Cohomology Theory of Topological Transformation Groups
Upload
others
View
4
Download
0
Embed Size (px)
344 x 292
429 x 357
514 x 422
599 x 487
Citation preview
Ergebnisse der Mathematik und ihrer Grenzgebiete Band 85
Herausgegeben von P. R. Halmos P. J. Hilton R. Remmert B.
Sz6kefalvi-Nagy
Unler Mitwirkung von L. V. Ahlfors R. Baer F. L. Bauer A. Dold J.
L. Doob S. Eilenberg K. W Gruenberg M. Kneser G. H. Muller M. M.
Postnikov B. Segre E. Sperner
GeschiiftifUhrender Herausgeber P. J. Hilton
WuYi Hsiang
Wu Yi Hsiang
AMS Subject Classification (1970): 57 Exx
ISBN-l3: 978-3-642-66054-2 DOl: 10.1007/978-3-642-66052-8
e-ISBN-13: 978-3-642-66052-8
Library of Congress Cataloging in Publication Data. Hsiang, Wu Vi,
1937-. Cohomology theory of topological transformation groups.
(Ergebnisse der Mathematik und ihrer Grenzgebiete; Bd. 85).
Bibliography: p. Includes index. 1. Transformation groups. 2.
Topological groups. 3. Homology theory. I. Title. II. Series.
QA613.7.H85. 514'.2. 75-5530. This work is su bject to copyright.
All rights are reserved, whether the whole or part of the material
is concerned. specifically those of translation, reprinting, re-use
of illustrations, broadcasting. reproduction by photocopying
machine or similar means. and storage in data banks. Under § 54 of
the German Copyright Law where copies are made for other than
private use, a fee is payable to the publisher, the amount of the
fee to be determined by agreement with the publisher. © by
Springer-Verlag Berlin Heidelberg 1975.
Softcover reprint of the hardcover 1 st edition 1975
Introduction
Historically, applications of algebraic topology to the study of
topological transformation groups were originated in the work of L.
E. 1. Brouwer on periodic transformations and, a little later, in
the beautiful fixed point theorem ofP. A. Smith for prime periodic
maps on homology spheres. Upon comparing the fixed point theorem of
Smith with its predecessors, the fixed point theorems of Brouwer
and Lefschetz, one finds that it is possible, at least for the case
of homology spheres, to upgrade the conclusion of mere existence
(or non-existence) to the actual determination of the homology type
of the fixed point set, if the map is assumed to be prime periodic.
The pioneer result of P. A. Smith clearly suggests a fruitful
general direction of studying topological transformation groups in
the framework of algebraic topology. Naturally, the immediate
problems following the Smith fixed point theorem are to generalize
it both in the direction of replacing the homology spheres by
spaces of more general topological types and in the direction of
replacing the group tlp by more general compact groups. It is
usually rather straightforward to deduce similar fixed point
theorems for actions of p-primary groups (or extensions of torus
groups by p-primary groups) directly from the corresponding fixed
point theorems for actions of the group 7l p • However, various
efforts to extend such fixed point theorems beyond p-pi·imary
groups (or extensions of torus by p-primary groups) all eventually
wound up with puz zling counter-examples [C 6, C 8, F 2, H 5]. On
the other hand, if the group is an elementary p-group, (i e., 7l;
or T'" if p =0), then a far-reaching generalization of the Smith
fixed point theorem, valid for all finite dimensional, locally
compact spaces, can be formulated and proved in the framework of
equivariant cohomology. (cf. Theorem (IV. i), § i, Ch. IV).
The basic setting for our approach using the cohomology theory in
compact topological transformation groups is the following
equivariant cohomology theory introduced by A. Borel [B 5]. Let G
be a compact Lie group and let X be a given G-space. Then the
equivariant cohomology of the G-space X is defined to be the
ordinary cohomology of the total space XG of the universal bundle,
X -+ XG -+ BG, with the given G-space as the typical fibre. The
reasons for adopting such an equivariant cohomology theory in terms
of the universal bundle con struction are roughly the
following:
(i) Intuitively and heuristically, the complexity of the G-action
on X will be reflected in the complexity of the associated
universal bundle, X -+ XG -+ BG; and the classical characteristic
class theory clearly demonstrates that cohomology
VI Introd uction
theory can then be used to detect the complexity of the bundle,
which, in turn, also detects the complexity of the· G-action on X
itself. Therefore, the above definition of equivariant cohomology
simply formalizes and also generalizes the classical characteristic
class theory to the study of the topology of general fibre
bundles.
(ii) From a technical standpoint, the above equivariant cohomology
theory naturally and successfully brings together the modern
theories of fibre bundles, spectral sequences and sheaves in a nice
convenient way. Hence, it not only possesses all the convenient
formal properties that one expects, but also is effec tively
computable.
Basic properties of this equivariant cohomology theory as well as
some fundamental general theorems such as the localization theorem
of Borel-Atiyah Segal type are formulated and proved in Chapter
III.
In Chapter IV, we shall proceed to investigate the relationship
between the geometric structures of a given G-space X and the
algebraic structures of its equivariant cohomology HJ(X). From the
viewpoint of transformation groups, those structures which are
usually summarized as the orbit structure are certainly the most
important geometric structures of a given G-space X. Hence, it is
almost imperative that one should investigate how much of the orbit
structure of X can actually be determined from the algebraic
structure of HJ (X). Examples of specific problems in this area
are: How much of the cohomology structure of the fixed point set,
H*(F), is determined by the algebraic structure of Ht(X)? Is it
possible to give a criterion for the existence of fixed points
purely in terms of HJ(X)? Suppose F=F(G,X)=0 (is empty), how does
one determine the set of maximal isotropy subgroups from H~(X)? In
the special case of elementary p-groups, we shall formulate various
commutative algebraic invariants of HJ(X), which are, then, proved
to be intimately related to the orbit structure of the G-space X.
No tice that there are general counter-examples for almost all
non-p-primary groups which clearly indicate the non-existence of a
general relationship between the orbit structure of X and the
algebraic structure of Hi; (X). Such a sharp contrast of behaviors
between transformations of elementary p-groups and transformations
ofnon-p-primary groups is probably one of the most profound as well
as fascinating facts in the cohomology theory of transformation
groups. In retrospect, this also explains why torus groups play
such a central role in the representation theory of compact
connected Lie groups, which, after all, is concerned with the
special case of linear transformation groups.
Methodologically, one of the central themes in the approach of this
book is that the cohomology theory of topological transformation
groups can be developed roughly along the same lines as the
classical linear representation theory of compact connected Lie
groups. A concise exposition of the theory of compact Lie groups
and their linear representations is given in Chapter II. In order
to present the theory of linear transformation groups as a
prototype of cohomology theory of topological transformation
groups, we purposely adopt a rather geometric approach, in which
the orbit structure of the adjoint action plays the central role.
Moreover, it will be clear from such an exposition that the
following two basic theorems constitute the foundation of linear
representation theory of compact connected Lie groups:
Introduction VII
(i) Structural splitting theorem for linear tori actions: every
complex linear representiation of a torus group always splits into
the sum of one-dimensional representations.
(ii) Maximal tori theorem of E. Cartan: the set of maximal tori
forms a single conjugacy class and G= U{gTg- 1 ; gEG}.
The first result classifies linear tori actions in terms of an
extremely simple invariant called the weight system and the second
result reduces the classification of linear actions of a compact
connected Lie group G to the restricted actions of its maximal
tori. Correspondingly, in the setting of the cohomology theory of
topological transformation groups, the above structural splitting
theorem for linear tori actions can be generalized into various
structural splitting theorems of the equivariant cohomology (cf.
Chapter IV), which can be considered as the generalized splitting
principle in the geometric theory of generalized charateristic
classes. Similar to the linear case, one may also combine the
structural splitting theorems with the maximal tori theorem to
define a (geometric) weight system for topological transformation
groups. Such a program is carried out explicitly for the special
cases of acyclic manifolds and cohomology spheres in Chapter V; and
for cohomology projective spaces in Chapter VI. Although the
geometric weight systems are no longer "complete invariants" for
topological transformation groups, they nevertheless determine the
cohomology aspects of orbit structures of the restricted actions of
maximal tori and, hence, also the orbit structure of the original
G-action to a great extent.
In Chapter VII, we apply the cohomology method to study
transformation groups on compact homogeneous spaces. Due to the
fact that compact homo geneous spaces encompass great varieties
oftopological types and that the study of transformation groups on
them is just getting started, there is an abundance of natural
problems in this area, but, so far, only a small number of testing
cases have been successfully settled and most of them are as yet
unpublished. Therefore, results in this chapter are rather
incomplete, and they should be considered only as beginning testing
cases that serve to indicate the existence of interesting problems
and deep results. In a paper soon to be published, I hope to give a
more systematic account of the applications of the cohomology
method to the study of trans formation groups on compact
homogeneous spaces.
This book is based on a course given at the University of
California, Berkeley. I am indebted to the participants of that
course, especially to Dr. T. Skjelbred who helped to prepare a
preliminary draft of Chapters III and IV.
Berkeley, in Spring 1975 Wu Yi Hsiang
Table of Contents
Chapter I. Generalities on Compact Lie Groups and G-Spaces .
1
§ 1. General Properties of Compact Topological Groups . . . 1 § 2.
Generalities of Fibre Bundles and Free G-Spaces . . . . . 6 § 3.
The Existence of Slice and its Consequences on General G-Space ....
10 § 4. General Theory of Compact Connected Lie Groups. . . . . . .
. . 13
Chapter II. Structural and Classification Theory of Compact Lie
Groups and Their Representations. . . . . . . . . 17
§ 1. Orbit Structure of the Adjoint Action. . . . . . 17 § 2.
Classification of Compact Connected Lie Groups. 23 § 3.
Classification of Irreducible Representations. . . 30
Chapter III. An Equivariant Cohomology Theory Related to Fibre
Bundle Theory. . . . . . . . . . . . . . . . . 33
§ 1. The Construction of H~(X) and its Formal Properties. 33 § 2.
Localization Theorem of Borel-Atiyah-Segal Type . . 39
Chapter IV. The Orbit Structure of a G-Space X and the Ideal
Theoretical Invariants of H~(X). . . . . . . . . . . . . . . . .
43
§ 1. Some Basic Fixed Point Theorems . . . . . . . . . . . . . . 43
§ 2. Torsions of Equivariant Cohomology and F-Varieties of G-Spaces
54 § 3. A Splitting Theorem for Poincare Duality Spaces. . . . . .
. . 65
Chapter V. The Splitting Principle and the Geometric Weight System
of Topological Transformation Groups on Acyclic Cohomology
Manifolds or Cohomology Spheres . . . . . . . . . . . . 70
§ 1. The Splitting Principle and the Geometric Weight System for
Actions on Acyclic Cohomology Manifolds . . . . . . . . . . . . . .
. . . 71
§ 2. Geometric Weight System and Orbit Structure. . . . . . . . . .
. 75 § 3. Classification of Principal Orbit Types for Actions of
Simple Compact
Lie Groups on Acyclic Cohomology Manifolds. . . . . . . . . . .
83
x Table of Contents
§ 4. Classification of Connected Principal Orbit Types for Actions
of (General) Compact Connected Lie Groups on Acyclic Cohomology
Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . .
91
§ 5. A Basis Fixed Point Theorem . . . . . . . . . . . . . . . . .
. 95 § 6. Low Dimensional Topological Representations of Compact
Connected
Lie Groups. . . . . . . . . . . . . . . . . . . . . 97 § 7.
Concluding Remarks Related to Geometric Weight System .....
101
Chapter VI. The Splitting Theorems and the Geometric Weight System
of Topological Transformation Groups on Cohomology Projective
Spaces ........................ 105
§ 1. Transformation Groups on Cohomology Complex Projective Spaces
. 106 § 2. Transformation Groups on Cohomology Quaternionic
Projective
Spaces ............................ 112 § 3. Structure Theorems for
Actions of 7lp-Tori on 7lp-Cohomology Pro
jective Spaces (p Odd Primes) . . . . . . . . . . . . . . . . . .
118 § 4. Structure Theorems for Actions of 7l2-Tori on
7l2-Cohomology Pro-
jective Spaces . . . . . . . . . . . . . . . . . . . . . . ..
121
Chapter VII. Transformation Groups on Compact Homogeneous Spaces
129
§ 1. Topological Transformation Groups on Spaces of the Rational
Homotopy Type of Product of Odd Spheres. . . . . 134
§ 2. Degree of Symmetry of Compact Homogeneous Spaces 148
References . . 160
Chapter I. Generalities on Compact Lie Groups and G-Spaces
This chapter will briefly review the general facts about compact
topological groups, fibre bundles, topological G-spaces and compact
Lie groups that are necessary for the subsequent development. Basic
concepts and definitions will be adequately explained; and
pro<?fs of some fundamental theorems will also be included
whenever short clear cut proofs are available.
§ 1. General Properties of Compact Topological Groups
Naturally, a topological group G consists of both a topological
structure and ·a group structure which are compatible in the sense
that the group structure is continuous with respect to the
topological structure. More precisely, the multi plication and
inversion mappings are both continuous: G x G-+G, (gl,g2)>-+gl .
g2; G-+G, g>-+g-l. Similarly, a Lie group (or rather a
differentiable group) G consists of both a differentiable structure
and a group structure which are compatible in the sense that the
multiplication and inversion mappings are both differentiable. For
many problems, the subclass of compact topological groups, or more
specifi cally, compact Lie groups plays ·an important role. In
this section, we shall sum marize the basic properties of compact
topological groups:
(A) Averaging and Haar Measure
Obviously, a finite group G (with discrete topology) is a rather
special example of compact topological group. Suppose qJ: G-+G 1(V)
is a given representation which represents G as a group oflinear
transformations on a vector space V. The following well known
"averaging method" is a natural, simple-minded way to show the
existence of an invariant inner product on V (with respect to the
action of G). Given an arbitrary inner product <x,y) on V, it is
clear that the following "averaged inner product" (x,y):
2 Chapter I. Compact Lie Groups and G-Spaces
is an invariant inner product on V with respect to the action of G.
And this is also the only effective way, so far, to prove the
complete reducibility of representations of finite groups.
Next let us consider the group of unit complex numbers Sl =
{ZE<C; Izi = 1}. TopologicalIy, it is a circle and it is one of
the simplest example of compact topological group with an infinite
number of elements. If we parametrize the circle group in the usual
way, i.e. Sl={e21ti8;O~e~1}, and f is a continuous function on Sl,
then the integration of f
S6f(e)de
is clearly a generalized "average value of f". Similarly, to any
linear representation cp:Sl-'>GI(V) and a given inner product
<x,y) on V, the folIowing "(generalized) averaged inner product"
is again invariant with respect to the given action of Sl on
V.
(x,y) = S6 <e21ti8 • x, e21ti8 • y) de.
As for a general compact group G, it is natural to blend together
the above two kind of "averagings". Namely, for a finite subset
A={aj}~G and a continuous function f(g), one defines the averaging
of f over A to .be
HeuristicalIy, it is reasonable to expect that {A ·f} wilI tend to
a constant function, 1(f), as a limit when A becomes more and more
dense in G. This is exactly the idea of Von Neumann in defining the
(invariant) Haar-integral on a compact group G. We state the result
as the folIowing theorem and refer to Pontrjgin's book "Topological
Groups" for a detail proof.
Theorem 1.1 (Existence and uniqueness of Haar integral). Let G be a
given compact topological group and C(G) be the space of
real-valued continuous functions of G. Then there exists a unique
continuous linear functional I:C(G)-,>IR. satisfying
(i) (left invariant): l(fa(g))=I(f) for all aEG, where fa (g) =
f(ag). (ii) (positive andnormalized): f~O=I(f)~O, and 1(1)=1.
The above invariant linear functional I can also be considered as
the integral with respect to the invariant measure-the Haar
measure-on G with total volume 1, i.e., 1(f)=SGf(g)dg.
Corollary (1.1.1). Every complex (resp. real) representation of
compact group G is equivalent to a unitary (resp. orthogonal)
represenation.
Corollary (1.1.2). Every complex (resp. real) representation cp ()[
a compact group G is completely reducible. Hence it decomposes
uniquely into the direct sum of irreducible representations.
§ 1. Compact Topological Groups 3
(B) Schur Lemma and Schur Orthogonality Relations
The Schur lemma is simply a neat reformulation of irreducibility
purely in terms of operators; however, it is extremely useful and
is of fundamental importance.
Schur Lemma. Let Sl' S2 he irreducible subsets of linear
transformations 011
Vj. Vz respectively and A: VI -+ V2 he a linear mapping such that
A· Sl = Sz . A (set-wise). Then either A =0 or A is
invertible.
Proof. It follows easily from A· SI = S2' A that Ker A and Irn(A)
are invariant subspaces of S I. S 2 respectively. Hence. the
irreducibility of Sl' S2 implies that either KerA=V1• i.e., A=O; or
KerA=(O} and Im(A)=V2 . i.e., A is inverti ble. 0
The following special form of Schur's lemma is often useful and
deserves special attention.
Special Form of Schur's Lemma. In the special case that SI = S2
(then of course VI = Vz). and the field F is algebraically closed
then there exists ;ooEF such that A=)oo·I.
Proof. A· SI = Sl . A=(A -;.1). SI = SI . (A -;.1) for any ;.EF. On
the other hand. since F is assumed to be algebraically closed,
there exists eigen-value lo E F of A, i.e., (A-}o1) is
non-invertible. Hence, by the above lemma (A-)'oI)=O, i.e., A =)'0
1. 0
As a direct consequence, we have the following simple but
fundamental theorem for representation of compact commutative
groups.
Theorem (1.2). Every irreducihle complex representation cp of a
cOl/Jpact commuta tive group G is one-dimensional.
Proof. It follows from the commutativity of G and the
irreducihility of cp( G) that for every gEG, cp(g)=A{?J)' I. for
some }.(g)E<C.
Thus, cp(G)~ p. I} = the set of scalar multiples. Hence, every
subspace is invariant and cp(G) is irreducible only when dim cp =
1. 0
Corollary (1.2.1) (Classification' of irreducible representation of
torus group). The set of equiwlence classes of irreducible complex
representations of a torus group of rank k, jk ~ IRk /71.\ is
naturally in 1 -1 correspondence with the set oj" integral linear
functional oj" IRk. Hence. it is also naturally in 1 -1
correspondence with elements oj" Hl(Tk,71.), i.e ..
where i E H l(SI, 71.) is the fundamental class with positil'e
orientation.
Definition. With respect to a chosen basis. let a unitary
representation cp: G-+ Urn) be given by its matrix form (p(g) =
(CPu(g)). Then CPij(g) are called the representation functions of
cp.
Example. In the well known classical case of G = Sl, the
representation function corresponding to the irreducible
representation (p: Sl-+SI with winding number
4 Chapter 1. Compact Lie Groups and G-Spaces
n is e21Cin6, and it is the fundamental fact of Fourier analysis
that the set of such representation functions, {e27tin6; n E Z}, of
all irreducible representations of Sl forms an orthonormal basis of
L2(Sl).
Combining the Schur lemma with the invariant integration, one has
the following orthogonality relations for representation functions
of a general compact group G.
Theorem (1.3) (Schur orthogonality relations). Let G be a given
compact group and cP, ljJ are two non-equivalent, irreducible,
complex representations of G. Then
(i) SG CPij(g) ljJkl(g)dg = <CPiig), ljJkl(g) =0,
(ii) SGCPiig)'CPkl(g)dg=<CPij,CPkl) =-d.1 ·bik·bjl · lmcp
Proof. Let V, W be the representation space of ljJ,cP and .2"(V,
W);:; V*@W be the space of all linear mappings of V into W. Define
the induced representation on .2"(V, W) by
g.A=cp(g).A.ljJ(g)-l=cp(g)A·ljJ(g)t, AE.2"(V, W).
T~en, it is cl~ar that A=SG(g·A)dg is an invariant element, i.e.,
g.A=cp(g) . A .ljJ(g)-l =A for all gEG. Hence, it follows from
Schur lemma that A is either 0 or invertible. But cP,ljJ are
non-equivalent, A must be 0 for all AE.2"(V, W). If we write down
the equation Ejk=O for the usual basis Ejk of .2"(V, W) in matrix
form, we get
Similarly, it follows from the special form of Schur's lemma
that
and suitable ABE <C. On the other hand, the invariance of trace
under conjugation enables us to compute AB as follows:
dimcp· AB =tr(.8) = SG tr(g B)dg = SG tr(B)dg =tr(B).
Hence, again, we have
(C) Characters: A Complete Invariant of Linear
Representations
Definition. X",(g) def tr(cp(g)) = L~= 1 CPii(g) is called the
character (function) of the complex representation cP of G.
§ 1. Compact Topological Groups 5
Observations. (i) Since tr(cp(g)) = the sum of eigen-values of
cp(g), Xc/g) may be considered as a linear-type invariant of
cp(g),
(ii) Xcp+ ",(g) = tr( cp + I/J)(g)) = tr(cp(g)) + tr(l/J(g)) =
Xcp(g) + X",(g), (iii) Xcp®",(g) = tr(cp@l/J(g)) = tr(cp(g))·
tr(l/J(g)) = Xcp(g)· X",(g), (iv) Xcp(gl ggil)=tr(cp(gl)CP·
(g)cp(gl)-I)=tr(cp(g))= Xcp(g)·
One of the most fundamental consequences of Theorem (1.3) is that
the charac ter function Xcp(g) completely classifies the
representation.
Theorem (1.3'). (i) The characters of two non-equivalent,
irreducible complex representations cp and I/J are orthogonal to
each other, <XCP' X",) = o.
(ii) A complex representation cp is irreducible if and only if the
norm of Xcp is 1, i. e.,
(iii) Two complex representation cp I' CP2 (not necessary
irreducible) are equivalent if and only if
Proof. Follows immediately from Theorem (1.3) and complete
reducibility.
Remarks. (i) Since the character function Xcp has the same value on
conjugate elements, Xcp can be viewed as a function on the set of
conjugacy classes GlAd. Furthermore if H c::; G is a closed
subgroup of G which intersects every conjugacy class of G,
then
(ii) The compactness of G implies that cp(g) are unitary [Cor.
