Francoise J.-P., Naber G.L., Tsun T.S. (editors) Encyclopedia of Mathematical Physics

Подождите немного. Документ загружается.

Symmetric Cones

Let us start with X = GL(n, R)=O( n). This manifold

corresponds to the classi cal theory of quadratic

forms: X can be realized as the manifold Sym

(n)of

symmetric positive matrices x  0 of order n

(corresponding to positive quadratic forms). Then

the transitivity of GL(n; R)onX corresponds to the

possibility to transform positive forms to a sum of

squares. The sufficiency of triang le matrices for such

transformations corresponds to the transitivity on

X = Sym

(n) of the parabolic subgroup P of (upper)

triangle matrices with positive diagonal elements. So

A is the group of diagonal matrices with positive

elements and the submanifold of diagonal matrices

in X parametrizes K-orbits. The general fact about

A-parametrization in this example is the classical

fact abou t the reduction of quadratic forms to

diagonal form by orthogonal transformations.

There are complex and quaternioni c versions

of this pictu re. The symmetric manifold

X = GL(n; C)=U(n) is realized as the manifold

Herm

(n) of positive complex Hermitian matrices

(forms) and similarly classical facts of linear algebra

on Hermitian quadratic forms are transformed into

geometrical statements on symmetric spaces. Let us

emphasize that we consider here the group GL(n; C)

as the real group. The same situation exists with the

manifold Herm

(H, n) of positive quaternionic

Hermitian matrices, which is the symmetric mani-

fold for the real group GL(n; H).

These three manifolds can be included in an

impressive geometrical structure. They all are con-

vex homogeneous cones V in linear spaces R

which

are self-dual (V = V



) relative to a bilinear form

h, i. Let us recall that



¼fx; hx; yi > 0; y 2 V n 0g

Here

V is the closure of V. So these three symmetric

manifolds are linear homogeneous self-dual cones.

There is only one more type of classical homo-

geneous self-dual cones – quadratic (Lorentzian)

cones

¼fx 2 R

nþ1

; x

 x

x

nþ1

> 0; x

> 0g

which is also called the future light cone (the

condition x

< 0 defines the past light cone). The

group of linear automorphisms of this cone is

SO(1, n)  R

; the first factor is the Lorentz group.

There is also one exceptional 27-dimensional

cone; it is possible to interpret this cone as the

cone of positive Hermitian matrices of third order

over Cayley numbers. There is a very nice structural

theory of homogeneous self-dual cones; it is con-

venient to develop this theory in the language of

Jordan algebras (Faraut and Koranyi 1994). Such

cones participate as elements of explicit construc-

tions of other classes of symmetric spaces (see

below).

Following Siegel, it is possible to connect with

homogeneous self-dual cones multidimensional ver-

sions of Euler integrals (- and B-functions) (Faraut

and Koranyi 1994). They have many applications,

including tho se to integral formulas for complex

symmetric domains.

Riemann Symmetric Manifolds of Rank 1

The first example of non-Euclidean geometry is

connected with the Riemann symmetric manifolds of

rank 1 – hyperbolic spaces; X = SO(1, n)=O(n) is the

hyperbolic space of dimension n. It can be realized

as the upper sheet of the two-sheeted hyperboloid:

 x

x

¼1; x

> 0

Pseudoorthogonal linear transformations from

SO(1, n) preserve this surface; they play the role of

hyperbolic motions. The equivalent realizati on is in

the real ball x

x

< 1 relative to the

projective transformati ons preserving this ball.

Another example of a Riemann symmetric mani-

fold of rank 1 is the complex hyperbolic symmetric

space X = SU(1; n)=U(n). It can similarly be realized

either as the hyperboloid

jz

jz

¼1

in C

nþ1

relative to pseudounitary linear transforma-

tions or as the complex ball jz

þþjz

< 1

relative to complex projective transformations pre-

serving it. There are also quaternionic hyperbolic

spaces which are realized as the quaternionic balls in

the quaternionic projective spaces. These three series

exhaust all classical Riemann symmet ric manifolds

of rank 1 of noncompact type. There is only one

exceptional symmetric manifold of rank 1: it has the

dimension 16 and can be interpreted as a two-

dimensional ball for Cayley number s.

Classical Symmetric Domains in C

(Cartan Domains)

Riemann symmetric manifolds of noncompact type

which admit an invariant complex structure also

have an invariant Hermitian form corresponding to

the Riemann metrics. For this reason, we will call

them noncompact Hermitian symmetric manifolds

(we considered above the compact Hermitian sym-

metric manifolds). They are Stein manifolds, but as

opposed to symmetric Stein manifolds, which we

considered above, they are homogeneous relative to

real groups. The condition for a Riemann symmetric

Classical Groups and Homogeneous Spaces 505

manifold X = G=K to be Hermitian is that K has an

one-dimensional center. All Hermitia n symmetric

manifolds of noncompact type can be realized as

bounded domains in C

(but, of course, not all their

holomorphic automorphisms extend in C

). In the

case of classical manifolds, these domains are called

Cartan’s domains: Cartan gave their explicit matrix

realizations.

The nature of groups of holomorphic automorph-

isms of symmetric domains X = G=K  C

explained by Cartan’s duality. Each such domain

(Hermitian symmetric mani fold of noncompact

type) admits an embedding in a Hermitian sym-

metric manifold of compact type X

such that the

complexification G

of G is the group of holo-

morphic automorphisms of X

(correspondingly,

D is an open G-orbit on X

). Moreover, X lies

inside a (Zariski open ) coordinate chart C

, which

is an orbit of a parabolic subgroup.

The simplest example is the complex ball CB

(complex hyperbolic space) imbedded in the com-

plex projective space CP

. The affine chart C

is the

orbit of the parabolic subgroup of affine transfor-

mations. Let us consider more complicated

examples.

