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An alternative <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c0210401.png" />-dimensional algebra, derived from the algebra of generalized quaternions via the Cayley–Dickson process (cf. [[Quaternion|Quaternion]] and [[Alternative rings and algebras|Alternative rings and algebras]]). The latter starts out from a given algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c0210402.png" /> to construct a new algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c0210403.png" /> (of twice the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c0210404.png" />) and is a generalization of the doubling process (see [[Hypercomplex number|Hypercomplex number]]). Namely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c0210405.png" /> be an algebra with a unit 1 over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c0210406.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c0210407.png" /> be some non-zero element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c0210408.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c0210409.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104010.png" />-linear mapping which is an involution, and such that
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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/c/c021/c021040/c02104011.png" /></td> </tr></table>
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An alternative  $  8 $-
 +
dimensional algebra, derived from the algebra of generalized quaternions via the Cayley–Dickson process (cf. [[Quaternion|Quaternion]] and [[Alternative rings and algebras|Alternative rings and algebras]]). The latter starts out from a given algebra  $  A $
 +
to construct a new algebra  $  A _ {1} $(
 +
of twice the dimension of  $  A $)
 +
and is a generalization of the doubling process (see [[Hypercomplex number|Hypercomplex number]]). Namely, let  $  A $
 +
be an algebra with a unit 1 over a field  $  F $,
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let  $  \delta $
 +
be some non-zero element of  $  F $,
 +
and let  $  x \rightarrow x  ^ {*} $
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be an  $  F $-
 +
linear mapping which is an involution, and such that
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 +
$$
 +
x + x  ^ {*}  = \
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\mathop{\rm tr} ( x)  \in  F,\ \
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xx  ^ {*}  = \
 +
n ( x)  \in  F.
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$$
  
 
The formula
 
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/c/c021/c021040/c02104012.png" /></td> </tr></table>
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$$
 +
( a _ {1} , a _ {2} )
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( b _ {1} , b _ {2} )  = \
 +
( a _ {1} b _ {1} -
 +
\delta b _ {2} a _ {2}  ^ {*} ,\
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a _ {1}  ^ {*} b _ {2} +
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b _ {1} a _ {2} )
 +
$$
  
now defines a multiplication operation on the direct sum of linear spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104013.png" />, relative to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104014.png" /> is an algebra. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104015.png" /> may be imbedded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104016.png" /> as a subalgebra: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104017.png" />, and the involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104018.png" /> extends to an involution in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104019.png" />:
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now defines a multiplication operation on the direct sum of linear spaces $  A _ {1} = A \oplus A $,  
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relative to which $  A _ {1} $
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is an algebra. The algebra $  A $
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may be imbedded in $  A _ {1} $
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as a subalgebra: $  x \rightarrow ( x, 0) $,  
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and the involution $  * $
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extends to an involution in $  A _ {1} $:
  
<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/c/c021/c021040/c02104020.png" /></td> </tr></table>
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$$
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( a _ {1} , a _ {2} )  ^ {*}  = \
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( a _ {1}  ^ {*} , - a _ {2} ).
 +
$$
  
 
Moreover,
 
Moreover,
  
<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/c/c021/c021040/c02104021.png" /></td> </tr></table>
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$$
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\mathop{\rm tr} ( a _ {1} , a _ {2} )  = \
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\mathop{\rm tr} ( a _ {1} ),\ \
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n ( a _ {1} , a _ {2} )  = \
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n ( a _ {1} ) + \delta n ( a _ {2} ).
 +
$$
  
The extension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104022.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104023.png" /> can be repeated resulting in an ascending chain of algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104024.png" />; the parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104025.png" /> need not be the same at each stage. If the Cayley–Dickson process is begun with an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104026.png" /> with basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104027.png" />, multiplication table
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The extension of $  A $
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to $  A _ {1} $
 +
can be repeated resulting in an ascending chain of algebras $  A \subset  A _ {1} \subset  A _ {2} \subset  \dots $;  
 +
the parameter $  \delta $
 +
need not be the same at each stage. If the Cayley–Dickson process is begun with an algebra $  A $
 +
with basis $  \{ 1, u \} $,  
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multiplication table
  
<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/c/c021/c021040/c02104028.png" /></td> </tr></table>
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$$
 +
u  ^ {2}  = u + \alpha ,\ \
 +
\alpha \in F,\ \
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4 \alpha + 1 \neq 0,
 +
$$
  
and involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104029.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104030.png" />, the first application of the process yields an algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104031.png" /> of generalized quaternions (an associative algebra of dimension 4), and the second — an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104032.png" />-dimensional algebra, known as a Cayley–Dickson algebra.
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and involution $  1  ^ {*} = 1 $,  
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$  u  ^ {*} = 1 - u $,  
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the first application of the process yields an algebra $  A _ {1} $
 +
of generalized quaternions (an associative algebra of dimension 4), and the second — an $  8 $-
 +
dimensional algebra, known as a Cayley–Dickson algebra.
  
