatro-Quaternions and the matrix representations of octonions."
We introduce a modified matrix product that allows one to construct
matrix representations for octonions and other Cayley-Dickson
Then we define a matrix algebra with two multiplications, one the
standard associative product and the other our new non-associative
product, and call the 2x2 matricies over the quaternions with
this dual-product ``quatro-quaternions'', of which octonions
are a natural subalgebra.
We describe these quatro-quaternions which have the interesting
property that for every quatro-quaternion that has the form of
an octonion, o = (A,B), the difference of squares of the two
products is a diagonal matrix of the commutators [A,B], [A,B*]
o x o - o.o = ( 0 , AB* - B*A
AB-BA , 0 )
where . is the standard associative matrix product,
and x is the new non-associative matrix product.
Then we obtain 4x4 and 8x8 matrix representations of octonions
over complex numbers and reals by replacing the quaternions with
complex and real matricies, and so obtain generalized algebras
M[x](4,C) and M[x](8,R), with the new matrix product x replacing
the standard . matrix product.
Finally, we show that our new 8x8 matrix algebra over the reals,
using this non-standard matrix product, is equivalent to O"x O,
the "product algebra" of two octonion type algebras.
[This final observation was inspired by the comment by Waterhouse
and others, on our last paper, that Hexpentaquaternions was just
the tensor product of the quaternion algebra with its opposite
algebra, and that also H(x)H == M(4,R), and this led us to conjecture
that a similar thing might be true and found, i.e. O" x O == M[x](8,R)]