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arXiv:cond-mat/0304363v2 28 Jan 2004

DFTT 10/03

Random matrix theory and symmetric spaces

M. Caselle1and U. Magnea2

1Department of Theoretical Physics, University of Torino

and INFN, Sez. di Torino

Via P. Giuria 1, I-10125 Torino, Italy

2Department of Mathematics, University of Torino

Via Carlo Alberto 10, I-10123 Torino, Italy

caselle@to.infn.it

magnea@dm.unito.it

Abstract

In this review we discuss the relationship between random matrix theories and symmetric

spaces. We show that the integration manifolds of random matrix theories, the eigenvalue

distribution, and the Dyson and boundary indices characterizing the ensembles are in strict

correspondence with symmetric spaces and the intrinsic characteristics of their restricted

root lattices. Several important results can be obtained from this identification. In par-

ticular the Cartan classification of triplets of symmetric spaces with positive, zero and

negative curvature gives rise to a new classification of random matrix ensembles. The re-

view is organized into two main parts. In Part I the theory of symmetric spaces is reviewed

with particular emphasis on the ideas relevant for appreciating the correspondence with

random matrix theories. In Part II we discuss various applications of symmetric spaces

to random matrix theories and in particular the new classification of disordered systems

derived from the classification of symmetric spaces. We also review how the mapping from

integrable Calogero–Sutherland models to symmetric spaces can be used in the theory of

random matrices, with particular consequences for quantum transport problems. We con-

clude indicating some interesting new directions of research based on these identifications.

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Contents

1Introduction4

2Lie groups and root spaces9

2.1Lie groups and manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . .9

2.2The tangent space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10

2.3Coset spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11

2.4The Lie algebra and the adjoint representation . . . . . . . . . . . . . . . .13

2.5Semisimple algebras and root spaces. . . . . . . . . . . . . . . . . . . . .15

2.6The Weyl chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21

2.7The simple root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .22

3Symmetric spaces23

3.1Involutive automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . .24

3.2The action of the group on the symmetric space . . . . . . . . . . . . . . .26

3.3 Radial coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27

3.4The metric on a Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5The algebraic structure of symmetric spaces . . . . . . . . . . . . . . . . .32

4Real forms of semisimple algebras 33

4.1The real forms of a complex algebra . . . . . . . . . . . . . . . . . . . . . .33

4.2The classification machinery . . . . . . . . . . . . . . . . . . . . . . . . . .37

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5The classification of symmetric spaces41

5.1 The curvature tensor and triplicity. . . . . . . . . . . . . . . . . . . . . . 42

5.2Restricted root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46

5.3Real forms of symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . .52

6Operators on symmetric spaces54

6.1Casimir operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55

6.2Laplace operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58

6.3Zonal spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .64

6.4The analog of Fourier transforms on symmetric spaces . . . . . . . . . . . .68

7Integrable models related to root systems72

7.1The root lattice structure of the CS models . . . . . . . . . . . . . . . . . .73

7.2Mapping to symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . 74

8 Random matrix theories and symmetric spaces 78

8.1Introduction to the theory of random matrices . . . . . . . . . . . . . . . . 78

8.1.1What is random matrix theory? . . . . . . . . . . . . . . . . . . . . 78

8.1.2Some of the applications of random matrix theory . . . . . . . . . . 79

8.1.3Why are random matrix models successful?. . . . . . . . . . . . .81

8.2The basics of matrix models . . . . . . . . . . . . . . . . . . . . . . . . . . 82

8.3Identification of the random matrix integration manifolds . . . . . . . . . .86

8.3.1Circular ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . .86

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8.3.2Gaussian ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

8.3.3Chiral ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . .96

8.3.4 Transfer matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . 100

8.3.5The DMPK equation . . . . . . . . . . . . . . . . . . . . . . . . . . 104

8.3.6BdG and p–wave ensembles . . . . . . . . . . . . . . . . . . . . . . 107

8.3.7S–matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

8.4Identification of the random matrix eigenvalues and universality indices . . 110

8.4.1Discussion of the Jacobians of various types of matrix ensembles . . 111

8.5Fokker–Planck equation and the Coulomb gas analogy . . . . . . . . . . . 115

8.5.1The Coulomb gas analogy . . . . . . . . . . . . . . . . . . . . . . . 116

8.5.2Connection with the Laplace–Beltrami operator . . . . . . . . . . . 118

8.5.3Random matrix theory description of parametric correlations . . . . 119

8.6A dictionary between random matrix ensembles and symmetric spaces . . . 119

9On the use of symmetric spaces in random matrix theory120

9.1Towards a classification of random matrix ensembles. . . . . . . . . . . . 121

