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are even functions of x, so the ambiguity in the

choice of sign in the eigenvalues plays no role. This

defines characteristic classes

ðVÞ2H

ðM; C Þ and

ðVÞ2H

ðM; C Þ

Summary of Formulas

We summarize below some of the formulas in terms

of characteristic classes:

1. c

() =

ﬃﬃﬃﬃﬃﬃﬃ

1

tr()

2

2. c

() =

8

{tr(

)  tr()

3. p

() = 

8

tr(

4. ch(V) = k þ c

 2c

þ



(V),

5. td(V)= 1 þ

þc

)

þ



(V),

A(V) = 1 

 4p

5760

þ



(V),

7. L(V) = 1 þ

 p

þ



(V),

8. td(V  W) = td(V)td(W),

A(V  W) =

A(V)

A(W),

10. L(V  W) = L(V)L(W).

The Euler Form

So far, this article has dealt with the structure groups

O(k) in the real setting and U(k) in the complex

setting. There is one final characteristic class which

arises from the structure group SO(k). Suppose k = 2n

is even. While a real antisymmetric matrix A of shape

2n  2n cannot be diagonalized, it can be put in block

off 2-diagonal form with blocks,

0 









The top Pontrjagin class p

(A) = x

x

is a perfect

square. The Euler class

ðAÞ :¼ x

x

is the square root of p

.IfV is an oriented vector

bundle of dimension 2n, then

ðVÞ2H

ðM; C Þ

is a well-defined characteristic class satisfying

(V)

= p

(V).

If V is the underlying real oriented vector bundle

of a complex vector bundle W,

ðVÞ¼c

ðWÞ

If M is an even-dimensional manifold, let e

(M):=

(TM). If we reverse the local orientation of M,

then e

(M) changes sign. Consequently, e

(M)isa

measure rather than an m-form; we can use the

Riemannian measure on M to regard e

(M)asa

scalar. Let R

ijkl

be the components of the curvature of

the Levi-Civita connection with respect to some local

orthonormal frame field; we adopt the convention

that R

1221

= 1 on the standard sphere S

in R

.If

I,J

:= (e

, e

) is the totally antisymmetric tensor, then

:¼

I; J

I;J

R

m1

ð8Þ

Let R := R

ijji

and 

:= R

ikkj

be the scalar curvature

and the Ricci tensor, respectively. Then

4

32

ðR

 4jj

þjRj

Characteristic Classes of Principal

Bundles

Let g be the Lie algebra of a compact Lie group G.

Let  : P!M be a principal G bundle over M. For

 2P, let



:¼ ker 



: T



P!T



M and H



:¼V



be the vertical and horizontal distributions of the

projection , respectively. We assume that the metric

on P is chosen to be G-invariant and such that





: H



! T



M is an isometry; thus,  is a Rieman-

nian submersion. If F is a tangent vector field on M,

let HF be the corresponding vertical lift. Let 

orthogonal projection on the distribution V. The

curvature is defined by

ðF

; F

Þ¼

½HðF

Þ; HðF

Þ

the horizontal distribution H is integrable if and only if

the curvature vanishes. Since the metric is G-invariant,

(F

, F

) is invariant under the group action. We may

use a local section s to P over a contractible coordinate

chart O to split 

1

O= OG. This permits us to

identify V with TG and to regard  as a g-valued

2-form. If we replace the section s byasection

s,then

=gg

1

changes by the adjoint action of G on g.

If V is a real or complex vector bundle over M,

we can put a fiber metric on V to reduce the

structure group to the orthogonal group O(r) in the

real setting or the unitary group U(r) in the complex

setting. Let P

be the associated frame bundle. A

Riemannian connection r on V induces an invariant

splitting of TP

= VH and defines a natural

Characteristic Classes 495

metric on P

; the curvature  of the connection r

defined here agrees with the definition previously.

Let Q(G) be the algebra of all polynomials on

g which are invariant under the adjoint action. If

Q 2Q(G), then Q() is well defined. One has

dQ() = 0. Furthermore, the de Rham cohomology

class Q(P):= [Q()] is independent of the particular

connection chosen. We have

QðUðkÞÞ ¼ C½c

; ...; c



QðSUðkÞÞ ¼ C½c

; ...; c



QðOð2kÞÞ ¼ C½p

; ...; p



QðOð2k þ 1ÞÞ ¼ C½p

; ...; p



QðSOð2kÞÞ ¼ C½p

; ...; p

; e

=e

¼ p

QðSOð2k þ 1ÞÞ ¼ C½p

; ...; p



Thus, for this category of groups, no new character-

istic classes ensue. Since the invariants are Lie-

algebra theoretic in nature,

QðSpinðkÞÞ ¼ Qð SOðkÞÞ

Other groups, of course, give rise to different

characteristic rings of invariants.

Acknowledgmnts

Research of P Gilkey was partially supported by

the MPI (Leipzig, Germany), that of R Ivanova by

the UHH Seed Money Grant, and of S Nikc

evic

MM 1646 (Serbia), DAAD (Germany), and Dierks

von Zweck Stiftung (Esen, Germany ).

See also: Cohomology Theories; Gerbes in Quantum

Field theory; Instantons: Topological Aspects; K-Theory;

Mathai-Quillen Formalism; Riemann Surfaces.

Further Reading

Besse AL (1987) Einstein manifolds. Ergebnisse der Mathematik

und ihrer Grenzgebiete (3) [Results in Mathematics and

Related Areas (3)], p. 10. Berlin: Springer-Verlag.

