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''of transformations of a differentiable manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757001.png" />''
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A family of diffeomorphisms from open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757002.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757003.png" /> that is closed under composition of mappings, transition to the inverse mapping, as well as under restriction and glueing of mappings. More precisely, a pseudo-group of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757004.png" /> of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757005.png" /> consists of local transformations, i.e. pairs of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757006.png" /> where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757007.png" /> is an open subset of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757008.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p0757009.png" /> is a [[Diffeomorphism|diffeomorphism]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570010.png" />, where it is moreover assumed that 1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570011.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570012.png" />; 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570013.png" /> implies <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570014.png" />; 3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570015.png" />; and 4) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570016.png" /> is a diffeomorphism from an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570017.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570018.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570019.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570020.png" /> are open sets in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570021.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570022.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570023.png" />. With necessary changes in 1)–4) one can also define pseudo-groups of transformations of an arbitrary topological space (cf. [[#References|[7]]]) or even of an arbitrary set. As a group of transformations, a pseudo-group of transformations determines an equivalence relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570024.png" />; the equivalence classes are called its orbits. A pseudo-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570025.png" /> of transformations of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570026.png" /> is called transitive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570027.png" /> is its only orbit, and is called primitive if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570028.png" /> does not admit non-trivial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570029.png" />-invariant foliations (otherwise the pseudo-group is called imprimitive).
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A pseudo-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570030.png" /> of transformations of a differentiable manifold is called a Lie pseudo-group of transformations defined by a system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570031.png" /> of partial differential equations if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570032.png" /> consists of exactly those local transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570033.png" /> that satisfy the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570034.png" />. E.g., the pseudo-group of conformal transformations of the plane is a Lie pseudo-group of transformations, determined by the Cauchy–Riemann equations (cf. [[Cauchy–Riemann conditions|Cauchy–Riemann conditions]]). The order of a Lie pseudo-group of transformations is the minimum order of its defining system of differential equations.
+
''of transformations of a differentiable manifold $  M $''
  
Examples of Lie pseudo-groups of transformations. a) The pseudo-group of all holomorphic local transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570035.png" />-dimensional complex space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570036.png" />.
+
A family of diffeomorphisms from open subsets of  $  M $
 +
into  $  M $
 +
that is closed under composition of mappings, transition to the inverse mapping, as well as under restriction and glueing of mappings. More precisely, a pseudo-group of transformations  $  \Gamma $
 +
of a manifold  $  M $
 +
consists of local transformations, i.e. pairs of the form  $  p =( D _ {p} , \overline{p}\; ) $
 +
where  $  D _ {p} $
 +
is an open subset of  $  M $
 +
and  $  \overline{p}\; $
 +
is a [[Diffeomorphism|diffeomorphism]]  $  D _ {p} \rightarrow M $,
 +
where it is moreover assumed that 1)  $  p , q \in \Gamma $
 +
implies  $  p \circ q = ( \overline{q}  ^ {- 1} ( D _ {p} \cap \overline{q} ( D _ {q} ) ) , \overline{p}\; \circ \overline{q}\; ) \in \Gamma $;
 +
2)  $  p \in \Gamma $
 +
implies  $  p  ^ {- 1} = ( \overline{p} ( D _ {p} ) , \overline{p} ^ {- 1} ) \in \Gamma $;
 +
3)  $  ( M ,  \mathop{\rm id} ) \in \Gamma $;
 +
and 4) if  $  \overline{p}\; $
 +
is a diffeomorphism from an open subset  $  D \subset  M $
 +
into  $  M $
 +
and  $  D = \cup _  \alpha  D _  \alpha  $,
 +
where  $  D _  \alpha  $
 +
are open sets in  $  M $,
 +
then  $  ( D , \overline{p} ) \in \Gamma \iff ( D _  \alpha  , \overline{p}\; \mid  _ {D _  \alpha  } ) \in \Gamma $
 +
for any  $  \alpha $.  
 +
With necessary changes in 1)–4) one can also define pseudo-groups of transformations of an arbitrary topological space (cf. [[#References|[7]]]) or even of an arbitrary set. As a group of transformations, a pseudo-group of transformations determines an equivalence relation on  $  M $;
 +
the equivalence classes are called its orbits. A pseudo-group  $  \Gamma $
 +
of transformations of a manifold  $  M $
 +
is called transitive if  $  M $
 +
is its only orbit, and is called primitive if  $  M $
 +
does not admit non-trivial  $  \Gamma $-invariant foliations (otherwise the pseudo-group is called imprimitive).
  
b) The pseudo-group of all holomorphic local transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570037.png" /> with constant [[Jacobian|Jacobian]].
+
A pseudo-group  $  \Gamma $
 +
of transformations of a differentiable manifold is called a Lie pseudo-group of transformations defined by a system  $  S $
 +
of partial differential equations if  $  \Gamma $
 +
consists of exactly those local transformations of $  M $
 +
that satisfy the system  $  S $.  
 +
E.g., the pseudo-group of conformal transformations of the plane is a Lie pseudo-group of transformations, determined by the Cauchy–Riemann equations (cf. [[Cauchy-Riemann equations]]). The order of a Lie pseudo-group of transformations is the minimum order of its defining system of differential equations.
  
c) The pseudo-group of all holomorphic local transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570038.png" /> with Jacobian 1.
+
Examples of Lie pseudo-groups of transformations. a) The pseudo-group of all holomorphic local transformations of $  n $-dimensional complex space  $  \mathbf C  ^ {n} $.
  
d) The Hamilton pseudo-group of all holomorphic local transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570039.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570040.png" /> even) preserving the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570041.png" />-form
+
b) The pseudo-group of all holomorphic local transformations of $  \mathbf C  ^ {n} $
 +
with constant [[Jacobian|Jacobian]].
  