9.2Symmetries of random matrix ensembles . . . . . . . . . . . . . . . . . . . 122

9.3Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

9.4 Use of symmetric spaces in quantum transport . . . . . . . . . . . . . . . . 127

9.4.1Exact solvability of the DMPK equation in the β = 2 case . . . . . 128

9.4.2 Asymptotic solutions in the β = 1,4 cases . . . . . . . . . . . . . . 133

9.4.3Magnetic dependence of the conductance . . . . . . . . . . . . . . . 133

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9.4.4 Density of states in disordered quantum wires. . . . . . . . . . . . . 135

10 Beyond symmetric spaces136

10.1 Non–Cartan parametrization of symmetric spaces and S–matrix ensembles 136

10.1.1 Non-Cartan parametrization of SU(N)/SO(N) . . . . . . . . . . . 138

10.2 Clustered solutions of the DMPK equation . . . . . . . . . . . . . . . . . . 141

10.3 Triplicity of the Weierstrass potential . . . . . . . . . . . . . . . . . . . . . 144

11 Summary and conclusion147

A Appendix: Zonal spherical functions151

A.1 The Itzykson–Zuber–Harish–Chandra integral . . . . . . . . . . . . . . . . 153

A.2 The Duistermaat–Heckman theorem . . . . . . . . . . . . . . . . . . . . . . 154

1Introduction

The study of symmetric spaces has recently attracted interest in various branches of physics,

ranging from condensed matter physics to lattice QCD. This is mainly due to the gradual

understanding during the past few years of the deep connection between random matrix

theories and symmetric spaces. Indeed, this connection is a rather old intuition, which

traces back to Dyson [1] and has subsequently been pursued by several authors, notably by

H¨ uffmann [2]. Recently it has led to several interesting results, like for instance a tentative

classification of the universality classes of disordered systems. The latter topic is the main

subject of this review.

The connection between random matrix theories and symmetric spaces is obtained simply

through the coset spaces defining the symmetry classes of the random matrix ensembles.

Although Dyson was the first to recognize that these coset spaces are symmetric spaces,

the subsequent emergence of new random matrix symmetry classes and their classification

in terms of Cartan’s symmetric spaces is relatively recent [3, 4, 5, 6, 7]. Since symmetric

spaces are rather well understood mathematical objects, the main outcome of such an

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identification is that several non–trivial results concerning the behavior of the random

matrix models, as well as the physical systems that these models are expected to describe,

can be obtained.

In this context an important tool, that will be discussed in the following, is a class of inte-

grable models named Calogero–Sutherland models [8]. In the early eighties, Olshanetsky

and Perelomov showed that also these models are in one–to–one correspondence with sym-

metric spaces through the reduced root systems of the latter [9]. Thanks to this chain of

identifications (random matrix ensemble – symmetric space – Calogero–Sutherland model)

several of the results obtained in the last twenty years within the framework of Calogero–

Sutherland models can also be applied to random matrix theories.

The aim of this review is to allow the reader to follow this chain of correspondences. To

this end we will devote the first half of the paper (sections 2 through 7) to the necessary

mathematical background and the second part (sections 8 through 10) to the applications

in random matrix theory. In particular, in the last section we discuss some open directions

of research. The reader who is not interested in the mathematical background could skip

the first part and go directly to the later sections where we list and discuss the main results.

This review is organized as follows:

The first five sections of Part I (sections 2–6) are devoted to an elementary introduction

to symmetric spaces. As mentioned in the Abstract, these sections consist of the material

presented in [10], which is a self–contained introductory review of symmetric spaces from

a mathematical point of view. The material on symmetric spaces should be accessible to

physicists with only elementary background in the theory of Lie groups. We have included

quite a few examples to illustrate all aspects of the material. In the last section of Part I,

section 7, we briefly introduce the Calogero–Sutherland models with particular emphasis

on their connection with symmetric spaces.

After this introductory material we then move on in Part II to random matrix theories

and their connection with symmetric spaces (section 8). Let us stress that this paper

is not intended as an introduction to random matrix theory, for which very good and

thorough references already exist [11, 26, 27, 28, 29]. In this review we will assume that the

reader is already acquainted with the topic, and we will only recall some basic information

(definitions of the various ensembles, main properties, and main physical applications).

