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The general framework for defining and studying the Hurwitz transformations is that of the Cayley–Dickson algebras (cf. also [[Cayley–Dickson algebra|Cayley–Dickson algebra]]). Familiar examples of Cayley–Dickson algebras are:
 
The general framework for defining and studying the Hurwitz transformations is that of the Cayley–Dickson algebras (cf. also [[Cayley–Dickson algebra|Cayley–Dickson algebra]]). Familiar examples of Cayley–Dickson algebras are:
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h1103701.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h1103702.png" />, the algebras of ordinary and hyperbolic complex numbers, respectively;
+
$  A ( - 1 ) \equiv \mathbf C $
 +
and $  A ( 1 ) \equiv \Omega $,  
 +
the algebras of ordinary and hyperbolic complex numbers, respectively;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h1103703.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h1103704.png" />, the algebras of ordinary (or Hamilton) and hyperbolic quaternions, respectively;
+
$  A ( - 1, - 1 ) \equiv \mathbf H $
 +
and $  A ( 1,1 ) \equiv \mathbf N _ {1} $,  
 +
the algebras of ordinary (or Hamilton) and hyperbolic quaternions, respectively;
  
<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h1103705.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h1103706.png" />, the algebras of ordinary (or Cayley) and hyperbolic octonions, respectively. Higher-dimensional Cayley–Dickson algebras over the real field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h1103707.png" /> are denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h1103708.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h1103709.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037010.png" />-tuple <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037011.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037012.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037013.png" />). Such a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037014.png" />-dimensional algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037015.png" />, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037016.png" />, may be constructed from an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037017.png" />-dimensional Cayley–Dickson algebra by a  "doubling"  process [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]]. (This process generalizes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037018.png" />.) For fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037019.png" />, the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037020.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037021.png" /> corresponds to ordinary (or elliptic) hypercomplex numbers (cf. also [[Hypercomplex number|Hypercomplex number]]), while the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037022.png" /> other algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037023.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037024.png" /> correspond to hyperbolic hypercomplex numbers. The Cayley–Dickson algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037025.png" /> is referred to as normed or pseudo-normed according to whether the [[Metric|metric]]
+
$  A ( - 1, - 1, - 1 ) \equiv \mathbf O $
 +
and $  A ( 1,1,1 ) \equiv \mathbf O  ^  \prime  $,  
 +
the algebras of ordinary (or Cayley) and hyperbolic octonions, respectively. Higher-dimensional Cayley–Dickson algebras over the real field $  \mathbf R $
 +
are denoted by $  A ( c ) $,  
 +
where $  c $
 +
is a $  p $-
 +
tuple $  c = ( c _ {1} \dots c _ {p} ) $
 +
with $  c _ {s} = \pm  1 $(
 +
$  s = 1 \dots p $).  
 +
Such a $  2m $-
 +
dimensional algebra $  A ( c ) $,  
 +
with $  2m = 2  ^ {p} $,  
 +
may be constructed from an $  m $-
 +
dimensional Cayley–Dickson algebra by a  "doubling"  process [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]]. (This process generalizes $  \mathbf C = \mathbf R + i \mathbf R $.)  
 +
For fixed $  p $,  
 +
the algebra $  A ( c ) $
 +
with $  \sum _ {s = 1 }  ^ {p} c _ {s} = - p $
 +
corresponds to ordinary (or elliptic) hypercomplex numbers (cf. also [[Hypercomplex number|Hypercomplex number]]), while the $  2m - 1 $
 +
other algebras $  A ( c ) $
 +
with $  \sum _ {s = 1 }  ^ {p} c _ {s} \neq - p $
 +
correspond to hyperbolic hypercomplex numbers. The Cayley–Dickson algebra $  A ( c ) $
 +
is referred to as normed or pseudo-normed according to whether the [[Metric|metric]]
  