Let X

be the Grassmannian Gr

(k; n), q = n 

k  p; we will use matrix homogeneous coordinates

Z – k  n matrices – for the description of the

symmetric domain. Then G

= SL(n; C). Let us take

its real form G = SU( k; q), k þ q = n. We fix a

Hermitian form H of the signature (k, q) and realize

G as the group of matrices preserving H:

gHg



¼H

Then X = X

k, q

= SU(k, q)=S(U(k)  U(q)) can be rea-

lized as the domain in the Grassmannian

ZHZ



0

so that this Hermitian matrix of order k must be

positive. It is essential that this condition is invariant

relative to multiplications of Z on nondegenerate

matrices u on the left and, therefore, it is a well-

defined condition in homogeneous coordinates.

Let us specify the choice of H:

0 E



Then the corresponding domain X

is defined in

inhomogeneous coordinates Z = (E

, z), z 2 Mat(k, q),

by the condition

zz



0

This matrix ball lies completely in the coordinate

chart C

. Its rank is equal to min (k, q). Thus, we

have the real ization of this Hermitian symmetric

space as a bounded domain in C

, N = kq. In the

case k = 1, we have the usual (scalar) complex ball.

Let us remark that the edge of the boundary

(Shilov’s boundary) is the compact symmetric space



¼ E

with the group of automorphisms S(U(k)  U(q))

(the isotropy subgroup of X). For k = q the edge

coincides with the set of unitary matrix U(k).

Different forms H of the signature (k, q) are

linearly equivalent and they correspond to different

(biholomorphically equivalent) realizations of this

Hermitian symmetric spaces. Let us, in the beginning,

set k = q; the inhomogeneous matrix coordinates are

square matrices of order k. Let us take the form

0iE

iE



Then, in inhomogeneous matrix coordinates, we

have the domain X

ðz z



Þ0

(complex matrices with positive skew-Hermitian

parts). This domain (but not its boundary) lies in

the chart. It has the structure of the tube domain

T = R

þ iV, n = k

, corresponding to the symmetric

cone of positive Hermitian matrices (we take the

space of such matrices as a real form of C

). The

group of affine transformations of the tube domain:

z 7! uzu



þ a; u 2 GLðk; CÞ; a 2 HermðkÞ

is transitive on X

; it is the parabolic subgroup in

SU(k, q).

The biholomorphic equivalency of the realizations

of X corresponding to different H is induced by the

equivalency of these forms. We have

¼H





;¼

ﬃﬃﬃ

iE



Then the transform Z 7!Z transforms X

in X

.In

inhomogeneous coordinates it is the fractional linear

matrix transform

z 7! iðz þ iE

1

ðz iE

It is the matrix version of the classical Cayley transform.

Similarly, we can write down the inverse transform.

If q 6¼ k, then there is also an analog of the tube

realization. Let r = q  k > 0 and

0iE

iE

00E

506 Classical Groups and Homogeneous Spaces

Let us represent the inhomogeneous coordinates

as z = (E

, w, u), w 2 Mat(k), u 2 Mat(k, r). Then the

domain X

is defined by the condition

ðw w



Þuu



0

This is an ex ample of Siegel domains of the second

kind (Pyatetskii-Shapiro 1969). This domain has a

transitive group of affine transformations:

ðw; uÞ7!ðw þa þ2ub



þbb



; u þbÞ

a 2 HermðkÞ; b 2 Matðk; rÞ

ðw; uÞ7!ðcwc



; cuÞ c 2 GL ðk; CÞ

This class of symmetric domains in Grassman-

nians is called Cartan’s domains of the first class.

There are similar constructions for minimal flag

domains (compact Hermitian symmetric spaces)

with other groups. Let us consider the Lagrangian

Grassmannian Gr

(k;2k) corresponding to the

form J above. Here G

= Sp(k, C). Its real form

G = Sp(k; R) can be realized as the subgroup

of complex symplectic matrices preserving a

Hermitian form H of the signature (k, k). In other

words, we intersect the domain s from the last

example with the Lagrangian Grassmannians. We

consider the coordinate chart with inhomogeneous

coordinates – symmetric matrices z 2 Sym(k). For

we have the domain of symmetric matrices z

with the condition

z



z 0; z ¼z

This bounded realization is called Siegel’s disk. For

the real form is the group of real symplectic

matrices and X

is the domain

=z ¼

ðz 



zÞ0; z ¼z

of complex symmetric matrices with positive ima-

ginary parts; it is called Siegel’s half-plane. This is

the third class of Cartan’s domains. There are Siegel

domains of second kind connecting with the cones

of positive symmetric matrices; some of them are

homogeneous, but they are never symmetric.

There are two more series of classical minimal flag

manifolds: the isotropic Grassmannians and quadrics.

They both contain the dual bounded symmetric

domains (Cartan’s domains of second and fourth

classes correspondingly). Some of these domains in

the isotropic Grassmannians admit the realizations as

tubes with the cone of positive Hermitian quaternionic

matrices and others as Siegel domains of the second

kind corresponding to the same cones.

Symmetric domains in quadrics can be realized as

tube domains with the Lorentzian (light) cones.

The corresponding tubes are called the future (past)

tube, depending on which light cone was taken.

Let us consider this construction. The group of

holomorphic automorphisms of these domains is

G = SO(2; n) – the conformal extension of the

Lorentz group. To realize this group, let us fix a

real symmetric matrix Q of signature (2, n) and the

group is the group of linear transformations preser-

ving simultaneously the quadratic symmetric and

Hermitian forms with this matrix Q:

Qg ¼Q; g



Qg ¼Q

The standard realization corresponds to the diagonal

matrix Q with the diagonal (1, 1, 1, ..., 1).