The main goal of this section is instead to discuss the identifications that give rise to the

close relationship between random matrix ensembles and symmetric spaces.

Section 9 is devoted to a discussion of some of the consequences of the above mentioned

identifications. In particular we will deduce, starting from the Cartan classification of

symmetric spaces, the analogous classification of random matrix ensembles. We discuss

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the symmetries of the ensembles in terms of the underlying restricted root system, and

see how the orthogonal polynomials belonging to a certain ensemble are determined by

the root multiplicities. In this section we also give some examples of how the connection

between random matrix ensembles on the one hand, and symmetric spaces and Calogero–

Sutherland models on the other hand, can be used to obtain new results in the theoretical

description of physical systems, more precisely in the theory of quantum transport.

The last section of the paper is devoted to some new results that show that the mathe-

matical tools discussed in this paper (or suitable generalizations of these) can be useful for

going beyond the symmetric space paradigm, and to explore some new connections between

random matrix theory, group theory, and differential geometry. Here we discuss clustered

solutions of the Dorokhov–Mello–Pereyra–Kumar equation, and then we go on to discuss

the most general Calogero–Sutherland potential, given by the Weierstrass P–function, and

show that it covers the three cases of symmetric spaces of positive, zero and negative cur-

vature. Finally, in the appendix we discuss some intriguing exact results for the so called

zonal spherical functions, which not only play an important role in our discussion, but are

also of great relevance in several other branches of physics.

There are some important and interesting topics that we will not review because of lack of

space and competence. For these we refer the reader to the existing literature. In particular

we shall not discuss:

• the supersymmetric approach to random matrix theories and in particular their clas-

sification in terms of supersymmetric spaces. Here we refer the reader to the original

paper by M. Zirnbauer [4], while a good introduction to the use of supersymmetry

in random matrix theory and a complete set of the relevant references can be found

in [12];

• the very interesting topic of phase transitions. For this we refer to the recent and

thorough review by G. Cicuta [13];

• the extension to two–dimensional models of the classification of symmetric spaces,

and more generally the methods of symmetric space analysis [14];

• the generalization of the classification of symmetric spaces to non–hermitean random

matrices [15] (see however a discussion in the concluding section 11);

• the so called q–ensembles [16];

• the two–matrix models [17] and multi–matrix models [18] and their continuum limit

generalization.

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The last item in the list given above is a very interesting topic, which has several physical

applications and would indeed deserve a separate review. The common feature of these

two– and multi–matrix models which is of relevance for the present review, is that they all

can be mapped onto suitably chosen Calogero–Sutherland systems. These models represent

a natural link to two classes of matrix theories which are of great importance in high energy

physics: on the one hand, the matrix models describing two–dimensional quantum gravity

(possibly coupled to matter) [19], and on the other hand, the matrix models pertaining

to large N QCD, which trace back to the original seminal works of ’t Hooft [20]. In

particular, a direct and explicit connection exists between multi–matrix models (the so

called Kazakov–Migdal models) for large N QCD [21] and the exactly solvable models of

two–dimensional QCD on the lattice [22].

The mapping of these models to Calogero–Sutherland systems of the type discussed in this

review can be found for instance in [23]. The relevance of these models, and in particular of

their Calogero–Sutherland mappings, for the condensed matter systems like those discussed

in the second part of this review, was first discussed in [24]. A recent review on this aspect,

and more generally on the use of Calogero–Sutherland models for low-dimensional models,

can be found in [25].

We will necessarily be rather sketchy in discussing the many important physical applica-

tions of the random matrix ensembles to be described in section 8. We refer the reader

to some excellent reviews that have appeared in the literature during the last few years:

the review by Beenakker [26] for the solid state physics applications, the review by Ver-

baarschot [27] for QCD–related applications, and [28, 29] for extensive reviews including a

historical outline.

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Part I

The theory of symmetric spaces has a long history in mathematics. In this first part of the

paper we will introduce the reader to some of the most fundamental concepts in the theory

of symmetric spaces. We have tried to keep the discussion as simple as possible without

assuming any previous familiarity of the reader with symmetric spaces. The review should

be particularly accessible to physicists. In the hope of addressing a wider audience, we

have almost completely avoided using concepts from differential geometry, and we have

presented the subject mostly from an algebraic point of view. In addition we have inserted

a large number of simple examples in the text, that will hopefully help the reader visualize

the ideas.

