**Cayley-Dickson Algebras:**

The Cayley-Dickson algebras are special real algebras satisfying

for If we let be the basis of then the algebra is generated by the relations

where Hence and is called the **quaternions** and is often denoted and are called the **octonions** and **sedenions** respectively and are respectively also denoted and

The Cayley-Dickson algebras can also be constructed inductively using a recursive unary operation (called conjugation). It proceeds as follows, let We define constructively. Let and define the following operations:

One can verify this structure is isomorphic to Next we let and define its structure

and addition remaining the same and the inner conjugation being complex conjugation. How does is this retain isomorphism to our original definition of Note the dimension is retained by the fact that

Hence using this as our inclusion, we show the original requirements that

We have

Similar computations can be done for and and for showing that

It turns out and beyond are not associative, so in our initial requirement that

we will clarify that we mean

We continue inductively be setting and defining product and conjugation as before (and componentwise addition). One can establish the following table of properties

where associativity and commutativity refer to the multiplication (the addition is always both). All of the above properties (save the division algebra structure) can at this point (albeit with some tediousness) be shown. If we can define its **real part** as

Note for we have

For we can define For we have

Similarly we can define For we have

Hence And similarly for we have

so Inductively, one can see that the norm on coincides with the euclidean norm in We can also note that the multiplicative inverse of an element is

Although it turns out that in the sedenions and above ( for ), there are zero divisors (for example, ), so the algebra fails to be a division algebra.

**Theorem 1. (Hurwitz’ Theorem)** If is a real normed division algebra with identity, then

**Corollary 2. (Frobenius’ Theorem)** If is an associative real division algebra, then

**Generalized Cross Products:**

The cross product of two elements can be written

There is a natural way to define a cross product of elements in by

However, our ultimate objective is to define vorticity in other dimensions. So we will want a binary cross product that we can apply to our differential operator and a vector field: We want to satisfy some conditions

These conditions are respectively called *bilinearity*, *orthogonality*, and *magnitude*. Hence we want to be the product in a real normed algebra. Moreover magnitude tells us that we want We also have in our Cayley-Dickson algebras. It turns out that the cross product in can be modeled in as follows. Define by

and by

Then one can verify that for

Note that in the cross product is trivial for all This is obvious if we require it to satisfy orthogonality—where the dot product is just multiplication. So the zero-product property implies the cross product must be It is compatible with our map as well (in this case ):

Hence we are left with one remaining cross product: the cross product in where, hereafter, is defined by

and is the imaginary map

The cross product of is defined by

We will now define

If is differentiable, then we define its curl by