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## The Lie Algebras su(N)

An Introduction Walter Pfeifer

112 pp, 46 line figures

Dr. Walter Pfeifer: Stapfenackerweg 9, CH 5034 Suhr, Switzerland

2003, revised 2008

### Contents

Contents 1

Preface 3

1 Lie algebras 4

1.1 Definition 4

1.2 Isomorphic Lie algebras 10

1.3 Operators and Functions 10

1.4 Representation of a Lie algebra 13

1.5 Reducible and irreducible representations, multiplets. 14

2 The Lie algebras su(N) 18

2.1 Hermitian matrices 18

2.2 Definition 19

2.3 Structure constants of su(N) 23

3 The Lie algebra su(2) 25

3.1 The generators of the su(2)-algebra. 25

3.2 Operators constituting the algebra su(2) 29

3.3 Multiplets of su(2) 32

3.4 Irreducible representations of the su(2)-algebra 36

3.5 Direct products of irreducible representations and their function sets. 38

3.6 Reduction of direct products of su(2)-representations and multiplets. 42

3.7 Graphical reduction of direct products of su(2)-multiplets. 49

4 The Lie algebra su(3) 51

4.1 The generators of the su(3)-algebra 51

4.2 Subalgebras of the su(3)-algebra. 53

4.3 Step operators and states in su(3) 54

4.4 Multiplets of su(3) 57

4.5 Individual states of the su(3)-multiplet and their multiplicities. 59

4.6 Dimension of the su(3)-multiplet. 66

4.7 The smallest su(3)-multiplets. 68

4.8 The fundamental multiplet of su(3). 69

4.9 The hypercharge Y. 71

4.10 Irreducible representations of the su(3) algebra. 73

4.11 Casimir operators. 76

4.12 The eigenvalue of the Casimir operator C1 in su(3). 77

4.13 Direct products of su(3)-multiplets. 79

4.14 Decomposition of direct products of multiplets by means of Young diagrams.
82

5 The Lie algebra su(4) 85

5.1 The generators of the su(4)-algebra, subalgebras. 85

5.2 Step operators and states in su(4). 90

5.3 Multiplets of su(4). 92

5.4 The charm C. 96

5.5 Direct products of su(4)-multiplets. 97

5.6 The Cartan–Weyl basis of su(4). 98

6 General properties of the su(N)-algebras 106

6.1 Elements of the su(N)-algebra. 106

6.2 Multiplets of su(N). 107

References 110

Index 111

### Preface

Lie algebras are not only an interesting
mathematical field but also efficient tools to analyze the properties of
physical systems. Concrete applications comprise the formulation of symmetries
of Hamiltonian systems, the description of atomic, molecular and nuclear
spectra, the physics of elementary particles and many others.

In particular,
Lie algebras very frequently appear and "there is hardly any
student of physics or mathematics who will never come across symbols like
and
" (Fuchs, Schweigert, 1997, p. XV). For instance, the algebra
describes angular momenta,
is related to harmonic oscillator properties (Talmi,
1993, p. 621) or to rotation properties of systems (Talmi, 1993, p. 797;
Pfeifer, 1998, p.113) and
represents states of elementary particles in the
quark model (Greiner, Müller, 1994, p. 367)

This book is mainly directed to
undergraduate students of physics or to interested physicists. It is conceived
to give directly a concrete idea of the
algebras and of their laws. The detailed
developments, the numerous references to preceding places, the figures and many
explicit calculations of matrices should enable the beginner to follow. Laws
which are given without proof are marked clearly and mostly checked with
numerical tests. Knowledge of linear algebra is a prerequisite. Many results are
obtained, which hold generally for (simple) Lie algebras. Therefore the text on
hand can make the lead-in to this field easier.

The structure of the contents is simple.
First, Lie algebras are defined and the
algebras are introduced starting from anti-Hermitian
matrices. In chapter 3 the
algebras, their multiplets and the direct
product of the multiplets are investigated. The treatment of the
multiplets in chapter 4 is more labour-intensive.
Casimir operators and methods to determine the dimensions of multiplets are
introduced. In chapter 5 the
multiplets are represented three-dimensionally.
Making use of the
algebra, the properties of the Cartan-Weyl basis
are demonstrated. Chapter 6 points to general relations of the
algebras.

Any reader who detects errors is urged to
contact the author via the email address mailbox@walterpfeifer.ch. Of course,
the author is willing to answer questions.

### 1. Lie algebra Lie algebras

In this chapter the Lie algebra is defined
and the properties of its underlying vector space are described. Discussing the
role of the basis elements of the algebra one is led to the structure constants,
to some of their symmetry properties and to their relationship to the adjoint
matrices. With these matrices the Killing form is constructed. As a natural
example for a Lie algebra, general square matrices are looked at. The notion of
"simplicity" is introduced.

Operators which constitute a Lie algebra
act on vector spaces of functions. By means of the corresponding expansion
coefficients, properties of these operators are shown. The matrices of the
expansion coefficients make up a representation of the algebra. If this
representation is reducible it can be transformed to the equivalent block
diagonal form. The functions which are assigned to an irreducible representation
form a multiplet.

