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Lie's theorem is one of the three classical theorems in the theory of Lie groups that describe the connection between a local Lie group (cf. [[Lie group, local|Lie group, local]]) and its Lie algebra. Lie's theorems are the foundations of the theory developed in the 19th century by S. Lie and his school (see ).

Let $  G $ 
be an $  r $ -
dimensional real effective local [[Lie transformation group|Lie transformation group]] of a domain $  \Omega \subset \mathbf R ^{n} $ , 
let $  e $ 
be the identity of $  G $ 
and suppose that in local coordinates in a neighbourhood of the set $  \{ e \} \times \Omega $ 
in $  G \times \Omega $ 
the action of $  G $ 
on $  \Omega $ 
is given by a system of analytic functions$$ \tag{1}
y _{i}   =   f _{i} ( g _{1} \dots g _{r} ; \ 
x _{1} \dots x _{n} ) ,  
i = 1 \dots n ,
 $$ 
where $  g = ( g _{1} \dots g _{r} ) \in G $ , 
$  x = ( x _{1} \dots x _{n} ) \in \Omega $ 
and $  g (x) = y = ( y _{1} \dots y _{n} ) \in \Omega $ . 
This action defines $  r $ 
analytic vector fields on $  \Omega $ ,
$$ \tag{2}
X _{i}   =   \sum _{j=1} ^ n
\xi _{ij} (x) 
\frac \partial {\partial x _{j}}
 ,  
i = 1 \dots r ,
 $$ 
where $  \xi _{ij} (x) = {\partial f _{j}} / {\partial g _{i}} ( e ,\  x) $ .


Lie's first theorem establishes that the functions $  f _{j} $ , 
$  j = 1 \dots n $ , 
which define the action of $  G $ 
are themselves defined by some auxiliary system of analytic functions $  \psi _{ki} (g) $ , 
$  k ,\  i = 1 \dots r $ , 
on $  G $ 
which satisfy the condition$$ \tag{3}
\psi _{ki} (e)   =   \delta _{ki} ,
 $$ 
where $  \delta _{ki} $ 
is the [[Kronecker symbol|Kronecker symbol]]. More precisely, $  ( \psi _{ki} (g) ) $ 
is the matrix of the differential of the right translation of $  G $ 
by the element $  g ^{-1} $ 
at the point $  g $ , 
and the system of functions (1) is precisely the solution of the system of equations$$ \tag{4}

\frac{\partial f _{j}}{\partial g _{i}}

( g ,\  x )   =   \sum _{k=1} ^ r
\xi _{kj} ( f ( g ,\  x ) )
\psi _{ki} (g) ,
 $$ 
$$ 
i   =   1 \dots r ;    j   =   1 \dots n ,
 $$ 
that satisfies the initial conditions $  f _{j} ( e ,\  x ) = x $ , 
$  j = 1 \dots n $ .


Lie's second theorem describes the properties of the functions $  \xi _{ij} (x) $ 
and $  \psi _{ki} (x) $ . 
Namely, the $  \xi _{ij} (x) $ 
satisfy the system of equations$$ \tag{5}
\sum _{k=1} ^ n
\left (
\xi _{ik}

\frac{\partial \xi _{jl}}{\partial x _{k}}

- \xi _{jk}

\frac{\partial \xi _{il}}{\partial x _{k}}

\right )
  =   \sum _{k=1} ^ r
c _{ij} ^{k} \xi _{kl} ,
 $$ 
$$ 
1   \leq   i ,\  j   \leq   r ,   1   \leq   l   \leq   n
 $$ (
this system is the condition that the system (4) is integrable), and the functions $  \psi _{ki} (g) $ 
satisfy the system of equations$$ 

\frac{\partial \psi _{kl}}{\partial g _{m}}
 -

\frac{\partial \psi _{km}}{\partial g _{l}}
   =  
\sum _ {i , j = 1} ^ r
c _{ji} ^{k} \psi _{im} \psi _{jl} ,  
1   \leq   k ,\  l ,\  m   \leq   r ,
 $$ 
where the $  c _{ij} ^{k} $ 
are certain constants. The relations (5) imply that the commutator ([[Lie bracket|Lie bracket]]) $  [ X _{i} ,\  X _{j} ] $ 
of two vector fields $  X _{i} $ 
and $  X _{j} $ 
is a linear combination of the fields $  X _{1} \dots X _{r} $ 
with constant coefficients:$$ \tag{6}
[ X _{i} ,\  X _{j} ]   =  
\sum _{k=1} ^ r
c _{ij} ^{k} X _{k} ,
 $$ 
that is, the linear hull $  \mathfrak g $ 
of the fields $  X _{1} \dots X _{r} $ 
is an algebra with respect to the Lie bracket.

