Difference between revisions of "Borel subgroup"
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− | <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</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> A. Borel, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–150</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, J. Tits, "Groupes réductifs" ''Publ. Math. IHES'' , '''27''' (1965) pp. 55–150 {{MR|0207712}} {{ZBL|0145.17402}} </TD></TR></table> |
Revision as of 10:02, 24 March 2012
A maximal connected solvable algebraic subgroup of a linear algebraic group . Thus, for instance, the subgroup of all non-singular upper-triangular matrices is a Borel subgroup in the general linear group
. A. Borel [1] was the first to carry out a systematic study of maximal connected solvable subgroups of algebraic groups. Borel subgroups can be characterized as minimal parabolic subgroups, i.e. algebraic subgroups
of the group
for which the quotient variety
is projective. All Borel subgroups of
are conjugate and, if the Borel subgroups
,
and the group
are defined over a field
,
and
are conjugate by an element of
. The intersection of any two Borel subgroups of a group
contains a maximal torus of
; if this intersection is a maximal torus, such Borel subgroups are said to be opposite. Opposite Borel subgroups exist in
if and only if
is a reductive group. If
is connected, it is the union of all its Borel subgroups, and any parabolic subgroup coincides with its normalizer in
. In such a case a Borel subgroup is maximal among all (and not only algebraic and connected) solvable subgroups of
. Nevertheless, maximal solvable subgroups in
which are not Borel subgroups usually exist. The commutator subgroup of a Borel subgroup
coincides with its unipotent part
, while the normalizer of
in
coincides with
. If the characteristic of the ground field is 0, and
is the Lie algebra of
, then the subalgebra
of
which is the Lie algebra of the Borel subgroup
of
is often referred to as a Borel subalgebra in
. The Borel subalgebras in
are its maximal solvable subalgebras. If
is defined over an arbitrary field
, the parabolic subgroups which are defined over
and are minimal for this property, play a role in the theory of algebraic groups over
similar to that of the Borel groups. For example, two such parabolic subgroups are conjugate by an element 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, J. Tits, "Groupes réductifs" Publ. Math. IHES , 27 (1965) pp. 55–150 MR0207712 Zbl 0145.17402 |
Borel subgroup. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Borel_subgroup&oldid=21818