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Flag Manifolds The Erwin Schrodinger International Boltzmanngasse ESI Institute for Mathematical Physics A Wien Austria Flag Manifolds DV Alekseevsky Vienna Preprint ESI January Supp orted by Federal Ministry of Science and Research Austria Available via httpwwwesiacat FLAG MANIFOLDS DVAlekseevsky Contents I Basic facts ab out Lie groups and Lie algebras Lie groups and Lie algebras Functor Lie Basic results ab out structure of Lie groups and Lie algebras Structure of complex semisimple Lie algebras Real forms of a complex semisimple Lie algebra Parab olic subalgebras of a complex semisimple Lie algebra I I Homogeneous manifolds I I I Flag manifolds as homogeneous spaces of a complex semisimple Lie group Flag manifolds of classical groups SL C SO C S p C n n n Flag manifolds of a semisimple Lie group G The action of a maximal compact subgroup of G on a ag manifold M GP IV Flag manifolds as a quotient of a compact Lie group T ro ots T chamb ers and T Weyl group of a ag manifold Realization of a ag manifold by an adjoint orbit Closed invariant forms on a ag manifold Decomp osition of the isotropy representation into irreducible submo dules Invariant complex structures and Kahler metrics Invariant KahlerEinstein metrics Painted Dynkin graphs and classication of ag manifolds Painted Dynkin graphs T chamb ers and Koszul numb ers Examples Typ eset by A ST X M E DVALEKSEEVSKY I Basic facts ab out Lie groups and Lie algebras Lie groups and Lie algebras Functor Lie Recall that a Lie group is a smo oth manifold G together with structure of a group such that the group op erations G G G g g g g i G G g g are smo oth mappings Here we will assume that either all ob jects are real and then G is a real Lie group or all ob jects are complex and holomorphic then G is a complex Lie group An example of Lie group is the general linear group GLV AutV ie the group of all n n automorphisms of a nite dimensional vector space V R or C If V is the arithmetic vector space n n k k R or C then the group GLk is denoted also by GL k and it is called the general matrix n group Any closed subgroup G of GLV is a Lie group called a linear group Example of linear groups are the following classical Lie groups The group SLV fA GLV det A g of unimo dular transformations and the group Aut V b fA GLV A b bA A bg of automorphisms of a non degenerate bilinear form b If b is a skewsymmetric then the group l n AutV S p V S p C V C is called symplectic group and it is connected For V k l n m the standard notation for symplectic group is S p k If b g is a symmetric form then the m group Aut V g O V is called the orthogonal group The connected comp onent of the unity is g g denoted by SO V fA O V det A g g g n or by SO C if V C n Lie algebra real or complex is a vector space over k R C with a bilinear op eration g g X Y X Y g which satises the Jacobi identity X Y Z Y Z X Z X Y X Y Z g Let G b e a Lie group Denote by L the left action of G on G that is the homomorphism L G Di G g L L g g g g g G g g The space L G X X g G X XG L g Lie G XG g of all L invariant vector elds on G is a subalgebra of the Lie algebra XG of vector elds with resp ect G to the Lie bracket X Y X Y X Y Y X Here a vector elds are considered as derivations of the algebra C G of smo oth functions on G and o means the comp osition of derivations Lie algebra of a Lie group G is called the tangent Lie algebra g of G FLAG MANIFOLDS Any L invariant vector eld X is dened by its value X in the p oint e G G e X X g R X g G g e g e where R g g g is the right multiplication In the case of a linear Lie group G X X g where g g e L G T G which dot stands for matrix multiplication The map X X denes isomorphism XG e e L G with the tangent space T G If G GLV is a linear allows identify the tangent Lie algebra XG e Lie group then the Lie bracket on g T G is the commutator e X Y X Y Y X X Y g l V End V The map G g Lie G denes a functor from category of Lie groups to category of Lie algebras It is called Lie functor and is very closed to b e an isomorphism of categories Indeed any Lie algebra g is the tangent Lie algebra of some simply connected Lie group G dened up to an isomorphism Any Lie group G with tangent Lie algebra g is isomorphic to the quotient G of G by a central discrete subgroup of G All Lie groups with the same tangent Lie algebra are lo cally isomorphic ie they have the same group op erations in some neighb orho o d of the unity Many prop erties of a Lie group G can b e describ ed in terms of prop erties of the tangent Lie algebra g For example Lie group G is solvable nilpotent commutative i the