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''local analytic group''
 
''local analytic group''
  
An [[Analytic manifold|analytic manifold]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l0586501.png" /> over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l0586502.png" /> that is complete with respect to some non-trivial absolute value, which is endowed with a distinguished element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l0586503.png" /> (the identity), an open subset <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l0586504.png" /> and a pair of analytic mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l0586505.png" /> of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l0586506.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l0586507.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l0586508.png" /> of the neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l0586509.png" /> into itself, for which:
+
An [[Analytic manifold|analytic manifold]] $  G $
 +
over a field $  k $
 +
that is complete with respect to some non-trivial absolute value, which is endowed with a distinguished element $  e $(
 +
the identity), an open subset $  U \ni e $
 +
and a pair of analytic mappings $  ( g , h ) \mapsto g h $
 +
of the manifold $  U \times U $
 +
into $  G $
 +
and $  g \mapsto g  ^ {-} 1 $
 +
of the neighbourhood $  U $
 +
into itself, for which:
  
1) in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865010.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865011.png" />;
+
1) in some neighbourhood of $  e $
 +
one has $  g e = e g $;
  
2) in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865012.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865013.png" />;
+
2) in some neighbourhood of $  e $
 +
one has $  e = g g  ^ {-} 1 = g  ^ {-} 1 g $;
  
3) for some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865014.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865015.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865016.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865017.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865018.png" /> are arbitrary elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865019.png" />.
+
3) for some neighbourhood $  U  ^  \prime  \subset  U $
 +
of $  e $
 +
one has $  U  ^  \prime  U  ^  \prime  \subset  U $
 +
and $  g ( hr) = ( gh) r $,  
 +
where $  g , h , r $
 +
are arbitrary elements of $  U  ^  \prime  $.
  
 
Local Lie groups first made their appearance in the work of S. Lie and his school (see [[#References|[1]]]) as local Lie transformation groups (cf. [[Lie transformation group|Lie transformation group]]).
 
Local Lie groups first made their appearance in the work of S. Lie and his school (see [[#References|[1]]]) as local Lie transformation groups (cf. [[Lie transformation group|Lie transformation group]]).
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865020.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865021.png" /> be two local Lie groups with identities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865023.png" />, respectively. A local homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865024.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865025.png" /> (denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865026.png" />) is an [[Analytic mapping|analytic mapping]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865027.png" /> of some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865028.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865029.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865031.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865032.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865033.png" /> in some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865035.png" />. The naturally defined composition of local homomorphisms is also a local homomorphism. Local homomorphisms <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865036.png" /> that coincide in some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865037.png" /> are said to be equivalent. If there are local homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865039.png" /> such that the compositions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865040.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865041.png" /> are equivalent to the identity mappings, then the local Lie groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865042.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865043.png" /> are said to be equivalent.
+
Let $  G _ {1} $
 +
and $  G _ {2} $
 +
be two local Lie groups with identities $  e _ {1} $
 +
and $  e _ {2} $,  
 +
respectively. A local homomorphism of $  G _ {1} $
 +
into $  G _ {2} $(
 +
denoted by $  f : G _ {1} \rightarrow G _ {2} $)  
 +
is an [[Analytic mapping|analytic mapping]] $  f : U \rightarrow G _ {2} $
 +
of some neighbourhood $  U \ni e _ {1} $
 +
in $  G _ {1} $
 +
for which $  f ( e _ {1} ) = e _ {2} $
 +
and $  f ( g h ) = f ( g) f ( h) $
 +
for $  g $
 +
and $  h $
 +
in some neighbourhood $  U _ {1} \subset  U $
 +
of $  e _ {1} $.  
 +
The naturally defined composition of local homomorphisms is also a local homomorphism. Local homomorphisms $  G _ {1} \rightarrow G _ {2} $
 +
that coincide in some neighbourhood of $  e _ {1} $
 +
are said to be equivalent. If there are local homomorphism $  f _ {1} : G _ {1} \rightarrow G _ {2} $
 +
and $  f _ {2} : G _ {2} \rightarrow G _ {1} $
 +
such that the compositions $  f _ {2} \circ f _ {1} $
 +
and $  f _ {1} \circ f _ {2} $
 +
are equivalent to the identity mappings, then the local Lie groups $  G _ {1} $
 +
and $  G _ {2} $
 +
are said to be equivalent.
  
