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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a0121001.png" /> of alternions of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a0121002.png" /> and of index <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a0121003.png" /> is an algebra of dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a0121004.png" /> over the field of real numbers, with unit element 1 and a system of generators <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a0121005.png" />, in which the multiplication satisfies the formula
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a0121006.png" /></td> </tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a0121007.png" />, the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a0121008.png" /> occurs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a0121009.png" /> times and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210010.png" /> occurs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210011.png" /> times, respectively. A base of the algebra is formed by the unit element and by elements of the form
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210012.png" /></td> </tr></table>
+
$$
 +
l _ {i} l _ {j}  = - l _ {j} l _ {i} ,
 +
\  l _ {i}  ^ {2}  = - \epsilon _ {i} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210013.png" />. In this base any alternion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210014.png" /> can be written as
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210015.png" /></td> </tr></table>
+
$$
 +
l _ {j _ {1}  } \dots
 +
l _ {j _ {k}  }  = l _ {j _ {1}  \dots j _ {k} } ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210016.png" /></td> </tr></table>
+
where  $  j _ {1} < \dots < j _ {k} $.  
 +
In this base any alternion  $  \alpha $
 +
can be written as
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210017.png" /> are real numbers. The alternion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210018.png" /> conjugate to the alternion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210019.png" /> is defined by the formula
+
$$
 +
\alpha  = a + \sum _ { i } a ^ {i} l _ {i} +
 +
\sum _ { i } \sum _ { j } a ^ {i j } l _ {i j }  + \dots +
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210020.png" /></td> </tr></table>
+
$$
 +
+
 +
a ^ {1 \dots (n-1) } l _ {1 \dots n - 1 }  ,
 +
$$
 +
 
 +
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
 
The following equalities hold
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210021.png" /></td> </tr></table>
+
$$
 +
\overline{ {\alpha + \beta }}\;  = \overline \alpha \; + \overline \beta \; ,
 +
\  \overline \alpha \; bar  = \alpha ,
 +
\  \overline{ {\alpha \beta }}\; = \overline \beta \; \overline \alpha \; .
 +
$$
  
The product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210022.png" /> is always a positive real number; the quantity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210023.png" /> is called the modulus of the alternion <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210024.png" />. If the number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210025.png" /> is taken as the distance between two alternions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210027.png" />, then the algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210030.png" />, are isometric to the Euclidean space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210031.png" /> and the pseudo-Euclidean spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210032.png" />, respectively. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210033.png" /> is isomorphic to the field of real numbers; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210034.png" /> is isomorphic to the field of complex numbers; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210035.png" /> is isomorphic to the algebra of double numbers; <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210036.png" /> is isomorphic to the skew-field of quaternions; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210037.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210038.png" /> are isomorphic to the so-called algebras of anti-quaternions. The elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210039.png" /> are the so-called Clifford numbers. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012100/a01210040.png" /> was studied by P. Dirac in the context of the spin of an electron.
+
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]]).
 
The algebras of alternions are special cases of Clifford algebras (cf. [[Clifford algebra|Clifford algebra]]).

Revision as of 16:10, 1 April 2020


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=45093
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