(1.1)] and hence diagonalizable. Therefore the equivalence class of
cp(g) is completely determined by the basic symmetric forms {oJ of
its eigen-values {A J On the other hand, since the set cp(G) is
closed under multiplication, the character Xcp(g)=trcp(g) =CT 1
(..1.) not only gives us CT 1 (..1.), but also
Xcp(gk)=tr(cp(gt)=L'i=I..1.~, which in turn gives back {CT j ,j=l,
... , n}. Roughly, this is the basic reason why such a simple,
linear-type invariant as the character already suffices to classify
them. One also sees that both the compactness and the group
structure play a vital role.
(D) Completeness Theorem of Peter-Weyl
Theorem (1.4) (Peter-Weyl). For a given compact topological group
G, let r be a complete collection of non-equivalent, irreducible
complex representation of G. Then the linear combinations of
{CPiig), CPEr} are uniformly dense in the space of continuous
functions on G, C(G). Or equivalently, {CPiig), CPEr} forms a basis
in the Hilbert space of L2-functions on G, L2(G).
We refer the reader to Pontrjgin's book for a standard proof of
this theorem.
Definition. A representation cp is called faithful if it is an
isomorphism into.
Corollary (1.4.1). Every compact Lie group has a faithful
representation.
6 Chapter 1. Compact Lie Groups and G-Spaces
Proof. Since {CPiig), CPEr} span C(G), they clearly seperate
points. Hence, it is a direct consequences of Peter-Weyl theorem
that
n {ker(cp); CPEr} = rid} = the identity subgroup.
Therefore n {kercp- U); CPEr} =0
for any neighborhood U of the identity, and it follows from the
compactness of G (or rather the compactness of (G- U)), that there
exist a finite subcollection CPl' ... , CPk Er with n~=l {kercpi-
U} =0.
n~= 1 {kercpJ = ker(cpl + ... + CPk) ~ U .
On the other hand, it is easy to show that for a small neighborhood
U in a Lie group, the only subgroup contained in U is the identity
subgroup. Hence
1. e., cP 1 + ... + CPk is a faithful representation of G. 0
Remark. (i) Since every closed subgroup of a Lie group is also a
Lie group, a compact topological group has a faithful
representation when and only when it is a Lie group.
(ii) For a compact group G with a given faithful representation, it
is not difficult to deduce the Peter-Weyl theorem of G by using the
Stone-Weistrass approximation theorem. Hence, for compact Lie
groups, the Peter-Weyl theorem is equivalent to the existence of
faithful representations.
§ 2. Generalities of Fibre Bundles and Free G-Spaces
(A) The Concept of G-spaces and Fibre Bundles
Definition. A topological transformation group consists of a
topological space X, a topological group G and a transformation of
G on X (i. e., a map G x X -> X; (g,x)>--*g' x with e· x=x,
gl' (g2' X)=(gl' g2)' x) which is continuous in the sense that the
map G x X -> X; (g, x)>--*g· x is continuous. Sometimes, we
also call such a structure a topological G-action on X, or simply a
topological G-space.
Examples. (i) The linear transformation groups of
GL(n, IR) on IR n, or GL(n, <C) on <cn ,
or their restrictions via some linear representations:
G ..!4 GL(n,IR) (or GL(n, <C))
are obviously topological transformation groups.
§ 2. Fibre Bundles 7
(ii) The isometry group of a Riemannian manifold, (equipped with
compact open topology) is a topological transformation group.
Intuitively, a fibre bundle consists of a projection p: B-X from
the bundle space B onto the base space X which is locally a product
but globally twisted. One of the simplest such example is the
Mobius band which is a twisted product of the circle Sl and the
interval I with an orientation reversion Z2-twist on I. Technically
and theoretically, it is important to pin down precisely what are
the permissable twistings. This is exactly where the so called
structural group G and the G-space structure on the fibre Y enter
into the formal definition of fibre bundles.
Definition. A fibre bundle consists of a bundle space B, a base
space X, a fibre Y with a given topological G-action and a
projection p: B-X together with compa tible local product
structures. Namely, there exists a family of open coverings {V;} of
X and local product structures, ¢/V;xY-p-l(V;)~B such that the
twistings between two overlabing local product structures are
provided by the given G-action on Y, i. e.,
where ¢jl¢;(X,y)=gji(X)'Y' xE(V;nV;), YEY and gji:V;nV;-G is
continuous.
Remark. The above definition involves a choice of coordinate
neighborhoods {V;} and local product structures {¢;}, hence it is
not intrinsic and there is an ob vious kind of equivalence
relation among two such structures. We refer to § 2,3 of Steenrod's
book [S 11] for a detail discussion on this matter.
(B) Principal Bundle and Principal Map
Definition (of bundle map). Let B,B' be two fibre bundles with the
same G-space Y as fibre. Then a continuous map h:B-B' is called a
bundle map if
(i) h carries each fibre Yx of B homeomorphically onto a fibre Yx'
of B', thus inducing a map Ii: X - X' with p' h = Ii p,
(ii) it is compatible with the twisting in the sense that
v,nhfJlX Y 0"", T,Y p-l(V;nli-l(U)) ~ p'-l(U)
8 Chapter!. Compact Lie Groups and G-Spaces
Remarks. (i) In the simplest case of the bundle Y -> {p t} over
a point, the operation of an element gEG, Y->Y:y~g·y, is clearly
a bundle map and conversely, every bundle map Y -> Y also
concides with the operation of some element of G. Hence, there is a
bijection between the set of all bundle maps of Y onto itself and
G=GIKo, where Ko={gEG;g·y=yforallYEY} is the ineffective
kernel.
(ii) It is obvious that the composition of bundle maps is a bundle
map.
Principal bundle. Let B..J4 X be a given fibre bundle with a given
G-space Y as typical fibre. Naturally, the set 13 of all bundle
maps of the simplest bundle Y -> {pt} into B..J4 X constitutes
an important structural invariant. As usual, we shall first try to
equip 13 with as many natural structures as possible. Since the
composition of two bundle maps is a bundle map, there is a natural
(right) action ofG on 13:13 x G-+B, (b,fj)~bog. Moreover, there is
a projection p:B->X by assigning each bundle map b to the
induced image in the base, i. e., p(b)=pb(Y)EX, and it is not
difficult to show that X~BIG (bijection).
Definition (Principal bundle). The principal bundle associated to a
bundle B ..J4 X is the set 13 of all bundle maps of Y ->{pt}
into B ..J4 X, equipped with the above right G-action and the
projection 13 L X ~ BIG.
Remark. It follows easily from the local product structure of B
..J4 X that 13 L X is also locally a product. Hence, it is easy to
equip 13 with a unique suitable topology so that the bijection X ~
BIG becomes a homeomorphism.
Definition. The evaluation map 13 x Y L B, (b,y)~b(y) is called the
principal map.
Remark. (i) In the case that G acts effectively on Y, i. e., Ko =
rid}, then G = G and 13 is a free (right) G-space and the above
principal map P:B x Y->B simply identifies B with the quotient
space BxGY=Bx Y/,,~", where (b,y)~(b.y-\g·y). In general 13 is a
free (right) G-space which can also be considered as a (right)
G-space naturally. Then again, the map P identifies 13 x G Y ~
B.
(ii) The principal map P indicates how to recover the bundle B..J4
X from the principal bundle 13 and the given G-space Y. Hence, for
a fixed G-space Y, the principal bundle 13 constitutes a complete
invariant.
(C) Intrinsic Definition of Bundles and Gleason Lemma
(XxG) has an obvious (right) G-action given by (X,g)·gl=(X,g·gll. A
free (right) G-space E is called locally trivial if to every point
xEEIG (orbit space of E) there exist a neighborhood U, such that
p-l(U) is isomorphic to the obvious (right) G-space U x G. In view
of the principal bundle and principal map, it is natural to give
the following intrinsic definition of fibre bundle:
Definition. A fibre bundle consists of a locally trivial free
(right) G-space 13 (called the principal bundle) and a (left)
G-space Y (called the fibre). Then the orbit space X=BIG is called
the base space; the quotient space B=BxGY=Bx Y/,,~", where (b, y) ~
(b g- 1, g y) is called the bundle space (or total space) and the
following map p is called the projection:
§ 2. Fibre Bundles 9
j- j, B=BxGY ~ B/G=X
where P1 is the projection onto the first factor and vertical map
are projections onto respective "orbit spaces".
In many useful cases, the structural group G is either
automatically a compact Lie group or can be reduced to a compact
Lie group (cf. (E)). In those cases, the following lemma of Gleason
adds a convenient bonus to the above definition, namely, the local
triviality condition holds automatically for any free G-space when
G is a compact Lie group.
Gleason Lemma. ~f G is a compact Lie group, then any free G-space
is locally trivial.
(We shall prove a more general form of this lemma in § 3.)
(D) Homotopy Lifting Property, Induced Bundles and Classifying
Bundles
For the purpose of homotopical investigations, the following direct
consequence of the local product structure of fibre bundles is of
fundamental importance.
Homotopy lifting property. Let B ~ X be a fibre bundle and K be a
finite polyhedron. Let fa: K -> B be a map and 1: K x I -> X
be a homotopy of p -fa = la. Then there exists a homotopy f: K x
1-> B covering l (i. e., p f =f), and f is stationary with
.r.
The proof of the above property is simple and straightforward by
using the local product structure of p: B -> X. However, it is
so fundamental that it led 1. P. Serre to axiomatize it in defining
fibre spaces [S 3J.
Induced bundle. Let B ~ X be a fibre bundle and f: X' ->X be a
map. Let B' be those points (b,x') of B x X' with p(b)=f(x') and
l,p' be the restriction of projections to B', i. e.,
Then, it is easy to check that B' ~ X' is a fibre bundle and .1 is
a bundle map. B' ~ X' is called the induced bundle of B ~ X with
respect to .f.
10 Chapter 1. Compact Lie Groups and G-Spaces
Classification Theorem. Let E'G ~ B'G be a principal G-bundle with
n,(E'G) = 0 for 0 ~ i ~ n, and K be a finite polyhedron of dim ~ n.
Then the operation of assign ing to each map f: K -+ B'G its
induced bundle sets up a bijection between homotopy classes of maps
of K into B'G and equivalence classes of principal G-bundles over
K.
We refer to § 19 of [S 11] and [M 2] for a proof of the above
theorem and the general existence of such classifying (universal)
bundle.
(E) Reduction and Extension of Structural Group
Let H s;; G be a closed subgroup of G and B ~ X and B' L... X be
respectively H-bundle and G-bundle over X (i.e., free H-space and
G-space with X as their common orbit space). If there exists an
equivariant map r:B-B' which commutes with projections, i.
e.,
B~B'
\} X
commutes and r(b· h) = r(b)· h, then B ~ X is called a reduction of
B' L... X and B' L... X is called an extension of B ~ X.
Observe that B' =B' xGG and hence B'jH =B' xG GjH. Therefore, a
reduction r:B->B' induces the following maps
BjH ~ B'jH=B'xGGjH
\! X
which gives a cross-section r·p-l:X-B'x GGjH. Conversely, suppose
s:X -+B' xG GjH =B'jH is a given cross-section. Then the inverse
image of s(X) of the projection Po:B'-+B'/H, i.e., B= P0 1(S(X)),
is clearly a free H-space with s(X)=X as its orbit space and the
inclusion B=P01(S(X))S;;B' is equivariant. Hence,
Proposition. For a given closed subgroup H S;; G, there is a 1 -1
correspondence between the reductions of a G-bundle B' -> X to
an H -bundle and the cross-sections of the associated GjH-bundle
B'jH =B' xG GjH -X.
