Borel fixed-point theorem
A connected solvable algebraic group acting regularly (cf. Algebraic group of transformations) on a non-empty complete algebraic variety
over an algebraically-closed field
has a fixed point in
. It follows from this theorem that Borel subgroups (cf. Borel subgroup) 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
: Let
be a complete variety defined over a field
on which a connected solvable
-split group
acts regularly, then the set of rational
-points
is either empty or it contains a point which is fixed with respect to
. Hence the generalization of the theorem of conjugation of Borel subgroup is: If the field
is perfect, the maximal connected solvable
-split subgroups of a connected
-defined algebraic group
are mutually conjugate by elements of the group of
-points of
[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 |
Borel fixed-point theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_fixed-point_theorem&oldid=21854