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In mathematics, a generalized flag variety (or simply flag variety) is a
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of '' ...
whose points are
flags A flag is a piece of fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design employ ...
in a finite-dimensional
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but ca ...
''V'' over a
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
F. When F is the real or complex numbers, a generalized flag variety is a
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...
or
complex manifold In differential geometry and complex geometry, a complex manifold is a manifold with an atlas of charts to the open unit disc in \mathbb^n, such that the transition maps are holomorphic. The term complex manifold is variously used to mean a c ...
, called a real or complex flag manifold. Flag varieties are naturally
projective varieties In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
. Flag varieties can be defined in various degrees of generality. A prototype is the variety of complete flags in a vector space ''V'' over a field F, which is a flag variety for the
special linear group In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the gener ...
over F. Other flag varieties arise by considering partial flags, or by restriction from the special linear group to subgroups such as the
symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gr ...
. For partial flags, one needs to specify the sequence of dimensions of the flags under consideration. For subgroups of the linear group, additional conditions must be imposed on the flags. In the most general sense, a generalized flag variety is defined to mean a projective homogeneous variety, that is, a
smooth Smooth may refer to: Mathematics * Smooth function, a function that is infinitely differentiable; used in calculus and topology * Smooth manifold, a differentiable manifold for which all the transition maps are smooth functions * Smooth algebraic ...
projective variety ''X'' over a field F with a
transitive action In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of a
reductive group In mathematics, a reductive group is a type of linear algebraic group over a field. One definition is that a connected linear algebraic group ''G'' over a perfect field is reductive if it has a representation with finite kernel which is a direct ...
''G'' (and smooth stabilizer subgroup; that is no restriction for F of characteristic zero). If ''X'' has an F-
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
, then it is isomorphic to ''G''/''P'' for some
parabolic subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
''P'' of ''G''. A projective homogeneous variety may also be realised as the orbit of a
highest weight In the mathematical field of representation theory, a weight of an algebra ''A'' over a field F is an algebra homomorphism from ''A'' to F, or equivalently, a one-dimensional representation of ''A'' over F. It is the algebra analogue of a multiplica ...
vector in a projectivized
representation Representation may refer to: Law and politics * Representation (politics), political activities undertaken by elected representatives, as well as other theories ** Representative democracy, type of democracy in which elected officials represent a ...
of ''G''. The complex projective homogeneous varieties are the
compact Compact as used in politics may refer broadly to a pact or treaty; in more specific cases it may refer to: * Interstate compact * Blood compact, an ancient ritual of the Philippines * Compact government, a type of colonial rule utilized in Briti ...
flat model spaces for Cartan geometries of parabolic type. They are homogeneous
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
s under any
maximal compact subgroup In mathematics, a maximal compact subgroup ''K'' of a topological group ''G'' is a subgroup ''K'' that is a compact space, in the subspace topology, and maximal amongst such subgroups. Maximal compact subgroups play an important role in the class ...
of ''G'', and they are precisely the coadjoint orbits of
compact Lie group In mathematics, a compact (topological) group is a topological group whose topology realizes it as a compact topological space (when an element of the group is operated on, the result is also within the group). Compact groups are a natural gen ...
s. Flag manifolds can be
symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
s. Over the complex numbers, the corresponding flag manifolds are the
Hermitian symmetric space In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian ...
s. Over the real numbers, an ''R''-space is a synonym for a real flag manifold and the corresponding symmetric spaces are called symmetric ''R''-spaces.


Flags in a vector space

A flag in a finite dimensional vector space ''V'' over a field F is an increasing sequence of subspaces, where "increasing" means each is a proper subspace of the next (see
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter m ...
): :\ = V_0 \sub V_1 \sub V_2 \sub \cdots \sub V_k = V. If we write the dim ''V''''i'' = ''d''''i'' then we have :0 = d_0 < d_1 < d_2 < \cdots < d_k = n, where ''n'' is the
dimension In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordin ...
of ''V''. Hence, we must have ''k'' ≤ ''n''. A flag is called a ''complete flag'' if ''d''''i'' = ''i'' for all ''i'', otherwise it is called a ''partial flag''. The ''signature'' of the flag is the sequence (''d''1, ..., ''d''''k''). A partial flag can be obtained from a complete flag by deleting some of the subspaces. Conversely, any partial flag can be completed (in many different ways) by inserting suitable subspaces.