Examples. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865044.png" /> be an [[Analytic group|analytic group]] with identity <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865045.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865046.png" /> an open neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865047.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865048.png" />. Then the analytic structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865049.png" /> induces an analytic structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865050.png" />, and the operations of multiplication and taking the inverse of an element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865051.png" /> convert <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865052.png" /> into a local Lie group (in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865053.png" /> itself can be regarded as a local Lie group). All local Lie groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865054.png" /> obtainable in this way from a fixed analytic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865055.png" /> are equivalent to one another.
+
Examples. Let $  \overline{G}\; $
 +
be an [[Analytic group|analytic group]] with identity $  e $
 +
and $  G $
 +
an open neighbourhood of $  e $
 +
in $  \overline{G}\; $.  
 +
Then the analytic structure on $  \overline{G}\; $
 +
induces an analytic structure on $  G $,  
 +
and the operations of multiplication and taking the inverse of an element in $  \overline{G}\; $
 +
convert $  G $
 +
into a local Lie group (in particular, $  \overline{G}\; $
 +
itself can be regarded as a local Lie group). All local Lie groups $  G $
 +
obtainable in this way from a fixed analytic group $  \overline{G}\; $
 +
are equivalent to one another.
  
 
One of the fundamental questions in the theory of Lie groups is the question of how general a character the example given above has, that is, whether every local Lie group is (up to equivalence) a neighbourhood of some analytic group. The answer to this question is affirmative (see [[#References|[2]]], [[#References|[3]]], [[#References|[4]]]; in the case of local Banach Lie groups the answer is negative, see [[#References|[4]]]).
 
One of the fundamental questions in the theory of Lie groups is the question of how general a character the example given above has, that is, whether every local Lie group is (up to equivalence) a neighbourhood of some analytic group. The answer to this question is affirmative (see [[#References|[2]]], [[#References|[3]]], [[#References|[4]]]; in the case of local Banach Lie groups the answer is negative, see [[#References|[4]]]).
  
The most important tool for studying local Lie groups is the correspondence between the local Lie group and its [[Lie algebra|Lie algebra]]. Namely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865056.png" /> be a local Lie group over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865057.png" /> and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865058.png" /> be the identity of it. The choice of a [[Chart|chart]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865059.png" /> of the analytic manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865060.png" /> at the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865061.png" /> makes it possible to identify some neighbourhood of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865062.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865063.png" /> with some neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865064.png" /> of the origin in the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865065.png" />-dimensional coordinate space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865066.png" />, so that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865067.png" /> becomes a local Lie group. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865068.png" /> be a neighbourhood of the origin in the local Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865069.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865070.png" /> a product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865071.png" /> is defined. Then, in coordinate form, multiplication in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865072.png" /> in the neighbourhood <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865073.png" /> is specified by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865074.png" /> analytic functions
+
The most important tool for studying local Lie groups is the correspondence between the local Lie group and its [[Lie algebra|Lie algebra]]. Namely, let $  G $
 +
be a local Lie group over a field $  k $
 +
and let $  e $
 +
be the identity of it. The choice of a [[Chart|chart]] $  c $
 +
of the analytic manifold $  G $
 +
at the point $  e $
 +
makes it possible to identify some neighbourhood of $  e $
 +
in $  G $
 +
with some neighbourhood $  U $
 +
of the origin in the $  n $-
 +
dimensional coordinate space $  k  ^ {n} $,  
 +
so that $  U $
 +
becomes a local Lie group. Let $  U _ {0} $
 +
be a neighbourhood of the origin in the local Lie group $  U $
 +
such that for any $  x , y \in U _ {0} $
 +
a product $  z = x y \in U $
 +
is defined. Then, in coordinate form, multiplication in $  U $
 +
in the neighbourhood $  U _ {0} $
 +
is specified by $  n $
 +
analytic functions
  