1.1 Definition

### What is a Lie algebra?

A Lie
algebra comprises the elements
, which may be general matrices with certain
properties (real/complex matrix elements, internal symmetries)
or linear operators etc. The elements can be combined in two different
ways. To come first, the elements of a Lie algebra must be able to form Lie
products
, which are also named Lie brackets
or commutators
. For square matrices
and
the well-known relation

(1.1.1)

defines a
commutator, which, of course, is again a matrix. On the other hand, if the
elements of a Lie algebra are operators or more general quantities the
commutator
has still to be defined, but the right hand side
of (1.1.1) need not to be satisfied. We want that, in addition to the formation
of the commutator, it must be possible to combine the elements of the Lie
algebra linearly, i.e., they constitute a vector space.

Thus, the
**definition of a Lie** algebra demands the following properties of its elements

a) the
commutator of two elements is again an element of the algebra

(1.1.2)

b) a
linear combination
of the elements
and
with the real or complex numbers
and
is again an element of the algebra i.e.

(1.1.3)

Therefore
the element 0 (zero) belongs to the algebra.

c) The
following linearity is postulated

(1.1.4)

d)
Interchanging both elements of a commutator results in the relation

. (1.1.5)

With
(1.1.5) and (1.1.4) one proves that also
holds. Of course, we have
.

e)
Finally the Jacobi identity
has to be satisfied as follows

. (1.1.6)

Using
(1.1.1) one shows that this identity holds for square matrices.

Note that
we don't demand that the commutators are associative i.e. the relation
is not true in general. Some authors choose
other logical sequences of postulates for the Lie algebra. For instance one can
define
and deduce (1.1.5) (Carter, 1995, p. 5).

f) In
addition to a) up to e) we demand that a Lie algebra has a finite dimension *n*
* *i.e. it comprises a set of
*n* linearly independent elements
, which act as a basis
, by which every element *x* of the algebra can be
represented like this

. (1.1.7)

In other
words, the algebra constitutes an *n*–dimensional vector space
. Sometimes the dimension is named order. If the coefficients
in (1.1.7) and
in (1.1.3) are real the algebra is named real
. In a complex or complexified
algebra the coefficients are complex.

**We
summarize** points a) up to f): *A Lie algebra is
a vector space with an alternate product satisfying the Jacobi condition.*

In
accordance with the definition point e) the basis elements
meet the **Jacobi identity**
(1.1.6). If this is the case, the arbitrary
elements

,
and
(1.1.8)

satisfy
the identity as well. We introduce a symbol for the Jacobi form

, (1.1.9)

which is
linear in *a*, *b*, *c*. Replacing these elements by *x*, *
y* and *z* and inserting the expressions for *x*, *y* and *z*
we obtain

. (1.1.10)

Since
every Jacobi form on the right hand side vanishes, it is proved that in order to
have the identity (1.1.6), it suffices to ask that the condition is satisfied by
the basis elements.

Clearly
the **choice of the basis**
is arbitrary. With a nonsingular matrix
, which contains the real or complex numbers
, a new basis
can be built this way

. (1.1.11)

The new
elements meet the Jacobi identity, which is shown in (1.1.10). It is a
well-known fact that the elements
are linearly independent if
do and if
is non-singular. Of course a change of the basis
of a complex (or real) Lie algebra by means of complex (or real) coefficients
in (1.1.11) restores the algebra and it keeps
its name.

### The structure constants

Due to
(1.1.2) the commutator of two basis elements belongs also to the algebra and,
following (1.1.7) it can be written like this

. (1.1.12)

The
coefficients
are called *structure constants*
, relative to the
-basis. They are not invariant under a transformation as per
(1.1.11) (see (1.1.17)). Given the set of basis elements, the structure
constants specify the Lie algebra completely. Of course, a Lie algebra with
complex structure constants is complex itself. Clearly, if the structure
constants are real, a real Lie algebra can be constructed on its basis.

The
commutator of the elements *x* and *y*, (1.1.8), can be expressed by
the basis elements as follows

(1.1.13)

Analogously the structure constants of the new basis
, (1.1.11), are determined this way

. (1.1.14)

Since
is nonsingular, the inverse matrix
exists and

holds, (1.1.15)

which we
insert in (1.1.14) like this

. (1.1.16)

Thus,
the structure constants of the new basis read

. (1.1.17)

We insert
(1.1.12) in the Jacoby identity (1.1.6) and obtain

(1.1.18)

Because
the basis elements
are linearly independent, we get *n*
equations for given values *i, j, k* like this

, *m* = 1, ... , *n*. (1.1.19)

We write
the antisymmetry relation (1.1.5) of commutators as
and insert equation (1.1.12), which yields
. Due to the linear independence of the basis elements we
obtain *n* equations for given values *i,k* of the following form

. (1.1.20)

In
Section 2.3 we will see that in
-algebras the structure constants relative to the
-basis are antisymmetric in all indices and not only in the
first two ones like in (1.1.20).