Any Cayley–Dickson algebra is an alternative, but non-associative, central simple algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104033.png" />; conversely, a simple alternative ring is either associative or a Cayley–Dickson algebra over its centre. The quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104034.png" /> in 8 variables defined on a Cayley–Dickson algebra (the 8 variables correspond to the basis elements) has the multiplicative property:
+
Any Cayley–Dickson algebra is an alternative, but non-associative, central simple algebra over $  F $;  
 +
conversely, a simple alternative ring is either associative or a Cayley–Dickson algebra over its centre. The quadratic form $  n ( x) $
 +
in 8 variables defined on a Cayley–Dickson algebra (the 8 variables correspond to the basis elements) has the multiplicative property:
  
<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/c/c021/c021040/c02104035.png" /></td> </tr></table>
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$$
 +
n ( xy)  = \
 +
n ( x) \cdot n ( y).
 +
$$
  
This establishes a connection between Cayley–Dickson algebras and the existence problem for compositions of quadratic forms. A Cayley–Dickson algebra is a division algebra if and only if the quadratic form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104036.png" /> (the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104037.png" />) does not represent the zero in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104038.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104039.png" /> is a field of characteristic other than 2, a Cayley–Dickson algebra has a basis <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104040.png" /> with the following multiplication table:''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104041.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104042.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104043.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104044.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104045.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104046.png" /></td> <td colname="8" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104047.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104048.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104049.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104050.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104051.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104052.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104053.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104054.png" /></td> <td colname="8" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104055.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104056.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104057.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104058.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104059.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104060.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104061.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104062.png" /></td> <td colname="8" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104063.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104064.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104065.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104066.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104067.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104068.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104069.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104070.png" /></td> <td colname="8" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104071.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104072.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104073.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104074.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104075.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104076.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104077.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104078.png" /></td> <td colname="8" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104079.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104080.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104081.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104082.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104083.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104084.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104085.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104086.png" /></td> <td colname="8" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104087.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104088.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104089.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104090.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104091.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104092.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104093.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104094.png" /></td> <td colname="8" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104095.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104096.png" /></td> <td colname="2" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104097.png" /></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104098.png" /></td> <td colname="4" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c02104099.png" /></td> <td colname="5" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040100.png" /></td> <td colname="6" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040101.png" /></td> <td colname="7" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040102.png" /></td> <td colname="8" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040103.png" /></td> </tr> </tbody> </table>
+
This establishes a connection between Cayley–Dickson algebras and the existence problem for compositions of quadratic forms. A Cayley–Dickson algebra is a division algebra if and only if the quadratic form $  n ( x) $(
 +