Since our aim in Part II will be to introduce the reader to the application of symmetric

spaces in physical integrable systems and random matrix models, we have chosen the

background material presented here with this in mind. Therefore we have put emphasis

not only on fundamental issues but on subjects that will be relevant in these applications

as well. Our treatment will be somewhat rigorous; however, we skip proofs that can be

found in the mathematical literature and concentrate on simple examples that illustrate

the concepts presented.The reader is referred to Helgason’s book [30] for a rigorous

treatment; however, this book may not be immediately accessible to physicists. For the

reader with little background in differential geometry we recommend the book by Gilmore

[31] (especially Chapter 9) for an introduction to symmetric spaces of exceptional clarity.

In section 2, after reviewing the basics about Lie groups, we will present some of the most

important properties of root systems. In section 3 we define symmetric spaces and discuss

their main characteristics, defining involutive automorphisms, spherical decomposition of

the group elements, and the metric on the Lie algebra. We also discuss the algebraic

structure of the coset space.

In section 4 we show how to obtain all the real forms of a complex semisimple Lie algebra.

The same techniques will then be used to classify the real forms of symmetric spaces in

section 5. In this section we also define the curvature of a symmetric space, and discuss

triplets of symmetric spaces with positive, zero and negative curvature, all corresponding

to the same symmetric subgroup. We will see why curved symmetric spaces arise from

semisimple groups, whereas the flat spaces are associated to non–semisimple groups. In

addition, in section 5 we will define restricted root systems. The restricted root systems are

associated to symmetric spaces, just like ordinary root systems are associated to groups.

As we will discuss in detail in Part II of this paper, they are key objects when considering

the integrability of Calogero–Sutherland models.

In section 6 we discuss Casimir and Laplace operators on symmetric spaces and men-

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tion some known properties of the eigenfunctions of the latter, so called zonal spherical

functions. These functions play a prominent role in many physical applications.

The introduction to symmetric spaces we present contains the basis for understanding the

developments to be discussed in more detail in Part II. The reader already familiar with

symmetric spaces is invited to start reading in the last section of Part I, section 7, where

we give a brief introduction to Calogero–Sutherland models.

2Lie groups and root spaces

In this introductory section we define the basic concepts relating to Lie groups. We will

build on the material presented here when we discuss symmetric spaces in the next section.

The reader with a solid background in group theory may want to skip most or all of this

section.

2.1Lie groups and manifolds

A manifold can be thought of as the generalization of a surface, but we do not in general

consider it as embedded in a higher–dimensional euclidean space. A short introduction to

differentiable manifolds can be found in ref. [32], and a more elaborate one in refs. [33] and

[34] (Ch. III). The points of an N–dimensional manifold can be labelled by real coordinates

(x1,...,xN). Suppose that we take an open set Uαof this manifold, and we introduce local

real coordinates on it. Let ψαbe the function that attaches N real coordinates to each

point in the open set Uα. Suppose now that the manifold is covered by overlapping open

sets, with local coordinates attached to each of them. If for each pair of open sets Uα, Uβ,

the fuction ψα◦ ψ−1

go smoothly from one coordinate system to another in this region. Then the manifold is

differentiable.

β

is differentiable in the overlap region Uα∩ Uβ, it means that we can

Consider a group G acting on a space V . We can think of G as being represented by

matrices, and of V as a space of vectors on which these matrices act. A group element

g ∈ G transforms the vector v ∈ V into gv = v′.

If G is a Lie group, it is also a differentiable manifold. The fact that a Lie group is a

differentiable manifold means that for two group elements g, g′∈ G, the product (g,g′) ∈

G × G → gg′∈ G and the inverse g → g−1are smooth (C∞) mappings, that is, these

mappings have continuous derivatives of all orders.

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Example: The space Rnis a smooth manifold and at the same time an abelian group.

The “product” of elements is addition (x,x′) → x + x′and the inverse of x is −x. These

operations are smooth.

Example: The set GL(n,R) of nonsingular real n × n matrices M, detM ?= 0, with

matrix multiplication (M,N) → MN and multiplicative matrix inverse M → M−1is a

non–abelian group manifold. Any such matrix can be represented as M = e

Xiare generators of the GL(n,R) algebra and tiare real parameters.