<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/h/h110/h110370/h11037026.png" /></td> </tr></table>
+
$$
 +
\eta = { \mathop{\rm diag} } ( 1, - c _ {1} , - c _ {2} , c _ {1} c _ {2} , - c _ {3} \dots ( - 1 )  ^ {p} c _ {1} c _ {2} \dots c _ {p} )
 +
$$
  
is Euclidean or pseudo-Euclidean (cf. also [[Euclidean space|Euclidean space]]; [[Pseudo-Euclidean space|Pseudo-Euclidean space]]). For each algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037027.png" /> there exist <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037028.png" /> anti-involutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037029.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037030.png" />, (the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037031.png" /> satisfies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037033.png" />). One of the anti-involutions is the mapping
+
is Euclidean or pseudo-Euclidean (cf. also [[Euclidean space|Euclidean space]]; [[Pseudo-Euclidean space|Pseudo-Euclidean space]]). For each algebra $  A ( c ) $
 +
there exist $  2m $
 +
anti-involutions $  j : {A ( c ) } \rightarrow {A ( c ) } $:  
 +
$  u \mapsto j ( u ) $,  
 +
(the mapping $  j $
 +
satisfies $  j ( uv ) = j ( v ) j ( u ) $
 +
and $  j ( j ( u ) ) = u $).  
 +
One of the anti-involutions is the mapping
  
<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/h/h110/h110370/h11037034.png" /></td> </tr></table>
+
$$
 +
j _ {0} : u \equiv ( u _ {0} \dots u _ {2m - 1 }  ) \in A ( c ) \mapsto
 +
$$
  
<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/h/h110/h110370/h11037035.png" /></td> </tr></table>
+
$$
 +
\mapsto
 +
j _ {0} ( u ) = ( u _ {0} , - u _ {1} \dots - u _ {2m - 1 }  ) \in A ( c ) ;
 +
$$
  
the remaining <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037036.png" /> anti-involutions correspond to anti-involutions of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037037.png" /> on the various <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037038.png" />-dimensional Cayley–Dickson subalgebras of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037039.png" />.
+
the remaining $  2m - 1 $
 +
anti-involutions correspond to anti-involutions of type $  j _ {0} $
 +
on the various $  m $-
 +
dimensional Cayley–Dickson subalgebras of $  A ( c ) $.
  
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037040.png" /> has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037041.png" /> real components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037042.png" />; these define a vector in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037043.png" /> and can thus be associated with a column vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037044.png" />. From the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037045.png" /> of two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037047.png" />, an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037048.png" /> (generalized) Hurwitz matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037049.png" /> is defined via <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037050.png" />.
+
An element $  u \in A ( c ) $
 +
has $  2m $
 +
real components $  ( u _ {0} \dots u _ {2m - 1 }  ) $;  
 +
these define a vector in $  \mathbf R ^ {2m } $
 +
and can thus be associated with a column vector $  \mathbf u \in \mathbf R ^ {2m \times 1 } $.  
 +
From the product $  w = uv $
 +
of two elements $  u \in A ( c ) $
 +
and $  v \in A ( c ) $,  
 +
an $  \mathbf R ^ {2m \times 2m } $(
 +
generalized) Hurwitz matrix $  H ( \mathbf u ) $
 +
is defined via $  \mathbf w = H ( \mathbf u ) \mathbf v $.
  
 
The application
 
The application
  
<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/h/h110/h110370/h11037051.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1 }
 +
d \mathbf u \in \mathbf R ^ {2m \times 1 } \mapsto \omega = 2 H ( \mathbf u ) \varepsilon d \mathbf u \in \mathbf R ^ {2m \times 1 } ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037052.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037053.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037054.png" />), defines a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037055.png" />, called Hurwitz transformation and denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037056.png" />. The row vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037057.png" /> consists of:
+
where $  \varepsilon = { \mathop{\rm diag} } ( 1, \varepsilon _ {1} \dots \varepsilon _ {2m - 1 }  ) $
 +
with $  \varepsilon _ {t} = \pm  1 $(
 +
$  t = 1 \dots 2m - 1 $),  
 +
defines a mapping $  \mathbf R ^ {2m } \rightarrow \mathbf R ^ {2m - n } $,  
 +
called Hurwitz transformation and denoted by $  T [ 1; c; \varepsilon ] $.  
 +
The row vector $  \omega $
 +
consists of:
  
i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037058.png" /> total differentials, leading to a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037059.png" />;
+
i) $  2m - n $
 +
total differentials, leading to a vector $  x \in \mathbf R ^ {2m - n } $;
  