Bott R and Tu LW (1982) Differential forms in algebraic

topology. Graduate Texts in Mathematics, p. 82. New York–

Berlin: Springer-Verlag.

Chern S (1944) A simple intrinsic proof of the Gauss–Bonnet

formula for closed Riemannian manifolds. Annals of Mathe-

matics 45: 747–752.

Chern S (1945) On the curvatura integra in a Riemannian

manifold. Annals of Mathematics 46: 674–684.

Conner PE and Floyd EE (1964) Differentiable periodic maps.

Ergebnisse der Mathematik und ihrer Grenzgebiete, N.F.,

Band 33. New York: Academic Press; Berlin–Go¨ ttingen–

Heidelberg: Springer-Verlag.

de Rham G (1950) Complexes a` automorphismes et home´omorphie

diffe´rentiable (French). Ann. Inst. Fourier Grenoble 2: 51–67.

Eguchi T, Gilkey PB, and Hanson AJ (1980) Gravitation, gauge

theories and differential geometry. Physics Reports 66: 213–393.

Eilenberg S and Steenrod N (1952) Foundations of Algebraic

Topology. Princeton, NJ: Princeton University Press.

Greub W, Halperin S, and Vanstone R (1972) Connections,

Curvature, and Cohomology. Vol. I: De Rham Cohomology

of Manifolds and Vector Bundles. Pure and Applied Mathe-

matics, vol. 47. New York–London: Academic Press.

Hirzebruch F (1956) Neue topologische Methoden in der

algebraischen Geometrie (German). Ergebnisse der Mathema-

tik und ihrer Grenzgebiete (N.F.), Heft 9. Berlin–Go¨ ttingen–

Heidelberg: Springer-Verlag.

Husemoller D (1966) Fibre Bundles. New York–London–Sydney:

McGraw-Hill.

Karoubi M (1978) K-theory. An introduction. Grundlehren der

Mathematischen Wissenschaften, Band 226. Berlin–New York:

Springer-Verlag.

Kobayashi S (1987) Differential Geometry of Complex Vector

Bundles. Publications of the Mathematical Society of Japan, 15.

Kanoˆ Memorial Lectures, 5. Princeton, NJ: Princeton University

Press; Tokyo: Iwanami Shoten.

Milnor JW and Stasheff JD (1974) Characteristic Classes. Annals

of Mathematics Studies, No. 76. Princeton, NJ: Princeton

University Press; Tokyo: University of Tokyo Press.

Steenrod NE (1962) Cohomology Operations. Lectures by NE

Steenrod written and revised by DBA Epstein. Annals of Mathe-

matics Studies, No. 50. Princeton, NJ: Princeton University Press.

Steenrod NE (1951) The Topology of Fibre Bundles. Princeton

Mathematical Series, vol. 14. Princeton, NJ: Princeton

University Press.

Stong RE (1968) Notes on Cobordism Theory. Mathematical

Notes. Princeton, NJ: Princeton University Press; Tokyo:

University of Tokyo Press.

Weyl H (1939) The Classical Groups. Their Invariants and

Representations. Princeton, NJ: Princeton University Press.

Chern–Simons Models: Rigorous Results

A N Sengupta, Louisiana State University,

Baton Rouge, LA, USA

Introduction

The relationship between topological invariants and

functional integrals from quantum Chern–Simons

theory discovered by Witten (1989) raised several

challenges for mathematicians. Most of the tremen-

dous amount of mathematical activity generated by

Witten’s discovery has been concerned primarily with

issues that arise after one has accepted the functional

integral as a formal object. This has left, as an

important challenge, the task of giving rigorous

meaning to the functional integrals themselves and to

rigorously derive their relation to topological invar-

iants. The present article will discuss efforts to put the

functional integral itself on a rigorous basis.

496 Chern–Simons Models: Rigorous Results

Chern–Simons Functional Integrals

We shall describe here the typical Chern–Simons

functional integral. For the purposes of this article,

we will confine ourselves to a simpler setting rather

than the most general possible one. In fact, we shall

work with fields over three-dimensional Euclidean

space R

(instead of a general 3-manifold).

The typical Chern–Simons functional integral is of

the form

iðk=4ÞS

ðAÞ

ðAÞ...W

ðAÞDA ½1

Our objective in this section will be to specify what

the terms in this formal integral mean. Very briefly,

the integration is with respect to a formal ‘‘Lebesgue

measure’’ on A, an infinite-dimensional space of

geometric objects A called connections over R

with

values in the Lie algebra LG of a group G . In the

first term in the integrand, in the exponent, k is a

real number, and S

(A) is the Chern–Simons action

for the connection A. Each term W

(A)isa

Wilson loop observable, the trace in some represen-

tation R

of the holonomy of the connection A

around the loop C

. The entire integral, formal

though it may be, provides an invar iant associated

with the system of loops C

, ..., C

Let G be a compact Lie group; for ease of

exposition, let us take G to be a closed, connected

subgroup of U(n). Thus, each element of G is an

n  n complex matrix g with g



g = I, the identity.

The Lie algebra LG consists of all n  n matrices A

which are skew-Hermitian, that is, satisfy A



= A,

and for which e

2 G for all real numbers t.OnLG

there is a convenient inner product g iven by

hA; Bi¼trðAB



This inner product is invariant under the conjuga-

tion action of the group G on its Lie algebra LG.

By a connection over R

we shall mean a C

1-form with values in LG. The set of all connections

is an affine (in our case, actually a linear) space A.If

A 2A, then define

ðAÞ¼

trðA ^ dA þ

A ^ A ^ AÞ½2

This is, up to constant multiple, the Chern–Simons

action functional.