?

itiXiwhere

2.2The tangent space

In each point of a differentiable manifold, we can define the tangent space. If a curve

through a point P in the manifold is parametrized by t ∈ R

xa(t) = xa(0) + λata = 1,...,N(2.1)

where P = (x1(0),...,xN(0)), then λ = (λ1,...,λN) = (˙ x1(0),..., ˙ xN(0)) is a tangent vector

at P. Here ˙ xa(0) =

tangent space. In particular, the tangent vectors to the coordinate curves (the curves

obtained by keeping all the coordinates fixed except one) through P are called the natural

basis for the tangent space.

d

dtxa(t)|t=0. The space spanned by all tangent vectors at P is the

Example: In euclidean 3–space the natural basis is {ˆ ex, ˆ ey, ˆ ez}. On a patch of the unit

2–sphere parametrized by polar coordinates it is {ˆ eθ, ˆ eφ}.

For a Lie group, the tangent space at the origin is spanned by the generators, that play the

role of (contravariant) vector fields (also called derivations), expressed in local coordinates

on the group manifold as X = Xa(x)∂a(for an introduction to differential geometry see

ref. [35], Ch. 5, or [34]). Here the partial derivatives ∂a=

field. That the generators span the tangent space at the origin can easily be seen from the

exponential map. Suppose X is a generator of a Lie group. The exponential map then

maps X onto etX, where t is a parameter. This mapping is a one–parameter subgroup,

and it defines a curve x(t) in the group manifold. The tangent vector of this curve at the

origin is then

∂

∂xaform a basis for the vector

d

dtetX|t=0= X(2.2)

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All the generators together span the tangent space at the origin (also called the identity

element).

2.3Coset spaces

The isotropy subgroup Gv0of a group G at the point v0∈ V is the subset of group elements

that leave v0fixed. The set of points that can be reached by applying elements g ∈ G to

v0is the orbit of G at v0, denoted Gv0. If Gv0= V for one point v0, then this is true for

every v ∈ V . We then say that G acts transitively on V .

In general, a symmetric space can be represented as a coset space. Suppose H is a subgroup

of a Lie group G. The coset space G/H is the set of subsets of G of the form gH, for g ∈ G.

G acts on this coset space: g1(gH) is the coset (g1g)H. We will refer to the elements of the

coset space by g instead of by gH, when the subgroup H is understood from the context,

because of the natural mapping described in the next paragraph. If g / ∈ H, gH corresponds

to a point on the manifold G/H away from the origin, whereas hH = H (h ∈ H) is the

identity element identified with the origin of the symmetric space. This point is the north

pole in the example below.

If G acts transitively on V , then V = Gv for any v ∈ V . Since the isotropy subgroup

Gv0leaves a fixed point v0invariant, gGv0v0= gv0= v ∈ V , we see that the action of the

group G on V defines a bijective action of elements of G/Gv0on V . Therefore the space

V on which G acts transitively, can be identified with G/Gv0, since there is one–to–one

correspondence between the elements of V and the elements of G/Gv0. There is a natural

mapping from the group element g onto the point gv0on the manifold.

Example: The SO(2) subgroup of SO(3) is the isotropy subgroup at the north pole of

a unit 2–sphere imbedded in 3–dimensional space, since it keeps the north pole fixed. On

the other hand, the north pole is mapped onto any point on the surface of the sphere by

elements of the coset SO(3)/SO(2). This can be seen from the explicit form of the coset

representatives. As we will see in eq. (3.20) in subsection 3.5, the general form of the

elements of the coset is

M = exp

?

0C

0−CT

?

=

? √I2− XXT

−XT

X

√1 − XTX

?

(2.3)

where C is the matrix

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C =

?

t2

t1

?

(2.4)

and t1, t2are real coordinates. I2in eq. (2.3) is the 2 ×2 unit matrix. For the coset space

SO(3)/SO(2), M is equal to

M = exp

?2

i=1

?

tiLi

?

,L1=1

2

0

0

0 −1 0

0

0

0

1

,

L2=1

2

0

0

0 1

0 0

−1 0 0

(2.5)

The third SO(3) generator

L3=1

2

01 0

−1 0 0

0 0 0

(2.6)

spans the algebra of the stability subgroup SO(2), that keeps the north pole fixed:

exp(t3L3)

0

0

1

=

0

0

1

(2.7)

The generators Li(i = 1,2,3) satisfy the SO(3) commutation relations [Li,Lj] =1

Note that since the Liand the tiare real, C†= CT.