<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/l/l058/l058650/l05865075.png" /></td> </tr></table>
+
$$
 +
z _ {i}  = f _ {i} ( x _ {1} \dots x _ {n} ; \
 +
y _ {1} \dots y _ {n} ) ,\ \
 +
i = 1 \dots n ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865078.png" /> are, respectively, the coordinates of the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865079.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865080.png" />. In a sufficiently small neighbourhood of the origin the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865081.png" /> is represented as the sum of a convergent power series (also denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865082.png" /> henceforth), and the presence in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865083.png" /> of an identity and the associative law is expressed by the following properties of these series, regarded as formal power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865084.png" /> variables:
+
where $  ( x _ {1} \dots x _ {n} ) $,  
 +
$  ( y _ {1} \dots y _ {n} ) $,  
 +
$  ( z _ {1} \dots z _ {n} ) $
 +
are, respectively, the coordinates of the points $  x , y \in U _ {0} $
 +
and $  z = x y \in U $.  
 +
In a sufficiently small neighbourhood of the origin the function $  f _ {i} $
 +
is represented as the sum of a convergent power series (also denoted by $  f _ {i} $
 +
henceforth), and the presence in $  U $
 +
of an identity and the associative law is expressed by the following properties of these series, regarded as formal power series in $  2n $
 +
variables:
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865085.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865086.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865087.png" />;
+
a) $  f _ {i} ( x _ {1} \dots x _ {n} ;  0 \dots 0 ) = x _ {i} $
 +
and $  f _ {i} ( 0 \dots 0; y _ {1} \dots y _ {n} ) = y _ {i} $
 +
for all $  i $;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865088.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865089.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865090.png" />.
+
b) $  f _ {i} ( u _ {1} \dots u _ {n; } f _ {1} ( v _ {1} \dots v _ {n} ;  w _ {1} \dots w _ {n} ) \dots f _ {n} ( v _ {1} \dots v _ {n} ;  w _ {1} \dots w _ {n} ) )= $
 +
$  f _ {i} ( f _ {1} ( u _ {1} \dots u _ {n; } v _ {1} \dots v _ {n} ) \dots f _ {n} ( u _ {1} \dots u _ {n; } v _ {1} \dots v _ {n} );  w _ {1} \dots w _ {n} ) $
 +
for all $  i $.
  
Properties a) and b) imply that the system of formal power series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865091.png" /> is a [[Formal group|formal group]]. In particular, the homogeneous component of degree 2 of each of the series <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865092.png" /> is a [[Bilinear form|bilinear form]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865093.png" />, that is, it has the form
+
Properties a) and b) imply that the system of formal power series $  F _ {c} = ( f _ {1} \dots f _ {n} ) $
 +
is a [[Formal group|formal group]]. In particular, the homogeneous component of degree 2 of each of the series $  f _ {i} $
 +
is a [[Bilinear form|bilinear form]] on $  k  ^ {n} $,  
 +
that is, it has the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865094.png" /></td> </tr></table>
+
$$
 +
\sum _ { j,l } b _ {jl}  ^ {i} x _ {j} y _ {l}  = \
 +
b _ {i} ( x , y ) ,\  x = ( x _ {1} \dots x _ {n} ) ,\ \
 +
y = ( y _ {1} \dots y _ {n} ) ,
 +
$$
  
which makes it possible to define a multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865095.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865096.png" /> according to the rule:
+
which makes it possible to define a multiplication $  [  , ] $
 +
on $  k  ^ {n} $
 +
according to the rule:
  
<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/l/l058/l058650/l05865097.png" /></td> </tr></table>
+
$$
 +
[ x , y ]  = ( b _ {1} ( x , y ) - b _ {1} ( y , x ) \dots b _ {n} ( x , y ) - b _ {n} ( y , x ) ) .
 +
$$
  