ii) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037060.png" /> one-forms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037061.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037062.png" />) which are not total differentials and which are taken to be equal to zero to account for the non-bijectivity of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037063.png" />: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037064.png" />. The integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037066.png" />, depends on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037067.png" />. The various possibilities for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037068.png" /> are:
+
ii) $  n $
 +
one-forms $  \omega _ {1} \dots \omega _ {n} $(
 +
when $  n \neq 0 $)  
 +
which are not total differentials and which are taken to be equal to zero to account for the non-bijectivity of the mapping $  \mathbf R ^ {2m } \rightarrow \mathbf R ^ {2m - n } $:  
 +
$  u \mapsto x $.  
 +
The integer $  n $,  
 +
0 \leq  n \leq  2m - 1 $,  
 +
depends on $  \varepsilon $.  
 +
The various possibilities for $  \varepsilon $
 +
are:
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037069.png" /> is the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037070.png" />-matrix;
+
1) $  \varepsilon $
 +
is the unit $  ( 2m \times 2m ) $-
 +
matrix;
  
2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037071.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037072.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037073.png" />) and thus corresponds to the anti-involution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037074.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037075.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037076.png" /> if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037077.png" />);
+
2) $  \varepsilon $
 +
is such that $  \varepsilon _ {t} = - 1 $(
 +
$  t = 1 \dots 2m - 1 $)  
 +
and thus corresponds to the anti-involution $  j _ {0} $
 +
of $  A ( c ) $(
 +
$  v = j _ {0} ( u ) $
 +
if and only if $  \mathbf v = \varepsilon \mathbf u $);
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037078.png" /> corresponds to one of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037079.png" /> remaining anti-involutions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037080.png" />;
+
3) $  j $
 +
corresponds to one of the $  2m - 1 $
 +
remaining anti-involutions of $  A ( c ) $;
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037081.png" /> is a matrix not listed in the other cases. Equation (a1) defines the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037082.png" /> as quadratic functions of the components of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037083.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037084.png" />, the vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037085.png" /> may also be generated from the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037086.png" /> that produces a column vector, with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037087.png" /> vanishing entries and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037088.png" /> non-vanishing entries, corresponding to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037089.png" />.
+
4) $  \varepsilon $
 +
is a matrix not listed in the other cases. Equation (a1) defines the components of $  x $
 +
as quadratic functions of the components of $  u $.  
 +
For $  2m = 2,4, 8 $,  
 +
the vector $  x $
 +
may also be generated from the product $  H ( \mathbf u ) \varepsilon \mathbf u $
 +
that produces a column vector, with $  n $
 +
vanishing entries and $  2m - n $
 +
non-vanishing entries, corresponding to $  x $.
  
Another type of Hurwitz transformation, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037090.png" />, is formally obtained by replacing <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037091.png" /> in (a1) or in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037092.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037093.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037094.png" />. This leads to non-quadratic transformations [[#References|[a4]]].
+
Another type of Hurwitz transformation, denoted by $  T [ N;c; \varepsilon ] $,  
 +
is formally obtained by replacing $  H ( \mathbf u ) $
 +
in (a1) or in $  H ( \mathbf u ) \varepsilon \mathbf u $
 +
by $  H ( \mathbf u )  ^ {N} $
 +
with $  N \in \mathbf Z $.  
 +
This leads to non-quadratic transformations [[#References|[a4]]].
  
The cases <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037095.png" /> deserve special attention, since they correspond to the Hurwitz factorization problem (the situations addressed in [[#References|[a5]]] concern <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037096.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037097.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037098.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h11037099.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370100.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370101.png" />). In these cases, the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370102.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370103.png" /> satisfies
+
The cases $  2m = 2,4,8 $
 +
deserve special attention, since they correspond to the Hurwitz factorization problem (the situations addressed in [[#References|[a5]]] concern $  c _ {1} = c _ {2} = c _ {3} = - 1 $
 +
for $  2m = 8 $,  
 +
$  c _ {1} = c _ {2} = - 1 $
 +
for $  2m = 4 $
 +
and $  c _ {1} = - 1 $
 +
for $  2m = 2 $).  
 +
In these cases, the $  ( 2m \times 2m ) $-
 +
matrix $  H ( \mathbf u ) $
 +
satisfies
  
