Borel subgroup: Difference between revisions
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==Definition== | ==Definition== | ||
A '''Borel subgroup''' of an [[algebraic group]] is a maximal [[connected algebraic group|connected]] [[solvable group|solvable]] [[closed subgroup]]. | A '''Borel subgroup''' of an [[algebraic group]] is a subgroup satisfying the following equivalent conditions: | ||
# It is maximal among [[connected algebraic group|connected]] [[defining ingredient::solvable group|solvable]] [[closed subgroup]]s of the whole group. | |||
# It is minimal among [[defining ingredient::parabolic subgroup]]s of the whole group, i.e., it is a closed subgroup such that the quotient is a [[complete variety]] but such that there is no smaller closed subgroup for which the quotient is a complete variety. | |||
==Facts== | ==Facts== | ||
When the base field is [[algebraically closed field|algebraically closed]] all the Borel subgroups form a single conjugacy class. In other words, upto conjugacy, we can talk of '''the''' Borel subgroup. | When the base field is [[algebraically closed field|algebraically closed]] all the Borel subgroups form a single conjugacy class. In other words, upto conjugacy, we can talk of '''the''' Borel subgroup. | ||
Revision as of 01:27, 1 January 2012
The following property can be evaluated for a closed subgroup of an algebraic group
Definition
A Borel subgroup of an algebraic group is a subgroup satisfying the following equivalent conditions:
- It is maximal among connected solvable closed subgroups of the whole group.
- It is minimal among parabolic subgroups of the whole group, i.e., it is a closed subgroup such that the quotient is a complete variety but such that there is no smaller closed subgroup for which the quotient is a complete variety.
Facts
When the base field is algebraically closed all the Borel subgroups form a single conjugacy class. In other words, upto conjugacy, we can talk of the Borel subgroup.