With respect to this multiplication <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865098.png" /> is a Lie algebra. The structure of a Lie algebra carries over to the tangent space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l05865099.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650100.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650101.png" /> by means of the chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650102.png" />, defined above, by the isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650103.png" />. The formal groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650104.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650105.png" /> defined by different charts are isomorphic, and the structure of a Lie algebra on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650106.png" /> does not depend on the choice of the chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650107.png" />. The Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650108.png" /> is called the Lie algebra of a local Lie group. For any local homomorphism of a local Lie group its differential at the identity is a homomorphism of Lie algebras, which implies that the correspondence between a local Lie group and its Lie algebra is functorial. In particular, equivalent local Lie groups have isomorphic Lie algebras.
+
With respect to this multiplication $  k  ^ {n} $
 +
is a Lie algebra. The structure of a Lie algebra carries over to the tangent space $  \mathfrak g $
 +
to $  G $
 +
at $  e $
 +
by means of the chart $  c $,  
 +
defined above, by the isomorphism $  g \rightarrow k  ^ {n} $.  
 +
The formal groups $  F _ {c} $
 +
and $  F _ {c  ^  \prime  } $
 +
defined by different charts are isomorphic, and the structure of a Lie algebra on $  \mathfrak g $
 +
does not depend on the choice of the chart $  c $.  
 +
The Lie algebra $  \mathfrak g $
 +
is called the Lie algebra of a local Lie group. For any local homomorphism of a local Lie group its differential at the identity is a homomorphism of Lie algebras, which implies that the correspondence between a local Lie group and its Lie algebra is functorial. In particular, equivalent local Lie groups have isomorphic Lie algebras.
  
If the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650109.png" /> has characteristic 0, then the construction given above, which goes back to Lie [[#References|[1]]], makes it possible to reduce the study of properties of local Lie groups to the study of the corresponding properties of their Lie algebras. In this case the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650110.png" /> determines the local Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650111.png" /> uniquely up to equivalence. Namely, the chart <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650112.png" /> can be chosen so that the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650113.png" /> in the local Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650114.png" /> is expressed as a convergent series (the so-called Campbell–Hausdorff series) of elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650115.png" /> obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650117.png" /> by means of the commutation operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650118.png" /> and multiplication by elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650119.png" /> (see [[Campbell–Hausdorff formula|Campbell–Hausdorff formula]]). Conversely, for an arbitrary finite-dimensional Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650120.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650121.png" /> the Campbell–Hausdorff series converges in some neighbourhood of the origin in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650122.png" /> and determines in this neighbourhood the structure of a local Lie group with Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650123.png" />. Thus, for any given Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650124.png" /> there is a unique (up to equivalence) local Lie group with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650125.png" /> as its Lie algebra. Moreover, every homomorphism of Lie algebras is induced by a unique homomorphism of the corresponding local Lie groups. In other words, the correspondence between a local Lie group and its Lie algebra defines an equivalence of the category of local Lie groups and the category of finite-dimensional Lie algebras over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650126.png" />. Moreover, the correspondence between a local Lie group and the corresponding formal group defines an equivalence of the category of local Lie groups and the category of formal groups over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650127.png" />.
+
If the field $  k $
 +
has characteristic 0, then the construction given above, which goes back to Lie [[#References|[1]]], makes it possible to reduce the study of properties of local Lie groups to the study of the corresponding properties of their Lie algebras. In this case the Lie algebra $  \mathfrak g $
 +
determines the local Lie group $  G $
 +
uniquely up to equivalence. Namely, the chart $  c $
 +
can be chosen so that the product $  x y $
 +
in the local Lie group $  U $
 +
is expressed as a convergent series (the so-called Campbell–Hausdorff series) of elements of $  k  ^ {n} $
 +
obtained from $  x $
 +
and $  y $
 +
by means of the commutation operation $  [  , ] $
 +
and multiplication by elements of $  k $(
 +
see [[Campbell–Hausdorff formula|Campbell–Hausdorff formula]]). Conversely, for an arbitrary finite-dimensional Lie algebra $  \mathfrak h $
 +
over $  k $
 +
the Campbell–Hausdorff series converges in some neighbourhood of the origin in $  \mathfrak h $
 +
and determines in this neighbourhood the structure of a local Lie group with Lie algebra $  \mathfrak h $.  
 +
Thus, for any given Lie algebra $  \mathfrak h $
 +
there is a unique (up to equivalence) local Lie group with $  \mathfrak h $
 +
as its Lie algebra. Moreover, every homomorphism of Lie algebras is induced by a unique homomorphism of the corresponding local Lie groups. In other words, the correspondence between a local Lie group and its Lie algebra defines an equivalence of the category of local Lie groups and the category of finite-dimensional Lie algebras over $  k $.  
 +
Moreover, the correspondence between a local Lie group and the corresponding formal group defines an equivalence of the category of local Lie groups and the category of formal groups over $  k $.
  