<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/h/h110/h110370/h110370104.png" /></td> </tr></table>
+
$$
 +
{} ^ {\textrm{ t } } H ( \mathbf u ) \eta H ( \mathbf u ) = ( {} ^ {\textrm{ t } } \mathbf u \eta \mathbf u ) \eta
 +
$$
  
and may be written in terms of elements of a [[Clifford algebra|Clifford algebra]] of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370105.png" />. As a consequence, the factorization property
+
and may be written in terms of elements of a [[Clifford algebra|Clifford algebra]] of degree $  2m - 1 $.  
 +
As a consequence, the factorization 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/h/h110/h110370/h110370106.png" /></td> </tr></table>
+
$$
 +
( {} ^ {\textrm{ t } } \mathbf w \eta \mathbf w ) = ( {} ^ {\textrm{ t } } \mathbf u \eta \mathbf u ) ( {} ^ {\textrm{ t } } \mathbf v \eta \mathbf v ) ,
 +
$$
  
for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370108.png" /> fixed, is satisfied by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370109.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370110.png" />.
+
for $  \mathbf u $
 +
and $  \mathbf v $
 +
fixed, is satisfied by $  \mathbf w = H ( \mathbf u ) \mathbf v $
 +
for $  2m = 2,4,8 $.
  
The geometric and group-theoretical properties of the transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370111.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370112.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370113.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370114.png" /> are well known. From the geometrical point of view, they correspond to fibrations on spheres and hyperboloids [[#References|[a3]]], [[#References|[a6]]]. From the point of view of group theory, they are associated to Lie algebras under constraints [[#References|[a7]]].
+
The geometric and group-theoretical properties of the transformations $  T [ 1; c; \varepsilon ] $
 +
for $  c = ( c _ {1} , c _ {2} ,c _ {3} ) $,
 +
$  c = ( c _ {1} , c _ {2} ) $
 +
and $  c = ( c _ {1} ) $
 +
are well known. From the geometrical point of view, they correspond to fibrations on spheres and hyperboloids [[#References|[a3]]], [[#References|[a6]]]. From the point of view of group theory, they are associated to Lie algebras under constraints [[#References|[a7]]].
  