Prototype: the complete flag variety

According to basic results of
linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrices. ...
, any two complete flags in an ''n''-dimensional vector space ''V'' over a field F are no different from each other from a geometric point of view. That is to say, the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
transitively on the set of all complete flags. Fix an ordered
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
for ''V'', identifying it with F''n'', whose general linear group is the group GL(''n'',F) of ''n'' × ''n'' invertible matrices. The standard flag associated with this basis is the one where the ''i''th subspace is spanned by the first ''i'' vectors of the basis. Relative to this basis, the stabilizer of the standard flag is the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of nonsingular lower triangular matrices, which we denote by ''B''''n''. The complete flag variety can therefore be written as a
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of '' ...
GL(''n'',F) / ''B''''n'', which shows in particular that it has dimension ''n''(''n''−1)/2 over F. Note that the multiples of the identity act trivially on all flags, and so one can restrict attention to the
special linear group In mathematics, the special linear group of degree ''n'' over a field ''F'' is the set of matrices with determinant 1, with the group operations of ordinary matrix multiplication and matrix inversion. This is the normal subgroup of the gener ...
SL(''n'',F) of matrices with determinant one, which is a semisimple algebraic group; the set of lower triangular matrices of determinant one is a
Borel subgroup In the theory of algebraic groups, a Borel subgroup of an algebraic group ''G'' is a maximal Zariski closed and connected solvable algebraic subgroup. For example, in the general linear group ''GLn'' (''n x n'' invertible matrices), the subgroup ...
. If the field F is the real or complex numbers we can introduce an
inner product In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
on ''V'' such that the chosen basis is
orthonormal In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal (or perpendicular along a line) unit vectors. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of uni ...
. Any complete flag then splits into a direct sum of one-dimensional subspaces by taking orthogonal complements. It follows that the complete flag manifold over the complex numbers is the
homogeneous space In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of '' ...
:U(n)/T^n where U(''n'') is the
unitary group In mathematics, the unitary group of degree ''n'', denoted U(''n''), is the group of unitary matrices, with the group operation of matrix multiplication. The unitary group is a subgroup of the general linear group . Hyperorthogonal group is ...
and T''n'' is the ''n''-torus of diagonal unitary matrices. There is a similar description over the real numbers with U(''n'') replaced by the orthogonal group O(''n''), and T''n'' by the diagonal orthogonal matrices (which have diagonal entries ±1).


Partial flag varieties

The partial flag variety : F(d_1,d_2,\ldots d_k, \mathbb F) is the space of all flags of signature (''d''1, ''d''2, ... ''d''''k'') in a vector space ''V'' of dimension ''n'' = ''d''''k'' over F. The complete flag variety is the special case that ''d''''i'' = ''i'' for all ''i''. When ''k''=2, this is a
Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...
of ''d''1-dimensional subspaces of ''V''. This is a homogeneous space for the general linear group ''G'' of ''V'' over F. To be explicit, take ''V'' = F''n'' so that ''G'' = GL(''n'',F). The stabilizer of a flag of nested subspaces ''V''''i'' of dimension ''d''''i'' can be taken to be the group of nonsingular
block Block or blocked may refer to: Arts, entertainment and media Broadcasting * Block programming, the result of a programming strategy in broadcasting * W242BX, a radio station licensed to Greenville, South Carolina, United States known as ''96.3 ...
lower triangular matrices, where the dimensions of the blocks are ''n''''i'' := ''d''''i'' − ''d''''i''−1 (with ''d''0 = 0). Restricting to matrices of determinant one, this is a parabolic subgroup ''P'' of SL(''n'',F), and thus the partial flag variety is isomorphic to the homogeneous space SL(''n'',F)/''P''. If F is the real or complex numbers, then an inner product can be used to split any flag into a direct sum, and so the partial flag variety is also isomorphic to the homogeneous space : U(n)/U(n_1)\times\cdots \times U(n_k) in the complex case, or : O(n)/O(n_1)\times\cdots\times O(n_k) in the real case.