Let A be a connection and consider a piecewise

smooth path

C : ½0; 1!R

With this one can associate a G-valued path [0,1] !

G : t 7!g(t) 2 G satisfying the differe ntial equation

ðtÞgðtÞ

1

¼AðC

ðtÞÞ

subject to the initial condition g(0) = I, the identity.

The path t 7!g (t) describes parallel transport along C

by the connection A.IfC is a loop then the final value

g(1) is the holonomy of A around C.IfR is a repre-

sentation of G on some finite-dimensional vector space

then the trace of R(g(1)) is the Wilson loop observable:

C;R

ðAÞ¼trðRðgð1ÞÞÞ ½3

Thus, we have specified the meaning of the terms

appearing in the formal integral [1], where

, ..., C

of eqn [1] form a link (a family of

nonintersecting, imbedded loops) in R

and

, ..., R

are finite-dimensional representations of

G. Witten showed that, at least for suitable values of

k, integrals of this form oug ht to produce topologi-

cal invariants, which he identified, for the link.

The integral [1] is problematic for several reasons.

First, there is no reasonable and useful analog of

Lebesgue measure on an infinite-dimensional space.

Even if one were to regularize this measure in some

simple way, one would run into the problem that the

measure would not live on the space of smooth

connections, and so the integrand would become

meaningless.

There are several different approaches to a

mathematical interpretation of [1]. The approach

that is often taken in practice is to simply ignore the

analytical problem and define the value of the

integral [1] to be what Witten’s calculations have

given. One approach, used, for instance, by Bar-

Natan (1995) is to expand the integrand in a series

and relate each individual integral in this expansion

separately to topological invariants. Discrete

approximation procedures to the continuum integral

have also been explored. In the abelian case, infinite-

dimensional oscillatory integral techniques have

been used to understand the functional integral.

Fro¨ hli ch and King (1999) showed the possibility of

interpreting parallel transport using ideas from

stochastic differential equations. Such an approach

has been used successfully in the case of two-

dimensional Yang–Mills theory, where the func-

tional integral actually corresponds to integration

with respect to a measure. In this article, we focus

on a method of understanding the normalized

Chern–Simons functional integral in terms

of infinite-dimensional distribution theory and

examining some ideas for understanding Wilson

loop expectation values in this setting.

Infinite Dimensional Distributions

Let (x

, x

) denote the usual coordinates on R

Gauge symmetry, an issue which will not be

examined here, may be used to simplify the problem

of the Chern–Simons integral. In particular, one

Chern–Simons Models: Rigorous Results 497

need only focus on connections which vanish in the

-direction, that is, connections of the form

A = A

þ A

. For such A , the triple wedge-

product term in the Chern–Simons action disap-

pears, and we are left with the quadratic expression:

ðAÞ¼

trðA ^ dAÞ½4

This is good, since the functional integral now

involves a quadratic exponent and so stands a good

chance of rigorous realization, just as Gaussian

measure can be given rigorous meaning in infinite

dimensions. However, in the Chern–Simons situa-

tion, there is no hope of actually getting a measure,

not even a complex measure.

The next best thing to a measure is a distribution

or ‘‘generalized function.’’ A distribution over a space

Y is a continuous linear functional on a topological

vector space of functions on Y. Thus, the objective is

to realize the Chern–Simons functional integral as a

continuous linear functional on some space of test

functions over A (more precisely, on an extension of

A). Before turning to the specific case of the Chern–

Simons integral, let us examine some elements of the

theory of infinite-dimensional distributions, in as

much as they are relevant to our needs.

Let us consider a Hilbert space E

, and a positive

Hilbert–Schmidt operator T on E

. For each integer

p  0, let E

= T

), which is a Hilbert space with

the inner product hx, yi

= hT

p

x, T

p

yi. Then we

have the chain of inclusions

E¼

p1

E

E

½5

with each inclusion E

pþ1

being Hilbert–

Schmidt. Let E

p

= E

be the topological dual of E

the space of continuous linear functionals on E

, and

let E

be the topological dual of E, where the latter is

given the topology generated by all the norms kk

Then we have the inclusions

’E

E

1

E

2

E

[

p0

p

½6

For each x 2E there is the evaluation map

x : E

!R :  7!(x). A very special case of a general

theorem of Minlos guarantees that on the dual E

there

is a measure  on the sigma algeba generated by all the

functions

x such that each

x is a Gaussian random

variable of mean zero and variance jx j

,thatis,

d ¼ e

t

jxj

for all x 2E and t 2 R. This measure  is the

standard Gaussian measure on E

for the infini te-

dimensional nuclear space E.

The inner products h, i

give rise to a nuclear space

structure on function spaces over E.LetU be the

algebra of functions on E

generated by the exponen-

tials e



,withx running over E and  over C. For each

p  0, there is an inner product hh, ii

on U such that



x

jxj

; e



y

jyj

DEDE

¼ e





hx;yi

½7

For p = 0 the left-hand side coincides with the L

()

inner product. Let [E]

be the Hilbert space

completion of U in the hh, ii

inner product. Then

½E

½E

¼ L

ðE

;Þ½8

Let [E] = \

p0

[E]

, equipped with topology from all

the norms kk

, and [E]

its topological dual.

Elements of [E]

, being continuous linear functionals

on the ‘‘test function space’’ [E], are called distribu-

tions over E, in the language of white-noise analysis.

