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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:

$  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 [[#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]]

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

is Euclidean or pseudo-Euclidean (cf. also [[Euclidean space|Euclidean space]]; [[Pseudo-Euclidean space|Pseudo-Euclidean space]]). For each algebra  $  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 [[#References|[a4]]].

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

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

and may be written in terms of elements of a [[Clifford algebra|Clifford algebra]] of degree  $  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 [[#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  $  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====
<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>