A solvable subgroup of the group (where is a finite-dimensional vector space over an algebraically closed field) has a normal subgroup of index at most , where depends only on , such that in there is a flag that is invariant with respect to . In other words, there is a basis in in which the elements of are written as triangular matrices. If is a connected closed subgroup of in the Zariski topology, then ; in this case the Lie–Kolchin theorem is a generalization of Lie's theorem, which was proved by S. Lie for complex connected (in the Euclidean topology) solvable Lie groups (see Lie group, solvable; Lie theorem). This assertion can also be considered as a special case of Borel's fixed-point theorem (cf. Borel fixed-point theorem).
The following analogue of the Lie–Kolchin theorem is true for an arbitrary field: A solvable group of matrices contains a normal subgroup of finite index whose commutator subgroup is nilpotent.
|||E.R. Kolchin, "Algebraic matrix groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations" Ann. of Math. (2) , 49 (1948) pp. 1–42|
|||A.I. [A.I. Mal'tsev] Mal'Avcev, "On certain classes of infinite soluble groups" Transl. Amer. Math. Soc. (2) , 2 (1956) pp. 1–21 Mat. Sb. , 28 (1951) pp. 567–588|
|||M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian) MR0551207 Zbl 0549.20001|
In Western literature the Lie–Kolchin theorem usually designates the more restricted version about connected closed subgroups of .
For the role of the Lie–Kolchin theorem in the Galois theory for ordinary linear differential equations see [a1].
|[a1]||I. Kaplansky, "An introduction to differential algebra" , Hermann (1957) MR0093654 Zbl 0083.03301|
|[a2]||J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 MR0396773 Zbl 0325.20039|
|[a3]||A. Borel, "Linear algebraic groups" , Benjamin (1969) pp. 283ff MR0251042 Zbl 0206.49801 Zbl 0186.33201|
Lie–Kolchin theorem. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Lie%E2%80%93Kolchin_theorem&oldid=24158