Cartan’s domains of the fourth class are connected

components of the manifold

ZQZ

¼0; ZQZ



> 0

where rows Z are homogeneous coordinates in the

projective space CP

nþ1

. In other words , we consider

a domain on the quadric in the projective space

(which is the complex flat conformal space CC

For the standard Q the domain will lie in the

coordinate chart; thus it is the bounded realization.

For the tube realization, we take

Q ¼

01 0

10 0

00E

Let Z = (z

, z

, w

, ..., w

), w = u þiv, q(s , t) = s



s

and we consider the affine

chart C

nþ1

= {z

= 1}. We have

ZQZ

¼2z

þ qðw; wÞ¼0

ZQZ



¼2<z

þ qðw;



wÞ> 0

The first condition gives 2<z

= q(v, v)  q(u, u) and

then the second condition gives the final description

of the considered set in C

qðv; vÞ¼v

v

 v

> 0; w ¼u þiv

as the union of the future and the past tubes



= {v

00}). The edge R

of these tubes (v = 0)

has the structure of the Minkowski space correspond-

ing to the form q. The parabolic subgroup is the affine

conformal group of the Minkowski space. It includes

the Poincare´ group and is transitive on tubes. The

complete group of holomorphic automorphisms of

tubes G = SO(2, n) is the group of all (not only affine)

conformal transformations of the Minkowski space.

The complete edge of these symmetric domains in the

quadric CC

is the conformal compactification of the

Minkowski space (a compact symmetric R-space with

the compact group S(O(2)  O(n)) on which the

noncompact group SO(2, n)alsoacts).

Classical Groups and Homogeneous Spaces 507

In addition to four Cartan’s classes of classical

domains there are two exceptional symmetric

domains in the dimensions 27 and 16 (dual to two

exceptional compact Hermitian symmetric spaces of

these dimensio ns). The first of them can be realized

as the tube domain corresponding to the exceptional

cone of positive Hermitian matrices with Cayley

numbers of order 3 (the dimension 27) and another

can be realized as a Siegel domain of the second

kind associated with the eight-dimensional future

tube. It is possible, using  -function of self-dual

homogeneous cones, to write explicit Bergman and

Cauchy–Szego integral formulas.

Noncompact Symmetric R-Spaces

There are several other interesting noncompact

symmetric manifolds. Let us mention the noncom-

pact symmetric R-spaces which are real forms of

complex symmetric domains. The typical example is

the domain of real square matrices x 2 Mat(k):

xx

0

The condition is that this symmetric matrix is

positive. It is the Riemann symmetric space with

the group G = SO(k, k). It can be embedded in the

real Grassmannian Gr

(k;2k) with the matrix

homogeneous coordinates X 2 Mat

(k,2k) and the

group SL(2k; R) acting of X by right mul tiplications.

Let

0 E



and SO(k, k) be the subgroup of matrices preserving

the quadratic form I

: gI

= I

. This group will

preserve the domain XI

 0 and, in the inho-

mogeneous coordinates X = (E

, x), x 2 Mat

(k), it

will be exactly the same as the domai n above. The

group SO(k, k) acts by matrix fractional linear

transformations. This domain is the real form on

Siegel’s ball. If we replace the form on

0 E



then we realize our symmetric manifold as the

domain

x þx

0

So, the symmetric part of the matrix x must be

positive. This realization is homogeneous relative

to the linear automorphisms: x 7!axa

þ b, a 2

GL(k; R), b = b

. A similar construction exists

for rectangular matrices.

Geometry of Isomorphisms in Small Dimensions

We connected above several local isomorphisms of

complex classical groups with some geometrical

facts. Let us mention now several similar examples

for real groups. We start from isomorphisms of

symmetric cones. The cone Sym

(2) of symmetric

positive matrices of second order is (linearly)

isomorphic to the future lig ht cone L(2). The

comparison of the groups of automorphisms gives

the local isomorphism

SLð2; RÞﬃSOð1; 2Þ

This isomorphism corresponds also to the isomorph-

ism of two classical realizations of hyperbolic plane –

of Poincare´ and Klein. Let us also mention that the

isomorphism SL(2, R) ﬃ SU(1, 1) corresponds to the

holomorphic equivalency of the disk and the upper

half-plane. The isomorphism Herm

(2) = L(3) corres-

ponds to the presentation of any Hermitian matrix of

the order 2 in Pauli’s coordinates,

z ¼

t x

þix

ix

t þx



Then, det z = t

 x

. To compare the

groups of automorphisms, we receive

SLð2; CÞﬃSOð1; 3Þ

Similarly, in the quaternionic case, the isomorphism

of the cones Herm

(2, H) give s the isomorphism

SLð2; HÞﬃSOð1; 5Þ

The linear isomorphism of cones produces the

holomorphic isomorphism of corresponding tubes

and their groups of holomorphic automorphisms. So

each of these three isomorphisms gives automati-

cally one more isomorphism. Let us give it for the

first two cones:

Spð2; RÞﬃSOð2; 2Þ; SUð2; 2ÞﬃSOð2; 3Þ

We just compared the descriptions of automorph-

isms of classical tubes from above.

Considering det (x) as the quadratic form of

signature (2, 2) on Mat(2) ’ R

, we obtain

SOð2; 2ÞﬃSLð2; RÞSLð2; RÞ

Each of local isomorphisms in the complex case

has different real forms which admit some geome-

trical interpretations. We mentioned above two real

forms of the isomorphism SL(4; C) ﬃ SO(6; C). The

isomorphism for SO(2, 2) admits another interpreta-

tion in the language of Plu¨ cker’s coordinates: points

of the quadric in RP

of the signature (2, 3) can be

interpreted as (complex) lines in CP

which lie on a

Hermitian quadric of the signature (2, 2) (Gindikin

508 Classical Groups and Homogeneous Spaces

1983). The isomorphism above for the group

SL(2, H) also corresponds to Hopf’s fibering of

on complex lines over the sphere S

or the

isomorphism S

and the quaternionic projective line

. In all these cases, isomorphisms of homo-

geneous manifolds intertwine the actions of locally

isomorphic groups.