the norm of $  x $)
 +
does not represent the zero in  $  F $.
 +
If  $  F $
 +
is a field of characteristic other than 2, a Cayley–Dickson algebra has a basis  $  \{ 1, u _ {1} \dots u _ {7} \} $
 +
with the following multiplication table:<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1"></td> <td colname="2" style="background-color:white;" colspan="1"> $  u _ {1} $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  u _ {2} $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  u _ {3} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  u _ {4} $
 +
</td> <td colname="6" style="background-color:white;" colspan="1"> $  u _ {5} $
 +
</td> <td colname="7" style="background-color:white;" colspan="1"> $  u _ {6} $
 +
</td> <td colname="8" style="background-color:white;" colspan="1"> $  u _ {7} $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  u _ {1} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1"> $  -\alpha $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  u _ {3} $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  -\alpha u _ {2} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  - u _ {5} $
 +
</td> <td colname="6" style="background-color:white;" colspan="1"> $  \alpha u _ {4} $
 +
</td> <td colname="7" style="background-color:white;" colspan="1"> $  - u _ {7} $
 +
</td> <td colname="8" style="background-color:white;" colspan="1"> $  \alpha u _ {6} $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  u _ {2} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1"> $  - u _ {3} $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  -\beta $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  \beta u _ {1} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  - u _ {6} $
 +
</td> <td colname="6" style="background-color:white;" colspan="1"> $  u _ {7} $
 +
</td> <td colname="7" style="background-color:white;" colspan="1"> $  \beta u _ {4} $
 +
</td> <td colname="8" style="background-color:white;" colspan="1"> $  -\beta u _ {5} $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  u _ {3} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1"> $  \alpha u _ {2} $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  -\beta u _ {1} $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  -\alpha\beta $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  - u _ {7} $
 +
</td> <td colname="6" style="background-color:white;" colspan="1"> $  -\alpha u _ {6} $
 +
</td> <td colname="7" style="background-color:white;" colspan="1"> $  \beta u _ {5} $
 +
</td> <td colname="8" style="background-color:white;" colspan="1"> $  \alpha\beta u _ {4} $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  u _ {4} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1"> $  u _ {5} $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  u _ {6} $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  u _ {7} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  -\gamma $
 +
</td> <td colname="6" style="background-color:white;" colspan="1"> $  -\gamma u _ {1} $
 +
</td> <td colname="7" style="background-color:white;" colspan="1"> $  -\gamma u _ {2} $
 +
</td> <td colname="8" style="background-color:white;" colspan="1"> $  -\gamma u _ {3} $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  u _ {5} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1"> $  -\alpha u _ {4} $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  - u _ {7} $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  \alpha u _ {6} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  \gamma u _ {1} $
 +
</td> <td colname="6" style="background-color:white;" colspan="1"> $  -\alpha\gamma $
 +
</td> <td colname="7" style="background-color:white;" colspan="1"> $  -\alpha u _ {3} $
 +
</td> <td colname="8" style="background-color:white;" colspan="1"> $  \alpha\gamma u _ {4} $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  u _ {6} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1"> $  u _ {7} $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  -\beta u _ {4} $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  -\beta u _ {5} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  \gamma u _ {2} $
 +
</td> <td colname="6" style="background-color:white;" colspan="1"> $  \gamma u _ {3} $
 +
</td> <td colname="7" style="background-color:white;" colspan="1"> $  -\beta\gamma $
 +
</td> <td colname="8" style="background-color:white;" colspan="1"> $  -\beta\gamma u _ {1} $
 +
</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"> $  u _ {7} $
 +
</td> <td colname="2" style="background-color:white;" colspan="1"> $  -\alpha u _ {6} $
 +
</td> <td colname="3" style="background-color:white;" colspan="1"> $  \beta u _ {5} $
 +
</td> <td colname="4" style="background-color:white;" colspan="1"> $  -\alpha \beta u _ {4} $
 +
</td> <td colname="5" style="background-color:white;" colspan="1"> $  \gamma u _ {3} $
 +
</td> <td colname="6" style="background-color:white;" colspan="1"> $  -\alpha \gamma u _ {2} $
 +
</td> <td colname="7" style="background-color:white;" colspan="1"> $  \beta\gamma u _ {1} $
 +
</td> <td colname="8" style="background-color:white;" colspan="1"> $  -\alpha\beta\gamma $
 +
</td> </tr> </tbody> </table>
  
 
</td></tr> </table>
 
</td></tr> </table>
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040104.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040105.png" />, and the involution is defined by the conditions
+
where $  \alpha , \beta , \gamma \in F $,
 +
$  \alpha \beta \gamma \neq 0 $,
 +
and the involution is defined by the conditions
 +
 
 +
$$
 +
1  ^ {*}  = 1,\ \
 +
u _ {i}  ^ {*}  = - u _ {i} ,\ \
 +
i = 1 \dots 7.
 +
$$
 +
 
 +
This algebra is denoted by  $  A ( \alpha , \beta , \gamma ) $.  
 +
The algebras  $  A ( \alpha , \beta , \gamma ) $
 +
and  $  A ( \alpha  ^  \prime  , \beta  ^  \prime  , \gamma  ^  \prime  ) $
 +
are isomorphic if and only if their quadratic forms  $  n ( x) $
 +
are equivalent. If  $  n ( x) $
 +
represents zero, the corresponding Cayley–Dickson algebra is isomorphic to  $  A (- 1, 1, 1) $,
 +
which is known as the Cayley splitting algebra, or the vector-matrix algebra. Its elements may be expressed as matrices
 +
 
