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 of alternions of order
and of index
is an algebra of dimension
over the field of real numbers, with unit element 1 and a system of generators
, in which the multiplication satisfies the formula
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where , the value
occurs
times and
occurs
times, respectively. A base of the algebra is formed by the unit element and by elements of the form
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where . In this base any alternion
can be written as
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where are real numbers. The alternion
conjugate to the alternion
is defined by the formula
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The following equalities hold
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The product is always a positive real number; the quantity
is called the modulus of the alternion
. If the number
is taken as the distance between two alternions
and
, then the algebras
and
,
, are isometric to the Euclidean space
and the pseudo-Euclidean spaces
, respectively. The algebra
is isomorphic to the field of real numbers;
is isomorphic to the field of complex numbers;
is isomorphic to the algebra of double numbers;
is isomorphic to the skew-field of quaternions; and
and
are isomorphic to the so-called algebras of anti-quaternions. The elements of
are the so-called Clifford numbers. The algebra
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) |
Alternion. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Alternion&oldid=15232