Pseudo-Riemann Symmetric Manifolds

We obtain the next broad class of homogeneous

manifolds if we preserve conditions that the group G

is a real semisimple one, the isotropy subgroup H is

involutive, but we remove the restriction that H

must be (maximal) compact. Such symmetric mani-

folds are often called semisimple pseudo-Riemann

symmetric manifolds (since there are also pseudo-

Riemann symmetric manifolds whose groups are not

semisimple). This class of spaces contains symmetric

Stein manifolds X

= G

. Each semisimple

symmetric manifold X = G=H admits complexifica-

tion as a symmetric Stein manifold. Each real

semisimple Lie group G is symmetric relative to

the group G  G.

The simplest family of semisimple symmetric

manifolds is the family of all hyperboloids of all

signatures

p;q

¼fx

þ  þ x

x

pþ1

  x

¼1g

with the groups SO(p, q). Their complexifications

are complex hyperboloids. There are two types

of Riemann manifolds in these families: compact

ones – spheres and noncompact ones – two-sheeted

hyperboloids; all others are pseudo-Riemann.

The Cartan duality holds for pseudo-Hermitian

symmetric manifolds: they are domains in compact

Hermitian symmetric manifolds (minimal flag mani-

folds) Z = G

. They are ope n orbits of real

forms G of the groups of holomorphic automorph-

isms G

. We construct examples of such manifolds

if we consider one of the above-describ ed realiza-

tions of noncompact Hermitian symmetric mani-

folds (through matrix homogeneous coordinates)

and replace the condition of positivity with the

condition that the symmetric (Hermitian) matrix in

the definition has a fixed nondegenerate signature

(i, k  i). We can call such pseudo-Hermitian sym-

metric manifold s satellites of Hermitian ones.

Correspondingly, we can consider nonconvex

tubes, for example, the set T of such symmetric

matrices whose imaginary parts have the signature

(i, n  i). This domain is linear homogeneous, but it

is not symmetric; to receive the symmetric manifold

we need to extend the nonconvex tube by a

manifold of smaller dimension (which plays a role

of infinity).

There are pseudo-Hermitian symmetric manifolds

which are not satellites of Hermitian ones. Let us

give an interesting example. The group SL(2p, R)

has two open orbits on the Grassmannian

(p;2p) which are both pseudo-Hermitian sym-

metric spaces. Let us consider as above the Stiefel

coordinates Z 2 Mat

(p,2p) and let Z = X þ iY.

Then the orbits are defined by the conditions

det



In the intersection with the coordinate chart

Z = (E, z), z 2 Mat

(p), z = x þ iy, we have the

conditions

det y00

Therefore, we obtain (nonconvex) tube domains in

= Mat

(p), N = p

, correspon ding to nonco nvex

homogeneous cones V



of real matrices with

positive (negative) determinants. These tubes do

not coincide with the symmetric manifolds which

include also some sets of small dimensions outside of

the coordinate chart (on ‘‘infinity’’). There are other

homogeneous nonconvex cones such that corre-

sponding tube domains are Zariski open parts of

pseudo-Hermitian symmetric spaces (D’Atri and

Gindikin 1993). Between these cones are cones of

nondegenerate skew-symmetric matrices, of skew-

Hermitian quaternionic matrices. We again observe

strong connections with classical mathematic s. Not

all pseudo-Hermitian symmetric manifolds admit

such tube realizations of dense parts. Analysis in

pseudo-Hermitian symmetric manifolds is very

interesting: we consider there instead of holo-

morphic functions



@-cohomology of some degree.

Geometric relations between different symmetric

manifolds are usually important for analytic applica-

tions since they can produce some nontrivial integral

transformations. In a broad sense, such transforms are

considered in integral geometry (Gelfand et al. 2003).

An important example is duality between some

compact Hermitian symmetric manifolds (when points

in one of them are interpreted as submanifolds in

another one). The simplest example is the projective

duality between dual copies of projective spaces or,

more generally, the realization of points of Grass-

mannians as projective planes. Such a duality can

induce a duality between orbits of real forms of groups.

In a special case, it can be a duality between Hermitian

and pseudo-Hermitian symmetric manifolds.

Here is one important example. Let us consider in

the projective space CP

2k1

the domain D which in

Classical Groups and Homogeneous Spaces 509

homogeneous coordinates – rows z = (z

, z

, ..., z

)

are defined by the equation zHz



> 0, where H

is a Hermitian form of the signature (k, k), for

example,

þ  þjz

jz

k þ1

  jz

> 0

This domain is (k  1)-pseudoconcave and it con-

tains (k  1)-dimensional complex compact cycles,

namely (k  1)-dimensional planes. The manifold of

these planes is exactly the domain X in the Grass-

mannian Gr

(k;2k) (of projective (k  1)-planes)

which is the noncompact Hermitian symmetric

space – the orbit of the group SU(k, k) (see above).

This picture is the geometrical basis for a deep

analytic construction. In the domain D the spaces

of (k  1)-dimensional



@-cohomology are infinite

dimensional for some coefficients. Their integration

on (k  1)-planes (the Penrose transform) gives

sections of corresponding vector bundles on X. The

images are described by differential equations –

generalized massless equations. The basic twistor

theory corresponds to k = 2 when X is isomorphic

to four-dimensional future tub e (see above).