Generalization to semisimple groups

The upper triangular matrices of determinant one are a Borel subgroup of SL(''n'',F), and hence the stabilizers of partial flags are parabolic subgroups. Furthermore, a partial flag is determined by the parabolic subgroup which stabilizes it. Hence, more generally, if ''G'' is a semisimple algebraic or
Lie group In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the ad ...
, then the (generalized) flag variety for ''G'' is ''G''/''P'' where ''P'' is a parabolic subgroup of ''G''. The correspondence between parabolic subgroups and generalized flag varieties allows each to be understood in terms of the other. The extension of the terminology "flag variety" is reasonable, because points of ''G''/''P'' can still be described using flags. When ''G'' is a classical group, such as a
symplectic group In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and for positive integer ''n'' and field F (usually C or R). The latter is called the compact symplectic gr ...
or
orthogonal group In mathematics, the orthogonal group in dimension , denoted , is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations. ...
, this is particularly transparent. If (''V'', ''ω'') is a
symplectic vector space In mathematics, a symplectic vector space is a vector space ''V'' over a field ''F'' (for example the real numbers R) equipped with a symplectic bilinear form. A symplectic bilinear form is a mapping that is ; Bilinear: Linear in each argument ...
then a partial flag in ''V'' is ''
isotropic Isotropy is uniformity in all orientations; it is derived . Precise definitions depend on the subject area. Exceptions, or inequalities, are frequently indicated by the prefix ' or ', hence ''anisotropy''. ''Anisotropy'' is also used to describe ...
'' if the symplectic form vanishes on proper subspaces of ''V'' in the flag. The stabilizer of an isotropic flag is a parabolic subgroup of the symplectic group Sp(''V'',''ω''). For orthogonal groups there is a similar picture, with a couple of complications. First, if F is not algebraically closed, then isotropic subspaces may not exist: for a general theory, one needs to use the
split orthogonal group In mathematics, the indefinite orthogonal group, is the Lie group of all linear transformations of an ''n''-dimensional real vector space that leave invariant a nondegenerate, symmetric bilinear form of signature , where . It is also called the p ...
s. Second, for vector spaces of even dimension 2''m'', isotropic subspaces of dimension ''m'' come in two flavours ("self-dual" and "anti-self-dual") and one needs to distinguish these to obtain a homogeneous space.


