# Category:Cayley-Dickson Construction

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This category contains results about **the Cayley-Dickson construction on $*$-algebras**.

Let $A = \struct {A_F, \oplus}$ be a $*$-algebra.

The **Cayley-Dickson Construction** on $A$ is the procedure which generates a new algebra $A'$ from $A$ as follows.

Let:

- $A' = \struct {A'_F, \oplus'} = \struct {A, \oplus}^2$

where $\struct {A, \oplus}^2$ denotes the Cartesian product of $\struct {A, \oplus}$ with itself.

Then $\oplus'$ and $*'$ are defined on $A'$ as follows:

- $\tuple {a, b} \oplus' \tuple {c, d} = \tuple {a \oplus c - d \oplus b^*, a^* \oplus d + c \oplus b}$
- ${\tuple {a, b}^*}' = \tuple {a^*, -b}$

where:

- $\tuple {a, b}, \tuple {c, d} \in A'$
- $a^*$ is the conjugation of $a \in A$.

## Pages in category "Cayley-Dickson Construction"

The following 6 pages are in this category, out of 6 total.