# Lie group, local

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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 ) 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 , , ; in the case of local Banach Lie groups the answer is negative, see ).

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 , 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 ).

How to Cite This Entry:
Lie group, local. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie_group,_local&oldid=47631
This article was adapted from an original article by V.L. Popov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article