Definition:Cayley-Dickson Construction

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Definition

Let $A = \left({A_F, \oplus}\right)$ be a $*$-algebra.


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

Let:

$A' = \left({A'_F, \oplus'}\right) = \left({A, \oplus}\right)^2$

where $\left({A, \oplus}\right)^2$ denotes the Cartesian product of $\left({A, \oplus}\right)$ with itself.


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

$\left({a, b}\right) \oplus' \left({c, d}\right) = \left({a \oplus c - d \oplus b^*, a^* \oplus d + c \oplus b}\right)$
${\left({a, b}\right)^*}' = \left({a^*, -b}\right)$

where:

$\left({a, b}\right), \left({c, d}\right) \in A'$
$a^*$ is the conjugation of $a \in A$.



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A part of this page has to be extracted as a theorem.

If $\dim \left({A_F}\right)$ is $d$, then $\dim \left({A'_F}\right)$ is $2 d$.


Also see


Source of Name

This entry was named for Arthur Cayley and Leonard Eugene Dickson.


Sources