<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/p/p075/p075700/p07570042.png" /></td> </tr></table>
+
c) The pseudo-group of all holomorphic local transformations of  $  \mathbf C  ^ {n} $
 +
with Jacobian 1.
  
e) The pseudo-group of all holomorphic local transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570043.png" /> preserving <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570044.png" /> up to constant factor.
+
d) The Hamilton pseudo-group of all holomorphic local transformations of $  \mathbf C  ^ {n} $ ($  n $ even) preserving the differential 2-form
  
f) The contact pseudo-group of all holomorphic local transformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570045.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570046.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570047.png" />) preserving the differential <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570048.png" />-form
+
$$
 +
\omega  =  d z  ^ {1} \wedge
 +
d z  ^ {2} + d z  ^ {3} \wedge
 +
d z  ^ {4} + \dots + d z  ^ {n- 1}
 +
\wedge d z  ^ {n} .
 +
$$
  
<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/p/p075/p075700/p07570049.png" /></td> </tr></table>
+
e) The pseudo-group of all holomorphic local transformations of  $  \mathbf C  ^ {n} $
 +
preserving  $  \omega $
 +
up to constant factor.
 +
 
 +
f) The contact pseudo-group of all holomorphic local transformations of  $  \mathbf C  ^ {n} $ ($  n = 2 m + 1 $,
 +
$  m \geq  1 $)
 +
preserving the differential 1-form
 +
 
 +
$$
 +
d z  ^ {n} +
 +
\sum _ { i= 1} ^ { m }
 +
( z  ^ {i}  d z  ^ {m+ i} -
 +
z  ^ {m+ i}  d z  ^ {i} )
 +
$$
  
 
up to a factor (which can be a function).
 
up to a factor (which can be a function).
Line 27: Line 87:
 
The order of the Lie pseudo-groups of Examples a), c)–f) is 1, while in b) the order is 2.
 
The order of the Lie pseudo-groups of Examples a), c)–f) is 1, while in b) the order is 2.
  
Any Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570050.png" /> of transformations of a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570051.png" /> determines a pseudo-group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570052.png" /> of transformations, consisting of the restrictions of the transformations from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570053.png" /> onto open subsets of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570054.png" />. A pseudo-group of transformations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570055.png" /> is called globalizable. E.g., a pseudo-group of local conformal transformations of the sphere <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570056.png" /> is globalizable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570057.png" /> and not globalizable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570058.png" />.
+
Any Lie group $  G $
 +
of transformations of a manifold $  M $
 +
determines a pseudo-group $  \Gamma ( G) $
 +
of transformations, consisting of the restrictions of the transformations from $  G $
 +
onto open subsets of $  M $.  
 +
A pseudo-group of transformations of the form $  \Gamma ( G) $
 +
is called globalizable. E.g., a pseudo-group of local conformal transformations of the sphere $  S  ^ {n} $
 +
is globalizable for $  n > 2 $
 +
and not globalizable for $  n = 2 $.
 +
 
 +
A Lie pseudo-group of transformations is said to be of finite type if there is a natural number  $  d $
 +
such that every local transformation  $  p \in \Gamma $
 +
is uniquely determined by its  $  d $-jet at some point  $  x \in D _ {p} $;
 +
the smallest such  $  d $
 +
is called the degree, or type, of  $  \Gamma $;
 +
if such a  $  d $
 +
does not exist, then  $  \Gamma $
 +
is called a pseudo-group of transformations of infinite type. The pseudo-groups of Examples a)–f) are primitive Lie pseudo-groups of transformations of infinite type.
  