§ 3. The Existence of Slice and its Consequences on General
G-Space
From now on, we shall always assume that G is a compact Lie group.
The following simple, useful fact was first noticed by Koszul [K 3]
and then further generalized and exploited by Montgomery-Yang,
Mostow and Palais. [M 4, M 9, P 1]
§ 3. Slices and G-Spaces 11
(A) Differentiable Slice
Theorem (1.5). Let M be a differentiable G-space, H = Gx be the
isotropy subgroup at a point xEM, and ({Jx be the local
representation oj H on normal vectors (w.r.t. a chosen invariant
metric) oj the orbit G(x)~ G/H at x. Then, the equivariant normal
bundle v(G(x)) oj G(x) is isomorphic to
where H acts on G as right translations and on lRk via ({Jr
Furthermore, Jor a sufficient small e>O, the exponential map
(w.r.t. a chosen invariant metric) is an equivariant diffeomorphism
oj the associated e-disc bundle oj a(({Jx): G xHD:--.G/H onto an
invariant tubular neighborhood oj G(x).
ProoJ. Let lR~ be the set of normal vectors to G(x) at x (w.r.t. a
chosen invariant metric, which always exists by means of
averaging). Then the induced G-action carries lR~ to lR~.x' It is
easy to check that gt'Vl=gZ'VZ if and only if (gt,v 1)
and (gz, Vz)E G x lR~ are equivalent in the usual sense, i.
e.,
~ G x T(M) ---> T(M)
Hence, the above theorem follows from a straightforward
verification. 0
(B) Gleason Lemma and the Existence oj Topological Slice
Lemma (3.1). IJ H is a closed subgroup oj G (a compact Lie group),
then there exists a finite dimensional linear G-action with a
vector v such that H = Gv is the isotropy subgroup oj v.
Proof. It follows directly from Peter-Weyl theorem and the
descending chain condition for closed subgroups of a compact Lie
group. We refer to the paper of Mostow [M 9J and Palais [P 1 J for
a detail proof. 0
Gleason Lemma (Generalized Form). IJ K is a compact invariant
subspace oj a G-space X and if J: K --. V is an equivariant map oj
K into a linear G-space, then J admits an equivariant extension j:
X --. V.
ProoJ. By Tietze's extension theorem, there is a continuous
extension J*: X --. V. Let J(x)= JGg- 1 J*(gx)dg. It is easy to
check that J is an equivariant extension of f.
Theorem (IS) (Existence of a slice). Let X be a topological G-space
and H = Gx'
Then there exists a subset S oj X such that (i) S is invariant
under H, (ii) G(S) = G x H S and is an invariant neighborhood oj
G(x) in X. Such a subset is called a slice at x.
12 Chapter I. Compact Lie Groups and G-Spaces
Proof. In the special case that X is a differentiable G-space, a
small normal disc D~(e) is clearly such a slice at x. In the
general topological case, we first equi variantly embed the orbit
G(x) ~ G/H into a suitable linear G-space V,f: G(x) ~G/H ~ G(v)s; V
(by lemma (3.1)). Then extend f to an equivariant map j: X --+ V.
Let S' be a slice at v (which exists because V is linear) and put S
= J -I (S'). Then it is easy to check that S is such a slice.
0
The above slice theorem has the following two consequences of basic
im portance:
(C) Equivariant Embedding
Theorem (1.6) (Mostow-Palais). If G is a compact Lie group and X is
a separable, metrizable G-space of finite dimension and with finite
orbit types, then X admits an equivariant imbedding into a linear
G-space.
Since we actually do not need the above theorem in any essential
way, we refer to [M 9J or Ch. VIII of [B 10J for a proof.
(D) Principal Orbit Type Theorem
For a given G-space X, the set of all isotropy subgroups
(D(X)={Gx;XEX} clearly divides into conjugacy classes which
correspond to the types of orbits in X. Observe that a homogeneous
space G/K can map equivariantly onto G/H if and only if K is
conjugate to a subgroup of H, i. e., gKg- I s;H for suitable gEG.
Hence, it is natural to introduce the following partial ordering
relation among the set of orbit types:
(D(X)= {Gx: XEX} = U(H;) (conjugacy classes), (Hi)?: (Hj)~;""Hi is
conjugate to a subgroup of Hj.
Theorem (1.7) (Montgomery-Samelson-Yang). Let M be a connected
manifold with a given differentiable G-action. Then, among the set
of orbit types (D(M) = U(H;), there is a unique maximal orbit type
(HI) such that
(i) (HI)?:(GJ for all XEM, (ii) the union of all orbits of
(H1)-type=M(Hd={x;GxE(H I)) is open dense
in M and the codimension of (M -M(Hd) is at least 1. (iii) F(HI,M)
intersects every orbit, (iv) the orbit space M(HjG is
connected.
Proof. Let M' =M/G be the orbit space and (D(M) be the set of orbit
types (with discrete topology and partial ordering). Let us
conSider the orbit type function Q:M'--+(D(M). It follows from the
differentiable slice theorem that x'=G(x) is a local maximal if and
only if ({Jx is a trivial representation (i. e., Gx acts trivially
on the slice Sx = D~, and hence Q is locally constant in a
neighborhood of those local maximal points x'. On the other hand,
suppose y' = G(y) is not a local maximal, then there are the
following two cases:
§ 4. Compact Connected Lie Groups 13
(1) codim(F(G)',S))~2, then Sy-F(Gy,Sy) is still connected and
hence if we remove those orbits of the same type as y' from the
neighborhood of y', S)Gy, the remaining set [Sy - F( Gv, S.vlJ/Gy
is still connected.
(2) codimF(G)"Sy)=1, then qJy:Gy ...... O(1)~~2 and acts on Sy as a
reflection with respect to the hyperplane F(Gy,S). Hence the orbit
space S)Gy is a half plane with the image of F( Gy, Sy) as
boundary. Therefore [Sy - F( Gv, SyJ/G,. is again connected.
Since M is assumed to be connected, M' = M /G must be also
connected. The above fact shows that the removing of all those
points y' which are not local maximals did not even separate the
space M' locally. Hence, the subspace M~ of all local maximal
points x' of M' is still connected. But Q is locally constant on
M~, Q must be in fact a constant on M~. That is, there is a unique
(local) maximal orbit type (Hi) and M~ = M(HJl/G is connected, open
dense in M'. The assertions (ii), (iii) also follow immediately.
0
Remark. (i) The above theorem also holds without modification for
topological G-action on cohomology manifold over ~. It also holds
for connected orbit types, (i. e., the types of connected isotropy
subgroups G~) for topological G-action on cohomology manifold over
<Q. Such generalizations will be proved when we are ready to
show the same fact about codim F(Gy,S) by cohomology method.
(ii) The unique maximal orbit type is called the principal orbit
type, and its corresponding isotropy subgroups are called principal
isotropy subgroups.
§ 4. General Theory of Compact Connected Lie Groups
(A) Adjoint Action and Adjoint Representation A one-parameter
subgroup of a Lie group G is a (differentiable) homomorphism, ~, of
the simplest Lie group JR 1 into G. Clearly, the right translations
of {~(t), tE JR l} exhibit a left-invariant (due to the
associativity of G) JR I-action on G. Hence, the velocity vector at
every point forms a left-invariant vector field X on G. Conversely,
if we integrate a left invariant vector field X, then it is easy to
show (by uniqueness and left invariance of integral curves) that
the integral curve passing through identity is a one-parameter
subgroup. Hence, there are the following bijections
{one-parameter subgroups} +-> {left invariant JRi-actions} t
t
{tangent vectors at identity} +-> (left invariant vector fields}
.
We identify them via the above bijections and call it the Lie
algebra, g, of G, which is a vector space with a bilinear bracket
product [X, YJ (of vector fields) satisfying the following Jacobi
identity:
[[X, YJ,ZJ + [[Y,ZJ,XJ + [[Z,XJ, YJ =0.
14 Chapter I. Compact Lie Groups and G-Spaces
Furthermore, It IS convenient and useful to organize all
one-parameter sub groups into a (universal) map:
Exp:g-+G
such that Exp(t X): IRl-+g-+G gives the one-parameter subgroup with
X as its velocity vector at the identity e.
Remark. (i) Recall that the geometric meaning of bracket product
[X, Y] of two vector fields X, Y is the following: If one "drifts"
successively in the flow X, Y,( - X) and then (- Y) each for a time
t, one ends up travelling approx imately t2. [X, Y]. Hence, in our
case of left-invariant vector fields, we have [X, Y]e = the
velocity vector (at e) of the curve
y(t) = Exp Vi X. Exp Vi y. Exp( - Vi X)· Exp( - Vi Y).
t2-[X. Y] (I)
(ii) If h: G1 -+G2 is a homomorphism of Lie groups, then the
differential of h at e clearly induces a homomorphism of their Lie
algebra dhe :g1 -+g2 ,
Definition. The conjugations represent G as a differentiable
transformation on itself via inner automorphisms, i.e., GxG-+G,
(gl,g2)>-+glg2g1 1• We shall call it the adjoint G-action on G
itself. The above adjoint action on G induces another adjoint
action on its Lie algebra g, which is a linear representation of G
into GL(g), Ad: G-+GL(g) (general linear group of g), called the
adjoint representa tion of G. Furthermore, the differential of Ad
at e, ad = Ade : 9 -+g /(g) is called the adjoint representation of
g. In terms of exponential maps, the above two adjoint
representations of G and 9 are respectively characterized by the
follow ing identities
(i) Expt· [Ad(ExpsX)· Y] =ExpsX· Expt Y·Exp( -sX), (ii) Ad(ExptX)·
Y = Exp(t·ad(X)). Y.
Proposition (4.1). ad(X)· Y = [X, Y] for all X, Y Eg.
Proof. It follows from the geometric interpretation of the bracket
product that
ExpsX ExptY·Exp( -sX)·Exp( -tY) = Expst· [X, Y]+O(st).
§ 4. Compact Connected Lie Groups
On the other hand, by the above definition, we have
Exp[(Exps ad(X) - I)· t Y] = Exp(st· ad(X)· Y)+O(s t)
II Exp(t Exps ad(X)· Y)· Exp( - t Y)+O(st)
II by (ii) by (i)
Expt[Ad(ExpsX). Y] Exp( -t Y) = ExpsX· ExptY· Exp( -sX)Exp(
-tY)
= Expst·[X, Y]+O(st).
Hence, if we take the limit s,t~O, we get ad(X)· Y = [X, Y].
0
(B) Cart an's Theorem of Maximal Tori and its Consequences
15
Theorem (1.8) CEo Cartan). Let G be a compact connected Lie group
and T be a given maximal torus. Then
(i) all maximal tori of G are mutually conjugate and they are
exactly those connected principal isotropy subgroups of the adjoint
action on G.