 
The Lie algebra can also be defined for any local Banach Lie group; the main result about the equivalence of the categories of local Lie groups and Lie algebras can be generalized to this case (see [[#References|[2]]]).
 
The Lie algebra can also be defined for any local Banach Lie group; the main result about the equivalence of the categories of local Lie groups and Lie algebras can be generalized to this case (see [[#References|[2]]]).
Line 43: Line 183:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lie,  F. Engel,  "Theorie der Transformationsgruppen" , '''1–3''' , Leipzig  (1888–1893)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.G. Chebotarev,  "The theory of Lie groups" , Moscow-Leningrad  (1940)  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Lie,  F. Engel,  "Theorie der Transformationsgruppen" , '''1–3''' , Leipzig  (1888–1893)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N. Bourbaki,  "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley  (1975)  (Translated from French)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L.S. Pontryagin,  "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  J.-P. Serre,  "Lie algebras and Lie groups" , Benjamin  (1965)  (Translated from French)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  N.G. Chebotarev,  "The theory of Lie groups" , Moscow-Leningrad  (1940)  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The equivalences of categories between local Lie groups, formal groups and Lie algebras over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650128.png" /> only hold for fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650129.png" /> of characteristic zero. In particular, for a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650130.png" /> of characteristic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650131.png" /> there are at least countably many non-isomorphic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650132.png" />-dimensional formal groups over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650133.png" />, while there is of course only one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650134.png" />-dimensional Lie algebra over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l058/l058650/l058650135.png" />.
+
The equivalences of categories between local Lie groups, formal groups and Lie algebras over a field $  k $
 +
only hold for fields $  k $
 +
of characteristic zero. In particular, for a field $  k $
 +
of characteristic $  p $
 +
there are at least countably many non-isomorphic $  1 $-
 +
dimensional formal groups over $  k $,  
 +
while there is of course only one $  1 $-
 +
dimensional Lie algebra over $  k $.

Latest revision as of 22:16, 5 June 2020


local analytic group

An analytic manifold $ G $ over a field $ k $ that is complete with respect to some non-trivial absolute value, which is endowed with a distinguished element $ e $( the identity), an open subset $ U \ni e $ and a pair of analytic mappings $ ( g , h ) \mapsto g h $ of the manifold $ U \times U $ into $ G $ and $ g \mapsto g ^ {-} 1 $ of the neighbourhood $ U $ into itself, for which:

1) in some neighbourhood of $ e $ one has $ g e = e g $;

2) in some neighbourhood of $ e $ one has $ e = g g ^ {-} 1 = g ^ {-} 1 g $;

3) for some neighbourhood $ U ^ \prime \subset U $ of $ e $ one has $ U ^ \prime U ^ \prime \subset U $ and $ g ( hr) = ( gh) r $, where $ g , h , r $ are arbitrary elements of $ U ^ \prime $.

Local Lie groups first made their appearance in the work of S. Lie and his school (see [1]) as local Lie transformation groups (cf. Lie transformation group).

Let $ G _ {1} $ and $ G _ {2} $ be two local Lie groups with identities $ e _ {1} $ and $ e _ {2} $, respectively. A local homomorphism of $ G _ {1} $ into $ G _ {2} $( denoted by $ f : G _ {1} \rightarrow G _ {2} $) is an analytic mapping $ f : U \rightarrow G _ {2} $ of some neighbourhood $ U \ni e _ {1} $ in $ G _ {1} $ for which $ f ( e _ {1} ) = e _ {2} $ and $ f ( g h ) = f ( g) f ( h) $ for $ g $ and $ h $ in some neighbourhood $ U _ {1} \subset U $ of $ e _ {1} $. The naturally defined composition of local homomorphisms is also a local homomorphism. Local homomorphisms $ G _ {1} \rightarrow G _ {2} $ that coincide in some neighbourhood of $ e _ {1} $ are said to be equivalent. If there are local homomorphism $ f _ {1} : G _ {1} \rightarrow G _ {2} $ and $ f _ {2} : G _ {2} \rightarrow G _ {1} $ such that the compositions $ f _ {2} \circ f _ {1} $ and $ f _ {1} \circ f _ {2} $ are equivalent to the identity mappings, then the local Lie groups $ G _ {1} $ and $ G _ {2} $ are said to be equivalent.