Similar dual realizations of Hermitian symmetric

manifolds exist only in special cases. The twistor

realization of four-dimensional future tube was

possible since the Grassmannian Gr

(2; 4) is iso-

morphic to the quadric in CP

. This does not work

for the future tubes of bigger dimensions but there is

another possibility (Gindikin 1998). Let us have the

quadric Q

n1

 CP

be defined in the homogeneous

coordinates by the equation

&ðzÞ¼ðz

ðz

  ðz

¼0

and z   is the bilinear form. As already mentioned,

the set of (nondegenerate) hyperplane sections

  z ¼0;2 C

nþ1

; & ðÞ¼1

of Q

n1

is the corresponding hyperboloid H

. Thus,

we have the duality between a flag manifold (the

quadric Q

n1

) and a symmetric Stein manifold (the

hyperboloid H

) with the same group SO(n þ 1, C);

they have different dimensions.

The group SO(1, n)hastwoorbitsonQ

n1

the r eal quadric Q

= {z 2 Q

n1

; =(z) = 0} and its

complement X = Q

n1

. Hyperplane sections

whichdonotintersectQ

(lie at X) correspond

such  2 H

that

&ð<ðzÞÞ > 0

This set has two connected components D



which

are biholomorphically equivalent to the future and

past tubes T



of the dimension n. Let us emphasize

that their group of automorphisms is SO(2, n)in

spite of the fact that this group acts neither on X

nor on H

. Such an extension of the symm etry

group is a very interesting phenomenon. It happens

for several other symmetric manifolds, but is not a

general fact. This geometrical construction gives a

possibility to construct a multidimensional version

of the Penrose transform from (n  2)-dimensional



@-cohomology with different coefficients into solu-

tions of massless equations on the future (past)

tubes.

The last duality is connected with some general

geometrical construction. We mentioned that each of

the Riemann symmetric manifolds X = G=K admits a

canonical embedding in the symmetric Stein manifold

= G

. It turns out that X has in X

a canonical

Stein neighborhood – the complex crown (X)such

that many analytic objects on X can be holomorphi-

cally extended on the crown (Gindikin 2002). For

example, all solutions of all invariant differential

equations on X (which are elliptic) admit such

holomorphic extension. In the last example, D

the crown of the Riemann symmetric space which is

defined, in H

, by the condition =() = 0, <(

) > 0.

Symmetric manifolds are distinguished from most

other homogeneous manifolds by a very rich

geometry which is a background for deep analytic

considerations. The re are several important nonsym-

metric homogeneous manifolds. We already men-

tioned flag manifol ds and Stein homogeneous

manifolds with complex semisimple Lie groups

which can be nonsymmetric. Pseudo-Riemann sym-

metric manifolds are ope n orbits of real groups on

compact Hermitian symmetric spaces. It turns out

that open orbits on other flag manifolds also

produce interesting homogeneous manifolds. Let

F = G

be a flag manifold. Flag domains are

open orbits of a real form G on F. Of course,

pseudo-Hermitian symmetric manifolds are a special

case of this construction. Let us consider a simple

example with G

= SL(3; C)andP – the triangle

group. Then points of F are pairs {a point z and a

line l passing through it}. Let G = SU(2; 1); it has

two open orbits on CP

: the complex ball D and its

complementary D

.OnF, the group G has three

open orbits (flag domains): in the first z 2 D, l is

arbitrary; in the second l  D

; in the third z 2 D

, l

intersects D. They are all 1-pseudoconcave. In one-

dimensional



@-cohomology of these flag domains

with coefficients in line bundles, are realized all

three discrete series of unitary representations of

SU(2, 1). For arbitrary semisimple Lie groups, all

discrete series of representations can also be realized



@-cohomology of flag domains. Crowns of

Riemann symmetric spaces which we just mentioned

parametrize cycles (complex compact submanifolds)

510 Classical Groups and Homogeneous Spaces

in flag domains. Some general version of the Penrose

transform connects through the integration along

cycles cohomology in flag domains with holo-

morphic solutions of some differential equations in

crowns (generalized mass less equations).

See also: Combinatorics: Overview; Compact Groups

and their Representations; Lie Groups: General Theory;

Pseudo-Riemannian Nilpotent Lie Groups; Several

Complex Variables: Compact Manifolds; Stability of

Minkowski Space; Symmetry Classes in Random Matrix

Theory; Twistor Theory: Some Applications; Twistors.

Further Reading

Akhiezer D (1990) Homogeneous complex manifolds. In: Gindikin

S and Henkin G (eds.) Several Complex Variables IV, vol. 10,

Encyclopaedia of Mathematical Science. New York: Springer.

D’Atri J and Gindikin S (1993) Siegel domain realization of

pseudo-Hermitian symmetric manifolds. Geometriae Dedicata

46: 91–126.

Faraut J and Koranyi A (1994) Analysis on Symmetric Cones.

Oxford: Oxford University Press.

Gelfand I, Gindikin S, and Graev M (2003) Selected Topics in

Integral Geometry. Providence, RI: American Mathematical

Society.

Gindikin S (1983) The complex universe of Roger Penrose.

Mathematical Intellingencer 5(1): 27–35.

Gindikin S (1998) SO(1; n)-twistors. Journal of Geometry and

Physics 26: 26–36.

Gindikin S (2002) Some remarks on complex crowns of real

symmetric spaces. Acta Mathematica Applicata 73(1–2): 95–101.

Helgasson S (1978) Differential Geometry, Lie Groups and

Symmetric Spaces. New York: Academic Press.

Helgasson S (1994) Geometric Analysis on Symmetric Spaces.

Providence, RI: American Mathematical Society.

Onishchik A and Vinberg E (1993) Lie Groups and Lie Algebras I

Foundations of Lie Theory. In: Onishchik A (ed.) Lie Groups and

Lie Algebras. Encyclopaedia of Mathematical Sciences, vol. 20.

