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)

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 $.