 +
$$
 +
\left \|
 +
 
 +
\begin{array}{cc}
 +
\alpha  & a  \\
 +
b  &\beta  \\
 +
\end{array}
 +
\
 +
\right \| ,
 +
$$
  
<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/c/c021/c021040/c021040106.png" /></td> </tr></table>
+
where  $  \alpha , \beta \in F $,
 +
$  a, b \in V $,
 +
with  $  V $
 +
a three-dimensional space over  $  F $
 +
with the usual definition of the scalar product  $  \langle  a, b \rangle $
 +
and vector product  $  a \times b $.  
 +
Matrix multiplication is defined by
  
This algebra is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040107.png" />. The algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040108.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040109.png" /> are isomorphic if and only if their quadratic forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040110.png" /> are equivalent. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040111.png" /> represents zero, the corresponding Cayley–Dickson algebra is isomorphic to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040112.png" />, which is known as the Cayley splitting algebra, or the vector-matrix algebra. Its elements may be expressed as matrices
+
$$
 +
\left \|
  
<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/c/c021/c021040/c021040113.png" /></td> </tr></table>
+
\begin{array}{cc}
 +
\alpha  & a  \\
 +
b  &\beta  \\
 +
\end{array}
 +
\
 +
\right \| \
 +
\left \|
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040115.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040116.png" /> a three-dimensional space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040117.png" /> with the usual definition of the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040118.png" /> and vector product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040119.png" />. Matrix multiplication is defined by
+
\begin{array}{cc}
 +
\gamma  & c \\
 +
d  &\delta  \\
 +
\end{array}
 +
\
 +
\right \|  = \
 +
\left \|
  
<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/c/c021/c021040/c021040120.png" /></td> </tr></table>
+
\begin{array}{cc}
 +
\alpha \gamma - \langle  a, d \rangle  &\alpha c + \delta a + b \times d  \\
 +
\gamma b + \beta d + a \times c  &\beta \delta - \langle  b, c \rangle  \\
 +
\end{array}
 +
\
 +
\right \| .
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040121.png" /> is the real field, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040122.png" /> is the algebra of [[Cayley numbers|Cayley numbers]] (a division algebra). Any Cayley–Dickson algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040123.png" /> is isomorphic to either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040124.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040125.png" />.
+
If $  F = \mathbf R $
 +
is the real field, then $  A ( 1, 1, 1) $
 +
is the algebra of [[Cayley numbers|Cayley numbers]] (a division algebra). Any Cayley–Dickson algebra over $  \mathbf R $
 +
is isomorphic to either $  A ( 1, 1, 1) $
 +
or $  A (- 1, 1, 1) $.
  
 
The construction of Cayley–Dickson algebras over an arbitrary field is due to L.E. Dickson, who also investigated their fundamental properties (see [[#References|[1]]], [[#References|[2]]]).
 
The construction of Cayley–Dickson algebras over an arbitrary field is due to L.E. Dickson, who also investigated their fundamental properties (see [[#References|[1]]], [[#References|[2]]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040126.png" /> be an alternative ring whose associative-commutative centre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040127.png" /> is distinct from zero and does not contain zero divisors; let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040128.png" /> be the field of fractions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040129.png" />. Then there is a natural imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040130.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040131.png" /> is a Cayley–Dickson algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040132.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c021/c021040/c021040133.png" /> is known as a Cayley–Dickson ring.
+
Let $  A $
 +
be an alternative ring whose associative-commutative centre $  C $
 +
is distinct from zero and does not contain zero divisors; let $  F $
 +
be the field of fractions of $  C $.  
 +
Then there is a natural imbedding $  A \rightarrow A \otimes _ {C} F $.  
 +
If $  A \otimes _ {C} F $
 +
is a Cayley–Dickson algebra over $  F $,  
 +
then $  A $
 +
is known as a Cayley–Dickson ring.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.E. Dickson,  "Linear algebras" , Cambridge Univ. Press  (1930)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.D. Schafer,  "An introduction to nonassociative algebras" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.A. Zhevlakov,  A.M. Slin'ko,  I.P. Shestakov,  A.I. Shirshov,  "Rings that are nearly associative" , Acad. Press  (1982)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.E. Dickson,  "Linear algebras" , Cambridge Univ. Press  (1930)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R.D. Schafer,  "An introduction to nonassociative algebras" , Acad. Press  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  K.A. Zhevlakov,  A.M. Slin'ko,  I.P. Shestakov,  A.I. Shirshov,  "Rings that are nearly associative" , Acad. Press  (1982)  (Translated from Russian)</TD></TR></table>