New York: Springer.

Pyatetskii-Shapiro I (1969) Automorphic Functions and Geometry

of Classical Domains. Amsterdam: Gordon and Breach.

Classical r-Matrices, Lie Bialgebras, and Poisson Lie Groups

M A Semenov-Tian-Shansky, Steklov Institute of

Mathematics, St. Petersburg, Russia, and Universite

de Bourgogne, Dijon, France

Introduction

The notion of ‘‘classical r-matrices’’ has emerged as

a by-product of the quantum inverse scattering

method (which was developed mainly by L D

Faddeev and his team in their work at the Steklov

Mathematical Institute in Leningrad); it has given a

new insight into the study of Hamiltonian structures

associated with classical integrable systems solvable

by the classical inverse scattering method and its

generalizations. Important classification results for

classical r-matrices are due to Belavin and Drinfeld.

Based on the initial results of Sklyanin, Drinfeld

introduced the important concepts of ‘‘Poisson Lie

groups’’ and ‘‘Lie bialgebras’’ which arise as a

semiclassical approximation in the study of quan-

tum groups.

A Poisson group is a Lie group G equipped

with a Poisson bracket such that t he multiplica-

tion m : G  G ! G is a Poisson mapping. A

Poisson bracket o n G with this property is called

multiplicative. M ore explicitly, let 

, 

be the

left and right translation ope rators in C

(G)by

an element x 2 G, 

’(y) = ’(xy), 

’(y) = ’(yx).

Multiplication in G is a Poisson mapping, if for

any ’, 2 C

(G), we have

f’; gðxyÞ¼f

’; 

gðyÞþf

’; 

gðxÞ½1

Note that in general, multiplicative brackets are

neither left nor right invariant; in other words, for

fixed x translation operators 

, 

do not preserve

Poisson brackets.

Multiplicative Poisson brackets naturally arise in the

study of integrable systems which admit the so-called

‘‘zero-curvature representation.’’ The study of zero-

curvature equations, and in particular, of the Poisson

properties of the associated monodromy map, was the

main source of nontrivial examples (associated with

classical r-matrices, classical Yang–Baxter equations,

and factorizable Lie bialgebras). The special class of

multiplicative Poisson brackets encountered in this

context is closely related to factorization problems in

Lie groups (in particular, the matrix Riemann pro-

blem); these problems represent the key tools in

constructing solutions of zero-curvature equations.

The equivalent definition of Poisson Lie groups

uses the dual language of ‘‘Hopf algebras.’’ Let

A = F(G) be the commutative algebra of (smooth)

functions on a Lie group G equipped with the

standard coproduct  : A !A  A

’ðx; yÞ¼’ðxyÞ;’2 FðGÞ; x; y 2 G

as usual, we identify the (topological) tensor product

F(G)  F(G ) with F(G  G). The multiplicative

Classical r-Matrices, Lie Bialgebras, and Poisson Lie Groups 511

Poisson bracket on G equips F(G) with the structure

of a Poisson–Hopf algebra, that is

f’; g¼f’;  g½2

Equation [2] is the starting point for the study of

relations between Poisson groups and quantum

groups. Following the general philosophy of defor-

mation quantization, we can look for a deformation

of the comm utative Hopf algebra A with the

deformation germ determined by the Poisson struc-

ture on A satisfying eqn [2]. The fundamental

theorem (conjectured by Drinfeld and proved by

Etingof and Kazhdan) asserts that any Poisson

algebra associated with a Poisson group admits a

formal quantization (in the category of Hopf

algebras).

Poisson Groups and Lie Bialgebras

Let G be a Lie group with Lie algebra g equipped

with a multiplicative Poisson bracket. Any Poisson

bracket is bilinear in differentials of functions; it is

convenient to expres s it by means of right- or left-

invariant differentials. For ’ 2 F(G) set

hr’ðxÞ; Xi¼ðd=dtÞ

t¼0

’ðe

xÞ;

’ðxÞ; Xi¼ðd=dtÞ

t¼0

’ðxe

Þ;

X 2 g; r’ðxÞ; r

’ðxÞ2g



Let us define the Poisson operator  : G !

Hom(g



, g) by setting

f’; gðxÞ¼hðxÞr’ðxÞ; r i½3

For a finite-dimensional Lie algebra, we can identify

Hom(g



, g) with g  g; the skew symmetry of

Poisson bracket implies that  2 g ^ g. By an abuse

of language, the same ident ification is traditionally

used for infinite-dimensional algebras (e.g., for loop

algebras) as well. Of course, in the latter case, the

corresponding Poisson tensors are represented by

singular kernels which do not lie in the algebraic

tensor product and should be regarded as

distributions.

Multiplicativity of Poisson bracket on G implies a

functional equation for 

ðxyÞ¼ðAd x  Ad xÞðyÞþðxÞ½4

which means that  is a 1-cocycle on G (with values

in g ^ g). By setting

ðXÞ¼



t¼0

ð e

Þ; X 2 g

we conclude from eqn [4] that  : g ! g ^ g is a

1-cocycle on g, that is,

ð½X; YÞ ¼ ½X  I þ I  X;ðYÞ

½Y  I þ I  Y;ðXÞ

Equation [4] implies that (e) = 0, that is, a multi-

plicative Poisson structure is identically zero at the

unit element. Its linearization at this point induces

the structure of a Lie algebra on the cotangent space



G ’ g



; namely, for any , 

2 g



, choose ’, ’

F(G) in such a way that r

’ = , r

’

= 

, and set

½; 





¼r

f’; ’

g½5

It is easy to see that h[, 

]



, Xi= h  ^ 

, (X)i,

which proves that the bracket is well defined,

while eqn [5] implies the Jacobi ident ity.