Examples. Let $ \overline{G}\; $ be an analytic group with identity $ e $ and $ G $ an open neighbourhood of $ e $ in $ \overline{G}\; $. Then the analytic structure on $ \overline{G}\; $ induces an analytic structure on $ G $, and the operations of multiplication and taking the inverse of an element in $ \overline{G}\; $ convert $ G $ into a local Lie group (in particular, $ \overline{G}\; $ itself can be regarded as a local Lie group). All local Lie groups $ G $ obtainable in this way from a fixed analytic group $ \overline{G}\; $ are equivalent to one another.

One of the fundamental questions in the theory of Lie groups is the question of how general a character the example given above has, that is, whether every local Lie group is (up to equivalence) a neighbourhood of some analytic group. The answer to this question is affirmative (see [2], [3], [4]; in the case of local Banach Lie groups the answer is negative, see [4]).

The most important tool for studying local Lie groups is the correspondence between the local Lie group and its Lie algebra. Namely, let $ G $ be a local Lie group over a field $ k $ and let $ e $ be the identity of it. The choice of a chart $ c $ of the analytic manifold $ G $ at the point $ e $ makes it possible to identify some neighbourhood of $ e $ in $ G $ with some neighbourhood $ U $ of the origin in the $ n $- dimensional coordinate space $ k ^ {n} $, so that $ U $ becomes a local Lie group. Let $ U _ {0} $ be a neighbourhood of the origin in the local Lie group $ U $ such that for any $ x , y \in U _ {0} $ a product $ z = x y \in U $ is defined. Then, in coordinate form, multiplication in $ U $ in the neighbourhood $ U _ {0} $ is specified by $ n $ analytic functions

$$ z _ {i} = f _ {i} ( x _ {1} \dots x _ {n} ; \ y _ {1} \dots y _ {n} ) ,\ \ i = 1 \dots n , $$

where $ ( x _ {1} \dots x _ {n} ) $, $ ( y _ {1} \dots y _ {n} ) $, $ ( z _ {1} \dots z _ {n} ) $ are, respectively, the coordinates of the points $ x , y \in U _ {0} $ and $ z = x y \in U $. In a sufficiently small neighbourhood of the origin the function $ f _ {i} $ is represented as the sum of a convergent power series (also denoted by $ f _ {i} $ henceforth), and the presence in $ U $ of an identity and the associative law is expressed by the following properties of these series, regarded as formal power series in $ 2n $ variables:

a) $ f _ {i} ( x _ {1} \dots x _ {n} ; 0 \dots 0 ) = x _ {i} $ and $ f _ {i} ( 0 \dots 0; y _ {1} \dots y _ {n} ) = y _ {i} $ for all $ i $;

b) $ f _ {i} ( u _ {1} \dots u _ {n; } f _ {1} ( v _ {1} \dots v _ {n} ; w _ {1} \dots w _ {n} ) \dots f _ {n} ( v _ {1} \dots v _ {n} ; w _ {1} \dots w _ {n} ) )= $ $ f _ {i} ( f _ {1} ( u _ {1} \dots u _ {n; } v _ {1} \dots v _ {n} ) \dots f _ {n} ( u _ {1} \dots u _ {n; } v _ {1} \dots v _ {n} ); w _ {1} \dots w _ {n} ) $ for all $ i $.

