View source for 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|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|Clifford algebra]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  B.A. Rozenfel'd,   "Non-Euclidean geometry" , Moscow  (1955)  (In Russian)</TD></TR></table>