Namespaces
Variants
Actions

Difference between revisions of "Borel fixed-point theorem"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (MR/ZBL numbers added)
(Tex done)
Line 1: Line 1:
A connected solvable algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b0170701.png" /> acting regularly (cf. [[Algebraic group of transformations|Algebraic group of transformations]]) on a non-empty complete algebraic variety <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b0170702.png" /> over an algebraically-closed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b0170703.png" /> has a fixed point in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b0170704.png" />. It follows from this theorem that Borel subgroups (cf. [[Borel subgroup|Borel subgroup]]) of algebraic groups are conjugate (The Borel–Morozov theorem). The theorem was demonstrated by A. Borel [[#References|[1]]]. Borel's theorem can be generalized to an arbitrary (not necessarily algebraically-closed) field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b0170705.png" />: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b0170706.png" /> be a complete variety defined over a field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b0170707.png" /> on which a connected solvable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b0170708.png" />-[[Split group|split group]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b0170709.png" /> acts regularly, then the set of rational <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b01707010.png" />-points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b01707011.png" /> is either empty or it contains a point which is fixed with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b01707012.png" />. Hence the generalization of the theorem of conjugation of Borel subgroup is: If the field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b01707013.png" /> is perfect, the maximal connected solvable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b01707014.png" />-split subgroups of a connected <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b01707015.png" />-defined algebraic group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b01707016.png" /> are mutually conjugate by elements of the group of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b01707017.png" />-points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017070/b01707018.png" /> [[#References|[2]]].
+
A connected solvable algebraic group $G$ acting regularly (cf. [[Algebraic group of transformations]]) on a non-empty complete algebraic variety $V$ over an algebraically-closed field $k$ has a fixed point in $V$. It follows from this theorem that [[Borel subgroup]]s of algebraic groups are conjugate (The Borel–Morozov theorem). The theorem was demonstrated by A. Borel [[#References|[1]]]. Borel's theorem can be generalized to an arbitrary (not necessarily algebraically-closed) field $k$: Let $V$ be a [[Complete algebraic variety|complete variety]] defined over a field $k$ on which a connected solvable $k$-[[split group]] $G$ acts regularly, then the set of rational $k$-points $V(k)$ is either empty or it contains a point which is fixed with respect to $G$. Hence the generalization of the theorem of conjugation of Borel subgroup is: If the field $k$ is [[Perfect field|perfect]], the maximal connected solvable $k$-split subgroups of a connected $k$-defined algebraic group $H$ are mutually conjugate by elements of the group of $k$-points of $H$ [[#References|[2]]].
  
 
====References====
 
====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Borel,  "Groupes linéaires algébriques"  ''Ann. of Math. (2)'' , '''64''' :  1  (1956)  pp. 20–82  {{MR|0093006}} {{ZBL|0070.26104}} </TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Borel,  "Linear algebraic groups" , Benjamin  (1969)  {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Morozov,  ''Dokl. Akad. Nauk SSSR'' , '''36''' :  3  (1942)  pp. 91–94    {{MR|}} {{ZBL|}} </TD></TR></table>
+
<table>
 +
<TR><TD valign="top">[1]</TD> <TD valign="top">  A. Borel,  "Groupes linéaires algébriques"  ''Ann. of Math. (2)'' , '''64''' :  1  (1956)  pp. 20–82  {{MR|0093006}} {{ZBL|0070.26104}} </TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  A. Borel,  "Linear algebraic groups" , Benjamin  (1969)  {{MR|0251042}} {{ZBL|0206.49801}} {{ZBL|0186.33201}} </TD></TR>
 +
<TR><TD valign="top">[3]</TD> <TD valign="top">  V.V. Morozov,  ''Dokl. Akad. Nauk SSSR'' , '''36''' :  3  (1942)  pp. 91–94    {{MR|}} {{ZBL|}} </TD></TR>
 +
</table>
 +
 
 +
{{TEX|done}}

Revision as of 19:18, 31 March 2018

A connected solvable algebraic group $G$ acting regularly (cf. Algebraic group of transformations) on a non-empty complete algebraic variety $V$ over an algebraically-closed field $k$ has a fixed point in $V$. It follows from this theorem that Borel subgroups of algebraic groups are conjugate (The Borel–Morozov theorem). The theorem was demonstrated by A. Borel [1]. Borel's theorem can be generalized to an arbitrary (not necessarily algebraically-closed) field $k$: Let $V$ be a complete variety defined over a field $k$ on which a connected solvable $k$-split group $G$ acts regularly, then the set of rational $k$-points $V(k)$ is either empty or it contains a point which is fixed with respect to $G$. Hence the generalization of the theorem of conjugation of Borel subgroup is: If the field $k$ is perfect, the maximal connected solvable $k$-split subgroups of a connected $k$-defined algebraic group $H$ are mutually conjugate by elements of the group of $k$-points of $H$ [2].

References

[1] A. Borel, "Groupes linéaires algébriques" Ann. of Math. (2) , 64 : 1 (1956) pp. 20–82 MR0093006 Zbl 0070.26104
[2] A. Borel, "Linear algebraic groups" , Benjamin (1969) MR0251042 Zbl 0206.49801 Zbl 0186.33201
[3] V.V. Morozov, Dokl. Akad. Nauk SSSR , 36 : 3 (1942) pp. 91–94
How to Cite This Entry:
Borel fixed-point theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_fixed-point_theorem&oldid=21854
This article was adapted from an original article by V.P. Platonov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article