A fundamental tool in the study of infinite-

dimensional distributions is the S-transform. This

generalizes the traditional Segal–Bargmann trans-

form from the L

-setting to the context of distribu-

tions. Let E

be the complexification of E. The inner

product h, i

on E extends to a complex-bilinear

pairing E

E

!C :(z, w) 7!z  w. The evaluation

pairing E

E!R also extends naturally to the

complexifications. For  a distribution belonging to

[E]

, define a function S on E by

SðzÞ¼ðc

for all z 2E

.Herec

is the coherent state function on

given by c

() = e

(z)(1=2)zz

. A fundamental and

useful result in white-noise analysis, due originally to

Potthoff and Streit, specifies the range of the transform

S and allows reconstruction of a distribution  from

the function S. Briefly, the range of S consists of

functions which are holomorphic, in an appropriate

sense, and have at most quadratic exponential growth.

In particular, this theorem implies that a function of the

form z 7!e

azz

, for any constant a,isintherangeof.

Rigorous Realization of Chern–Simons

Integrals

We return to the Chern–Simons context. As men-

tioned earlier, gauge symmetry may be invoked to

reduce the space of connections to the smaller space:

E¼X  X ½9

where X = S(R

)  LG is the space of rapidly

decreasing functions with values in the Lie algebra

LG. Let

¼

1

498 Chern–Simons Models: Rigorous Results

as a linear operator on L

), T

= T

3

 I the

induced operator on L

)  LG, and T = T

 T

Then, as described in the preceding section, we have

the space E and its dual E

. There is then the

standard Gaussian measure  on E

, and the space

[E]

of distributions over E

The normalized Chern–Simons integral may be

viewed as a linear functional



: F 7!

iðk=4ÞS

ðAÞ

FðAÞ DA½10

where N is a ‘‘normalizing’’ factor. Rigorous mean-

ing can be given to this by first formally working out

what the S-transform of 

ought to be. Calcula-

tion shows that S is indeed a holomorphic function

on E

of qua dratic growth. The Potthoff–Streit

theorem then implies that 

does exist as a

distribution in the space [E]

. Let us examine this

in some more detail.

As before, we take A to be of the form

A = A

þ A

, with the component A

equal

to 0. Integration by parts shows that

4

ðAÞ¼

2

trðA

Þdvol ½11

A formal computation reveals that S(

)(j) should

be given by

exp

2i

tr j

1





½12

where j = (j

, j

), and

1

f ðxÞ¼

ds½1

ð1;x



ðsÞ1

½x

;1Þ

ðsÞ fðx

; x

; sÞ

The Potthoff–Streit criterion implies the existence of

a distribution 

, whose S-transform is given by the

above expression.

The distribution 

is, however, not a suff i-

ciently powerful object to allow determination of

the Wilson loop expectatio ns that one would really

like to have. For instance, 

does not live on the

space of smooth connections and so the meaning of

parallel transport needs to be defined. The state of

knowledge, at the rigorous level, at this point is still

evolving, with progres s reported by A. Hahn. We

describe some ideas for the Wilson loop expecta-

tions in the following.

The strategy for defining parallel transport along

a path is to smear out the path by means of bump

functions and essentially replace the path by a path

of test functions in E. The description g iven here is

mainly for the case of abelian G. Choose first a C

non-negative bump function on R

, vanishing

outside the unit ball and having L

norm equal to 1.

For >0, let



be the scaled bump function given



(x) = 

3

(x=). Next, for a smooth loop

[0, 1] !l(t) = (l

(t), l

(t)), let l



(t) =



( l(t)),

the scaled bump function centered now at the path

point l(t). Now consider a generalized connection

A = (A

, A

) 2E

.Set



ðtÞ¼A

ðl



ðtÞÞ l

ðtÞ

þ A

ðl



ðtÞÞ l

ðtÞ

½13

The equation of parallel transport can be reformu-

lated as a differential equation for a matrix-valued

path t 7!P



(t) satisfying



ðtÞþB



ðtÞP



ðtÞ¼0 ½14

and the initial condition P



(t) = I. With this smear-

ing, one can consider functions of the form



ðL; AÞ¼

i¼1

trðP



ðAÞÞ ½15

for a link L consisting of loops l

, ..., l

, instead of

the classical Wilson loop variable.

At this stage, it woul d be natural to consider

taking #0in(W



(L)). However, this is still

problematic. A further regularization is needed,

roughly corresponding to the geometric notion of

framing. In the definition of 

, alteration is made

to the quadratic form Q(j, j) in the exponent which

appears in the expression for S( 

), replacing it

with Q( j, 



j), where {

}

s>0

is a family of suitable

diffeomorphisms of R

, with 

being the identity.

In a sense, this splits a single loop l into l and a

neighboring loop 

 l. At the end, one has to take

s#0. The resulting limiting value is the expected

link-invariant. We shall not go into the case of

nonabelian G, which is more complex, for which

work continues to be in progress.

Infinite-dimensional distributions can be used to

formulate a rigorous theory for normalized Chern–

Simons functional integrals. The more specific ques-

tions raised by the Wilson-loop integrals in this setting

opens up new problems for further developments in

the distribution theory, connecting geometry, topol-

ogy, and infinite-dimensional analysis.

Acknowledgments

This research is supported by US NSF grant DMS-

0201683.

See also: BF Theories; Feynman Path Integrals;

Fractional Quantum Hall Effect; Knot Theory and

Physics; Large-N and Topological Strings; Large-N

Dualities; Quantum 3-Manifold Invariants; Quantum Hall

Effect; Spin Foams; String Field Theory; Topological

Quantum Field Theory: Overview; Twistor Theory: Some

Applications.