2ǫijkLk.

In (2.3), M is a general representative of the coset SO(3)/SO(2). By expanding the

exponential we see that the explicit form of M is

M =

1 + (t2)2(cos√

t1t2(cos√

−t2sin√

(t1)2+(t2)2−1)

(t1)2+(t2)2

t1t2(cos√

1 + (t1)2(cos√

−t1sin√

(t1)2+(t2)2−1)

(t1)2+(t2)2

t2sin√

t1sin√

?

(t1)2+(t2)2

√

√

(t1)2+(t2)2

(t1)2+(t2)2

(t1)2+(t2)2−1)

(t1)2+(t2)2

(t1)2+(t2)2−1)

(t1)2+(t2)2

(t1)2+(t2)2

(t1)2+(t2)2

√

(t1)2+(t2)2

(t1)2+(t2)2

√

(t1)2+(t2)2

cos

(t1)2+ (t2)2

(2.8)

Thus the matrix X =

?

x

y

?

is given in terms of the components of C by (cf. eq. (3.21)):

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X =

?

x

y

?

=

t2sin√

t1sin√

(t1)2+(t2)2

√

√

(t1)2+(t2)2

(t1)2+(t2)2

(t1)2+(t2)2

(2.9)

Defining now z = cos

of the 2–sphere:

?

(t1)2+ (t2)2, we see that the variables x, y, z satisfy the equation

x2+ y2+ z2= 1 (2.10)

When the coset space representative M acts on the north pole it is easily seen that the

orbit is all of the 2–sphere:

M

0

0

1

=

. . x

. . y

. . z

0

0

1

=

x

y

z

(2.11)

This shows that there is one–to–one correspondence between the elements of the coset and

the points of the 2–sphere. The coset SO(3)/SO(2) can therefore be identified with a unit

2–sphere imbedded in 3–dimensional space.

2.4The Lie algebra and the adjoint representation

A Lie algebra G is a vector space over a field F. Multiplication in the Lie algebra is given

by the bracket [X,Y ]. It has the following properties:

[1] If X, Y ∈ G, then [X,Y ] ∈ G,

[2] [X,αY + βZ] = α[X,Y ] + β[X,Z] for α, β ∈ F,

[3] [X,Y ] = −[Y,X],

[4] [X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y ]] = 0 (the Jacobi identity).

The algebra G generates a group through the exponential mapping. A general group

element is

M = exp

??

i

tiXi

?

;ti∈ F, Xi∈ G(2.12)

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We define a mapping adX from the Lie algebra to itself by adX : Y → [X,Y ]. The

mapping X → adX is a representation of the Lie algebra called the adjoint representation.

It is easy to check that it is an automorphism: it follows from the Jacobi identity that

[adXi,adXj] = ad[Xi,Xj]. Suppose we choose a basis {Xi} for G. Then

adXi(Xj) = [Xi,Xj] = Ck

ijXk

(2.13)

where we sum over k. The Ck

transform as mixed tensor components. They define the matrix (Mi)jk= Cj

with the adjoint representation of Xi. One can show that there exists a basis for any

complex semisimple algebra in which the structure constants are real. This means the

adjoint representation is real. Note that the dimension of the adjoint representation is

equal to the dimension of the group.

ijare called structure constants. Under a change of basis, they

ikassociated

Example: Let’s construct the adjoint representation of SU(2). The generators in the

defining representation are

J3=1

2

?

1

0 −1

0

?

,J±=1

2

??

0 1

1 0

?

± i

?

0 −i

i0

??

(2.14)

and the commutation relations are

[J3,J±] = ±J±,[J+,J−] = 2J3

(2.15)

The structure constants are therefore C+

and the adjoint representation is given by (M3)++ = 1, (M3)−− = −1, (M+)+3 = −1,

(M+)3−= 2, (M−)−3= 1, (M−)3+= −2, and all other matrix elements equal to 0:

3+= −C+

+3= −C−

3−= C−

−3= 1, C3

+−= −C3

−+= 2

M3=

0 0

0 1

0 0 −1

0

0

,

M+=

00 2

−1 0 0

00 0

,

M−=

0 −2 0

00

10

0

0

,

(2.16)

These representation matrices are real, have the same dimension as the group, and satisfy

the SU(2) commutation relations [M3,M±] = ±M±, [M+,M−] = 2M3.

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