# 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).

How to Cite This Entry:
Alternion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alternion&oldid=45099
This article was adapted from an original article by N.N. Vil'yams (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article