Chern–Simons Models: Rigorous Results 499

Further Reading

Albeverio S, Hahn A, and Sengupta AN (2003) Chern–Simons

theory, Hida distributions, and state models. Infinite Dimen-

sional Analysis Quantum Probability and Related Topics

6: 65–81.

Albeveri o S and Scha¨fer J (1994) Abelian Chern–Simons

theory and linking n umbers via oscillatory integrals.

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Classical Groups and Homogeneous Spaces

S Gindikin, Rutgers University, Piscataway, NJ, USA

Classical groups are Lie groups corresponding to

three classical geometries – linear, metric, and

symplectic. Let us start with the complex field C.

We consider the linear space C

and the gro up

GL(n; C) of its automorphisms – nondegenerate

(invertible) linear transformations. The complex

linear metric space is the space C

endowed by a

nondegenerate symmetric bilinear form; the orthogo-

nal group O(n; C) is the subgroup in GL(n; C)of

automorphisms of this structure. If, for n = 2l,we

replace the symmetric form by a nondegenerate skew-

symmetric form, we obtain the linear symplectic

space and the group Sp(l; C) of its automorphisms –

the symplectic group.

A fundamental observation of nineteenth century

geometry was that the transfer from the complex

field to the real one, gives not only three corres-

ponding groups for R but a much reacher collection

of real forms of complex classical groups: unita ry,

pseudounitary, pseudoorthog onal, etc. (see below).

Classical geometries correspond to homogeneous

manifolds with classical groups of transformations.

Geometers understood that this produces a very

reach world of non-Euclidean geometries, including

the first example of non-Euclidean geometry –

hyperbolic geometry. Some classical algebraic the-

ories through such an approach obtain a geometrical

interpretation (see below the consideration of the

cone of symmetric positive form s). Between classical

manifolds there are Minkowski space, Grassman-

nians, and multidimensional analogs of the disk and

the half-plane. A substantial part of this theory is a

matrix geometry, which serves as a background for

matrix analysis. A ric h geometry on classical

manifolds with many symmetries is a background

for a rich multidimensional analysis with many

explicit formulas. Classical geometries, starting with

Minkowski geometry, have appeared in some

problems of mathematical physics.

A crucial technical fact is the embedding of the

classical groups in the class of semisimple Lie groups;

it gives a very strong unified method to work with

semisimple groups and corresponding geometries – the

method of roots. Nevertheless, some special realiza-

tions and constructions for classical groups can also be

very useful. A very impressive example is the twistors

of Penrose, where an initial construction is the

realization of points of four-dimensional Minkowski

space as lines in three-dimensional complex projective

space. We mention below some general facts about

semisimple groups and homogeneous manifolds, but

the focus will be on special possibilities for the classical

groups. The class of simple Lie groups contains,

besides the classical groups, only a finite number of

exceptional groups which are also very interesting and

are connected, in particular, with noncommutative

and nonassociative geometries; they have applications

to mathematical physics.

500 Classical Groups and Homogeneous Spaces

Complex Groups and Homogeneous

Manifolds

Complex Classical Groups

The complete linear group GL(n; C) is the group of

nongenerate matrices g of order n (det g 6¼ 0) and the

special linear group SL(n; C) is its subgroup of

matrices with the determinant equal 1 (unimodular

condition). The unimodular condition kills the one-

dimensional center, perhaps, leaving only a finite

center. We realize the direct products of several copies

of complete linear groups with different dimensions,

for example, GL(k; C)  GL(l; C), as the groups of the

blockdiagonal nondegenerate matrices. The letter S

always means that we take matrices with determinant

1. So the notation S(L(k; C)  L(l; C)) means that we

take blockdiagonal matrices with blocks of sizes k, l

and with the determinant 1.

Let I be a nondegenerate symmetric matrix of

order n; then the orthogonal group O(n; C) is the

subgroup in GL(n; C) of matrices preserving the

corresponding symmetric form so that

Ig ¼I

These matrices can have the determinant 1. The

special orthogonal group SO(n; C) is the subgroup

of orthogonal mat rices with determinant 1. Differ-

ent I’s give isomorphic orthogonal groups since they

are all linearly equivalent. If we take as I the unit

matrix E = E

, then we receive the group of

orthogonal matrices in the classi cal sense: g

g = E.

If n = 2l and we replace in this definition the

symmetric matrix I by a nondegenerate skew-

symmetric matrix J, we obtain the symplectic

group Sp(l; C). Again, different J’s give isomorphic

groups. The typical example of J is

J ¼

0 E

E



It is convenient then to represent matrices g as

g ¼



where the blocks A, B, C, D are matrices of order l.

Then the symplectic condition is that A

D 

A = E and matrices A

C and D

B are symmetric.

If C = 0 then D = (A

)

1

and A

1

B is a symmetric

matrix. In this way, we have in Sp(l; C) a subgroup

P of blocktriangular matrices of a very simple

structure; it is an example of subgroups which are

called parabolic.

There are two principal classes of homogeneous

spaces with complex semisimple Lie groups: flag

manifolds and Stein manifolds.

Flag Manifolds

These homogeneous spaces F = G=P with semi-

simple (in our case with classical) groups G have

parabolic subgroups P as the isotropy subgroups.

