Difference between revisions of "Lie-Kolchin theorem"
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| + | A solvable subgroup $ G $ | ||
| + | of the group $ \mathop{\rm GL} ( V) $( | ||
| + | where $ V $ | ||
| + | is a finite-dimensional vector space over an algebraically closed field) has a normal subgroup $ G _ {1} $ | ||
| + | of index at most $ \rho $, | ||
| + | where $ \rho $ | ||
| + | depends only on $ \mathop{\rm dim} V $, | ||
| + | such that in $ V $ | ||
| + | there is a [[Flag|flag]] $ F = \{ V _ {i} \} $ | ||
| + | that is invariant with respect to $ G _ {1} $. | ||
| + | In other words, there is a basis in $ V $ | ||
| + | in which the elements of $ G _ {1} $ | ||
| + | are written as triangular matrices. If $ G $ | ||
| + | is a connected closed subgroup of $ \mathop{\rm GL} ( V) $ | ||
| + | in the Zariski topology, then $ G _ {1} = G $; | ||
| + | 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 group, solvable]]; [[Lie theorem|Lie theorem]]). This assertion can also be considered as a special case of Borel's fixed-point theorem (cf. [[Borel fixed-point theorem|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 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. | ||
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====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> | + | <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> |
| − | |||
| − | |||
====Comments==== | ====Comments==== | ||
| − | In Western literature the Lie–Kolchin theorem usually designates the more restricted version about connected closed subgroups of | + | In Western literature the Lie–Kolchin theorem usually designates the more restricted version about connected closed subgroups of $ \mathop{\rm GL} ( V) $. |
For the role of the Lie–Kolchin theorem in the Galois theory for ordinary linear differential equations see [[#References|[a1]]]. | For the role of the Lie–Kolchin theorem in the Galois theory for ordinary linear differential equations see [[#References|[a1]]]. | ||
====References==== | ====References==== | ||
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> | + | <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> |
Latest revision as of 22:16, 5 June 2020
A solvable subgroup $ G $
of the group $ \mathop{\rm GL} ( V) $(
where $ V $
is a finite-dimensional vector space over an algebraically closed field) has a normal subgroup $ G _ {1} $
of index at most $ \rho $,
where $ \rho $
depends only on $ \mathop{\rm dim} V $,
such that in $ V $
there is a flag $ F = \{ V _ {i} \} $
that is invariant with respect to $ G _ {1} $.
In other words, there is a basis in $ V $
in which the elements of $ G _ {1} $
are written as triangular matrices. If $ G $
is a connected closed subgroup of $ \mathop{\rm GL} ( V) $
in the Zariski topology, then $ G _ {1} = G $;
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 $ \mathop{\rm GL} ( V) $.
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
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