A Lie pseudo-group of transformations is said to be of finite type if there is a natural number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570059.png" /> such that every local transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570060.png" /> is uniquely determined by its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570061.png" />-jet at some point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570062.png" />; the smallest such <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570063.png" /> is called the degree, or type, of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570064.png" />; if such a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570065.png" /> does not exist, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570066.png" /> is called a pseudo-group of transformations of infinite type. The pseudo-groups of Examples a)–f) are primitive Lie pseudo-groups of transformations of infinite type.
+
Let  $  \Gamma $
 +
be a transitive Lie pseudo-group of transformations of an  $  n $-dimensional manifold  $  M $
 +
and let  $  G  ^ {r} ( \Gamma ) $
 +
be the family of all  $  r $-jets of the local transformations in  $  \Gamma $
 +
that preserve a point  $  O \in M $,
 +
i.e. those  $  p \in \Gamma $
 +
for which  $  O \in D _ {p} $
 +
and  $  \overline{p} ( O) = O $.  
 +
The set  $  G  ^ {r} ( \Gamma ) $,
 +
endowed with the natural structure of a Lie group, is called the $  r $-th order isotropy group of  $  \Gamma $ ($  G  ^ {1} ( \Gamma ) $
 +
is also called the linear isotropy group of $  \Gamma $).  
 +
The Lie algebra  $  \mathfrak g  ^ {r} ( \Gamma ) $
 +
of  $  \Gamma  ^ {r} ( \Gamma ) $
 +
can be naturally imbedded in the Lie algebra of  $  r $-jets of vector fields on  $  M $
 +
at  $  O $.  
 +
If  $  \Gamma $
 +
is a Lie pseudo-group of transformations of order one, then the kernel  $  G  ^ {( r)} ( \Gamma ) $
 +
of the natural homomorphism  $  G  ^ {r+ 1} ( \Gamma ) \rightarrow G  ^ {r} ( \Gamma ) $
 +
depends, for any  $  r \geq  1 $,
 +
only on the linear isotropy group  $  G  ^ {1} ( \Gamma ) $,
 +
and is called its  $  r $-th extension. A Lie pseudo-group of transformations $  \Gamma $
 +
of order one is of finite type $  d $
 +
if and only if
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570067.png" /> be a transitive Lie pseudo-group of transformations of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570068.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570069.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570070.png" /> be the family of all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570071.png" />-jets of the local transformations in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570072.png" /> that preserve a point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570073.png" />, i.e. those <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570074.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570075.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570076.png" />. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570077.png" />, endowed with the natural structure of a Lie group, is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570079.png" />-th order isotropy group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570080.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570081.png" /> is also called the linear isotropy group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570082.png" />). The Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570083.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570084.png" /> can be naturally imbedded in the Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570085.png" />-jets of vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570086.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570087.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570088.png" /> is a Lie pseudo-group of transformations of order one, then the kernel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570089.png" /> of the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570090.png" /> depends, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570091.png" />, only on the linear isotropy group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570092.png" />, and is called its <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570093.png" />-th extension. A Lie pseudo-group of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570094.png" /> of order one is of finite type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570095.png" /> if and only if
+
$$
 +
\mathop{\rm dim}  G  ^ {( d- 1)}
 +
( \Gamma )  \neq  0 \ \
 +
\textrm{ and } \ \
 +
\mathop{\rm dim}  G  ^ {( d)}
 +
( \Gamma ) = 0 .
 +
$$
  
<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/p/p075/p075700/p07570096.png" /></td> </tr></table>
+
If, moreover,  $  G  ^ {1} ( \Gamma ) $
 +
is irreducible, then  $  d \leq  2 $ (cf. ). A Lie pseudo-group of transformations  $  \Gamma $
 +
of order one is a pseudo-group of transformations of finite type only if, and in the complex case if and only if, the Lie algebra  $  \mathfrak g  ^ {1} $
 +
does not contain endomorphisms of rank 1 (cf. [[#References|[10]]]). Such linear Lie algebras are called elliptic.
  
If, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570097.png" /> is irreducible, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570098.png" /> (cf. ). A Lie pseudo-group of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p07570099.png" /> of order one is a pseudo-group of transformations of finite type only if, and in the complex case if and only if, the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700100.png" /> does not contain endomorphisms of rank 1 (cf. [[#References|[10]]]). Such linear Lie algebras are called elliptic.
+
One has calculated the Lie algebras of all extensions  $  G  ^ {( r)} ( \Gamma ) $,  
 +
$  r \geq  1 $,  
 +
where  $  \Gamma $
 +
is a Lie pseudo-group of transformations of order one, in terms of the linear isotropy algebra. More precisely, the Lie algebra $  \mathfrak g  ^ {( r)} ( \Gamma ) $
 +
of  $  G  ^ {( r)} ( \Gamma ) $
 +
consists of the  $  ( r+ 1) $-jets of vector fields on  $  M $
 +
at  $  O $
 +
having, in some local coordinate system  $  ( x  ^ {1} \dots x  ^ {n} ) $,
 +
the form
  
One has calculated the Lie algebras of all extensions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700101.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700102.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700103.png" /> is a Lie pseudo-group of transformations of order one, in terms of the linear isotropy algebra. More precisely, the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700104.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700105.png" /> consists of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700106.png" />-jets of vector fields on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700107.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700108.png" /> having, in some local coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700109.png" />, the form
+
$$
 +
\sum v _ {i _ {0}  \dots i _ {r} }  ^ {i} x ^ {i _ {0} } \dots
 +
x ^ {i _ {r} }
  
<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/p/p075/p075700/p075700110.png" /></td> </tr></table>
+
\frac \partial {\partial  x  ^ {i} }
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700111.png" /> is an arbitrary tensor that is symmetric with respect to the lower indices and that satisfies the condition: For any fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700112.png" /> the matrix
+
where $  v _ {i _ {0}  \dots i _ {r} }  ^ {i} $
 +
is an arbitrary tensor that is symmetric with respect to the lower indices and that satisfies the condition: For any fixed $  i _ {1} \dots i _ {r} $
 +
the matrix
  