(ii) T intersects with every conjugacy class (i. e., orbit of
adjoint action) or equivalently G=U{gTg-:-1;gEG}.
Proof. The above theorem is a direct consequence of Theorem (1.7)
and the fact that maximal tori are exactly those connected
principal isotropy subgroups of the adjoint action on G. Since
principal orbits are everywhere dense and the adjoint action on 9
is precisely the local linear action around the identity eE G, it
is equivalent but technically simpler to prove that the maximal
tori are the connected principal isotropy subgroups of the adjoint
action on g. The follow ing lemma provides an effective way of
computing the principal isotropy sub groups of a linear
action.
Lemma (4.1). Let ljJ be a linear G-action and ({Jx be the slice
representation of Gx at x (cf. § 3-A). Then
Proof. It is clear that the local representation of Gx at origin is
ljJ I Gx ' while that of x is ({Jx (:normal part)+(AdGIGx-AdG
)(:tangent part). But the fixed point set F(Gx ' V) is a linear
subspace and hence obviously connected. Therefore the local
representations of Gx at origin and at x must be equal. 0
Let X be an arbitrary element of g, Gf be the connected isotropy
subgroup of X and gx be the Lie algebra of Gx . It is easy to show
the existence of at least a maximal torus T;2 Exp t X, hence Gx
contains at least one maximal torus for any X E g. On the other
hand, it follows from the above lemma that Gf is a connected
principal isotropy subgroup if and only if
16 Chapter I. Compact Lie Groups and G-Spaces
is a trivial representation which (in view of the fact Gx contains
at least a maxi mal torus), is equivalent to saying that Gf itself
is a maximal torus. 0
Corollary (1.8.1). Exp: g --+ G is onto for a compact connected Lie
group.
Proof. Let gEG be an arbitrary element then there exists gl EG such
that glgg11ET or gEg1 1 Tg1 and hence, the above corollary follows
from the obvious fact that Exponential map, is onto for a
torus.
Corollary (1.8.2). Gx (w. r. t. the adjoint action on g) is
connected for every X Eg.
Proof. Observe that Gx = centralizor of the torus subgroup T(X)
generated by {ExptX}. We claim that Gx = the union of all maximal
tori containing T(X) and hence connected. Let gEGx be an arbitrary
element of Gx . Then g, T(X) generates a compact abelian subgroup
which is, in general, a cyclic group prod uet with a torus, i.e.,
<g, T(X) =Zh x T1. It is easy to show that Zh x Tl =<gl) is a
topological cyclic group. Hence, by the above theorem, there exists
a maximal torus T2<gl)=<g,T(X), i.e., gET2T(X). 0
Corollary (1.8.3). Let Tr;;;,G be a maximal torus of a compact
connected Lie group G. Then two representations q>, IjJ of G are
equivalent if and only if their restrictions to T, q> I T and
IjJ I T are equivalent.
Proof. q>-1jJ <::> X",=X", <::>
X",IT=X",IT=X"'IT=X",IT<::> q>IT-IjJIT (for character
functions take constant value on each conjugacy class and T
intersects every conjugacy class). 0
Definition. The dimension of maximal tori of a compact Lie group G
is called the rank of G.
(C) Root System and Weight System
Definition. For a given complex representation IjJ of a compact
connected Lie group G and a fixed maximal torus T, it follows from
the above corollary and Schur lemma that IjJ I T is a complete
invariant and
. l/t I T = IjJ 1 ff)1jJ 2 ff) .. . ff)1jJ n
splits uniquely into the sum of one-dimensional representations.
Hence, the collection of integral linear functionals wjEHl(T,Z)
corresponding to IjJj forms a complete invariant of 1jJ, called the
weight system of 1jJ, and denoted by Q(IjJ).
Definition. The (non-zero) weight system of the complexification of
the adjoint representation of a compact connected Lie group G is
called the root system of G, and is denoted by LI(G), i.e.,
LI(G)=Q(AdG®<C)-{those zero weight vectors}.
Remark. The exclusion of zero weights in LI(G) is purely for
notational con venience.
Chapter II. Structural and Classification Theory of Compact Lie
Groups and Their Representations
This chapter consists of a concise exposition of the structure and
classification theories of compact Lie groups and their
representations from the geometric viewpoint of transformation
groups. An explicit and neat understanding of the orbit structure
of the adjoint action of a compact Lie group G plays a central role
in the classification theory developed by E. Cartan and H. Weyl.
This more geometric approach is actually more natural and
straightforward than the usual Lie-algebra-theoretical approach.
Furthermore, such an approach will also provide us with valuable
examples and insight for later investigation of top ological
transformation groups.
§ 1. Orbit Structure of the Adjoint Action
(A) The simplest fundamental examples: SU(2) or SO(3)
The special unitary group of rank 2, SU (2) consists of those 2 x 2
matrices
u = G -;) with det(u) = lal 2 + Ibl 2 = 1. It can also be
identified with the group
of unit quaternions, Sp(1)={q=a+j-b,lqI2=lalz+lblz=1}, which is
geo metrically the unit sphere S3 (1) s; IR 4. The center of Sp(
1) consists of {± 1 } and the adjoint representation of Sp(l) maps
Sp(t)/{ ± l} isomorphically onto SO(3), the rotation group of
euclidean 3-space. Geometrically, the adjoint action on Sp(1)=S3(1)
is simply the SO(3)-rotation with the real axis as the fixed axis.'
Since the total volume of the unit 3-sphere is 2nz, the normalized
Haar measure on Sp(l) is the usual measure on S3(1) modified by a
constant 1/2n 2 •
Now let us apply the above understanding of the geometric structure
of Spit) = SU (2) to the classification of irreducible complex
representations of Sp(l).
(i) Let 1j;1: Sp(t)~SU(2) be the usual complex linear action of
SU(2) on ([z = {(ZI , Z 2)}' Then the above linear substitutions
induce a complex linear action, Ij;k' of Sp(1)~SU(2) on the space
of degree k homogeneous polynomials PkS;([[ZI'ZZ] for each
k;;;,O.
(ii) Let Tl={a=e2ni8} = unit complexess;Sp(l) (unit quaternions).
Then
Tl is a maximal torus ofSp(l) and Ij;I(a)=(~ ~~1} i.e.,
Ij;t(a),zj =a·z1 , Ij;I(a)'zz=a~l·z2'
18 Chapter II. Classification Theory of Compact Lie Groups
Therefore, for the basis {z~, ... , z~ -[ z~, ... , z~} of Pk
,
Theorem (11.1). {'fk' k;?: O} constitutes a complete collection of
(non-equivalent) irreducible representations of Sp(l) =
SU(2).
Proof. Let d(J be the usual measure on SJ(1) and f be a function of
G = Sp(1) constant on every conjugacy class (i. e., a central
function). Then, we have the following integration formula
SGf(g)dg = J, Ss3f(g)d(J = ~ S612 f(e 21ti O).4n sin2(2ne)·2nde 2n-
2n
= ~ J6f(e21tiO)'le21tie - e- 2rri012. de = ~ ST'f(t)'IQ(tW dt
where "dg" and "dt" are the normalized Haar measure of G=Sp(l) and
TI respectively and Q(t)=(t-t) for t=e21ti OE TI.
(i) Applying the above formula to f(g) = Xk(g)· Xk(g), one
gets
SG Xk' Xk dg = ~ SrI Xk(t)· Xk(t)· Q(t)· Q(t) dt
= ~ SrIIXk(t)· Q(tW dt
= ~ Srlltk+ 1 _ t-(k+ 1)1 2 dt = 1.
Hence, it follows from Theorem (1.3') that I/Ik are irreducible for
all k;?:O. (ii) Notice that diml/lk=(k+ 1) are different for
different k, it is obvious
that {l/Ik,k;?:O} form a non-equivalent family. On the other hand,
it is a well known fact in Fourier series that
forms a basis of the sub-space of odd functions in L2 (T 1). Hence,
{l/Ik,k;?:O} is already a complete family in the sense that any
irreducible complex representa tion, 1/1, of Sp(1) must be
equivalent to one of I/Ik' For otherwise. it follows from Theorem
(1.3') that
<X'fr' Xk) L2(G) = SG XI/I' Xk dg
=~Srx"f(t)Q(t)'Xk(t)'Q(t)dt=O for all k.
§ 1. Orbit Structure of Adjoint Action 19
Since XIjJ(t)· Q(t) is a continuous odd-function, the above
orthogonal relations for the whole basis {Xk(t)· Q(t)} imply that
XIjJ(t)· Q(t) == 0, which is obviously a contra diction, for
XI//(e) = dim IjJ # O. 0
Remark. (i) The weight system of IPk form a string leading from the
highest weight vector k D to - k D, i. e.,
Q(ljJk) = {kO,(k-2)D, ... , (k-2f)D, ... , -kD}.
(ii) IjJk factor through SO(3), i. e., Ker(ljJk) = {± 1}, if and
only if k is even. And they forms a complete family of irreducible
representations of SO(3).
(iii) In either case of Sp(1), or SOC}), the weight system of any
representation consists of at least one non-negative weight which
is less than the positive root, i.e., O~w<ct.
Corollary (H.i.l). Every non-commutative compact connected Lie
group of rank one is either isomorphic to Sp(1) or to SO(3).
Proof. Observe that the Lie algebras of Sp(1) and SO(3) are
isomorphic and can be expressed in terms of basis 91=(H,X,Y) with
[H,X]=Y, [H,Y]=-X. and [X, Y]=H. Now, let G be a non-commutative
compact connected Lie group of rank one and 9 be its Lie algebra.
Let Tl S G be an arbitrary, fixed maximal torus of G, and AdlT I be
the restriction of the adjoint representation to Tl. Then
AdITI=1+LJ=1(Xj and 9=lRl +LJ=llR;j
where lR I is the Lie algebra of Tl and (Xj: T I ---+SO(2) is a
homomorphism with winding number nj>O; nl~"'~ns' Let 9~=lRl+lR;,
and H' the integral base vector of lR I, i. e., 7l. H' = Ker{lR I
exp ) T I), and X', Y' be an orthogonal basis of lR;,. Then, by
definition,
(E H') (cos2nnl t (Xl xpt· = .
. sm2nnl t -sin 2n 111 t) .
cos2nnl t
Hence, by Proposition (4.1) of § 1.4-A, we have
[H',X'] =nl Y' and [H', Y'] = -nl X'.
The fact that G is of rank one implies that [A, B] = 0 if and only
if A, Bare linearly dependent in 9. Hence [X', yI] #0, and
[H', [X', Y']] = - [X', [Y',H']] - [Y', [H',X']]
= - [X',n l X'] - [Y', nl Y'] = 0
imply that [X', Y'] = A' H' for a suitable ;e # O. Then, it is easy
to modify the basis {H',X', Y'} by suitable constant to obtain a
basis {HI,XI, Yl} of 9~ so that [HI ,Xl] = Yl , [HI' Yl ] = -Xl'
[Xl' Yl] =Hl · Hence 9~ ~ 91 as Lie algebra and
20 Chapter II. Classification Theory of Compact Lie Groups
consequently, the subgroup G~ with g; as its Lie algebra is
isomorphic to either Sp(1) or SO(3). We claim that G~ = G. For
otherwise, it follows by applying the above theorem to the isotropy
representation of G~ on tangent vectors of GIG; that its weight
system Q={cxz, ... ,cxJ must contain at least one
0:s;cxj<!X1
(cf. Remark (iii)) which is a contradiction to the choice that !X!
