Flag manifold
In mathematics , a flag manifold is the space of complete flags in a vector space or, more generally, the quotient of a semi-simple algebraic group after a Borel subgroup . Flag manifolds are projective varieties .
Flag manifold of a vector space
A complete flag in a finite-dimensional (real or complex) vector space is a sequence
of subspaces of with and , so that each subspace is properly contained in the following, i. H.
and so that
for applies, especially so .
The general linear group acts transitively on the set of all complete flags, the stabilizers of a flag are conjugate to the group of invertible upper triangular matrices . So there is a bijection between and the set of all complete flags. Therefore will
referred to as the flag manifold .
The canonical embedding in the product of Graßmann manifolds
makes the flag manifold a differentiable manifold and (by means of the Plücker embedding of the Graßmann manifolds) a projective variety.
Generalized flag manifolds
Let it be a semi-simple Lie group and a Borel group , i. H. a minimal parabolic subset of . Then the homogeneous space is called a generalized plume manifold . If is an algebraic group, is a projective variety.
The above examples of the flag manifolds of a vector space are obtained for or and the subgroup of the invertible upper triangular matrices.
literature
- Charles Ehresmann : Sur la topologie de certains espaces homogènes. Ann. of Math. (2) 35 (1934), no. 2, 396-443.
- Shiing-Shen Chern : On the characteristic classes of complex sphere bundles and algebraic varieties. Amer. J. Math. 75, (1953). 565-597.
- Armand Borel : Cohomologie des espaces homogènes. Séminaire Bourbaki, Vol. 1, Exp. 45, 371-378, Soc. Math. France, Paris, 1995.
- DV Alekseevsky: Flag manifolds. 11th Yugoslav Geometrical Seminar (Divčibare, 1996). E.g. Rad. Mat. Inst. Beograd. (NS) 6 (14): 3-35 (1997). online (PDF)
Web links
- Flag Manifold (MathWorld)
- Flag Variety (nLab)
- What are flag manifolds and why are they interesting? (Australian Mathematical Society Web Site - the Gazette)
- Brion: Lectures on the geometry of flag varieties