# Lie group, local

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)

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 $k$.