<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/p/p075/p075700/p075700113.png" /></td> </tr></table>
+
$$
 +
\| v _ {j , i _ {1}  \dots i _ {r} }  ^ {i} \| _ {i , j = 1 }  ^ {n}
 +
$$
  
belongs to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700114.png" />, relative to some coordinate system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700115.png" />.
+
belongs to $  \mathfrak g  ^ {1} ( \Gamma ) $,  
 +
relative to some coordinate system $  ( x  ^ {i} ) $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700116.png" /> be an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700117.png" />-dimensional differentiable manifold over the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700118.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700119.png" />. Every transitive Lie pseudo-group of transformations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700120.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700121.png" /> on a manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700122.png" /> coincides with the pseudo-group of all local automorphism of some <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700123.png" />-structure (cf. [[G-structure(2)|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700124.png" />-structure]]) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700125.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075700/p075700126.png" /> (Cartan's first fundamental theorem). The first classification of all primitive Lie pseudo-groups of infinite type was obtained by E. Cartan . According to his theorem, every primitive Lie pseudo-group of transformations of infinite type, consisting of holomorphic local transformations, is locally isomorphic to one of the pseudo-groups of Examples a)–f). This theorem has been repeatedly proved; its modern proofs lead to the study of certain filtered Lie algebras (cf. [[#References|[9]]]). The classification of these filtered Lie algebras can be given on the basis of the classification of simple graded Lie algebras (cf. [[#References|[3]]]). The classification of primitive pseudo-groups of transformations has also been obtained in the real case, and the condition of analyticity of the action of the pseudo-group of transformations has been replaced by the weaker condition of infinite differentiability (cf. [[#References|[8]]], [[#References|[9]]]). One has constructed certain abstract models of transitive Lie pseudo-groups, which came to play the same role in the theory of pseudo-groups of transformations of infinite type as do abstract Lie groups in the finite-dimensional case (cf. , [[#References|[9]]]).
+
Let $  M $
 +
be an $  n $-dimensional differentiable manifold over the field $  K = \mathbf R $
 +
or $  \mathbf C $.  
 +
Every transitive Lie pseudo-group of transformations $  \Gamma $
 +
of order $  k $
 +
on a manifold $  M $
 +
coincides with the pseudo-group of all local automorphism of some $  G  ^ {k} ( \Gamma ) $-structure (cf. [[G-structure| $  G $-structure]]) of order $  k $
 +
on $  M $ (Cartan's first fundamental theorem). The first classification of all primitive Lie pseudo-groups of infinite type was obtained by E. Cartan . According to his theorem, every primitive Lie pseudo-group of transformations of infinite type, consisting of holomorphic local transformations, is locally isomorphic to one of the pseudo-groups of Examples a)–f). This theorem has been repeatedly proved; its modern proofs lead to the study of certain filtered Lie algebras (cf. [[#References|[9]]]). The classification of these filtered Lie algebras can be given on the basis of the classification of simple graded Lie algebras (cf. [[#References|[3]]]). The classification of primitive pseudo-groups of transformations has also been obtained in the real case, and the condition of analyticity of the action of the pseudo-group of transformations has been replaced by the weaker condition of infinite differentiability (cf. [[#References|[8]]], [[#References|[9]]]). One has constructed certain abstract models of transitive Lie pseudo-groups, which came to play the same role in the theory of pseudo-groups of transformations of infinite type as do abstract Lie groups in the finite-dimensional case (cf. , [[#References|[9]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) {{MR|0193578}} {{ZBL|0129.13102}} </TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> E. Cartan, "Sur la structure des groupes infinis de transformations" , ''Oeuvres complètes'' , '''2''' , Gauthier-Villars (1953) pp. 571–624 {{MR|1509054}} {{MR|1509040}} {{ZBL|36.0223.03}} {{ZBL|35.0176.04}} </TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> E. Cartan, "Sur la structure des groupes infinis de transformations" , ''Oeuvres complètes'' , '''2''' , Gauthier-Villars (1953) pp. 625–714 {{MR|1509054}} {{MR|1509040}} {{ZBL|36.0223.03}} {{ZBL|35.0176.04}} </TD></TR><TR><TD valign="top">[2c]</TD> <TD valign="top"> E. Cartan, "Les groupes de transformations continus, infinis, simples" , ''Oeuvres complètes'' , '''2''' , Gauthier-Villars (1953) pp. 857–925 {{MR|1509105}} {{ZBL|40.0193.02}} {{ZBL|38.0194.01}} </TD></TR><TR><TD valign="top">[2d]</TD> <TD valign="top"> E. Cartan, "Les groupes de transformations continus, infinis, simples" , ''Oeuvres complètes'' , '''2''' , Gauthier-Villars (1953) pp. 1335–1384 {{MR|1509105}} {{ZBL|40.0193.02}} {{ZBL|38.0194.01}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V. Guillemin, "Infinite dimensional primitive Lie algebras" ''J. Diff. Geom.'' , '''4''' : 3 (1970) pp. 257–282 {{MR|0268233}} {{ZBL|0223.17007}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972) {{MR|0355886}} {{ZBL|0246.53031}} </TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top"> S. Kobayashi, T. Nagano, "On