:S;!Xj for all 2:S;j:S;s. 0
(B) An Elementary Structural Theorem for Groups Generated by
Reflections
Let M be a connected differentiable manifold. A diffeomorphism I'
of M onto itself is called a reflection if 1'2 = identity, the
fixed point set of r, F(r) is of co dimension one and (M - F(r))
consists of two connected components inter changed by r.
Theorem (11.2). Let r be a finite group generated by reflections
and ~ = {rj }
be the set of all reflections in r. Then r acts simply transitively
on the connected components of (M - U {F (r); rj E f!lt}) (called
chambers). The closure of an ar bitrary . chamber Co is a
fundamental domain of r in the sense that every point xEM is
conjugate to exactly one point Xo=Y'XECo '
Proof. Observe that y·F(r)=F(y·r·i)-l) and hence U{F(r),riEf!lt} is
invariant under r, and consequently (M - U{F(r), IjE..:.qf}) is
also invariant. Suppose Ci
and Cj are two chambers with (C;r\ F(r) n C) of co dimension one.
Then it is obvious that r(CJ=Cj. Since the deletion of all
intersections of different hyper planes {F(r;) n F(r), ri "" rjE
f!lt}, (which is of codimension :s; 2), fails to disconnect M, it
is not difficult to show that r acts transivitely on the set of all
chambers, i. e., r·Co=(M-U{F(r),rjE.g)!}). Hence
r·Co=(M-U{F(r)})=M. We refer to [H 8J for a detail proof that Co
intersects every r-orbit exactly once, i. e., Co is a fundamental
domain. 0
Corollary (11.2.1). Let ~o be the set of those reflections in r
with their hyperplanes F(r) have codimension one intersection with
Co. Then r is already generated by those reflections {rEf!lt o},
called a simple system of generators of r.
(C) Weyl Group and a Fundamental Reduction
Definition. Since all maximal tori of a compact connected Lie group
G are con jugate, we may arbitrarily choose a maximal torus T~ G
and define the Weyl group of G, W(G) =N(T)/T, where N(T) is the
normalizor of T in G. Clearly, the conjugations induce an action of
W(G) on T and consequently also on its Lie algebra 1), the Cartan
subalgebra. The importance of the Weyl group lies in the following
fundamental reduction.
Theorem (11.3). Let G be a compact connected Lie group, T a maximal
torus of G, L1 (G) the root system of G, and W the Weyl group of G.
Then
(i) the multiplicity of each root cx E L1 (G) is one and k· cx E L1
(G) !ff k = ± 1, (ii) the action of W on 1) (resp. on T) is a group
generated by reflections
{ra' CXEL1(G)}, where r, is the reflection w.r.t. the hyperplane
H,=Ker(1) ~ JR.!) (resp. Ta = Ker(T ~ U(l ))),
§ 1. Orbit Structure of Adjoint Action 21
(iii) Wx = W(GJ and the following inclusions induce bijections on
their respective orbit spaces, i. e.,
i~j l)jW ~ gjAd
j j TjW ~ GjAd
Proof. (i) Let Y,,=KerO{T ~ U(l)) and Ga=NO(TJ, the connected
normalizor of Ta.' G = GJY". Then it is clear that Ga is of rank
one where root system L1 (Ga)
consists of precisely all those roots in L1 (G) proportionate to
rt.. However, it follows from Corollary (11.1.1) of §1-A that
L1(Ga)={±rt.}, hence the multiplicity ofrt. must be 1 and
krt.EL1(G) iff k = ± 1.
(ii) Clearly, F(Ta,g)=ga (the Lie algebra of Ga=N°(TJ=Z(Y,,)) and
the induced action of Ga on ga is the adjoint action, AdG ' which
is, effectively the rotation of Ga on ga with Ha (co dim 3 in ga'
codim 1 in l») as fixed point set. Hence, it is not difficult to
see that F(Ga, g) =F{ra,F(T, g)) =F(ra, l») = Ha, where ra is the
reflection that generates W(Ga)~Z2.
(iii) Consider the Weyl group W as an (effective) transformation
group on l) =F(T, g), W contains the above reflections ra, for each
pair of roots ±rt.EL1(G). We claim that W is, in fact, generated by
the above reflections {ra}. In order to show that, let W' be the
subgroup generated by {ra, ±rt.EL1(G)}. Since the set L1(G) is
invariant under W, it is clear that (l)-U{Ha}) is also invariant
under W, consequently, W permutes its connected components, i.e.,
the (Weyl) chambers. Notice that W' permutes the set of chambers
simply transitively. Hence, we need only to show that W also
permutes the set of chambers simply transitively (for then, ord
W=ord W'=the number of chambers). Suppose the contrary. Then there
exists an element aE W of order p (prime) such that a(Co)= Co.
Hence, it follows from a theorem of P. A. Smith (and the acyclicity
of Co) that there exists X E Co fixed under a and consequently, Gx
is disconnected, (for G~ = T, GxjG~ 2{a}) which is a contradiction
to Corollary (1.8.2) of§ I.4-B. Hence W= W' and is generated by the
reflections {ra}.
(iv) It follows directly from the principal orbit theorem that the
fixed point set of a principal isotropy subgroup, F(T, g) = l)
(resp. F(T, G) = T) intersects every orbit. Namely, the following
inclusions induces surjections of the respective orbit
spaces:
l)
I GjAd
On the other hand, the injectivity of the above maps simply means
that F{T, G(X))=N(T, G)jN(T, Gx )= W(G)jW(Gx) which is a direct
consequence of
22 Chapter II. Classification Theory of Compact Lie Groups
maximal tori theorem. Finally, we remark that
Definition. The orbit space g/ Ad ~ l)/W is called the Weyl chamber
of 9 and the orbit space G/Ad~ T/W is called the Cm·tan polyhedron.
In view of Theorem (II.2), it is also customary to identify the
above orbit spaces with anyone of their respective fundamental
domains.
(D) The Volume Function and Weyl Integration Formula
If one equips G with a bi-invariant Riemannian metric with total
volume 1, and lis a central function on G, then the Haar integral
SG/(g)dg can be reduced to the following weighted integration
Sp/(t)p(t)dt =! Sr/(t)p(t)dt w
(w=ord(W(G)), and T consists ofwP's)
over the Cartan polyhedron P with p(t)=q-dim volume of G(t),
q=dimG/T. Now let us compute the above volume function pet): T -+R
as follows. Let us equip G/T with the homogeneous Riemannian metric
such that the metric on the tangent space at the base point is the
restriction of metric on 9 to l)-L. For a given tE T, we have
UI UI
and it is clear that p(t)=vol(G(t))=vol(G/T) ·IThe Jacobian of let)
at el. Since the tangent space at the base point eE G/T, l)\
decomposes into invariant root spaces W.r.t. dl(t)le:
one has det(dl(t))e = TIml+ det(dl~(t))e' In view of the
computation for the rank one case, it is not difficult to see that
det (d I,U)) = constant· sin 2 (n aCt)) and hence
p(t)=co' det(dl(t))e=C' TIml+ sin2 (n· a(t))=c'Q(t)· Q(t)
Remark. It is useful to note that Q(t) is antisymmetric w.r.t. W,
I.e., Q(O"(t)) =sign(O")·Q(t). Then it is easy to show
§ 2. Classification of Compact Lie Groups 23
where c5 =t L'E1 + r:t.. Use the above expression of Q(t). it is
easy to determine the constant c' as follows
S 1 S '2 c' 2) 1 = G 1, dg = - T c 'I'Q(t)1 dt = - (IQ(t)IL2 (T) W
W
Hence, we have the following important formula,
Weyl Integration Formula. For a central function f on G, one
has
SGf(g)dg = ~ h.f(t)IQ(tWdt w
(A) Cartan-Killing Form and Characterization of Compact Lie
Algebras
Definition. A (real) Lie algebra g is called a compact Lie algebra
if it is the Lie algebra of a compact Lie group G,
Obviously, a compact Lie algebra, g, has an inner product invariant
under the adjoint action of G. Namely,
(Ad(ExptX)·Y,Ad(ExptX)·Z)=(Y,Z), forall X,Y,ZEg.
The above identity is clearly equivalent to its differentiated
version;
d . - (Ad(Exp t Xl' Y, Ad(Exp t X), Z), 0 =([X, Y],Z) +(y, [X,Z])
=0, dt
An important, simple consequence of the above fact is the
following:
Theorem (UA). A compact Lie algebra 9 decomposes uniquely into the
sum of its center go and normal simple subalgebras: 9 = go + L
gj'
Proof. It follows easily from the above identity that the
perpendicular space f.L of an arbitrary normal subalgebra f is also
a normal subalgebra, and 9 = f + f.L (as Lie algebras), for XEg,
YEt, ZEf.L
[X, Y] d=>(Y, [X,Z])= -([X, Y],Z)=O=[X,Z] Ef.L. 0
In view of the above theorem, we shall from now on assume that 9 is
itself simple, although most of the following discussions are also
valid (with some
24 Chapter II. Classification Theory of Compact Lie Groups
obvious modifications) for general compact Lie algebras, or at
least for semi simple compact Lie algebras.
Since a linear subspace f of a Lie algebra 9 is invariant
(w.r.t.Ad) if and only iff is a normal subalgebra, the adjoint
action of G on its Lie algebra 9 is irreducible if and only if G
(resp. g) is simple. Hence, for a simple compact Lie algebra g, two
invariant symmetric bilinear forms on 9 are proportional, i. e., Bl
(X, Y) =kB2 (X, Y). On the other hand, the following Cartan-Killing
form
B(X, Y) def Tr(adx · ad y )
is clearly an intrinsically defined, symmetric bilinear form, and
consequently, also invariant (w.r.t. inner automorphisms of g).
Hence, the above Cartan-Killing form B(X, Y) is proportional to an
invariant inner product on g. Note that adx is anti-symmetric and
the eigenvalues of adx are imaginary, therefore the eigen values
of (adx)2 are negative, and B(X, X) =tr(adx)2 <0. Hence, B(X, Y)
is negative definite, and it is natural to consider (X, Y) = - B(X,
Y) as the intrinsic inner product on g. In fact,it is not difficult
to prove the following characterization of compact Lie
algebras:
Theorem (11.5). A simple (or semi-simple) real Lie algebra 9 is
compact if and only if its Cartan-Killing form B(X, Y) is negative
definite.
(B) System of Simple Roots and Dynkin Diagram
Let 9 be a simple compact Lie algebra, 1) be an arbitrarily chosen
but fixed Cartan subalgebra, and LI s;;1)* (the dual space of 1))
be the root system of g. Let (X, Y) = - B(X, Y) = - tr adx . adx be
the intrinsic inner product on 9 and respectively the induced inner
products on 1) and 1)*. Then the Weyl group W acts on 1) (resp. on
1)*) as an orthogonal transformation group generated by the
reflections {ra, ±IXELI}. Note that since W permutes the chambers
simply transitively, it is convenient to choose an arbitrary but
fixed chamber Co and then define the positivity of roots as
follows:
IX >O<O>IX(Co) >0 ; IJ(ELI [positivity w.r.t. Col
Then it is clear that LI splits into the disjoint union of positive
roots LI + and negative roots LI-, i. e., LI = LI + U LI-,
and
Definition. Let nELI+ be the subset of those positive roots whose
hyperplanes have codimension one intersections with Co. Then it is
clear that n is the minimal subset of LI + with Co = naelt {1);}. n
is called the system of simple roots (w. r. t. Co).
