# Alternion

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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)
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