filtered Lie algebras and geometric structures I" ''J. Math. Mech.'' , '''13''' : 5 (1964) pp. 875–907 {{MR|0168704}} {{ZBL|0142.19504}} </TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top"> S. Kobayashi, T. Nagano, "On filtered Lie algebras and geometric structures III" ''J. Math. Mech.'' , '''14''' : 5 (1965) pp. 679–706 {{MR|0188364}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6a]</TD> <TD valign="top"> M. Kuranishi, "On the local theory of continuous infinite pseudo groups I" ''Nagoya Math. J.'' , '''15''' (1959) pp. 225–260 {{MR|0116071}} {{ZBL|0212.56501}} </TD></TR><TR><TD valign="top">[6b]</TD> <TD valign="top"> M. Kuranishi, "On the local theory of continuous infinite pseudo groups II" ''Nagoya Math. J.'' , '''19''' (1961) pp. 55–91 {{MR|0142694}} {{ZBL|0212.56501}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> P. Libermann, "Pseudogroupes infinitésimaux attachées aux pseudogroupes de Lie" ''Bull. Soc. Math. France'' , '''87''' : 4 (1959) pp. 409–425 {{MR|123279}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> S. Shnider, "The classification of real primitive infinite Lie algebras" ''J. Diff. Geom.'' , '''4''' : 1 (1970) pp. 81–89 {{MR|0285574}} {{ZBL|0244.17014}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> I.M. Singer, S. Sternberg, "The infinite groups of Lie and Cartan. I. The transitive groups" ''J. d'Anal. Math.'' , '''15''' (1965) pp. 1–114 {{MR|0217822}} {{ZBL|0277.58008}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> R.L. Wilson, "Irreducible Lie algebras of infinite type" ''Proc. Amer. Math. Soc.'' , '''29''' : 2 (1971) pp. 243–249 {{MR|0277582}} {{ZBL|0216.07401}} </TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) {{MR|0193578}} {{ZBL|0129.13102}} </TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top"> E. Cartan, "Sur la structure des groupes infinis de transformations" , ''Oeuvres complètes'' , '''2''' , Gauthier-Villars (1953) pp. 571–624 {{MR|1509054}} {{MR|1509040}} {{ZBL|36.0223.03}} {{ZBL|35.0176.04}} </TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top"> E. Cartan, "Sur la structure des groupes infinis de transformations" , ''Oeuvres complètes'' , '''2''' , Gauthier-Villars (1953) pp. 625–714 {{MR|1509054}} {{MR|1509040}} {{ZBL|36.0223.03}} {{ZBL|35.0176.04}} </TD></TR><TR><TD valign="top">[2c]</TD> <TD valign="top"> E. Cartan, "Les groupes de transformations continus, infinis, simples" , ''Oeuvres complètes'' , '''2''' , Gauthier-Villars (1953) pp. 857–925 {{MR|1509105}} {{ZBL|40.0193.02}} {{ZBL|38.0194.01}} </TD></TR><TR><TD valign="top">[2d]</TD> <TD valign="top"> E. Cartan, "Les groupes de transformations continus, infinis, simples" , ''Oeuvres complètes'' , '''2''' , Gauthier-Villars (1953) pp. 1335–1384 {{MR|1509105}} {{ZBL|40.0193.02}} {{ZBL|38.0194.01}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> V. Guillemin, "Infinite dimensional primitive Lie algebras" ''J. Diff. Geom.'' , '''4''' : 3 (1970) pp. 257–282 {{MR|0268233}} {{ZBL|0223.17007}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972) {{MR|0355886}} {{ZBL|0246.53031}} </TD></TR><TR><TD valign="top">[5a]</TD> <TD valign="top"> S. Kobayashi, T. Nagano, "On filtered Lie algebras and geometric structures I" ''J. Math. Mech.'' , '''13''' : 5 (1964) pp. 875–907 {{MR|0168704}} {{ZBL|0142.19504}} </TD></TR><TR><TD valign="top">[5b]</TD> <TD valign="top"> S. Kobayashi, T. Nagano, "On filtered Lie algebras and geometric structures III" ''J. Math. Mech.'' , '''14''' : 5 (1965) pp. 679–706 {{MR|0188364}} {{ZBL|}} </TD></TR><TR><TD valign="top">[6a]</TD> <TD valign="top"> M. Kuranishi, "On the local theory of continuous infinite pseudo groups I" ''Nagoya Math. J.'' , '''15''' (1959) pp. 225–260 {{MR|0116071}} {{ZBL|0212.56501}} </TD></TR><TR><TD valign="top">[6b]</TD> <TD valign="top"> M. Kuranishi, "On the local theory of continuous infinite pseudo groups II" ''Nagoya Math. J.'' , '''19''' (1961) pp. 55–91 {{MR|0142694}} {{ZBL|0212.56501}} </TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top"> P. Libermann, "Pseudogroupes infinitésimaux attachées aux pseudogroupes de Lie" ''Bull. Soc. Math. France'' , '''87''' : 4 (1959) pp. 409–425 {{MR|123279}} {{ZBL|}} </TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top"> S. Shnider, "The classification of real primitive infinite Lie algebras" ''J. Diff. Geom.'' , '''4''' : 1 (1970) pp. 81–89 {{MR|0285574}} {{ZBL|0244.17014}} </TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top"> I.M. Singer, S. Sternberg, "The infinite groups of Lie and Cartan. I. The transitive groups" ''J. d'Anal. Math.'' , '''15''' (1965) pp. 1–114 {{MR|0217822}} {{ZBL|0277.58008}} </TD></TR><TR><TD valign="top">[10]</TD> <TD valign="top"> R.L. Wilson, "Irreducible Lie algebras of infinite type" ''Proc. Amer. Math. Soc.'' , '''29''' : 2 (1971) pp. 243–249 {{MR|0277582}} {{ZBL|0216.07401}} </TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Albert, P. Molino, "Pseudogroupes de Lie transitifs" , '''I–II''' , Hermann (1984–1987) {{MR|0904048}} {{MR|0770061}} {{ZBL|0682.53003}} {{ZBL|0563.53027}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.F. Pommaret, "Systems of partial differential equations and Lie pseudogroups" , Gordon &amp; Breach (1978) {{MR|0517402}} {{ZBL|0418.35028}} {{ZBL|0401.58006}} </TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> C. Albert, P. Molino, "Pseudogroupes de Lie transitifs" , '''I–II''' , Hermann (1984–1987) {{MR|0904048}} {{MR|0770061}} {{ZBL|0682.53003}} {{ZBL|0563.53027}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.F. Pommaret, "Systems of partial differential equations and Lie pseudogroups" , Gordon &amp; Breach (1978) {{MR|0517402}} {{ZBL|0418.35028}} {{ZBL|0401.58006}} </TD></TR></table>