Some typical examples of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370115.png" /> are as follows. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370116.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370117.png" /> is the unit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370118.png" />-matrix, corresponds to the Levi-Civita transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370119.png" /> used in the restricted [[Three-body problem|three-body problem]] of classical mechanics [[#References|[a8]]]. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370120.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370121.png" />, corresponds to the [[Kustaanheimo–Stiefel transformation|Kustaanheimo–Stiefel transformation]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370122.png" /> used in the regularization of the Kepler problem [[#References|[a9]]] and associated to the [[Hopf fibration|Hopf fibration]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370123.png" /> of fibre <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370124.png" /> [[#References|[a10]]]. The case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370125.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370126.png" />, corresponds to the Fock (stereographic) projection <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370127.png" /> used in the quantum mechanical problem of the hydrogen atom [[#References|[a11]]]. More generally, Hurwitz transformations are useful in number theory and in theoretical physics (classical and quantum mechanics, quantum field theory, local gauge symmetries). In particular, they can be useful for transforming a dynamical system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370128.png" /> into a dynamical system in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370129.png" /> subject to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h110/h110370/h110370130.png" /> constraints. (Under such a transformation, the coupling constant of one system is exchanged with the energy of the other.)
+
Some typical examples of $  T [ N; c; \varepsilon ] $
 +
are as follows. The case $  T [ 1; ( - 1 ) ; \mathbf I ] $,  
 +
where $  \mathbf I $
 +
is the unit $  ( 2 \times 2 ) $-
 +
matrix, corresponds to the Levi-Civita transformation $  \mathbf R  ^ {2} \rightarrow \mathbf R  ^ {2} $
 +
used in the restricted [[Three-body problem|three-body problem]] of classical mechanics [[#References|[a8]]]. The case $  T [ 1; ( - 1, - 1 ) ; \varepsilon ] $,  
 +
where $  \varepsilon = { \mathop{\rm diag} } ( 1, - 1, 1, 1 ) $,  
 +
corresponds to the [[Kustaanheimo–Stiefel transformation|Kustaanheimo–Stiefel transformation]] $  \mathbf R  ^ {4} \rightarrow \mathbf R  ^ {3} $
 +
used in the regularization of the Kepler problem [[#References|[a9]]] and associated to the [[Hopf fibration|Hopf fibration]] $  \mathbf S  ^ {3} \rightarrow \mathbf S  ^ {2} $
 +
of fibre $  \mathbf S  ^ {1} $[[#References|[a10]]]. The case $  T [ - 1; ( - 1, - 1 ) ; \varepsilon ] $,  
 +
where $  \varepsilon = { \mathop{\rm diag} } ( 1, 1, 1, - 1 ) $,  
 +
corresponds to the Fock (stereographic) projection $  \mathbf R  ^ {4} \rightarrow \mathbf S  ^ {3} $
 +
used in the quantum mechanical problem of the hydrogen atom [[#References|[a11]]]. More generally, Hurwitz transformations are useful in number theory and in theoretical physics (classical and quantum mechanics, quantum field theory, local gauge symmetries). In particular, they can be useful for transforming a dynamical system in $  \mathbf R ^ {2m } $
 +
into a dynamical system in $  \mathbf R ^ {2m - n } $
 +
subject to $  n $
 +
constraints. (Under such a transformation, the coupling constant of one system is exchanged with the energy of the other.)
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.E. Dickson,  "On quaternions and their generalization and the history of the eight square theorem"  ''Ann. of Math.'' , '''20'''  (1919)  pp. 155</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.P. Wene,  "A construction relating Clifford algebras and Cayley–Dickson algebras"  ''J. Math. Phys.'' , '''25'''  (1984)  pp. 2351</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Lambert,  M. Kibler,  "An algebraic and geometric approach to non-bijective quadratic transformations"  ''J. Phys. A: Math. Gen.'' , '''21'''  (1988)  pp. 307</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Kibler,  P. Labastie,  "Transformations generalizing the Levi-Civita, Kustaanheimo–Stiefel and Fock transformations"  Y. Saint-Aubin (ed.)  L. Vinet (ed.) , ''Group Theoretical Methods in Physics'' , World Sci.  (1989)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Hurwitz,  "Über die Komposition der quadratischen Formen von beliebig vielen Variablen"  ''Nachr. K. Gesellschaft Wissenschaft. Göttingen''  (1898)  pp. 309</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I.V. Polubarinov,  "On the application of Hopf fiber bundles in quantum theory" , ''Report E2-84-607'' , JINR: Dubna (Russia)  (1984)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Kibler,  P. Winternitz,  "Lie algebras under constraints and non-bijective transformations"  ''J. Phys. A: Math. Gen.'' , '''21'''  (1988)  pp. 1787</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  T. Levi-Civita,  "Sur la régularisation du problème des trois corps"  ''Acta Math.'' , '''42'''  (1918)  pp. 99</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  P. Kustaanheimo,  E. Stiefel,  "Perturbation theory of Kepler motion based on spinor regularization"  ''J. Reine Angew. Math.'' , '''218'''  (1965)  pp. 204</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  H. Hopf,  "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche"  ''Math. Ann.'' , '''104'''  (1931)  pp. 637</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  V. Fock,  "Zur Theorie des Wasserstoffatoms"  ''Z. Phys.'' , '''98'''  (1935)  pp. 145</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  L.E. Dickson,  "On quaternions and their generalization and the history of the eight square theorem"  ''Ann. of Math.'' , '''20'''  (1919)  pp. 155</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  G.P. Wene,  "A construction relating Clifford algebras and Cayley–Dickson algebras"  ''J. Math. Phys.'' , '''25'''  (1984)  pp. 2351</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  D. Lambert,  M. Kibler,  "An algebraic and geometric approach to non-bijective quadratic transformations"  ''J. Phys. A: Math. Gen.'' , '''21'''  (1988)  pp. 307</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Kibler,  P. Labastie,  "Transformations generalizing the Levi-Civita, Kustaanheimo–Stiefel and Fock transformations"  Y. Saint-Aubin (ed.)  L. Vinet (ed.) , ''Group Theoretical Methods in Physics'' , World Sci.  (1989)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  A. Hurwitz,  "Über die Komposition der quadratischen Formen von beliebig vielen Variablen"  ''Nachr. K. Gesellschaft Wissenschaft. Göttingen''  (1898)  pp. 309</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  I.V. Polubarinov,  "On the application of Hopf fiber bundles in quantum theory" , ''Report E2-84-607'' , JINR: Dubna (Russia)  (1984)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  M. Kibler,  P. Winternitz,  "Lie algebras under constraints and non-bijective transformations"  ''J. Phys. A: Math. Gen.'' , '''21'''  (1988)  pp. 1787</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  T. Levi-Civita,  "Sur la régularisation du problème des trois corps"  ''Acta Math.'' , '''42'''  (1918)  pp. 99</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  P. Kustaanheimo,  E. Stiefel,  "Perturbation theory of Kepler motion based on spinor regularization"  ''J. Reine Angew. Math.'' , '''218'''  (1965)  pp. 204</TD></TR><TR><TD valign="top">[a10]</TD> <TD valign="top">  H. Hopf,  "Über die Abbildungen der dreidimensionalen Sphäre auf die Kugelfläche"  ''Math. Ann.'' , '''104'''  (1931)  pp. 637</TD></TR><TR><TD valign="top">[a11]</TD> <TD valign="top">  V. Fock,  "Zur Theorie des Wasserstoffatoms"  ''Z. Phys.'' , '''98'''  (1935)  pp. 145</TD></TR></table>

