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''local analytic group''

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  $  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 [[#References|[1]]]) as local Lie transformation groups (cf. [[Lie transformation group|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|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|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]]]).

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

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

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|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

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

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 [[#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]]]).

====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>

====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 $.