Latest revision as of 18:13, 20 January 2022


of transformations of a differentiable manifold $ M $

A family of diffeomorphisms from open subsets of $ M $ into $ M $ that is closed under composition of mappings, transition to the inverse mapping, as well as under restriction and glueing of mappings. More precisely, a pseudo-group of transformations $ \Gamma $ of a manifold $ M $ consists of local transformations, i.e. pairs of the form $ p =( D _ {p} , \overline{p}\; ) $ where $ D _ {p} $ is an open subset of $ M $ and $ \overline{p}\; $ is a diffeomorphism $ D _ {p} \rightarrow M $, where it is moreover assumed that 1) $ p , q \in \Gamma $ implies $ p \circ q = ( \overline{q} ^ {- 1} ( D _ {p} \cap \overline{q} ( D _ {q} ) ) , \overline{p}\; \circ \overline{q}\; ) \in \Gamma $; 2) $ p \in \Gamma $ implies $ p ^ {- 1} = ( \overline{p} ( D _ {p} ) , \overline{p} ^ {- 1} ) \in \Gamma $; 3) $ ( M , \mathop{\rm id} ) \in \Gamma $; and 4) if $ \overline{p}\; $ is a diffeomorphism from an open subset $ D \subset M $ into $ M $ and $ D = \cup _ \alpha D _ \alpha $, where $ D _ \alpha $ are open sets in $ M $, then $ ( D , \overline{p} ) \in \Gamma \iff ( D _ \alpha , \overline{p}\; \mid _ {D _ \alpha } ) \in \Gamma $ for any $ \alpha $. With necessary changes in 1)–4) one can also define pseudo-groups of transformations of an arbitrary topological space (cf. [7]) or even of an arbitrary set. As a group of transformations, a pseudo-group of transformations determines an equivalence relation on $ M $; the equivalence classes are called its orbits. A pseudo-group $ \Gamma $ of transformations of a manifold $ M $ is called transitive if $ M $ is its only orbit, and is called primitive if $ M $ does not admit non-trivial $ \Gamma $-invariant foliations (otherwise the pseudo-group is called imprimitive).

A pseudo-group $ \Gamma $ of transformations of a differentiable manifold is called a Lie pseudo-group of transformations defined by a system $ S $ of partial differential equations if $ \Gamma $ consists of exactly those local transformations of $ M $ that satisfy the system $ S $. E.g., the pseudo-group of conformal transformations of the plane is a Lie pseudo-group of transformations, determined by the Cauchy–Riemann equations (cf. Cauchy-Riemann equations). The order of a Lie pseudo-group of transformations is the minimum order of its defining system of differential equations.