Latest revision as of 22:11, 5 June 2020


The general framework for defining and studying the Hurwitz transformations is that of the Cayley–Dickson algebras (cf. also Cayley–Dickson algebra). Familiar examples of Cayley–Dickson algebras are:

$ A ( - 1 ) \equiv \mathbf C $ and $ A ( 1 ) \equiv \Omega $, the algebras of ordinary and hyperbolic complex numbers, respectively;

$ A ( - 1, - 1 ) \equiv \mathbf H $ and $ A ( 1,1 ) \equiv \mathbf N _ {1} $, the algebras of ordinary (or Hamilton) and hyperbolic quaternions, respectively;

$ A ( - 1, - 1, - 1 ) \equiv \mathbf O $ and $ A ( 1,1,1 ) \equiv \mathbf O ^ \prime $, the algebras of ordinary (or Cayley) and hyperbolic octonions, respectively. Higher-dimensional Cayley–Dickson algebras over the real field $ \mathbf R $ are denoted by $ A ( c ) $, where $ c $ is a $ p $- tuple $ c = ( c _ {1} \dots c _ {p} ) $ with $ c _ {s} = \pm 1 $( $ s = 1 \dots p $). Such a $ 2m $- dimensional algebra $ A ( c ) $, with $ 2m = 2 ^ {p} $, may be constructed from an $ m $- dimensional Cayley–Dickson algebra by a "doubling" process [a1], [a2], [a3]. (This process generalizes $ \mathbf C = \mathbf R + i \mathbf R $.) For fixed $ p $, the algebra $ A ( c ) $ with $ \sum _ {s = 1 } ^ {p} c _ {s} = - p $ corresponds to ordinary (or elliptic) hypercomplex numbers (cf. also Hypercomplex number), while the $ 2m - 1 $ other algebras $ A ( c ) $ with $ \sum _ {s = 1 } ^ {p} c _ {s} \neq - p $ correspond to hyperbolic hypercomplex numbers. The Cayley–Dickson algebra $ A ( c ) $ is referred to as normed or pseudo-normed according to whether the metric

$$ \eta = { \mathop{\rm diag} } ( 1, - c _ {1} , - c _ {2} , c _ {1} c _ {2} , - c _ {3} \dots ( - 1 ) ^ {p} c _ {1} c _ {2} \dots c _ {p} ) $$

is Euclidean or pseudo-Euclidean (cf. also Euclidean space; Pseudo-Euclidean space). For each algebra $ A ( c ) $ there exist $ 2m $ anti-involutions $ j : {A ( c ) } \rightarrow {A ( c ) } $: $ u \mapsto j ( u ) $, (the mapping $ j $ satisfies $ j ( uv ) = j ( v ) j ( u ) $ and $ j ( j ( u ) ) = u $). One of the anti-involutions is the mapping

$$ j _ {0} : u \equiv ( u _ {0} \dots u _ {2m - 1 } ) \in A ( c ) \mapsto $$

$$ \mapsto j _ {0} ( u ) = ( u _ {0} , - u _ {1} \dots - u _ {2m - 1 } ) \in A ( c ) ; $$

the remaining $ 2m - 1 $ anti-involutions correspond to anti-involutions of type $ j _ {0} $ on the various $ m $- dimensional Cayley–Dickson subalgebras of $ A ( c ) $.