Cohomology

If ''G'' is a compact, connected Lie group, it contains a
maximal torus In the mathematical theory of compact Lie groups a special role is played by torus subgroups, in particular by the maximal torus subgroups. A torus in a compact Lie group ''G'' is a compact, connected, abelian Lie subgroup of ''G'' (and therefore ...
''T'' and the space ''G''/''T'' of left cosets with the
quotient topology In topology and related areas of mathematics, the quotient space of a topological space under a given equivalence relation is a new topological space constructed by endowing the quotient set of the original topological space with the quotient t ...
is a compact real manifold. If ''H'' is any other closed, connected subgroup of ''G'' containing ''T'', then ''G''/''H'' is another compact real manifold. (Both are actually complex homogeneous spaces in a canonical way through complexification.) The presence of a complex structure and cellular (co)homology make it easy to see that the
cohomology ring In mathematics, specifically algebraic topology, the cohomology ring of a topological space ''X'' is a ring formed from the cohomology groups of ''X'' together with the cup product serving as the ring multiplication. Here 'cohomology' is usually u ...
of ''G''/''H'' is concentrated in even degrees, but in fact, something much stronger can be said. Because ''G'' → ''G/H'' is a principal ''H''-bundle, there exists a classifying map ''G''/''H'' → ''BH'' with target the
classifying space In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
''BH''. If we replace ''G''/''H'' with the homotopy quotient ''G''''H'' in the sequence ''G'' → ''G/H'' → ''BH'', we obtain a principal ''G''-bundle called the Borel fibration of the right multiplication action of ''H'' on ''G'', and we can use the cohomological
Serre spectral sequence In mathematics, the Serre spectral sequence (sometimes Leray–Serre spectral sequence to acknowledge earlier work of Jean Leray in the Leray spectral sequence) is an important tool in algebraic topology. It expresses, in the language of homologica ...
of this bundle to understand the fiber-restriction
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same ...
''H''*(''G''/''H'') → ''H''*(''G'') and the characteristic map ''H''*(''BH'') → ''H''*(''G''/''H''), so called because its image, the ''characteristic subring'' of ''H''*(''G''/''H''), carries the characteristic classes of the original bundle ''H'' → ''G'' → ''G''/''H''. Let us now restrict our coefficient ring to be a field ''k'' of characteristic zero, so that, by Hopf's theorem, ''H''*(''G'') is an
exterior algebra In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
on generators of odd degree (the subspace of primitive elements). It follows that the edge homomorphisms :E_^ \to E_^ of the spectral sequence must eventually take the space of primitive elements in the left column ''H''*(''G'') of the page ''E''2 bijectively into the bottom row ''H''*(''BH''): we know ''G'' and ''H'' have the same rank, so if the collection of edge homomorphisms were ''not'' full rank on the primitive subspace, then the image of the bottom row ''H''*(''BH'') in the final page ''H''*(''G''/''H'') of the sequence would be infinite-dimensional as a ''k''-vector space, which is impossible, for instance by cellular cohomology again, because a compact homogeneous space admits a finite CW structure. Thus the ring map ''H''*(''G''/''H'') → ''H''*(''G'') is trivial in this case, and the characteristic map is surjective, so that ''H''*(''G''/''H'') is a quotient of ''H''*(''BH''). The kernel of the map is the ideal generated by the images of primitive elements under the edge homomorphisms, which is also the ideal generated by positive-degree elements in the image of the canonical map ''H''*(''BG'') → ''H''*(''BH'') induced by the inclusion of ''H'' in ''G''. The map ''H''*(''BG'') → ''H''*(''BT'') is injective, and likewise for ''H'', with image the subring ''H''*(''BT'')''W''(''G'') of elements invariant under the action of the
Weyl group In mathematics, in particular the theory of Lie algebras, the Weyl group (named after Hermann Weyl) of a root system Φ is a subgroup of the isometry group of that root system. Specifically, it is the subgroup which is generated by reflections t ...
, so one finally obtains the concise description :H^*(G/H) \cong H^*(BT)^/\big(\widetilde^*(BT)^\big), where \widetilde H^* denotes positive-degree elements and the parentheses the generation of an ideal. For example, for the complete complex flag manifold ''U''(''n'')/''T''''n'', one has :H^*\big(U(n)/T^n\big) \cong \mathbb _1,\ldots,t_n(\sigma_1,\ldots,\sigma_n), where the ''t''''j'' are of degree 2 and the σ''j'' are the first ''n''
elementary symmetric polynomials In mathematics, specifically in commutative algebra, the elementary symmetric polynomials are one type of basic building block for symmetric polynomials, in the sense that any symmetric polynomial can be expressed as a polynomial in elementary sy ...
in the variables ''t''''j''. For a more concrete example, take ''n'' = 2, so that ''U''(''2'')/ 'U''(1) × ''U''(1)is the complex
Grassmannian In mathematics, the Grassmannian is a space that parameterizes all -dimensional linear subspaces of the -dimensional vector space . For example, the Grassmannian is the space of lines through the origin in , so it is the same as the projective ...
Gr(1,\mathbb2) ≈ \mathbb''P''1 ≈ ''S''2. Then we expect the cohomology ring to be an exterior algebra on a generator of degree two (the
fundamental class In mathematics, the fundamental class is a homology class 'M''associated to a connected orientable compact manifold of dimension ''n'', which corresponds to the generator of the homology group H_n(M,\partial M;\mathbf)\cong\mathbf . The fundam ...
), and indeed, :H^*\big(U(2)/T^2\big) \cong \mathbb _1,t_2(t_1 + t_2, t_1 t_2) \cong \mathbb _1(t_1^2), as hoped.


Highest weight orbits and projective homogeneous varieties

If ''G'' is a semisimple algebraic group (or Lie group) and ''V'' is a (finite dimensional) highest weight representation of ''G'', then the highest weight space is a point in the
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generall ...
P(''V'') and its orbit under the action of ''G'' is a
projective algebraic variety In algebraic geometry, a projective variety over an algebraically closed field ''k'' is a subset of some projective ''n''-space \mathbb^n over ''k'' that is the zero-locus of some finite family of homogeneous polynomials of ''n'' + 1 variables wi ...
. This variety is a (generalized) flag variety, and furthermore, every (generalized) flag variety for ''G'' arises in this way.
Armand Borel Armand Borel (21 May 1923 – 11 August 2003) was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993. He worked in al ...
showed that this characterizes the flag varieties of a general semisimple algebraic group ''G'': they are precisely the complete homogeneous spaces of ''G'', or equivalently (in this context), the projective homogeneous ''G''-varieties.