The group G = GL(n; C) transitively acts on the

flag manifolds F(n

, ..., n

), 0 < n

< < n

< n,

whose elements are (n

, ..., n

)-flags – sequences of

embedded subspaces in C

of the dimensions

, ..., n

). The isotropy subgroup P = P (n

, ..., n

)

is the subgroup of blocktriangle matrices with the

diagonal blocks of sizes k

, ..., k

rþ1

, k

= (n



j1

), n

= 0, n

rþ1

= n. The flag manifolds are com-

pact complex manifolds. The matrices proportional

to the unit matrix E

act trivially and we can

consider instead of the action of G = GL(n; C) the

transitive action of G = SL(n; C).

Let us pay particular attention to two extremal

cases. The first one is the case of the maximal

flag manifold when we have the sequence of

all integers (1, 2, 3, ..., n  1) – complete flags; the

subgroup P in this case is called Borelian. Another

case is minimal flag manifolds with r = 1 (for them

the unipotent radical of the parabolic subgroups is

commutative). Then in the case of SL(n; C) the

sequence has only one element n

= k < n and we

have Grassmannian manifolds Gr

(k; n) = F(k)of

k-dimensional subspaces in C

.Ifk = 1ork = n  1,

we obtain the dual realizations of the complex

projective space CP

n1

. We can interpret points

of Gr

(k; n)alsoas(k  1)-dimensional planes in

n1

We can define points of the projective space

n1

by homogeneous coordi nates – as the

equivalency classes (z  cz, z 2 C

n {0}, c 2 C n 0).

For the Grassmannians we can similarly use matrix

homogeneous coordinates (Stiefel’s coordi nates):

classes of (k  n)-matrices Z 2 Mat(k, n)ofthe

maximal rank k relative to the equivalency

Z  uZ; u 2 GLðk; CÞ

The rows of a matrix Z correspond to a base in

subspace with the homogeneous coordinate Z; the

left multiplication on a matrix u replaces this base,

but does not change the subspace. The group

GL(n; C) acts by right multiplications:

Z 7!Zg

and this action preserves the equivalency classes.

Suppose k  n  k and the left k-minor of Z is not

zero. Such matrices give the dense coordinate chart

k(nk)

: we can pick in the equivalency classes the

representatives (E

, z), z 2 Mat(k, n  k), and con-

sider the matrices z as (inhomogeneous) local

coordinates. In the inhomogeneous coordinates the

Classical Groups and Homogeneous Spaces 501

action of the group has a matrix fractional linear

form: let

g ¼



A 2 MatðkÞ; D 2 Mat ðn  kÞ;

B 2 Matðk; n  kÞ; C 2 Matðn  k; kÞ

Then we have the transformation in inhomogeneous

coordinates:

z 7!ðA þ zCÞ

1

ðB þ zDÞ

The condition C = 0 defines the parabolic sub-

group which has affine action in inhomogeneous

coordinates which is transitive in the coordinate

chart. In such a way the Grassmannian is a

compactification of C

k(nk)

(realized as a space of

k  (n  k) matrices). If n = 2k, we can consider it as

the compactification of the space of square matrices

z of order k with the flat generalized conformal

structure defined by translations of the isotropy cone

{det z = 0}.

There are similar constructions of flag manifolds

for other classical groups. We will consider only the

minimal flag manifolds. For O(2k; C) we consider

the isotropic Grassmannian Gr

(2k; C) of isotropic

k-subspaces relative to the symmetric form I.We

take the matrix realization of Gr

(k;2k), using

Stiefel’s homogeneous coordinates, and add the

matrix equation

ZIZ

¼0

which is well defined in the homogeneous coordi-

nates (compatible with the equivalency classes) and

defines isotropic subspaces relative to I. This matrix

cone is preserved by the subgroup O(2k; C) 

GL(2k; C) corre sponding to the matrix I .Ifwe

take the symmetric matrix

I ¼

0 E



then in inhomogeneous coordinates (z is a square

k-matrix) this equation is transformed into the

condition that the matrix z i s skew-symmetric. So,

in a natural sense, the isotropic Grassmannian is

the compactification o f the linear space of skew-

symmetric matrices Alt(k) = C

, N = k(k  1)=2.

A similar constru ction makes sense for the

symplectic group: if we replace the symmetric form

I with the skew-symmetric form J, we obtain the

equation of the matrix cone representing the

Lagrangian Grassmannian Gr

(k;2k) of Lagrangian

subspaces in 2k-dimensional linear symplectic space.

If we were to choose J as above, then in the

(inhomogeneous) coordinate chart we obtain the

condition that the matrix z is symmetric. Thus, we

have the (dense) coordinate chart on the Lagrangian

Grassmannian C

= Sym( k), N = k(k þ 1)=2 – the

linear space of symmetric matrices.

There is one more type of minimal flag manifolds

for the orthogonal group SO(n; C) – the quadric Q

in the projective space:

IðzÞ¼zIz

¼0

where rows z 2 C

n{0} represent, in homogeneous

coordinates, points in C P

n1

.IfI = E

we have the

equation (z

)

þþ(z

)

= 0. This quadric is the

complex compact conformal flat manifold

, N = n  2; it is the compactification of C

endowed with the flat conformal structure corre-

sponding to the quadratic isotropic cone. The

parabolic group is generated by linear conformal

transformations and translations. On the quadric Q

the conformal structure is defined by intersections of

tangent spaces with Q. Apparently, this structure is

invariant relative to the natural action of SO(n; C).