An element $ u \in A ( c ) $ has $ 2m $ real components $ ( u _ {0} \dots u _ {2m - 1 } ) $; these define a vector in $ \mathbf R ^ {2m } $ and can thus be associated with a column vector $ \mathbf u \in \mathbf R ^ {2m \times 1 } $. From the product $ w = uv $ of two elements $ u \in A ( c ) $ and $ v \in A ( c ) $, an $ \mathbf R ^ {2m \times 2m } $( generalized) Hurwitz matrix $ H ( \mathbf u ) $ is defined via $ \mathbf w = H ( \mathbf u ) \mathbf v $.

The application

$$ \tag{a1 } d \mathbf u \in \mathbf R ^ {2m \times 1 } \mapsto \omega = 2 H ( \mathbf u ) \varepsilon d \mathbf u \in \mathbf R ^ {2m \times 1 } , $$

where $ \varepsilon = { \mathop{\rm diag} } ( 1, \varepsilon _ {1} \dots \varepsilon _ {2m - 1 } ) $ with $ \varepsilon _ {t} = \pm 1 $( $ t = 1 \dots 2m - 1 $), defines a mapping $ \mathbf R ^ {2m } \rightarrow \mathbf R ^ {2m - n } $, called Hurwitz transformation and denoted by $ T [ 1; c; \varepsilon ] $. The row vector $ \omega $ consists of:

i) $ 2m - n $ total differentials, leading to a vector $ x \in \mathbf R ^ {2m - n } $;

ii) $ n $ one-forms $ \omega _ {1} \dots \omega _ {n} $( when $ n \neq 0 $) which are not total differentials and which are taken to be equal to zero to account for the non-bijectivity of the mapping $ \mathbf R ^ {2m } \rightarrow \mathbf R ^ {2m - n } $: $ u \mapsto x $. The integer $ n $, $ 0 \leq n \leq 2m - 1 $, depends on $ \varepsilon $. The various possibilities for $ \varepsilon $ are:

1) $ \varepsilon $ is the unit $ ( 2m \times 2m ) $- matrix;

2) $ \varepsilon $ is such that $ \varepsilon _ {t} = - 1 $( $ t = 1 \dots 2m - 1 $) and thus corresponds to the anti-involution $ j _ {0} $ of $ A ( c ) $( $ v = j _ {0} ( u ) $ if and only if $ \mathbf v = \varepsilon \mathbf u $);

3) $ j $ corresponds to one of the $ 2m - 1 $ remaining anti-involutions of $ A ( c ) $;

4) $ \varepsilon $ is a matrix not listed in the other cases. Equation (a1) defines the components of $ x $ as quadratic functions of the components of $ u $. For $ 2m = 2,4, 8 $, the vector $ x $ may also be generated from the product $ H ( \mathbf u ) \varepsilon \mathbf u $ that produces a column vector, with $ n $ vanishing entries and $ 2m - n $ non-vanishing entries, corresponding to $ x $.

Another type of Hurwitz transformation, denoted by $ T [ N;c; \varepsilon ] $, is formally obtained by replacing $ H ( \mathbf u ) $ in (a1) or in $ H ( \mathbf u ) \varepsilon \mathbf u $ by $ H ( \mathbf u ) ^ {N} $ with $ N \in \mathbf Z $. This leads to non-quadratic transformations [a4].

The cases $ 2m = 2,4,8 $ deserve special attention, since they correspond to the Hurwitz factorization problem (the situations addressed in [a5] concern $ c _ {1} = c _ {2} = c _ {3} = - 1 $ for $ 2m = 8 $, $ c _ {1} = c _ {2} = - 1 $ for $ 2m = 4 $ and $ c _ {1} = - 1 $ for $ 2m = 2 $). In these cases, the $ ( 2m \times 2m ) $- matrix $ H ( \mathbf u ) $ satisfies

$$ {} ^ {\textrm{ t } } H ( \mathbf u ) \eta H ( \mathbf u ) = ( {} ^ {\textrm{ t } } \mathbf u \eta \mathbf u ) \eta $$

and may be written in terms of elements of a Clifford algebra of degree $ 2m - 1 $. As a consequence, the factorization property

$$ ( {} ^ {\textrm{ t } } \mathbf w \eta \mathbf w ) = ( {} ^ {\textrm{ t } } \mathbf u \eta \mathbf u ) ( {} ^ {\textrm{ t } } \mathbf v \eta \mathbf v ) , $$

for $ \mathbf u $ and $ \mathbf v $ fixed, is satisfied by $ \mathbf w = H ( \mathbf u ) \mathbf v $ for $ 2m = 2,4,8 $.

