Lie theorem

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) 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 be an -dimensional real effective local Lie transformation group of a domain , let be the identity of and suppose that in local coordinates in a neighbourhood of the set in the action of on is given by a system of analytic functions

(1)

where , and . This action defines analytic vector fields on ,

(2)

where .

Lie's first theorem establishes that the functions , , which define the action of are themselves defined by some auxiliary system of analytic functions , , on which satisfy the condition

(3)

where is the Kronecker symbol. More precisely, is the matrix of the differential of the right translation of by the element at the point , and the system of functions (1) is precisely the solution of the system of equations

(4)

that satisfies the initial conditions , .

Lie's second theorem describes the properties of the functions and . Namely, the satisfy the system of equations

(5)

(this system is the condition that the system (4) is integrable), and the functions satisfy the system of equations

where the are certain constants. The relations (5) imply that the commutator (Lie bracket) of two vector fields and is a linear combination of the fields with constant coefficients:

(6)

that is, the linear hull of the fields is an algebra with respect to the Lie bracket.

The converse of Lie's first and second theorems is the following: If the functions give a solution of (4) in which the matrix 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 satisfy the following relations:

(7)

that is, is a Lie algebra. The converse of the third theorem is important: If the are any constants satisfying (7), then there is a system of vector fields 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 over or , of a global Lie group with Lie algebra (see Lie algebra of an analytic group).

Lie's theorem on solvable Lie algebras: Let be a linear representation of a finite-dimensional solvable Lie algebra (cf. Lie algebra, solvable) in a vector space over an algebraically closed field of characteristic 0; then there is a basis of in which all the operators of 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.

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]	C. Chevalley, "Théorie des groupes de Lie" , 3 , Hermann (1955)
[6]	N.G. Chebotarev, "The theory of Lie groups" , Moscow-Leningrad (1940) (In Russian)

Comments

For part 1) of the main article above see also Frobenius theorem on Pfaffian systems.

References

[a1]	A. Borel, "Linear algebraic groups" , Benjamin (1969)
[a2]	V.S. Varadarajan, "Lie groups, Lie algebras, and their representations" , Prentice-Hall (1974)
[a3]	S. Helgason, "Differential geometry, Lie groups, and symmetric spaces" , Acad. Press (1978) pp. 121