Definition 1 Let g, g



be a pair of linear spaces set

in duality; (g, g



) is called a Lie bialgebra if both g

and g



are Lie algebras and the mapping  : g !g 

g which is dual to the commutator map [ , ]



: g







is a 1-cocycle on g.

Thus if G is a Poisson–Lie group, the pair (g, g



)is

a Lie bialgebra (called the ‘‘tangent Lie bialgebra’’ of

G). Poisson–Lie groups form a category in which the

morphisms are Lie group homomorphisms, which

are also Poisson mappings. A morphism

(g, g



) V (h, h



) in the category of Lie bialgebras is

a Lie algebra homomorphism g ! h such that the

dual map h



! g



is again a Lie algebra homo-

morphism. It is easy to see that morphisms of

Poisson groups induce morphisms of their tangent

bialgebras. The converse is also true.

Theorem 1

(i) Let (g, g



) be a Lie bialgebra, G a connected,

simply connected Lie group with Lie algebra g.

There is a unique multiplicative Poisson bracket

on G such that (g, g



) is its tangent Lie bialgebra.

(ii) Morphisms of Lie bialgebras induce Poisson

mappings of the corresponding Poisson groups.

Basically, the theorem asserts that a Poisson

tensor is uniquely restored from the infinitesimal

cocycle on the corresponding Lie algebra; moreover,

the obstruction for the Jacobi identity vanishes

globally if this is true for its infinitesimal part at

the unit element of the group.

It is important to observe that Lie bialgebras

possess a remarkable symmetry: if (g, g



) is a Lie

bialgebra, the same is true for (g



, g). Hence, the

dual group G



(which corresponds to g



) also carries

a multiplicative Poisson bracket. The duality theory

for Lie bialgebras, based on the key notion of the

Drinfeld double, is discussed in the next section.

512 Classical r-Matrices, Lie Bialgebras, and Poisson Lie Groups

Classical r-Matrices and Special

Classes of Lie Bialgebras

The genera l classification problem for Lie bialgebras

is unfeasible (e.g., classification of abelian Lie

bialgebras includes classification of all Lie algebras).

In applications, one mainly deals with important

special classes of Lie bialgebras, of which factoriz-

able Lie bialgebras are probably the most important.

In a sense, this class may be regarded as exhaustive,

since (as explained below) any Lie bialgebra is

canonically embedded into a factorizable one.

Various other special classes discussed in literature

are ‘‘coboundary bialgebras,’’ ‘‘triangular bialge-

bras,’’ and ‘‘quasitriangular bialgebras.’’

The Lie bialgebra (g, g



, ) is cal led a coboundary

bialgebra if the cobracket  is a trivial 1-cocycle on g,

that is,

ð XÞ¼½X  I þ I  X; r for all X 2 g ½6

the constant element r 2 g ^ g is called the ‘‘classical

r-matrix.’’ If g is semisimple, H

(g, V) = 0 for any

g-module V by the classical Whitehead theorem, and

hence all Lie bialgebra structures on g are of

coboundary type. The associated Lie bracket on g



is given by the formula

½; 





¼ ad



r  

 ad



r

  ½7

where we identified r 2 g ^ g with a skew-symmetric

linear operator r : g



!g. The restrictions imposed

on r by the Jacobi identity are formulated in terms

of the so-called ‘‘Yang–Baxter tensor’’ [[r, r]] 2 g ^

g ^ g, which is a quadratic expression in r. To define

it, let us mark different factors in tensor products,

for example, g  g  g, by fixed number s 1, 2, 3, ...

which indicate their place; for simplicity, we assume

that g is embedded in an associative algebra A with a

unit. The embeddings are defined as

; i

: g  g !AAA

by setting i

(XY)=XY I, and similarly

in other cases. For a 2 g  g, we put i

(a) = a

etc. Set

½½r; r ¼ ½r

; r

þ½r

; r

þ½r

; r

½8

The commu tators in the RHS are computed in the

associative algebra AAA; it is easy to check

that the result does not depend on the choice of the

embedding g ,!A.

Proposition 1 The Jacobi ident ity for [,]



is valid if

and only if [[r, r]] is ad g-invariant, that is, if

½X  I  I þ I  X  I þ I  I  X; ½½r ¼ 0

for all X 2 g

A coboundary Lie bialgebra with [[r, r]] 2 (^

is called ‘‘quasitriangular’’; it is called ‘‘triangular’’

if r satisf ies the classical Yang–Baxter equation

[[r, r]] = 0. (Both terms come from another name of

the classical Yang–Baxter equation, the ‘‘classical

triangle equation.’’)

When a Lie algebra g admits a nondegenerate

invariant inner product, the class of quasitriangular

Lie bialgebra structures on g admits an important

specialization. Let g  g



’ g  g be the natural

isomorphism induced by the inner product. Let I 2

g  g



be the canonical element; its image t 2 g  g

under this isomorphism is called the ‘‘tensor

Casimir element.’’ Clearly, t 2 (S

and, more-

over, [t

, t

] 2 (^

. When g is semisimple, the

mapping (S

!(^

: s 7![ s

, s

] is an iso-

morphism; in particular, if g is simple, both spaces

are one dimensional and generated by a tensor

Casimir (which is unique up to a scalar multiple). A

Lie bialgebra (g, r) is called factorizable if r 2 g ^ g

satisfies the mod ified classical Yang–Baxter

equation

½r; r¼c½t

; t

; c ¼ const 6¼ 0 ½9

The convenient normalization is c = 1=4 (it can be

achieved by an appropriate normalization of r).

Instead of dealing with the modified Yang–Baxter

equation, we may relax the antisymmetry condition

imposed on r. Set r



= r  (1=2) t 2 g  g. Since t

is ad g-invariant, the symmetric part of r



drops

out from the cobracket; on the other hand, one

has [[r



, r



]] = 0. Regarding r



as a linear operator,



2 Hom( g



, g), we get the following important

result:

Proposition 2 Let (g, g



) be a factorizable Lie

bialgebra.

