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parabolic subgroup

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Definition

Given a linear algebraic group GG (i.e. an algebraic subgroup of the general linear group GL(n,k)GL(n,k) where kk is a field), a subgroup PGP\subset G is said to be parabolic if it is closed in Zariski topology and the quotient variety G/PG/P is projective. A minimal (with respect to inclusion) parabolic subgroup of a linear algebraic group is called a Borel subgroup; in fact, given a Borel subgroup BB, any closed subgroup PBP\supset B is parabolic.

The Cartan geometry of parabolic subgroup inclusions is parabolic geometry.

References

Revised on December 4, 2014 20:06:39 by Urs Schreiber (195.113.31.253)