# Difference between revisions of "Cayley-Dickson algebra"

An alternative $8$- dimensional algebra, derived from the algebra of generalized quaternions via the Cayley–Dickson process (cf. Quaternion and Alternative rings and algebras). The latter starts out from a given algebra $A$ to construct a new algebra $A _ {1}$( of twice the dimension of $A$) and is a generalization of the doubling process (see Hypercomplex number). Namely, let $A$ be an algebra with a unit 1 over a field $F$, let $\delta$ be some non-zero element of $F$, and let $x \rightarrow x ^ {*}$ be an $F$- linear mapping which is an involution, and such that

$$x + x ^ {*} = \ \mathop{\rm tr} ( x) \in F,\ \ xx ^ {*} = \ n ( x) \in F.$$

The formula

$$( a _ {1} , a _ {2} ) ( b _ {1} , b _ {2} ) = \ ( a _ {1} b _ {1} - \delta b _ {2} a _ {2} ^ {*} ,\ a _ {1} ^ {*} b _ {2} + b _ {1} a _ {2} )$$

now defines a multiplication operation on the direct sum of linear spaces $A _ {1} = A \oplus A$, relative to which $A _ {1}$ is an algebra. The algebra $A$ may be imbedded in $A _ {1}$ as a subalgebra: $x \rightarrow ( x, 0)$, and the involution $*$ extends to an involution in $A _ {1}$:

$$( a _ {1} , a _ {2} ) ^ {*} = \ ( a _ {1} ^ {*} , - a _ {2} ).$$

Moreover,

$$\mathop{\rm tr} ( a _ {1} , a _ {2} ) = \ \mathop{\rm tr} ( a _ {1} ),\ \ n ( a _ {1} , a _ {2} ) = \ n ( a _ {1} ) + \delta n ( a _ {2} ).$$

The extension of $A$ to $A _ {1}$ can be repeated resulting in an ascending chain of algebras $A \subset A _ {1} \subset A _ {2} \subset \dots$; the parameter $\delta$ need not be the same at each stage. If the Cayley–Dickson process is begun with an algebra $A$ with basis $\{ 1, u \}$, multiplication table

$$u ^ {2} = u + \alpha ,\ \ \alpha \in F,\ \ 4 \alpha + 1 \neq 0,$$

and involution $1 ^ {*} = 1$, $u ^ {*} = 1 - u$, the first application of the process yields an algebra $A _ {1}$ of generalized quaternions (an associative algebra of dimension 4), and the second — an $8$- dimensional algebra, known as a Cayley–Dickson algebra.

Any Cayley–Dickson algebra is an alternative, but non-associative, central simple algebra over $F$; conversely, a simple alternative ring is either associative or a Cayley–Dickson algebra over its centre. The quadratic form $n ( x)$ in 8 variables defined on a Cayley–Dickson algebra (the 8 variables correspond to the basis elements) has the multiplicative property:

$$n ( xy) = \ n ( x) \cdot n ( y).$$

This establishes a connection between Cayley–Dickson algebras and the existence problem for compositions of quadratic forms. A Cayley–Dickson algebra is a division algebra if and only if the quadratic form $n ( x)$( the norm of $x$) does not represent the zero in $F$. If $F$ is a field of characteristic other than 2, a Cayley–Dickson algebra has a basis $\{ 1, u _ {1} \dots u _ {7} \}$

with the following multiplication table:

<tbody> </tbody>
 $u _ {1}$ $u _ {2}$ $u _ {3}$ $u _ {4}$ $u _ {5}$ $u _ {6}$ $u _ {7}$ $u _ {1}$ $-\alpha$ $u _ {3}$ $-\alpha u _ {2}$ $- u _ {5}$ $\alpha u _ {4}$ $- u _ {7}$ $\alpha u _ {6}$ $u _ {2}$ $- u _ {3}$ $-\beta$ $\beta u _ {1}$ $- u _ {6}$ $u _ {7}$ $\beta u _ {4}$ $-\beta u _ {5}$ $u _ {3}$ $\alpha u _ {2}$ $-\beta u _ {1}$ $-\alpha\beta$ $- u _ {7}$ $-\alpha u _ {6}$ $\beta u _ {5}$ $\alpha\beta u _ {4}$ $u _ {4}$ $u _ {5}$ $u _ {6}$ $u _ {7}$ $-\gamma$ $-\gamma u _ {1}$ $-\gamma u _ {2}$ $-\gamma u _ {3}$ $u _ {5}$ $-\alpha u _ {4}$ $- u _ {7}$ $\alpha u _ {6}$ $\gamma u _ {1}$ $-\alpha\gamma$ $-\alpha u _ {3}$ $\alpha\gamma u _ {4}$ $u _ {6}$ $u _ {7}$ $-\beta u _ {4}$ $-\beta u _ {5}$ $\gamma u _ {2}$ $\gamma u _ {3}$ $-\beta\gamma$ $-\beta\gamma u _ {1}$ $u _ {7}$ $-\alpha u _ {6}$ $\beta u _ {5}$ $-\alpha \beta u _ {4}$ $\gamma u _ {3}$ $-\alpha \gamma u _ {2}$ $\beta\gamma u _ {1}$ $-\alpha\beta\gamma$

where $\alpha , \beta , \gamma \in F$, $\alpha \beta \gamma \neq 0$, and the involution is defined by the conditions

$$1 ^ {*} = 1,\ \ u _ {i} ^ {*} = - u _ {i} ,\ \ i = 1 \dots 7.$$

This algebra is denoted by $A ( \alpha , \beta , \gamma )$. The algebras $A ( \alpha , \beta , \gamma )$ and $A ( \alpha ^ \prime , \beta ^ \prime , \gamma ^ \prime )$ are isomorphic if and only if their quadratic forms $n ( x)$ are equivalent. If $n ( x)$ represents zero, the corresponding Cayley–Dickson algebra is isomorphic to $A (- 1, 1, 1)$, which is known as the Cayley splitting algebra, or the vector-matrix algebra. Its elements may be expressed as matrices

$$\left \| \begin{array}{cc} \alpha & a \\ b &\beta \\ \end{array} \ \right \| ,$$

where $\alpha , \beta \in F$, $a, b \in V$, with $V$ a three-dimensional space over $F$ with the usual definition of the scalar product $\langle a, b \rangle$ and vector product $a \times b$. Matrix multiplication is defined by

$$\left \| \begin{array}{cc} \alpha & a \\ b &\beta \\ \end{array} \ \right \| \ \left \| \begin{array}{cc} \gamma & c \\ d &\delta \\ \end{array} \ \right \| = \ \left \| \begin{array}{cc} \alpha \gamma - \langle a, d \rangle &\alpha c + \delta a + b \times d \\ \gamma b + \beta d + a \times c &\beta \delta - \langle b, c \rangle \\ \end{array} \ \right \| .$$

If $F = \mathbf R$ is the real field, then $A ( 1, 1, 1)$ is the algebra of Cayley numbers (a division algebra). Any Cayley–Dickson algebra over $\mathbf R$ is isomorphic to either $A ( 1, 1, 1)$ or $A (- 1, 1, 1)$.

The construction of Cayley–Dickson algebras over an arbitrary field is due to L.E. Dickson, who also investigated their fundamental properties (see [1], [2]).

Let $A$ be an alternative ring whose associative-commutative centre $C$ is distinct from zero and does not contain zero divisors; let $F$ be the field of fractions of $C$. Then there is a natural imbedding $A \rightarrow A \otimes _ {C} F$. If $A \otimes _ {C} F$ is a Cayley–Dickson algebra over $F$, then $A$ is known as a Cayley–Dickson ring.

#### References

 [1] L.E. Dickson, "Linear algebras" , Cambridge Univ. Press (1930) [2] R.D. Schafer, "An introduction to nonassociative algebras" , Acad. Press (1966) [3] K.A. Zhevlakov, A.M. Slin'ko, I.P. Shestakov, A.I. Shirshov, "Rings that are nearly associative" , Acad. Press (1982) (Translated from Russian)
How to Cite This Entry:
Cayley-Dickson algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cayley-Dickson_algebra&oldid=11507
This article was adapted from an original article by E.N. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article