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)