Classical Stein Manifolds

Such homogeneous complex manifolds X = G=H have

complex reductive isotropy subgroups H.Contraryto

the flag manifolds which are compact, these manifolds

are Stein ones and there are many holomorphic

functions on them. The typical examples for

G = GL(n; C ) are homogeneous spaces S(k

, ...,

rþ1

), n = k

þþk

rþ1

, for which the isotropy sub-

groups are blockdiagonal matrices with the blocks of

sizes k

, ..., k

rþ1

. Then points of the manifold can be

realized as generic sets of subspaces L

 C

dim L

= k

,1 j  r þ 1 or, what is equivalent, gen-

eric sets of (k

 1)-dimensional planes in CP

n1

.Since

the isotropy subgroup of such a homogeneous space is a

subgroup of the parabolic subgroup P(n

, ..., n

= n

 n

j1

, we have the natural fibering S(k

, ...,

rþ1

) ! F(n

, ..., n

) (it is simple to see this geo-

metrically: the ith subspace of a flag in the base is the

direct sum of first i subspaces representing a point in

the fiber). This is a convenient tool to apply

complex analysis on S to the compact manifold F

where there are no nontrivial holomorphic functions.

Let us emphasize that such a connection exists only

for special classes of classical Stein manifolds.

Let us pay special attention to the subclass of

symmetric Stein manifolds. For such manifolds X,the

isotropy subgroup H is fixed relative to a holomorphic

involutive automorphism of G. Complex semisimple

Lie groups G (including classical ones) are symmetric

Stein manifolds relative to the action of their square

G  G by left and right multiplications.

502 Classical Groups and Homogeneous Spaces

Classical Stein manifolds for SL(n; C) considered

above are symmetric if r = 1 and we have the

manifold of pairs of subspaces of complimentary

dimensions intersecting only on {0}. The simplest

example is the manifold of pairs of different points

of the projective line CP

. Let us point out again

that the transition to the generic pairs of points

transforms the compact complex manifol d without

nonconstant holomorphic functions into a Stein

manifold with a large collection of holomorphic

functions.

Some other examples of symmetric Stein mani-

folds are connected with classical geometry and

linear algebra. The affine hyperboloid in C

QðzÞ¼1

is a symmetric space for G = O(n; C), H = O(n  1; C).

We can compare it with the projective quadric

Q(z) = 0 which is a minimal flag manifold. Let us

remark that there is a duality here: it is possible to

interpret points of the hyperboloid of dimension n

as generic hyperplane sections of the pro jective

quadric of dimension n  1.

The space X of complex symmetric matrices of

order n with determinant 1 is symmetric for the

group SL(n; C) which acts by the changes of

variables in the corresponding quadratic forms:

z 7!g

zg; g 2 SLðn; CÞ

The transitive action reflects the possibility of

transforming such a form into a sum of squares.

The isotropy subgroup is SO(n; C).

The Stein symmetric manifold X = SO(n; C)=

S(O(k; C)  O(n  k; C)) is realized as the manifold

of k-dimensional subspaces in C

on which the

restriction of the principal symmetric form I is

nondegenerate.

Isomorphisms in Small Dimensions

Isomorphisms of classical groups in small dimen-

sions produce isomorphisms of some classical

homogeneous manifolds. Such isomorphisms were

very important in the history of geometry; below are

a few examples. We will consider local isomorph-

isms (up to a finite center). We have SL(2; C) ﬃ

SO(3; C). Let us realize C

as the space of symmetric

matrices z of order 2. Then, as we remarked above,

the two-dimensional submanifold X of matrices

with determinant 1 is the symmetric Stein manifold

for the group SL(2; C). On the other hand, we can

take det z as the quad ratic symmetric form I in C

;

then X is the hyperboloid for this form and the

action of SL(2; C) on symmetric matrices gives the

orthogonal transformations relative to this form I.

Similarly, we can interpret the local isomorphism

SO(4; C) ﬃ SL(2; C)  SL(2; C). We realize C

as the

space of square matrices z of order 2 with the

symmetric quadratic form I(z, z) = det (z). Then left

and right multiplications of z on unim odular

matrices (z 7!uzv, u, v 2 SL(2; C)) induce orthogonal

transforms for the form I and any orthogonal

transform can be represented in such a form (one

can see it by the calculation of dimensions).

The local isomorphism SL(4; C) ﬃ SO(6; C)hasa

slightly more complicated nature. Let us consider the

Grassmannian Gr

(2; 4) of lines in the projective

space CP

with 2  4 matrices Z as matrix homo-

geneous coordinates. Let p

, i < j, be the minors of Z

with ith and jth columns. They are called Plu¨cker

coordinates on Gr

(2; 4): the equivalency class of

Z is defined by the sequence of six numbers

p = (p

,1 i < j  j) 6¼ (0, ..., 0) up to a constant

factor. Thus, we have an imbedding of Gr

(2; 4) in the

projective space CP

.Theimagewillbethequadric

 p

þ p

¼0

Thus, we have the isomorphism of two flag manifolds

and the action of SL(4; C) on the Grassmannian

transforms in orthogonal transformations of four-

dimensional quadric in CP

. The Plu¨ cker coordinates

can be defined for any Grassmannian, but they do not

produce in other cases some isomorphisms with other

flag manifolds; nevertheless, they realize them as

intersections of quadrics in projective spaces.

Compact Classical

Homogeneous Manifolds

Compact classical groups U(n), SU(n), O(n), SO(n),

Sp(l) are maximal compact subgroups in the corre-

sponding classical complex groups GL(n; C), SL(n; C ),

O(n; C), SO(n; C), Sp(l; C). This condition defines

them up to an isomorphism. They are fixed subgroups

of some antiholomorphic involutive automorphisms.