Examples of Lie pseudo-groups of transformations. a) The pseudo-group of all holomorphic local transformations of $ n $-dimensional complex space $ \mathbf C ^ {n} $.

b) The pseudo-group of all holomorphic local transformations of $ \mathbf C ^ {n} $ with constant Jacobian.

c) The pseudo-group of all holomorphic local transformations of $ \mathbf C ^ {n} $ with Jacobian 1.

d) The Hamilton pseudo-group of all holomorphic local transformations of $ \mathbf C ^ {n} $ ($ n $ even) preserving the differential 2-form

$$ \omega = d z ^ {1} \wedge d z ^ {2} + d z ^ {3} \wedge d z ^ {4} + \dots + d z ^ {n- 1} \wedge d z ^ {n} . $$

e) The pseudo-group of all holomorphic local transformations of $ \mathbf C ^ {n} $ preserving $ \omega $ up to constant factor.

f) The contact pseudo-group of all holomorphic local transformations of $ \mathbf C ^ {n} $ ($ n = 2 m + 1 $, $ m \geq 1 $) preserving the differential 1-form

$$ d z ^ {n} + \sum _ { i= 1} ^ { m } ( z ^ {i} d z ^ {m+ i} - z ^ {m+ i} d z ^ {i} ) $$

up to a factor (which can be a function).

g) The real analogues of the complex pseudo-groups of transformations of Examples a)–f).

The order of the Lie pseudo-groups of Examples a), c)–f) is 1, while in b) the order is 2.

Any Lie group $ G $ of transformations of a manifold $ M $ determines a pseudo-group $ \Gamma ( G) $ of transformations, consisting of the restrictions of the transformations from $ G $ onto open subsets of $ M $. A pseudo-group of transformations of the form $ \Gamma ( G) $ is called globalizable. E.g., a pseudo-group of local conformal transformations of the sphere $ S ^ {n} $ is globalizable for $ n > 2 $ and not globalizable for $ n = 2 $.

A Lie pseudo-group of transformations is said to be of finite type if there is a natural number $ d $ such that every local transformation $ p \in \Gamma $ is uniquely determined by its $ d $-jet at some point $ x \in D _ {p} $; the smallest such $ d $ is called the degree, or type, of $ \Gamma $; if such a $ d $ does not exist, then $ \Gamma $ is called a pseudo-group of transformations of infinite type. The pseudo-groups of Examples a)–f) are primitive Lie pseudo-groups of transformations of infinite type.

Let $ \Gamma $ be a transitive Lie pseudo-group of transformations of an $ n $-dimensional manifold $ M $ and let $ G ^ {r} ( \Gamma ) $ be the family of all $ r $-jets of the local transformations in $ \Gamma $ that preserve a point $ O \in M $, i.e. those $ p \in \Gamma $ for which $ O \in D _ {p} $ and $ \overline{p} ( O) = O $. The set $ G ^ {r} ( \Gamma ) $, endowed with the natural structure of a Lie group, is called the $ r $-th order isotropy group of $ \Gamma $ ($ G ^ {1} ( \Gamma ) $ is also called the linear isotropy group of $ \Gamma $). The Lie algebra $ \mathfrak g ^ {r} ( \Gamma ) $ of $ \Gamma ^ {r} ( \Gamma ) $ can be naturally imbedded in the Lie algebra of $ r $-jets of vector fields on $ M $ at $ O $. If $ \Gamma $ is a Lie pseudo-group of transformations of order one, then the kernel $ G ^ {( r)} ( \Gamma ) $ of the natural homomorphism $ G ^ {r+ 1} ( \Gamma ) \rightarrow G ^ {r} ( \Gamma ) $ depends, for any $ r \geq 1 $, only on the linear isotropy group $ G ^ {1} ( \Gamma ) $, and is called its $ r $-th extension. A Lie pseudo-group of transformations $ \Gamma $ of order one is of finite type $ d $ if and only if

$$ \mathop{\rm dim} G ^ {( d- 1)} ( \Gamma ) \neq 0 \ \ \textrm{ and } \ \ \mathop{\rm dim} G ^ {( d)} ( \Gamma ) = 0 . $$

If, moreover, $ G ^ {1} ( \Gamma ) $ is irreducible, then $ d \leq 2 $ (cf. ). A Lie pseudo-group of transformations $ \Gamma $ of order one is a pseudo-group of transformations of finite type only if, and in the complex case if and only if, the Lie algebra $ \mathfrak g ^ {1} $ does not contain endomorphisms of rank 1 (cf. [10]). Such linear Lie algebras are called elliptic.

One has calculated the Lie algebras of all extensions $ G ^ {( r)} ( \Gamma ) $, $ r \geq 1 $, where $ \Gamma $ is a Lie pseudo-group of transformations of order one, in terms of the linear isotropy algebra. More precisely, the Lie algebra $ \mathfrak g ^ {( r)} ( \Gamma ) $ of $ G ^ {( r)} ( \Gamma ) $ consists of the $ ( r+ 1) $-jets of vector fields on $ M $ at $ O $ having, in some local coordinate system $ ( x ^ {1} \dots x ^ {n} ) $, the form

$$ \sum v _ {i _ {0} \dots i _ {r} } ^ {i} x ^ {i _ {0} } \dots x ^ {i _ {r} } \frac \partial {\partial x ^ {i} } , $$

where $ v _ {i _ {0} \dots i _ {r} } ^ {i} $ is an arbitrary tensor that is symmetric with respect to the lower indices and that satisfies the condition: For any fixed $ i _ {1} \dots i _ {r} $ the matrix

$$ \| v _ {j , i _ {1} \dots i _ {r} } ^ {i} \| _ {i , j = 1 } ^ {n} $$

belongs to $ \mathfrak g ^ {1} ( \Gamma ) $, relative to some coordinate system $ ( x ^ {i} ) $.