(i) The mappings r



2 Hom( g



, g) are Lie algebra

homomorphisms; moreover , r



= r



(ii) The combined mapping

: g



! g  g : X 7!ðr

X; r



XÞ

is a Lie algebra embedding.

(iii) Any X 2 g admits a unique decomposition

X = X

 X



with (X

, X



) 2 Im i

The additive decomposition in a factorizab le Lie

bialgebra gives rise to a multiplicative factorization

problem in the associated Lie grou p. Namely, i

may

be extended to a Lie group embedding i

: G



!G 

G and any x 2 G , which is sufficiently close to the

unit element, admits a decomposition x = x

1



with (x

, x



) 2 Im i

Any Lie bialgebra (g, g



) admits a canonical

embedding into a larger Lie bialgebra (called its

Classical r-Matrices, Lie Bialgebras, and Poisson Lie Groups 513

‘‘double’’) which is already factorizable. Namely, set

d = g  g



as a linear space and equip it with the

natural inner product,

X; FðÞ; X

; F

ðÞhihi¼F; X

hiþF

; Xhi½10

Theorem 2

(i) There exists a unique structure of the Lie algebra

on d such that:(a) g, g



 d are Lie subalgebras.

(b) The inner product [10] is invariant.

(ii) Let P

, P



be the projection operators onto

g, g



 d parallel to the complementary sub-

algebra. Set r

= P

, r



= P



; then (d, r



) is a

factorizable Lie bialgebra.

(iii) The inclusion map ( g, g



) V (d, d



) is a homo-

morphism of Lie bialgebras and the dual inclusion

map (g



, g) V (d, d



) is an antihomomorphism.

Conversely, let a be a Lie algebra equipped with a

nondegenerate invariant inner product, a



 a its Lie

subalgebras such that (i) a



are isotropic with respect

to inner product, (ii) a = a

. þa



as a linear space.

The triple (a, a

, a



)iscalleda‘‘Manintriple.’’Let



be the projection operators onto a



in this

decomposition. Set r



= P



.Then(a, r



)isa

factorizable Lie bialgebra; moreover, a

and a



are

set into duality by the inner product in a and inherit

the structure of a Lie bialgebra, and a is their double.

If (g, g



) is itself a factorizable Lie bialgebra, its

double admits a simple explicit description. Set

d = g  g (direct sum of Lie algebras); let us equip

d with the inner product

hhðX; X

Þ; ðY; Y

Þii ¼ hX; YihY; Y

Let g



 d be the diagonal subalgebra; we identify



with the embedded subalgebra i



)  d.

Proposition 3

(i)(d, g



, i



)) is a Manin triple.

(ii) As a Lie algebra, d = g  g is isomorphic to the

double of g.

Key examples of factorizable Lie bialgebras are

associated with semisimple Lie algebras and their

loop algebras.

1. Let k be a compact semisimple Lie algebra: g = k

its complexification regarded as a real Lie algebra,

 2 Aut g the Cartan involution which fixes k,and

g = k  p the associated Cartan decomposition.

Fix a real split Cartan subalgebra a  p and the

associated Iwasawa decomposition g = k. þa. þn;

put s = a. þn.LetB be the complex Killing form

on g; let us equip g with the real inner product

(X, Y) = Im B(X, Y), then (g, k, s) is a Manin

triple. Hence, any compact semisimple Lie group

K carries a natural Poisson structure; its double

G = D(K) is the complex group G = K

(regarded

as a real Lie group). The associated factorization

problem in G is the Iwasawa decomposition

G = KAN, which exists globally.

2. Let g be a real split semisimple Lie algebra, h its

Cartan subalgebra, and 

a system of positive

roots. Fix an invariant inner product on g which

is positive on h, and let {e



;  2

} be the root

vectors normalized in such a way that



, e



) = 1. Let



2

R  e



Fix an orthonormal basis {H

}inh; let P



, P

be the projection operators onto n



, h in the

Bruhat decomposition g = n



. þh. þn

. The

standard Lie bialgebra structure on g is given

by the r-matrices r



= P



. In tensor

notation,



¼

2



^ e





 H

½11

Let b



= h. þn



be the opposite Borel subalge-

bras; the inner product in g sets them into

duality, and (b

, b



) is a Lie sub-bialgebra

in (g, g



). Let G be the connected, simply

connected Lie group associated with g, B



its opposite Borel subgroups which corres-

pond to b



. Let p : B



’ H be the

canonical projection. The associated factoriza-

tion problem in G, g = b

1



,(b

, b



) 2 B





, p(b

) = p(b



)

1

, is closely related to the

Bruhat decomposition; it is solvable for all g in

the open Bruhat cell B



 G.

3. Let Lg = g  C((z)) be the loop algebra of a finite

dimensional semisimple Lie algebra g, as usual we

denote the ring of formal Laurent series by C((z)).

Put L g

= g  C [[z]], Lg



= g  z

1

C[z

1

]. Fix an

invariant inner product on g and equip Lg with

the inner product

hhX; Yii ¼ Res

z¼0

hXðzÞ; YðzÞidz

Then (Lg, Lg

, Lg



) is a Manin triple. The associa-

ted classical r-matrix is called ‘‘rational r-matrix’’; in

tensor notation, it is represented by a singular kernel

rðz; z

Þ¼

z  z

where t 2 g  g is the tensor Casimir, which is

essentially the Cauchy kernel.

4. Let us assume that g = sl(n); in this case, the loop

algebra Lg admits a nontrivial decomposition

514 Classical r-Matrices, Lie Bialgebras, and Poisson Lie Groups