Theorem (11.6). (i) Let IX, [3 E LI and p, q ~ 0 be the respective
largest integers such that ([3+plX), ([3-qlX)ELI. Then ([3+jlX)ELI
for -q~j~p and 2([3,IJ()/(IX,IJ() =(q-p).
§ 2. Classification of Compact Lie Groups 25
(ii) Let 7rc,1+ be the system of simple roots. Then a;=I=a j E7r
implies (a;,a)::::;O and 7r forms a basis of 9* such that every
roots f3 E ,1 is an integral linear combination of simple roots
with uniform sign, i. e.,
(iii) Let 91,92 be two simple compact Lie algebra ,11' ,12 and 7r
l' 7r2 be respectively their root systems and systems of simple
roots. Then 91 ~ 92=>7r 1 and 7r2 isometric, and an isometry of
7r 1,7r2 can be uniquely extended to an isometry of ,11,,12'
Proof. (i) Since ,1 is invariant under the Weyl group W, it is
obvious that (f3 + pa, a)
ra(f3 + p a) = (f3 + pa) - 2 ( . a = (f3 - q a). Hence, one has
2(f3, a)/(a, a) = (q - ]i). a, a)
The fact that (f3+ja)E,1 for -q::::;j::::;p follows directly from
Theorem (II.1) of ~ 1-A applies to AdG I Ga'
(ii) Observe that if a positive root a can be decomposed into the
sum of two other positive roots, a=a 1 +a2 , then the condition
a(h);?!O is already implied by the conditions a1(h);?!0 and
az(h);?!O, and hence can be omitted from the above expression of Co
as intersection of half spaces. Therefore, it is easy to see that
7r S; ,1 + are exactly those indecomposable positive roots. Let a;
=1= a j E 7r be two simple roots. Then (a j - a)¢, ,1, for
otherwise, either (aj - a)E ,1 + =>aj
=(aj-aj)+lXj is decomposable or (a j -a;}E,1+ and IXj=(lXj-lXj)+aj
is de composable. Hence, it follows from (i) that q =0 and 2(lXj ,
a)!(a j , a) = (q - p) = - p::::; 0, i. e., (a j , a)::::; O. Then,
it is a simple fact of linear algebra that positivity of a j and
(aj,a)::::;O for all 1::::;i::::;j::::;r=>7r={a 1, ... ,ar }
linearly independent. Therefore every positive (resp. negative)
root f3 E ,1 + can be expressed uniquely as linear combination of a
j E7r with non-negative (resp. non-positive) integral coefficients.
The fact that 7r spans 9* follows easily from the fact that 9 has
no center.
(iii) Since all Cartan subalgebras of a compact Lie algebra are
conjugate to each other, one may modify the given isomorphism I:
91->92 by a suitable inner automorphism so that I (91) = 92'
Hence it follows directly from the de finition of root system and
the intrinsic inner product that I induces an isometry of ,11 onto
,12' Furthermore, since. W acts simply transitively on the set of
chambers and the choice of a system of simple roots, 7r, is in 1 -1
correspondence with the choice of a chamber, it is clear that W
also permutes simply transitively among different systems of simple
roots. Hence, after a suitable modification by a con jugation of
an element of W, we have the induced isometry maps 7r1 onto
7r2.
Finally, it is an easy consequence of (i) that ,1 is completely
determined by the metric property of 7r, and hence an isometry of
7rl onto 7r2 extends uniquely to an isometry of ,11 onto ,12'
0
Dynkin diagram.
h 2(aj,a) 2(a j.a) ... . Observe t at ---) . --'- < 4 Implies
that the mteger 2(a j,IX)I(a j ,a;) IS
(aj,a; (aj,a) either 0, -1, - 2 or - 3, which geometrically
corresponds to the cases that the angle between a; and aj is 90",
120°, 135°, or 1500 respectively. Therefore, it is convienent to
record the metric property of the system of simple roots 7r in
terms of the following diagram:
26 Chapter II. Classification Theory of Compact Lie Groups
Symbolically, we represent each simple root by a point and we join
two points by a single, or double, or triple bond if the angle
between the respective simple roots is 120°, or 135°, or 150°.
Moreover, in the case of double or triple bond, i.e., 2(a
i,a)/(ai,ai ) = -2, or -3 and 2(ai,a)/(aj,a) = -1, the two simple
roots are not of equal length. Hence it is natural to use directed
bond (~ or -) to indicate which one is longer than the other. Such
a diagram is called the Dynkin diagram of the system of simple
roots n (or of the Lie algebra g).
Proposition. The Dynkin diagram of a simple compact Lie algebra is
connected. In general, there is a 1-1 correspondence between the
connected components of its Dynkin diagram and the simple normal
subalgebras of a compact Lie algebra g.
Proof. Suppose n' is a connected component of nand n = n' + nil.
Then it follows from definition that n'l. nil. Hence, by the above
theorem, (a',a")=O if a',a" are respectively linear combinations of
n', nil, and therefore a' + a" rf,1 (for 2(a',a")/
(a',a')=(q-p)=-p=O) which in turn implies [Xa"Xa,,] =0. Then, it is
easy to show that the subalgebra generated by X a" a' E < n') is
a normal subalgebra and the proposition follows. 0
A connected Dynkin diagram is called geometrically feasible if
there exists a set of vectors with the metric property indicated by
the given diagram. Of course, for the purpose of such a purely
geometric consideration, only the angles are essential and it is
not difficult to see that a necessary and sufficient condition for
a set of vectors {a l' ... , ar } with preassigned angles is
geometrically feasible is that ILtja)2 = (2)jaj, Ltjaj)~O forany
tjE1R, (and =0 only when Ltjaj=O). Hence, it is rather elementary
to prove the following:
Theorem (11.7). There are only the following geometrically feasible
connected Dynkin diagram:
An> n ~ 1; 0---0--0 •.. 0----0
Bn> n ~ 2; 0---0--0 ••. o~o
C n' n ~ 3; 0---0--0 .• ' 0<=0
Dn>n~4; 0----0' .'.~
E6:~;E7:~;E8:~
We refer the reader to § 5, Ch. IV of Jacobson's Lie Algebra for a
standard proof of the above elementary theorem.
(C) Chevalley Basis and Classification Theorem
In the case of compact Lie algebras, the classification theorem of
Cartan-Killing can be simply stated as follows:
§ 2. Classification of Compact Lie Groups 27
Theorem (11.8) (Cartan-Killing). The map oj assigning a semi-simple
compact Lie algebra 9 to its Dynkin diagram D(g) is a bijection
between the set oj isomorphic classes oj semi-simple compact Lie
algebras and the set oj all geometrically Jeasible Dynkin diagrams.
(cf. Theorem (II.6).
We shall prove the above theorem as follows:
Let 9 be a semi-simple compact Lie algebra gc = 9 ® CC be its
complexification. Let l) be a Cartan subalgebra and Ll be the root
system of g. Then, by definition, we have the following
decomposition of gc w. r. t. AdT (or adl}):
where CCa={XEgc; [H,X]=ia(H)·X for HEl}} and are one-dimensional.
If one restrict the adjoint action to those subgroups, {G,,;
aELl+}, next to the maximal torus T, then the above root-space
decomposition are strung into invariant subspaces of AdG« (resp.
adg.) as follows:
It is, then, straightforward to verify the following properties of
root-spaces decomposition:
(i) For a,pELl,[CCa,CCp]=CCa+P if we set CCa+p=O for the case
a+p~Ll.
Proof. [H, [Xa'Xp]] = [[H,Xa],Xp] + [Xa' [H,Xp]] =i(a+ P)(H)·
[Xa'Xp] , H El}, XaECCa, XpECCp.
and thefact that a + PELl = [Xa,Xpl;60 follows from the fact that
(I)= _qCCp+ jJ is irreducible w. r. 1. adg«.
(ii) For each root aELl, let H~El} be such that (H~,H)=a(H) for all
HEl} and Ha=2H:J(a, a). Then each Ha is an integral linear
combination of Hj =Ha; where {ai' ... , ar } =n is a system of
simple roots.
Proof. Write rj for rajE W. Then·
.H~=H~ - 2(a j ,a) H'. rJ I I (a. a.) J'
J' J
and hence
since {rj} generates W, and a=w'a j for suitable WE Wand ajEn, (ii)
follows.
(iii) By looking at ga® CC=l}a® CC+ga® CC=l}a® CC+({Ha} ®
CC+CC.+CC_.), it is obvious that [Xa,X_a]=A.·Ha#O for Xa,X- a
non-zero vectors of CCot
and CC- a respectively. Moreover, the following identity
28 Chapter II. Classification Theory of Compact Lie Groups
implies that [Xa'X -a] =i(Xa'X _J·H~=!(Xa'X _J(ex,ex)iHa, (Xa,X
-a)o6O. In fact, it is not difficult to show that there exist Xa'X
-a such that X",=X -a and (Xa,X_a)=2/lexI2, i.e., [Xa,X_a]=iHa.
[Two such pairs differ by a factor of eiO, i.e., {eiO X a , e- iO X
-,J]
(iv) Let {Xa,exEL1} be so chosen that Xa=X-a and [Xa,X-a]=iHa.
Define Na,p by [Xa,Xp]=N",pXa+p if (ex+P)EL1 and Na,p=O if
(ex+P)rf;L1. Then one has the following properties of the
structural coefficients N a, p'
(a) N -a, _p=Na,p and Na,p= -Np,a: Obvious from definition.
Na,p_ Np,y_ Ny,a (b) For a triangle of roots ex+P+y=O, one has W -
lexl2 - IPI2 '
Proof. 2 ~Ig = (NaP X _y,Xy)=([Xa,Xp],Xy)
[ ] N~ =-(Xp, XaXy )=Nya(Xp,X _p)=2 IPI2 '
(c) Suppose ex,P,y,b,EL1 and ex+P+y+b=O but no two are
proportional. Ibl 2
Then [Xa, [Xp,Xy]] =Npy[Xa'Xp+y] =Npy·Na,p+yX -o=N pyNoa IP +y12 .
X -0'
Hence, it follows from the Jacobi identity that
2 lex+Pl 2 ( 2. (d) INapl =p(q+1) IPI2 = q+1) ,
(where ex,PEL1 and P+jexEL1, -q~j~p~1).
Proof. [X -a [Xa'Xp]] = NaP [X -aXa+p]
=Nap·N-a,a+PXp apply (b) to (-ex),(-P), (ex+P)]
IPI 2
=Nap·N -p,-a lex+PI2' Xp
_ 2 Ifil2 --INapl 'lex+PI2XP [by(a)].
