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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
1 Introduction4
2 Lie groups and root spaces9
2.1 Lie groups and manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . .9
2.2 The tangent space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10
2.3 Coset spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11
2.4 The Lie algebra and the adjoint representation . . . . . . . . . . . . . . . .13
2.5 Semisimple algebras and root spaces. . . . . . . . . . . . . . . . . . . . .15
2.6 The Weyl chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .21
2.7 The simple root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . .22
3 Symmetric spaces 23
3.1 Involutive automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . .24
3.2 The action of the group on the symmetric space . . . . . . . . . . . . . . .26
3.3 Radial coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27
3.4 The metric on a Lie algebra . . . . . . . . . . . . . . . . . . . . . . . . . .30
3.5 The algebraic structure of symmetric spaces . . . . . . . . . . . . . . . . .32
4 Real forms of semisimple algebras33
4.1 The real forms of a complex algebra . . . . . . . . . . . . . . . . . . . . . .33
4.2 The classification machinery . . . . . . . . . . . . . . . . . . . . . . . . . .37
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5 The classification of symmetric spaces 41
5.1 The curvature tensor and triplicity . . . . . . . . . . . . . . . . . . . . . .42
5.2 Restricted root systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .46
5.3 Real forms of symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . .52
6 Operators on symmetric spaces 54
6.1 Casimir operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .55
6.2 Laplace operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .58
6.3 Zonal spherical functions . . . . . . . . . . . . . . . . . . . . . . . . . . . .64
6.4 The analog of Fourier transforms on symmetric spaces . . . . . . . . . . . .68
7 Integrable models related to root systems72
7.1 The root lattice structure of the CS models . . . . . . . . . . . . . . . . . .73
7.2 Mapping to symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . .74
8 Random matrix theories and symmetric spaces 78
8.1 Introduction to the theory of random matrices . . . . . . . . . . . . . . . .78
8.1.1 What is random matrix theory? . . . . . . . . . . . . . . . . . . . .78
8.1.2 Some of the applications of random matrix theory . . . . . . . . . .79
8.1.3 Why are random matrix models successful?. . . . . . . . . . . . . 81
8.2 The basics of matrix models . . . . . . . . . . . . . . . . . . . . . . . . . .82
8.3 Identification of the random matrix integration manifolds . . . . . . . . . .86
8.3.1 Circular ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . .86
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8.3.2 Gaussian ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . .90
8.3.3 Chiral ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . .96
8.3.4 Transfer matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . 100
8.3.5 The DMPK equation . . . . . . . . . . . . . . . . . . . . . . . . . . 104
8.3.6 BdG and p–wave ensembles . . . . . . . . . . . . . . . . . . . . . . 107
8.3.7 S–matrix ensembles . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8.4 Identification of the random matrix eigenvalues and universality indices . . 110
8.4.1 Discussion of the Jacobians of various types of matrix ensembles . . 111
8.5 Fokker–Planck equation and the Coulomb gas analogy . . . . . . . . . . . 115
8.5.1 The Coulomb gas analogy . . . . . . . . . . . . . . . . . . . . . . . 116
8.5.2 Connection with the Laplace–Beltrami operator . . . . . . . . . . . 118
8.5.3 Random matrix theory description of parametric correlations . . . . 119
8.6 A dictionary between random matrix ensembles and symmetric spaces . . . 119
9On the use of symmetric spaces in random matrix theory 120
9.1Towards a classification of random matrix ensembles . . . . . . . . . . . . 121
9.2 Symmetries of random matrix ensembles . . . . . . . . . . . . . . . . . . . 122
9.3 Orthogonal polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
9.4Use of symmetric spaces in quantum transport . . . . . . . . . . . . . . . . 127
9.4.1Exact solvability of the DMPK equation in the β = 2 case . . . . . 128
9.4.2Asymptotic solutions in the β = 1,4 cases . . . . . . . . . . . . . . 133
9.4.3 Magnetic 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
1 Introduction
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
0 1 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.
2.4 The 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
0 0 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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