The geometric and group-theoretical properties of the transformations $ T [ 1; c; \varepsilon ] $ for $ c = ( c _ {1} , c _ {2} ,c _ {3} ) $, $ c = ( c _ {1} , c _ {2} ) $ and $ c = ( c _ {1} ) $ are well known. From the geometrical point of view, they correspond to fibrations on spheres and hyperboloids [a3], [a6]. From the point of view of group theory, they are associated to Lie algebras under constraints [a7].

Some typical examples of $ T [ N; c; \varepsilon ] $ are as follows. The case $ T [ 1; ( - 1 ) ; \mathbf I ] $, where $ \mathbf I $ is the unit $ ( 2 \times 2 ) $- matrix, corresponds to the Levi-Civita transformation $ \mathbf R ^ {2} \rightarrow \mathbf R ^ {2} $ used in the restricted three-body problem of classical mechanics [a8]. The case $ T [ 1; ( - 1, - 1 ) ; \varepsilon ] $, where $ \varepsilon = { \mathop{\rm diag} } ( 1, - 1, 1, 1 ) $, corresponds to the Kustaanheimo–Stiefel transformation $ \mathbf R ^ {4} \rightarrow \mathbf R ^ {3} $ used in the regularization of the Kepler problem [a9] and associated to the Hopf fibration $ \mathbf S ^ {3} \rightarrow \mathbf S ^ {2} $ of fibre $ \mathbf S ^ {1} $[a10]. The case $ T [ - 1; ( - 1, - 1 ) ; \varepsilon ] $, where $ \varepsilon = { \mathop{\rm diag} } ( 1, 1, 1, - 1 ) $, corresponds to the Fock (stereographic) projection $ \mathbf R ^ {4} \rightarrow \mathbf S ^ {3} $ used in the quantum mechanical problem of the hydrogen atom [a11]. More generally, Hurwitz transformations are useful in number theory and in theoretical physics (classical and quantum mechanics, quantum field theory, local gauge symmetries). In particular, they can be useful for transforming a dynamical system in $ \mathbf R ^ {2m } $ into a dynamical system in $ \mathbf R ^ {2m - n } $ subject to $ n $ constraints. (Under such a transformation, the coupling constant of one system is exchanged with the energy of the other.)

References

[a1] L.E. Dickson, "On quaternions and their generalization and the history of the eight square theorem" Ann. of Math. , 20 (1919) pp. 155
[a2] G.P. Wene, "A construction relating Clifford algebras and Cayley–Dickson algebras" J. Math. Phys. , 25 (1984) pp. 2351
[a3] D. Lambert, M. Kibler, "An algebraic and geometric approach to non-bijective quadratic transformations" J. Phys. A: Math. Gen. , 21 (1988) pp. 307
[a4] M. Kibler, P. Labastie, "Transformations generalizing the Levi-Civita, Kustaanheimo–Stiefel and Fock transformations" Y. Saint-Aubin (ed.) L. Vinet (ed.) , Group Theoretical Methods in Physics , World Sci. (1989)
[a5] A. Hurwitz, "Über die Komposition der quadratischen Formen von beliebig vielen Variablen" Nachr. K. Gesellschaft Wissenschaft. Göttingen (1898) pp. 309
[a6] I.V. Polubarinov, "On the application of Hopf fiber bundles in quantum theory" , Report E2-84-607 , JINR: Dubna (Russia) (1984)
[a7] M. Kibler, P. Winternitz, "Lie algebras under constraints and non-bijective transformations" J. Phys. A: Math. Gen. , 21 (1988) pp. 1787
[a8] T. Levi-Civita, "Sur la régularisation du problème des trois corps" Acta Math. , 42 (1918) pp. 99
[a9] P. Kustaanheimo, E. Stiefel, "Perturbation theory of Kepler motion based on spinor regularization" J. Reine Angew. Math. , 218 (1965) pp. 204
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How to Cite This Entry:
Hurwitz transformation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hurwitz_transformation&oldid=47281
This article was adapted from an original article by M. Kibler (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article