Let $ M $ be an $ n $-dimensional differentiable manifold over the field $ K = \mathbf R $ or $ \mathbf C $. Every transitive Lie pseudo-group of transformations $ \Gamma $ of order $ k $ on a manifold $ M $ coincides with the pseudo-group of all local automorphism of some $ G ^ {k} ( \Gamma ) $-structure (cf. $ G $-structure) of order $ k $ on $ M $ (Cartan's first fundamental theorem). The first classification of all primitive Lie pseudo-groups of infinite type was obtained by E. Cartan . According to his theorem, every primitive Lie pseudo-group of transformations of infinite type, consisting of holomorphic local transformations, is locally isomorphic to one of the pseudo-groups of Examples a)–f). This theorem has been repeatedly proved; its modern proofs lead to the study of certain filtered Lie algebras (cf. [9]). The classification of these filtered Lie algebras can be given on the basis of the classification of simple graded Lie algebras (cf. [3]). The classification of primitive pseudo-groups of transformations has also been obtained in the real case, and the condition of analyticity of the action of the pseudo-group of transformations has been replaced by the weaker condition of infinite differentiability (cf. [8], [9]). One has constructed certain abstract models of transitive Lie pseudo-groups, which came to play the same role in the theory of pseudo-groups of transformations of infinite type as do abstract Lie groups in the finite-dimensional case (cf. , [9]).

References

[1] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) MR0193578 Zbl 0129.13102
[2a] E. Cartan, "Sur la structure des groupes infinis de transformations" , Oeuvres complètes , 2 , Gauthier-Villars (1953) pp. 571–624 MR1509054 MR1509040 Zbl 36.0223.03 Zbl 35.0176.04
[2b] E. Cartan, "Sur la structure des groupes infinis de transformations" , Oeuvres complètes , 2 , Gauthier-Villars (1953) pp. 625–714 MR1509054 MR1509040 Zbl 36.0223.03 Zbl 35.0176.04
[2c] E. Cartan, "Les groupes de transformations continus, infinis, simples" , Oeuvres complètes , 2 , Gauthier-Villars (1953) pp. 857–925 MR1509105 Zbl 40.0193.02 Zbl 38.0194.01
[2d] E. Cartan, "Les groupes de transformations continus, infinis, simples" , Oeuvres complètes , 2 , Gauthier-Villars (1953) pp. 1335–1384 MR1509105 Zbl 40.0193.02 Zbl 38.0194.01
[3] V. Guillemin, "Infinite dimensional primitive Lie algebras" J. Diff. Geom. , 4 : 3 (1970) pp. 257–282 MR0268233 Zbl 0223.17007
[4] S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972) MR0355886 Zbl 0246.53031
[5a] S. Kobayashi, T. Nagano, "On filtered Lie algebras and geometric structures I" J. Math. Mech. , 13 : 5 (1964) pp. 875–907 MR0168704 Zbl 0142.19504
[5b] S. Kobayashi, T. Nagano, "On filtered Lie algebras and geometric structures III" J. Math. Mech. , 14 : 5 (1965) pp. 679–706 MR0188364
[6a] M. Kuranishi, "On the local theory of continuous infinite pseudo groups I" Nagoya Math. J. , 15 (1959) pp. 225–260 MR0116071 Zbl 0212.56501
[6b] M. Kuranishi, "On the local theory of continuous infinite pseudo groups II" Nagoya Math. J. , 19 (1961) pp. 55–91 MR0142694 Zbl 0212.56501
[7] P. Libermann, "Pseudogroupes infinitésimaux attachées aux pseudogroupes de Lie" Bull. Soc. Math. France , 87 : 4 (1959) pp. 409–425 MR123279
[8] S. Shnider, "The classification of real primitive infinite Lie algebras" J. Diff. Geom. , 4 : 1 (1970) pp. 81–89 MR0285574 Zbl 0244.17014
[9] I.M. Singer, S. Sternberg, "The infinite groups of Lie and Cartan. I. The transitive groups" J. d'Anal. Math. , 15 (1965) pp. 1–114 MR0217822 Zbl 0277.58008
[10] R.L. Wilson, "Irreducible Lie algebras of infinite type" Proc. Amer. Math. Soc. , 29 : 2 (1971) pp. 243–249 MR0277582 Zbl 0216.07401

Comments

References

[a1] C. Albert, P. Molino, "Pseudogroupes de Lie transitifs" , I–II , Hermann (1984–1987) MR0904048 MR0770061 Zbl 0682.53003 Zbl 0563.53027
[a2] J.F. Pommaret, "Systems of partial differential equations and Lie pseudogroups" , Gordon & Breach (1978) MR0517402 Zbl 0418.35028 Zbl 0401.58006
How to Cite This Entry:
Pseudo-group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-group&oldid=24540
This article was adapted from an original article by E.B. Vinberg (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article