Alternion

A hypercomplex number. Alternions may be considered as a generalization of the complex numbers, double numbers (cf. Double and dual numbers) and quaternions. The algebra $ {} ^ {l} A _ {n} $ of alternions of order $ n $ and of index $ l $ is an algebra of dimension $ 2 ^ {n-1} $ over the field of real numbers, with unit element 1 and a system of generators $ l _ {1} \dots l _ {n-1} $, in which the multiplication satisfies the formula

$$ l _ {i} l _ {j} = - l _ {j} l _ {i} , \ l _ {i} ^ {2} = - \epsilon _ {i} , $$

where $ \epsilon _ {i} = \pm 1 $, the value $ -1 $ occurs $ l $ times and $ +1 $ occurs $ n - l - 1 $ times, respectively. A base of the algebra is formed by the unit element and by elements of the form

$$ l _ {j _ {1} } \dots l _ {j _ {k} } = l _ {j _ {1} \dots j _ {k} } , $$

where $ j _ {1} < \dots < j _ {k} $. In this base any alternion $ \alpha $ can be written as

$$ \alpha = a + \sum _ { i } a ^ {i} l _ {i} + \sum _ { i } \sum _ { j } a ^ {i j } l _ {i j } + \dots + $$

$$ + a ^ {1 \dots (n-1) } l _ {1 \dots n - 1 } , $$

where $ a, a ^ {i} \dots a ^ {1 \dots (n-1) } $ are real numbers. The alternion $ \overline \alpha \; $ conjugate to the alternion $ \alpha $ is defined by the formula

$$ \overline \alpha \; = \sum _ { k } ( - 1 ) ^ {k ( k + 1 ) / 2 } a ^ {i _ {1} \dots i _ {k} } l _ {i _ {1} } \dots l _ {i _ {k} } . $$

The following equalities hold

$$ \overline{ {\alpha + \beta }}\; = \overline \alpha \; + \overline \beta \; , \ \overline \alpha \; bar = \alpha , \ \overline{ {\alpha \beta }}\; = \overline \beta \; \overline \alpha \; . $$

The product $ \overline \alpha \; \alpha $ is always a positive real number; the quantity $ | \alpha | = \sqrt {\overline \alpha \; \alpha } $ is called the modulus of the alternion $ \alpha $. If the number $ | \beta - \alpha | $ is taken as the distance between two alternions $ \alpha $ and $ \beta $, then the algebras $ {} ^ {0} A _ {n} $ and $ {} ^ {l} A _ {n} $, $ l > 0 $, are isometric to the Euclidean space $ \mathbf R ^ {2 ^ {n-1} } $ and the pseudo-Euclidean spaces $ {} ^ {l} \mathbf R ^ {2} ^ {n-1 } $, respectively. The algebra $ {} ^ {0} A _ {1} $ is isomorphic to the field of real numbers; $ {} ^ {0} A _ {2} $ is isomorphic to the field of complex numbers; $ {} ^ {1} A _ {2} $ is isomorphic to the algebra of double numbers; $ {} ^ {0} A _ {3} $ is isomorphic to the skew-field of quaternions; and $ {} ^ {1} A _ {3} $ and $ {} ^ {2} A _ {3} $ are isomorphic to the so-called algebras of anti-quaternions. The elements of $ {} ^ {0} A _ {n} $ are the so-called Clifford numbers. The algebra $ {} ^ {4} A _ {5} $ was studied by P. Dirac in the context of the spin of an electron.

The algebras of alternions are special cases of Clifford algebras (cf. Clifford algebra).

References

[1]	B.A. Rozenfel'd, "Non-Euclidean geometry" , Moscow (1955) (In Russian)