Properties a) and b) imply that the system of formal power series $ F _ {c} = ( f _ {1} \dots f _ {n} ) $ is a formal group. In particular, the homogeneous component of degree 2 of each of the series $ f _ {i} $ is a bilinear form on $ k ^ {n} $, that is, it has the form

$$ \sum _ { j,l } b _ {jl} ^ {i} x _ {j} y _ {l} = \ b _ {i} ( x , y ) ,\ x = ( x _ {1} \dots x _ {n} ) ,\ \ y = ( y _ {1} \dots y _ {n} ) , $$

which makes it possible to define a multiplication $ [ , ] $ on $ k ^ {n} $ according to the rule:

$$ [ x , y ] = ( b _ {1} ( x , y ) - b _ {1} ( y , x ) \dots b _ {n} ( x , y ) - b _ {n} ( y , x ) ) . $$

With respect to this multiplication $ k ^ {n} $ is a Lie algebra. The structure of a Lie algebra carries over to the tangent space $ \mathfrak g $ to $ G $ at $ e $ by means of the chart $ c $, defined above, by the isomorphism $ g \rightarrow k ^ {n} $. The formal groups $ F _ {c} $ and $ F _ {c ^ \prime } $ defined by different charts are isomorphic, and the structure of a Lie algebra on $ \mathfrak g $ does not depend on the choice of the chart $ c $. The Lie algebra $ \mathfrak g $ is called the Lie algebra of a local Lie group. For any local homomorphism of a local Lie group its differential at the identity is a homomorphism of Lie algebras, which implies that the correspondence between a local Lie group and its Lie algebra is functorial. In particular, equivalent local Lie groups have isomorphic Lie algebras.

If the field $ k $ has characteristic 0, then the construction given above, which goes back to Lie [1], makes it possible to reduce the study of properties of local Lie groups to the study of the corresponding properties of their Lie algebras. In this case the Lie algebra $ \mathfrak g $ determines the local Lie group $ G $ uniquely up to equivalence. Namely, the chart $ c $ can be chosen so that the product $ x y $ in the local Lie group $ U $ is expressed as a convergent series (the so-called Campbell–Hausdorff series) of elements of $ k ^ {n} $ obtained from $ x $ and $ y $ by means of the commutation operation $ [ , ] $ and multiplication by elements of $ k $( see Campbell–Hausdorff formula). Conversely, for an arbitrary finite-dimensional Lie algebra $ \mathfrak h $ over $ k $ the Campbell–Hausdorff series converges in some neighbourhood of the origin in $ \mathfrak h $ and determines in this neighbourhood the structure of a local Lie group with Lie algebra $ \mathfrak h $. Thus, for any given Lie algebra $ \mathfrak h $ there is a unique (up to equivalence) local Lie group with $ \mathfrak h $ as its Lie algebra. Moreover, every homomorphism of Lie algebras is induced by a unique homomorphism of the corresponding local Lie groups. In other words, the correspondence between a local Lie group and its Lie algebra defines an equivalence of the category of local Lie groups and the category of finite-dimensional Lie algebras over $ k $. Moreover, the correspondence between a local Lie group and the corresponding formal group defines an equivalence of the category of local Lie groups and the category of formal groups over $ k $.

The Lie algebra can also be defined for any local Banach Lie group; the main result about the equivalence of the categories of local Lie groups and Lie algebras can be generalized to this case (see [2]).

References

[1] S. Lie, F. Engel, "Theorie der Transformationsgruppen" , 1–3 , Leipzig (1888–1893)
[2] N. Bourbaki, "Elements of mathematics. Lie groups and Lie algebras" , Addison-Wesley (1975) (Translated from French)
[3] L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian)
[4] J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French)
[5] N.G. Chebotarev, "The theory of Lie groups" , Moscow-Leningrad (1940) (In Russian)

Comments

The equivalences of categories between local Lie groups, formal groups and Lie algebras over a field $ k $ only hold for fields $ k $ of characteristic zero. In particular, for a field $ k $ of characteristic $ p $ there are at least countably many non-isomorphic $ 1 $- dimensional formal groups over $ k $, while there is of course only one $ 1 $- dimensional Lie algebra over $ k $.

How to Cite This Entry:
Lie group, local. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_group,_local&oldid=12719
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article