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Difference between revisions of "Lie-Kolchin theorem"

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====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.I. Kargapolov,   J.I. [Yu.I. Merzlyakov] Merzljakov,   "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian)</TD></TR></table>
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<table><TR><TD valign="top">[1]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> 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</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top"> M.I. Kargapolov, J.I. [Yu.I. Merzlyakov] Merzljakov, "Fundamentals of the theory of groups" , Springer (1979) (Translated from Russian) {{MR|0551207}} {{ZBL|0549.20001}} </TD></TR></table>
  
  
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====References====
 
====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I. Kaplansky,   "An introduction to differential algebra" , Hermann (1957)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.E. Humphreys,   "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Borel,   "Linear algebraic groups" , Benjamin (1969) pp. 283ff</TD></TR></table>
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<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> I. Kaplansky, "An introduction to differential algebra" , Hermann (1957) {{MR|0093654}} {{ZBL|0083.03301}} </TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> J.E. Humphreys, "Linear algebraic groups" , Springer (1975) pp. Sect. 35.1 {{MR|0396773}} {{ZBL|0325.20039}} </TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top"> A. Borel, "Linear algebraic groups" , Benjamin (1969) pp. 283ff {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR></table>

Revision as of 10:54, 1 April 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) MR0551207 Zbl 0549.20001


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) 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
How to Cite This Entry:
Lie-Kolchin theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lie-Kolchin_theorem&oldid=22741
This article was adapted from an original article by V.V. Gorbatsevich (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article