Let be a normed -algebra, an algebra admitting an involution (http://planetmath.org/Involution2) , over a commutative ring with . The Cayley-Dickson construction is a way of enlarging to a new algebra, , extending the as well as the norm operations in , such that is a subalgebra of .
Define to be the module (external) direct sum of with itself:
Therefore, addition in is defined by addition componentwise in each copy of . Next, let be a unit in and define three additional operations:
(Multiplication) , where is the involution on ,
(Extended involution) , and
(Extended Norm) .
One readily checks that the multiplication is bilinear, since the involution (on ) is linear. Therefore, is an algebra.
Furthermore, since the extended involution is clearly bijective and linear, and that
this extended involution is well-defined and so is in addition a -algebra.
Finally, to see that is a normed -algebra, we identify as the first component of , then becomes a subalgebra of and elements of the form can now be written simply as . Now, the extended norm
where in the subsequent terms of the above equation array is the norm on given by . The fact that the , together with the equality show that the extended norm on is well-defined. Thus, is a normed -algebra.
The normed -algebra , together with the invertible element , is called the Cayley-Dickson algebra, , obtained from .
If has a unity 1, then so does and its unity is . Furthermore, write , we check that, . Therefore, and we can identify the second component of with and write elements of as for .
It is not hard to see that and . We are now able to write
where each element has a unique expression .
Properties. Let will be general elements of .
Examples. All examples considered below have ground ring the reals .
More generally, given any field , any algebra obtained by applying the Cayley-Dickson construction twice to is called a quaternion algebra over , of which is an example. In other words, a quaternion algebra has the form
where each . Any algebra obtained by applying the Cayley-Dickson construction three times to is called a Cayley algebra, of which is an example. In other words, a Cayley algebra has the form
where each . A Cayley algebra is an octonion algebra when .
- 1 Richard D. Schafer, An Introduction to Nonassociative Algebras, Dover Publications, (1995).
|Date of creation||2013-03-22 14:54:11|
|Last modified on||2013-03-22 14:54:11|
|Last modified by||CWoo (3771)|