Difference between revisions of "Lie-Kolchin theorem"
(Importing text file) |
Ulf Rehmann (talk | contribs) m (moved Lie–Kolchin theorem to Lie-Kolchin theorem: ascii title) |
(No difference)
| |
Revision as of 18:53, 24 March 2012
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.
The Lie–Kolchin theorem was proved by E.R. Kolchin [1] (for connected groups) and A.I. Mal'tsev [2] (in the general formulation). It is also sometimes called the Kolchin–Mal'tsev theorem.
References
| [1] | 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 |
| [2] | 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 |
| [3] | M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian) |
Comments
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].
References
| [a1] | I. Kaplansky, "An introduction to differential algebra" , Hermann (1957) |
| [a2] | J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 |
| [a3] | A. Borel, "Linear algebraic groups" , Benjamin (1969) pp. 283ff |
Lie-Kolchin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie-Kolchin_theorem&oldid=22741