Latest revision as of 16:43, 4 June 2020


An alternative $ 8 $- dimensional algebra, derived from the algebra of generalized quaternions via the Cayley–Dickson process (cf. Quaternion and Alternative rings and algebras). The latter starts out from a given algebra $ A $ to construct a new algebra $ A _ {1} $( of twice the dimension of $ A $) and is a generalization of the doubling process (see Hypercomplex number). Namely, let $ A $ be an algebra with a unit 1 over a field $ F $, let $ \delta $ be some non-zero element of $ F $, and let $ x \rightarrow x ^ {*} $ be an $ F $- linear mapping which is an involution, and such that

$$ x + x ^ {*} = \ \mathop{\rm tr} ( x) \in F,\ \ xx ^ {*} = \ n ( x) \in F. $$

The formula

$$ ( a _ {1} , a _ {2} ) ( b _ {1} , b _ {2} ) = \ ( a _ {1} b _ {1} - \delta b _ {2} a _ {2} ^ {*} ,\ a _ {1} ^ {*} b _ {2} + b _ {1} a _ {2} ) $$

now defines a multiplication operation on the direct sum of linear spaces $ A _ {1} = A \oplus A $, relative to which $ A _ {1} $ is an algebra. The algebra $ A $ may be imbedded in $ A _ {1} $ as a subalgebra: $ x \rightarrow ( x, 0) $, and the involution $ * $ extends to an involution in $ A _ {1} $:

$$ ( a _ {1} , a _ {2} ) ^ {*} = \ ( a _ {1} ^ {*} , - a _ {2} ). $$

Moreover,

$$ \mathop{\rm tr} ( a _ {1} , a _ {2} ) = \ \mathop{\rm tr} ( a _ {1} ),\ \ n ( a _ {1} , a _ {2} ) = \ n ( a _ {1} ) + \delta n ( a _ {2} ). $$

The extension of $ A $ to $ A _ {1} $ can be repeated resulting in an ascending chain of algebras $ A \subset A _ {1} \subset A _ {2} \subset \dots $; the parameter $ \delta $ need not be the same at each stage. If the Cayley–Dickson process is begun with an algebra $ A $ with basis $ \{ 1, u \} $, multiplication table

$$ u ^ {2} = u + \alpha ,\ \ \alpha \in F,\ \ 4 \alpha + 1 \neq 0, $$

and involution $ 1 ^ {*} = 1 $, $ u ^ {*} = 1 - u $, the first application of the process yields an algebra $ A _ {1} $ of generalized quaternions (an associative algebra of dimension 4), and the second — an $ 8 $- dimensional algebra, known as a Cayley–Dickson algebra.

Any Cayley–Dickson algebra is an alternative, but non-associative, central simple algebra over $ F $; conversely, a simple alternative ring is either associative or a Cayley–Dickson algebra over its centre. The quadratic form $ n ( x) $ in 8 variables defined on a Cayley–Dickson algebra (the 8 variables correspond to the basis elements) has the multiplicative property:

$$ n ( xy) = \ n ( x) \cdot n ( y). $$

This establishes a connection between Cayley–Dickson algebras and the existence problem for compositions of quadratic forms. A Cayley–Dickson algebra is a division algebra if and only if the quadratic form $ n ( x) $( the norm of $ x $) does not represent the zero in $ F $. If $ F $ is a field of characteristic other than 2, a Cayley–Dickson algebra has a basis $ \{ 1, u _ {1} \dots u _ {7} \} $

with the following multiplication table:

<tbody> </tbody>
$ u _ {1} $ $ u _ {2} $ $ u _ {3} $ $ u _ {4} $ $ u _ {5} $ $ u _ {6} $ $ u _ {7} $
$ u _ {1} $ $ -\alpha $ $ u _ {3} $ $ -\alpha u _ {2} $ $ - u _ {5} $ $ \alpha u _ {4} $ $ - u _ {7} $ $ \alpha u _ {6} $
$ u _ {2} $ $ - u _ {3} $ $ -\beta $ $ \beta u _ {1} $ $ - u _ {6} $ $ u _ {7} $ $ \beta u _ {4} $ $ -\beta u _ {5} $
$ u _ {3} $ $ \alpha u _ {2} $ $ -\beta u _ {1} $ $ -\alpha\beta $ $ - u _ {7} $ $ -\alpha u _ {6} $ $ \beta u _ {5} $ $ \alpha\beta u _ {4} $
$ u _ {4} $ $ u _ {5} $ $ u _ {6} $ $ u _ {7} $ $ -\gamma $ $ -\gamma u _ {1} $ $ -\gamma u _ {2} $ $ -\gamma u _ {3} $
$ u _ {5} $ $ -\alpha u _ {4} $ $ - u _ {7} $ $ \alpha u _ {6} $ $ \gamma u _ {1} $ $ -\alpha\gamma $ $ -\alpha u _ {3} $ $ \alpha\gamma u _ {4} $
$ u _ {6} $ $ u _ {7} $ $ -\beta u _ {4} $ $ -\beta u _ {5} $ $ \gamma u _ {2} $ $ \gamma u _ {3} $ $ -\beta\gamma $ $ -\beta\gamma u _ {1} $
$ u _ {7} $ $ -\alpha u _ {6} $ $ \beta u _ {5} $ $ -\alpha \beta u _ {4} $ $ \gamma u _ {3} $ $ -\alpha \gamma u _ {2} $ $ \beta\gamma u _ {1} $ $ -\alpha\beta\gamma $

where $ \alpha , \beta , \gamma \in F $, $ \alpha \beta \gamma \neq 0 $, and the involution is defined by the conditions

$$ 1 ^ {*} = 1,\ \ u _ {i} ^ {*} = - u _ {i} ,\ \ i = 1 \dots 7. $$

This algebra is denoted by $ A ( \alpha , \beta , \gamma ) $. The algebras $ A ( \alpha , \beta , \gamma ) $ and $ A ( \alpha ^ \prime , \beta ^ \prime , \gamma ^ \prime ) $ are isomorphic if and only if their quadratic forms $ n ( x) $ are equivalent. If $ n ( x) $ represents zero, the corresponding Cayley–Dickson algebra is isomorphic to $ A (- 1, 1, 1) $, which is known as the Cayley splitting algebra, or the vector-matrix algebra. Its elements may be expressed as matrices

$$ \left \| \begin{array}{cc} \alpha & a \\ b &\beta \\ \end{array} \ \right \| , $$

where $ \alpha , \beta \in F $, $ a, b \in V $, with $ V $ a three-dimensional space over $ F $ with the usual definition of the scalar product $ \langle a, b \rangle $ and vector product $ a \times b $. Matrix multiplication is defined by

$$ \left \| \begin{array}{cc} \alpha & a \\ b &\beta \\ \end{array} \ \right \| \ \left \| \begin{array}{cc} \gamma & c \\ d &\delta \\ \end{array} \ \right \| = \ \left \| \begin{array}{cc} \alpha \gamma - \langle a, d \rangle &\alpha c + \delta a + b \times d \\ \gamma b + \beta d + a \times c &\beta \delta - \langle b, c \rangle \\ \end{array} \ \right \| . $$

If $ F = \mathbf R $ is the real field, then $ A ( 1, 1, 1) $ is the algebra of Cayley numbers (a division algebra). Any Cayley–Dickson algebra over $ \mathbf R $ is isomorphic to either $ A ( 1, 1, 1) $ or $ A (- 1, 1, 1) $.

The construction of Cayley–Dickson algebras over an arbitrary field is due to L.E. Dickson, who also investigated their fundamental properties (see [1], [2]).

Let $ A $ be an alternative ring whose associative-commutative centre $ C $ is distinct from zero and does not contain zero divisors; let $ F $ be the field of fractions of $ C $. Then there is a natural imbedding $ A \rightarrow A \otimes _ {C} F $. If $ A \otimes _ {C} F $ is a Cayley–Dickson algebra over $ F $, then $ A $ is known as a Cayley–Dickson ring.

References

[1] L.E. Dickson, "Linear algebras" , Cambridge Univ. Press (1930)
[2] R.D. Schafer, "An introduction to nonassociative algebras" , Acad. Press (1966)
[3] K.A. Zhevlakov, A.M. Slin'ko, I.P. Shestakov, A.I. Shirshov, "Rings that are nearly associative" , Acad. Press (1982) (Translated from Russian)
How to Cite This Entry:
Cayley-Dickson algebra. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cayley-Dickson_algebra&oldid=46286
This article was adapted from an original article by E.N. Kuz'min (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article