On the other hand, it is a simple fact of the Ga (or rather, gJ
representation on (L<Cp+ja) that [X_ a[Xa,Yp]]=-p(q+1)Yp for any
YpE<Cp. Hence, one has
o -!::urthermore, let Tap~TanTp, Gap=N (TaP) and Gap = Gap/Tap, It
is clear that Gap is of rank 2 and (GaP) cons~ts of all those roots
of L1 w~ich are linear com bination of ex, p. Since ex + P E L1 (G
aP)' the Dynkin diagram of G must be connected
§ 2. Classification of Compact Lie Groups 29
and hence either 0---0, or 0=0, or 0$0. Then it is a simple matter
(though a little
tedious) to check in the above three cases that p(q + 1) ICXI;I~12
= (q + 1)2.
Theorem (U.8') (Chevalley). Let 9 be a compact semi-simple Lie
algebra and gc = g® <C be its complexification; gc = l) ® <C
+ LaELl <Ca. Then, it is possible to choose XaE<Ca such
that
X a= X -a; [Xa' X -J = iHa = integral linear combination of iHaj
,
[Xa'Xp] = ±(q + l)Xa+p.
Hence the above {X a' CXE .1} together with {iH aj; cx jE n} form a
basis oj' gc such that all the structural constants are integral.
It is called the Chevalley basis of gcr.
Remarks (i) {Haj,CXjEn}u {(Xa+X -a)' i(Xa-X -a); cxE.1+} forms a
basis of g. (ii) The above theorem clearly implies the "if part"
ofthe classification theorem,
namely, .1l~.12=gl~g2 (orresp. glc~g2cl- (iii) Since the existence
of Lie algebras of the classical types, i. e., An' Bn, Cn'
D,P
is a well known fact, one need only to show the existence of a
simple Lie algebra; for each of the five exceptional types. In view
of the above explicit basis and structural constants, it is a
matter of straightforward verification.
Proof of the Chevalley theorem. Since IN apl2 =(q + 1)2, one need
only to show that it is possible to choose Xx so ..!hat Na,li ~re
all real numbers. Note that two pairs {Xa' X -a} and {X~, X'-a}
with Xa =X -a' X~ = X'-x and (Xa' X -a) =(X~, X'-a) =2/10:1 2
differ by a factor of eiO, i. e., X~ = eiO . X x and X_ a = e - iO
X _ ct' It is natural to begin with an arbitrary basis {Xa'X -a;
O:E.1+} with Xa=X -z and (Xa,X _a)=2/10:1 2
and then inductively adjust each pair by suitable factor of eiO to
make N a. p all real. Let .1p = {O:E.1; - P < 0: < p}, pE
.1+. We may assume that Na,pEIR for o:,{J,(cx + {J)E.1p and proceed
to prove the induction step that N a, p E IR for 0:, {J, (0: + If)
ELl p u { ± p } . If p can not be expressed as the sum of two
vectors of .1 p , then we don't have to adjust {X p' X _ p},
Otherwise, let p = 0: + {J be such an expression with smallest 0:.
We simply adjust {Xp,X _,,} so that N a•1i is real. (In fact, there
are exact two ways by making NaP> 0 or < 0 respectively).
Suppose ), + p = 0: + (J = p is another such expression. Then
o:+{J+(-),)+(-p)=O and it follows from (c) of (iv) that NAIL is
also real. This completes the induction step and the theorem
follows by induction. D
(D) A Theorem of Weyl and the Determination of Z(G) for Simple
Connected G
Theorem (H.9) (Weyl). Let G be a semi-simple compact connected Lie
group. Then the simply connected (or universal) covering group G of
G is also compact.
Proof Suppose the contrary. Then ker(G-->G) S Z(G) is an
infinite discrete abelian subgroup of G. Hence it is easy to see
that there are compact covering groups G1 ; G-->Gj-->G, with
Z(G 1) of arbitrary large finite order, which clearly contradicts
the fact that ord (Z (G I)) ~ the number of vertices in the Cartan
polyhedron of G1 (obviously bounded). D
30 Chapter II. Classification Theory of Compact Lie Groups
Finally, for the sake of reference, we list the Dynkin diagram of
the Cart an polydera together with the centers of those simple,
compact, simply connected Lie groups as follows:
>-"'0=>0
~ ~Z(Dn)={~:+Z2 if n even
if n odd
~~ Z(E,) Z"C(E,) ~~ Z(E,)~Z, ~ Z(Es)={id}.
Remark. (i) In the above diagram, each dot represesents a "wall" of
the Cart an polyhedron, they are respectively rxiH)~O, (ljEn and
f3(H)::;;' 1 where f3 is the highest root which is represented by
the dark dot.
(ii) Let x be a vertex of the Cart an polyhedron and x be the
opposite wall of x. Then the Dynkin diagram D(Gx ) of the
centralizor of x, Gx , is exactly the one obtained by removing the
dot of x from the above diagram of Cartan polyhedron. Hence, Z (G)
is in 1 -1 correspondence with those vertices with C( G) - {x} = D(
G).
§ 3. Classification of Irreducible Representations
(A) Classification Theorem (11.10) (Cartan-Weyl). Let G be a simply
connected, semi-simple compact Lie group, 9 be its Lie algebra, l)
be a Cartan subalgebra and Wbe the Weyl group of G. Also let LI be
the root system and n be the system of simple roots (w. r. t. a
fixed ordering). Let IjI be an irreducible complex representation
of G and Q(IjI) be the weight system of 1jI. We shall denote the
largest weight vector in Q(IjI) (w.r.t. the fixed ordering) by A",
and call it the highest weight ofljl. Then,
(i) The multiplicity of A", is one, and any two irreducible complex
representations 1jI, <p are equivalent iff their highest weights
are the same, i. e., IjI ~ <p ¢> A", = A<p.
(ii) The character of IjI can be given in terms of A", by the
following formula of Weyl:
I det(u) e21ti<Y(A",H)(t)
x",(t) = <Yf d t( ) 2"i<Y.l(t) , where i5=tI"EJ+ rx. <YEW
e u e
(iii) A vector A E 9* can be realized as the highest weight of an
irreducible complex representation iff 2(A,rx)/(rxj,rx)=qj are
non-negative integers for rxjEn.
§ 3. Classification of Irreducible Representations 31
Proof Let X",(g) be the character function of Ij; and X",(t) be its
restriction to the maximal torus T (with l) as its Lie algebra).
Let mew) be the multiplicity of w in Q(Ij;). Then, by definition,
x",(t) = Lm(w)e21tiw (l).
In view of the Weyl integration formula (cf. § 1-0), it is natural
to try to deter mine the function x",(t). Q(t)
=x",(tHLuewdetae21tiUdU»). Note that X",(t) is symmetric and Q(t)
is anti-symmetric and hence X",(t). Q(t) is anti-symmetric (w.r.t.
to W-action). If one expands an anti-symmetric function f in terms
of linear combination of L2-basis of L2(T) consists of
representation functions, it is easy to see that
f =" c .(" det(a) e21tiu , v(t») L.,veCo v L.,O'E W
where v runs through weight vectors in the positive Weyl chamber
Co. Hence,
x",(t)· Q(t) = {m(A",)' e21tiA .,U) + ... }. {Luewdet(a) e21tiu,
d(t)}
= m(A",)' Luewdet(a)· e21tiu (A.,H)(t) + possible more terms.
Now the irreducibility of Ij; (cf. Theorem (1.3'), § 1-C, Ch. I)
implies that
f - 1f 2 1 2 1 = G x",(g)· x",(g)dg = - T Ix",(t)Q(t)1 dt = -
IIx",(t)· Q(t)IIL2(T) w w
= ! IIm(A",)' Luewdet(a)e21tiu(A.,H)(t) + '''IIL2(T) (by Schur
orthogonality)
1 f 2 } 2 =-\m(A",) ·W+"· ~m(A",) . w
Hence, one must have m(A",) = 1 and X",(t)·Q(t) =
Luewdet(a)e21tiu(A.,H)(t) which is exactly the Weyl character
formula. Since the character is a complete invariant and the above
Weyl character formula gives an explicit expression of X",(t) in
turms of the highest weight A"" it is obvious that Ij; ~ <p~ A",
= A",. There fore, (i) and (ii) are completely proved; (iii) is a
direct consequence of the com pleteness theorem of Peter-Weyl.
0
. n (A", +£5,0() Corollary (11.10.1). dlmlj; = ~eLl+ (£5,0()
.
Proof Observe that dim Ij; = X", (id) = X",(O). However, the above
formula of X",(t) reduces to a meaningless form of % if one simply
substitute zero into it. Hence, we shall instead use the formula to
compute
dimlj; =limt_Ox",(t).
For this purpose, it is convient to identify l)* with l) via the
inner product and rewrite the Weyl formula as
L det(a)e21ti (U(A H),t) X(t) = Ldet(a)e21ti (ud,t)
32 Chapter II. Classification Theory of Compact Lie Groups
Notice that
I det(a) e2rri (<1(A +<1).s") = L det(O") e2rri (,,·".s(A
+<1»
= Q(s· (A + 8)) = TIaELI + 2i sin (n <1X,s(A + 8)).
(Cf. §1-D) Hence
d · ,I l' (-) l' Q(s·(A+8)) 1m.;! = Ims~oX s·(j = Ims~o
Q(s.b)
-TI r sinn<lX,s(A+b)_TI <1X,A+b) 0 - aELI+ Ims~o . < ~) -
aELI+ < ~) .
SIn n IX, S' u IX, u
Chapter III. An Equivariant Cohomology Theory Related to Fibre
Bundle Theory
In the application of cohomology theory to the study of topological
transformation groups, a natural and convenient formalism is to
define an equivariant cohomology theory for the category of
G-spaces which effectively reflects the cohomological behavior of
both the space and the G-action. Following an idea of A. Borel [cf.
B 10], we shall define the equivariant cohomology of a G-space X to
be the ordinary cohomology of the total space X G of the universal
bundle, X -+ X G -+ EG, with the given G-space X as its typical
fibre, namely
The rationale of adopting the above equivariant cohomology theory
in terms of the universal bundle construction is roughly the
following:
(i) Intuitively and heuristically, the complexity of the G-action
on X will be reflected in the complexity of the associated
universal bundle X -+ X G -+ BG ,
e. g., the associated universal bundle is trivial if and only if
the G-action on X is trivial. And the classical obstruction theory,
especially the characteristic classes theory, clearly demonstrates
that cohomology theory can then be used to detect the complexity of
X G -+ BG , which, in turn, reflects the complexity of the G-action
itself.
(ii) Technically, it is not difficult to see that such an
equivariant cohomology theory not only possesses convenient formal
properties but is also effectively computable.
§ 1. The Construction of H6(X) and its Formal Properties
(A) The Construction of A. Borel
Let X be a given G-space and EG -+ BG be the universal G-bundle.
Then the total space X G of the associated universal bundle with X
as fibre may be regarded as: the orbit space of EG x X
XG =EG xG X =(EG x X)/G
34 Chapter III. Equivariant Cohomology Theory
where the G-action is given by g·(e,x)=(eg-1,gx). Since the two
projections are obviously equivariant, one has the following
commutative diagram:
BG +-( _..::.",--' - XG _....::":.:..2 ---+1 X IG
Next suppose that Yis a K-space, h:G~K is a homomorphism and f:X~Y
is an h-equivariant map, i.e., f(g·x)=h(g)-f(x). Then, it is
LOAD MORE