Symmetric spaces

Let ''G'' be a semisimple Lie group with maximal compact subgroup ''K''. Then ''K'' acts transitively on any conjugacy class of parabolic subgroups, and hence the generalized flag variety ''G''/''P'' is a compact homogeneous
Riemannian manifold In differential geometry, a Riemannian manifold or Riemannian space , so called after the German mathematician Bernhard Riemann, is a real, smooth manifold ''M'' equipped with a positive-definite inner product ''g'p'' on the tangent space ''T ...
''K''/(''K''∩''P'') with isometry group ''K''. Furthermore, if ''G'' is a complex Lie group, ''G''/''P'' is a homogeneous
Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnold ...
. Turning this around, the Riemannian homogeneous spaces :''M'' = ''K''/(''K''∩''P'') admit a strictly larger Lie group of transformations, namely ''G''. Specializing to the case that ''M'' is a
symmetric space In mathematics, a symmetric space is a Riemannian manifold (or more generally, a pseudo-Riemannian manifold) whose group of symmetries contains an inversion symmetry about every point. This can be studied with the tools of Riemannian geometry, ...
, this observation yields all symmetric spaces admitting such a larger symmetry group, and these spaces have been classified by Kobayashi and Nagano. If ''G'' is a complex Lie group, the symmetric spaces ''M'' arising in this way are the compact
Hermitian symmetric space In mathematics, a Hermitian symmetric space is a Hermitian manifold which at every point has an inversion symmetry preserving the Hermitian structure. First studied by Élie Cartan, they form a natural generalization of the notion of Riemannian ...
s: ''K'' is the isometry group, and ''G'' is the biholomorphism group of ''M''. Over the real numbers, a real flag manifold is also called an R-space, and the R-spaces which are Riemannian symmetric spaces under ''K'' are known as symmetric R-spaces. The symmetric R-spaces which are not Hermitian symmetric are obtained by taking ''G'' to be a
real form In mathematics, the notion of a real form relates objects defined over the field of real and complex numbers. A real Lie algebra ''g''0 is called a real form of a complex Lie algebra ''g'' if ''g'' is the complexification of ''g''0: : \mathfra ...
of the biholomorphism group ''G''c of a Hermitian symmetric space ''G''c/''P''c such that ''P'' := ''P''c∩''G'' is a parabolic subgroup of ''G''. Examples include
projective space In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generall ...
s (with ''G'' the group of
projective transformation In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In gener ...
s) and
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ...
s (with ''G'' the group of
conformal transformation In mathematics, a conformal map is a function that locally preserves angles, but not necessarily lengths. More formally, let U and V be open subsets of \mathbb^n. A function f:U\to V is called conformal (or angle-preserving) at a point u_0\i ...
s).


See also

*
Parabolic Lie algebra In algebra, a parabolic Lie algebra \mathfrak p is a subalgebra of a semisimple Lie algebra \mathfrak g satisfying one of the following two conditions: * \mathfrak p contains a maximal solvable subalgebra (a Borel subalgebra) of \mathfrak g; * the ...
*
Bruhat decomposition In mathematics, the Bruhat decomposition (introduced by François Bruhat for classical groups and by Claude Chevalley in general) ''G'' = ''BWB'' of certain algebraic groups ''G'' into cells can be regarded as a general expression of the principl ...


References

* Robert J. Baston and Michael G. Eastwood, ''The Penrose Transform: its Interaction with Representation Theory'', Oxford University Press, 1989. * Jürgen Berndt,
Lie group actions on manifolds
', Lecture notes, Tokyo, 2002. * Jürgen Berndt, Sergio Console and Carlos Olmos,
Submanifolds and Holonomy
', Chapman & Hall/CRC Press, 2003. * Michel Brion,

', Lecture notes, Varsovie, 2003. * James E. Humphreys,
Linear Algebraic Groups
', Graduate Texts in Mathematics, 21, Springer-Verlag, 1972. * S. Kobayashi and T. Nagano, ''On filtered Lie algebras and geometric structures'' I, II, J. Math. Mech. 13 (1964), 875–907, 14 (1965) 513–521. {{Authority control Differential geometry Algebraic homogeneous spaces