tangent Lie algebra g is solvable nilpotent commutative Lie group G is simple ie has no non discrete normal subgroup semisimple ie has no non discrete commutative normal subgroup reductive ie has a normal semisimple subgroup S with commutative quotient GS i the tangent Lie algebra g is simple has no prop er ideal semisimple has no prop er commutative ideal or reductive is a direct sum of commutative and semisimple Lie algebras There are corresp ondence b etween subalgebras resp ideals of a Lie algebra g and virtual subgroups resp virtual normal subgroups of a Lie group G with Lie G g Subgroup H of a Lie group G is called virtual if it is an immersed submanifold of G Such subgroup H has a structure of Lie group but H is not necessary a closed subgroup of G even if it is simple Exercise Prove that the classical Lie groups SLV SO V S p V has tangent Lie algebras g sl V fA g l V tr A g so V fA sl V g Ax y g x Ay x y V g g sp V fA sl V Ax y x Ay x y V g Prove that all of them are simple with the exception SO V where V C or V R and g has g signature or Basic results ab out structure of Lie groups and Lie algebras We collect here some useful general results ab out Lie groups and Lie algebras LeviMaltsev theorem Any Lie algebra g can b e decomp osed into a sum g s r s r of a maximal semisimple subalgebra s and the radical r maximal solvable ideal Any two maximal semisimple subalgebras are conjugated by an automorphism of g Any connected Lie group G can b e decomp osed into a pro duct G S R of a maximal semisimple subgroup S and the radical R ie the maximal solvable normal subgroup such that S R is a discrete subgroup of G Any two maximal semisimple subgroups of G are conjugated by an automorphism of G Any semisimple resp ectively reductive Lie algebra g is the direct sum of noncommutative simple ideals g g g k DVALEKSEEVSKY resp ectively simple ideals and the center Any simply connected connected semisimple Lie group G is the direct pro duct of simple connected normal subgroups G G G k Any simply connected connected reductive Lie group G is the direct pro duct of simple Lie groups p q G i and the connected comp onent of the center connected Lie group Z G R T where i q T S S z q is the q torus G Z G G G k Ado theorem Any Lie algebra g over R C admits exact linear representation that is a homomorphism g g l V into general linear Lie algebra g l V End V with trivial kernel For any simply connected connected Lie group G there exist a discrete central subgroup such that G admits an exact linear representation G GLV monomorphism into GLV Let G b e a Lie group and denote by Ad the natural homomorphism of G into the group AutG of automorphisms of G given by Ad g g g g g g G g Since any automorphism of G preserves the unity e the group Ad AutG acts on the tangent space G g T G by linear transformations and this action preserves the Lie bracket on g e Ad X Ad Y Ad X Y g G X Y T M g g g e If G GLV is a linear Lie group Ad X g X g g where denotes the matrix multiplication We get a representation Ad g Ad Aut g GLg g which is called the adjoint representation of a Lie group G It induces the adjoint representation ad of the tangent Lie algebra g Lie G by derivations of g ad g Derg X ad ad Y X Y X Y g X X Remark that the kernel of Ad coincides with Z G the center of G and ker ad Z g the center of g Dene a symmetric bilinear form B on a Lie algebra g by B X Y tr ad ad X Y g X Y It is called the Kil ling form The adjoint action of a Lie group G with Lie G g preserves B B Ad X Ad Y B X Y g G X Y g g g In terms of the adjoint action of the Lie algebra this condition can b e written as B ad X Y B X ad Y X Y Z g Z Z FLAG MANIFOLDS This condition follows from previous relation if we remark that any vector Z g generate an parametric d subgroup g exp tZ of G with tangent vector Z at e g Z Then to get it is sucient to t t dt t dierentiate with g g with resp ect to t and put t t The following useful criterion of solvability and semisimplicity of a Lie algebra g is due to Cartan A Lie algebra g is solvable if the commutant g g span fX Y X Y gg is in the kernel of B ie B X Y Z X Y Z g A Lie algebra g is semisimple if the Killing form B is non degenerate Structure of complex semisimple Lie algebras Cartan decomp osition of a semisimple Lie algebra Let now g b e a semisimple Lie algebra Since Z g the adjoint representation ad X ad X ad Y X Y is exact and we can identify g with linear Lie algebra ad derg glg Remark X g that the connected group of automorphisms Ad G Aut g has g as Lie algebra It is called the adjoint group Dene a Cartan subalgebra h of g as a
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