The converse of Lie's first and second theorems is the following: If the functions $  f _{1} \dots f _{n} $ 
give a solution of (4) in which the matrix $  \| \xi _{ij} \| $ 
has maximal rank and if (3) and (5) are satisfied, then (1) determines a local effective Lie transformation group. This local group is generated by the one-parameter transformation groups given by (2).

Lie's third theorem asserts that the constants $  c _{ij} ^{k} $ 
satisfy the following relations:$$ \tag{7}
\left . 
\begin{array}{c}
c _{ij} ^{k}   =   - c _{ji} ^{k} ,  \\
\sum _{l=1} ^ r
( c _{il} ^{m} c _{jk} ^{l} +
c _{kl} ^{m} c _{ij} ^{l} +
c _{jl} ^{m} c _{ki} ^{l} )   =   0 ,  
1 \leq i ,\  j ,\  k ,\  l ,\  m \leq r ,  \\
\end{array}
 \right \}
 $$ 
that is, $  \mathfrak g $ 
is a Lie algebra. The converse of the third theorem is important: If the $  c _{ij} ^{k} $ 
are any constants satisfying (7), then there is a system of vector fields $  X _{1} \dots X _{r} $ 
satisfying (6), and these vector fields arise by means of the construction described above from some local Lie transformation group (in other words, every finite-dimensional Lie algebra is the Lie algebra of some local Lie transformation group). Lie's third theorem is sometimes (see , for example) taken to be the assertion about the existence, for every finite-dimensional Lie algebra $  \mathfrak g $ 
over $  \mathbf R $ 
or $  \mathbf C $ , 
of a global Lie group with Lie algebra $  \mathfrak g $ (
see [[Lie algebra of an analytic group|Lie algebra of an analytic group]]).

Lie's theorem on solvable Lie algebras: Let $  \phi $ 
be a [[Linear representation|linear representation]] of a finite-dimensional solvable Lie algebra (cf. [[Lie algebra, solvable|Lie algebra, solvable]]) $  \mathfrak b $ 
in a vector space $  V $ 
over an algebraically closed field of characteristic 0; then there is a basis of $  V $ 
in which all the operators $  X $ 
of $  \phi ( \mathfrak b ) $ 
are written as upper triangular matrices. A similar assertion is true for a linear continuous representation of a connected topological solvable group in a finite-dimensional complex vector space (the group-theoretic analogue of Lie's theorem); the assumption that the group is connected is essential. A version of the group-theoretic analogue of Lie's theorem is known as the [[Lie–Kolchin theorem|Lie–Kolchin theorem]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> S. Lie, F. Engel, "Theorie der Transformationsgruppen" , '''1–3''' , Leipzig (1888–1893) {{MR|1510035}} {{ZBL|50.0002.01}} {{ZBL|53.0026.07}} {{ZBL|25.0626.01}} {{ZBL|25.0623.01}} {{ZBL|23.0376.01}} {{ZBL|22.0372.01}} {{ZBL|23.0364.01}} {{ZBL|20.0368.01}} {{ZBL|21.0356.02}} {{ZBL|16.0326.01}} {{ZBL|12.0292.01}} {{ZBL|11.0258.02}} {{ZBL|10.0260.01}} {{ZBL|10.0258.01}} </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) {{MR|0682756}} {{ZBL|0319.17002}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> L.S. Pontryagin, "Topological groups" , Princeton Univ. Press (1958) (Translated from Russian) {{MR|0201557}} {{ZBL|0022.17104}} </TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top"> J.-P. Serre, "Lie algebras and Lie groups" , Benjamin (1965) (Translated from French) {{MR|0218496}} {{ZBL|0132.27803}} </TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top"> C. Chevalley, "Théorie des groupes de Lie" , '''3''' , Hermann (1955) {{MR|0068552}} {{ZBL|0186.33104}} {{ZBL|0054.01303}} {{ZBL|0063.00843}} </TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top"> N.G. Chebotarev, "The theory of Lie groups" , Moscow-Leningrad (1940) (In Russian) {{MR|}} {{ZBL|}} </TD></TR></table>



====Comments====
For part 1) of the main article above see also [[Frobenius theorem on Pfaffian systems|Frobenius theorem on Pfaffian systems]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> V.S. Varadarajan, "Lie groups, Lie algebras, and their representations" , Prentice-Hall (1974) {{MR|0376938}} {{ZBL|0371.22001}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) pp. 121 {{MR|0514561}} {{ZBL|0451.53038}} </TD></TR></table>