Editing Linear algebraic group (section)

==Borel subgroups==
The '''[[Borel subgroup]]s''' are important for the structure theory of linear algebraic groups. For a linear algebraic group ''G'' over an algebraically closed field ''k'', a Borel subgroup of ''G'' means a maximal smooth connected solvable subgroup. For example, one Borel subgroup of ''GL''(''n'') is the subgroup ''B'' of [[upper triangular matrix|upper-triangular matrices]] (all entries below the diagonal are zero).

A basic result of the theory is that any two Borel subgroups of a connected group ''G'' over an algebraically closed field ''k'' are conjugate by some element of ''G''(''k'').<ref>Borel (1991), Theorem 11.1.</ref> (A standard proof uses the [[Borel fixed-point theorem]]: for a connected solvable group ''G'' acting on a [[proper variety]] ''X'' over an algebraically closed field ''k'', there is a ''k''-point in ''X'' which is fixed by the action of ''G''.) The conjugacy of Borel subgroups in ''GL''(''n'') amounts to the [[Lie–Kolchin theorem]]: every smooth connected solvable subgroup of ''GL''(''n'') is conjugate to a subgroup of the upper-triangular subgroup in ''GL''(''n'').

For an arbitrary field ''k'', a Borel subgroup ''B'' of ''G'' is defined to be a subgroup over ''k'' such that, over an algebraic closure <math>\overline k</math> of ''k'', <math>B_{\overline k}</math> is a Borel subgroup of <math>G_{\overline k}</math>. Thus ''G'' may or may not have a Borel subgroup over ''k''.

For a closed subgroup scheme ''H'' of ''G'', the [[quotient space (linear algebra)|quotient space]] ''G''/''H'' is a smooth [[quasi-projective]] scheme over ''k''.<ref>Milne (2017), Theorems 7.18 and 8.43.</ref> A smooth subgroup ''P'' of a connected group ''G'' is called '''parabolic''' if ''G''/''P'' is [[projective variety|projective]] over ''k'' (or equivalently, proper over ''k''). An important property of Borel subgroups ''B'' is that ''G''/''B'' is a projective variety, called the '''flag variety''' of ''G''. That is, Borel subgroups are parabolic subgroups. More precisely, for ''k'' algebraically closed, the Borel subgroups are exactly the minimal parabolic subgroups of ''G''; conversely, every subgroup containing a Borel subgroup is parabolic.<ref>Borel (1991), Corollary 11.2.</ref> So one can list all parabolic subgroups of ''G'' (up to conjugation by ''G''(''k'')) by listing all the linear algebraic subgroups of ''G'' that contain a fixed Borel subgroup. For example, the subgroups ''P'' ⊂ ''GL''(3) over ''k'' that contain the Borel subgroup ''B'' of upper-triangular matrices are ''B'' itself, the whole group ''GL''(3), and the intermediate subgroups
:<math>\left \{ \begin{bmatrix}
 * & * & * \\
 0 & * & * \\
0 & * & *
\end{bmatrix} \right \}</math> and <math>\left \{ \begin{bmatrix}
 * & * & * \\
 * & * & * \\
0 & 0 & *
\end{bmatrix} \right \}.</math>
The corresponding '''[[generalized flag variety|projective homogeneous varieties]]''' ''GL''(3)/''P'' are (respectively): the '''flag manifold''' of all chains of linear subspaces
:<math>0\subset V_1\subset V_2\subset A^3_k</math>
with ''V''<sub>''i''</sub> of dimension ''i''; a point; the '''[[projective space]]''' '''P'''<sup>2</sup> of lines (1-dimensional [[linear subspace]]s) in ''A''<sup>3</sup>; and the dual projective space '''P'''<sup>2</sup> of planes in ''A''<sup>3</sup>.