The unitary groups U(n) and SU(n) are the groups

of unitary matrices (g



g = E,) correspondingly, of

unitary matrices with determinant 1. As the compact

orthogonal group we can take the intersection U(n) \

O(n; C). For the standard form I, it will be the group of

real orthogonal matrices: g

g = E (so the involution in

O(n; C) is the conjugation g 7!



g). Similarly, we can

take Sp(l) = SU(2l) \ Sp(l; C) (then the involution is

g 7!J



gJ ).

Compact classical groups act on compact homo-

geneous Riemann manifolds. There are two mech-

anisms connecting compact and complex

homogeneous manifol ds. We observe the first

possibility in the case of flag manifolds which are

Classical Groups and Homogeneous Spaces 503

compact. We considered them so far relative to the

action of complex (noncompact) groups. It turns out

that on the flag manifold F = G=P the maximal

compact subgroup U  G continues to be transitive:

so we can consider flag manifolds also as being

homogeneous with compact groups. Then F = U=C,

where C is the centralizer of a torus in U. There is a

Ka¨ hler metric on F, invariant relative to U. Thus, G

is the group of all automorphisms of F as the

complex manifold, but U is the group of its

automorphisms as the Ka¨ hler manifold. It defines

two sides of geometry of flag manifolds: complex

and Ka¨ hler. Flag manifolds are the only compact

homogeneous Ka¨ hler manifolds with semisimple Lie

groups (the class of all compact Ka¨ hler manifolds

also con tains locally flat compact manifolds –

toruses). In the example considered above we have

F(n

, ..., n

) = SU(n)=S(U(k

) U(k

)). In the lan-

guage of Stiefel (homogeneous) coordinates, we fix a

positive Hermitian form in C

and characterize

subspaces by orthonormal bases. For r = 1 we have

Grassmannians Gr

(k; n), in particular the projec-

tive space CP

n1

which we consider relative to the

action of the unitary groups. Relative to this action

they are Hermitian symmetric spaces. In the case of

minimal flag manifolds for other groups the action

of maximal compact subgroups also defines on them

the structure of compact Hermitian symmetric

spaces. Let us emphasize that relative to noncom-

pact groups of biholo morphic automorphisms G,

the minimal flag manifolds (including the Grass-

mannians) are not symmetric.

In the case of homogeneous Stein manifolds

X = G=H, the picture is different: the maximal

compact subgroups have no open orbits. There are

totally real orbits which are the compact forms of

X: X

= G

, where G

and H

are compact

forms of G and H, respectively. It is the canonical

embedding of compact homogeneous manifolds

in their complexifications. The important special

case is the embedding of compact symmetric

manifolds in the Stein symmetric manifolds – their

complexifications.

For compact symmetric manifolds X = U=K the

groups U, K are compact Lie groups and elements

of K are fixed for an involutive automorphism 

such that K contains the connected component of

the subgroup of all fixed elements of . This

possibility to connect several symmetric manifolds

with one involution is illustrated by the next

example. The sphere S

n1

 R

is the symmetric

space SO(n)=SO(n  1); the real projective space

n1

is SO(n)=O(n  1). Here SO(n  1) is the

connected component of O(n  1) and S

n1

is a

double covering of RP

n1

. A few more examples, the

real Grassmannian Gr

(k; n)ofk-subspaces in R

can be defined as SO(n)=S(O(k)  O(n  k)). This

representation corresponds to the characterization

of subspaces by orthonormal bases. The considera-

tion of arbitrary bases defines the action of the

larger group GL(n; R)onGr

(k; n). Relative to this

action, the real Grassmannian is not symmetric since

the isotropy subgroup is parabo lic and is not

involutive. Such a possibility to extend the group is

typical for a class of compact symmetric manifolds

called symmetric R-spaces. They are real forms of

Hermitian compact symmetric manifolds (minimal

flag manifolds). Let us also mention compact

symmetric spaces SU(n)=SO(n), which is the compact

form of the space of unimodular symmetric matrices

and can be presented by the submanifold of unitary

matrices in it. Also, all compact Lie groups G are

symmetric spaces relative to the action of G  G.

Noncompact Riemannian

Symmetric Manifolds

This class of symmetric manifolds has the strongest

connections with classical mathematics. Let us

consider noncompact real semisimple Lie groups –

real forms of complex semisimple Lie groups. They

correspond to antiholomorphic involutions in com-

plex groups.

Between real forms of SL(C, n) there are real and

quaternionic unimodular groups SL(R, n), SL(H, n)

and pseudounitary groups SU(p, q) of complex

matrices preserving a Hermitian form H of the

signature (p, q). The complex orthogonal group has

as real forms, in particular, pseudoorthogonal

groups SO(p, q) of real matrices preserving a

quadratic form of the signature (p, q).

Let G be a real simple Lie group and K be its

maximal compact subgroup. Then X = G=K is a

Riemann symmetric manifold of noncompact type;

K is defined by an involutive automorphism of G.

Therefore, in irreducible situation there is a corre-

spondence between noncompact Riemann sym-

metric manifolds and real simple noncompact Lie

groups. K-orbits on X are parametrized by points of

the orbit on X of a maximal abelian subgroup A –

the Cartan subgroup of the symmetric space X. Its

dimension l is the important invar iant of X – its

rank. The algebraic base for geometry of X is the

Iwasawa decomposition

G ¼KAN

where N is a maximal unipotent subgroup (in a

natural sense compatible with A). Then the para-

bolic subgroup P = AN is transitive on X.

504 Classical Groups and Homogeneous Spaces