### The adjoint matrices

Here we
introduce the **adjoint** or
-**matrices**. As per (1.1.2) the commutator of an
arbitrary element
of a Lie algebra with basis elements
must be a linear combination of the basis
elements similarly to (1.1.12). Therefore we write

(1.1.21)

The
coefficients
are elements of the
-matrix
. It is easy to show that it is linear in the arguments like
this:

(1.1.22)

and the
following commutator relation holds:

. (1.1.23)

replacing
by
in (1.1.21) yields

. (1.1.24)

Comparing
with (1.1.12) we obtain

. (1.1.25)

We will
come back to the adjoint matrices in Sections 1.4 and 3.1.

These
-matrices appear also in the so-called "**Killing form**",
which plays an important part in the analysis of the structure of algebras.

### The Killing form

The "Killing form"
corresponding to any two elements
and
of a Lie algebra is defined by

.
(1.1.26)

The symbol
denotes the trace of the matrix product. It can
be shown that the Killing form
is symmetric and bilinear in the elements
and
. We write the Killing form of the basis elements
and
of a Lie algebra

.
(1.1.27)

With (1.1.25) we have

. (1.1.28)

In Sections 2.3 and 3.1 we will deal with
the Killing form of
and
, respectively.

### Simplicity

From the theoretical point of view it is
important to know whether a Lie algebra is *simple*. We give the
appropriate definitions concisely.

*A Lie algebra is said to be simple if it
is not Abelian and does not possess a proper invariant Lie subalgebra.*

The terms used here are defined as follows:

- A Lie algebra
is said to be *Abelian* if
for all
. Thus, in an Abelian Lie algebra all the structure constants are zero.

- A *subalgebra*
of a Lie algebra
is a subset of elements of
that
themselves form a Lie algebra with the same commutator and field as that
of
. This implies that
is real if
is real and
is complex if
is complex.

- A subalgebra
is said to be proper if at least one element of
is not
contained in
.

- A subalgebra
of a Lie algebra
is said to be *invariant* if
for all and
. An invariant subalgebra is also named *ideal*.

We will meet simple Lie algebras in
Sections 2.2 and 4.2.

### Example

Obviously, **square N-dimensional**

**matrices**( -matrices, sometimes called matrices of rank

*N*) constitute a Lie algebra. The commutators and linear combinations are again

*N*-dimensional matrices and the conditions a) up to e) are satisfied. The basis matrices for matrices with complex matrix elements can be chosen like this, where :

(1.1.29)

The set (1.1.29) can be written using the
square matrices
, which shows the value 1 at the position
and zeros elsewhere. The canonical basis
forms the basis (1.1.29).

Using these
basis elements the definition point f) is met
with real coefficients
in (1.1.7). In a word, the complex matrices of
order
form a real vector space of dimension
. On the other hand, the same vector space of complex
-matrices can be constructed using complex coefficients and
the basis
. The resulting complex algebra is named
and it has the dimension
. In Section 2.2 we will interrelate it with
.

### 1.2 Isomorphic Lie algebras

A Lie algebra with elements
is isomorphic to an other Lie algebra with
elements
if an unambiguous mapping
exists which is reversible, i.e.
. It has to meet the relations

(1.2.1)

for all
scalars
and
.

The
structure of both Lie algebras is identical and we expect that both have the
same dimension *n*. Supposed
are the basis elements of the first Lie algebra,
then
is a basis of the isomorphic Lie algebra. Of
course, two isomorphic Lie algebras have the same structure constants.

Here the **Ado**** theorem** is given without proof:

*Every
abstract Lie algebra is isomorphic to a Lie algebra of matrices with the
commutator defined as in equation* (1.1.1).

Consequently, the properties of Lie algebras can be studied by investigating the
relatively vivid matrix algebras.

### 1.3 Operators and Functions

**The
general set-up**

In
addition to the definition points a) up to f) of section 1.1 we demand that
operators which form a Lie algebra act linearly on functions. These functions or
states or "kets" make up a *d*-dimensional vector space with linearly
independent basis
functions
or
. That is, an arbitrary function
of this space can be written as

(1.3.1)

with real
or complex coefficients
.

We demand
that the operators of the Lie algebra
acting on a function produce a transformation
of the function so that the result lies still in the original vector space. If
we let act the element ** x** of the Lie algebra on the basis function
, it yields the following linear combination of basis
functions

. (1.3.2)

Notice the
sequence of the indices in
. Making use of (1.3.1) we can write down the action of
** x** on an arbitrary function
like this

, (1.3.3)

which is a
consequence of the linearity of ** x**. In section 1.4 we will see that
is linear in

**, i.e. . Therefore equation (1.3.3) results in**

*x*
. (1.3.4)

We see
that the action of the operators constituting a Lie algebra on the functions is
mainly described by the coefficients
.

Furthermore we demand that *the vector space of the functions is an inner
product**
space*. That is to
say, for every pair of functions
and
an inner product
is defined with the following well-known
properties

i.
(complex conjugate)

ii.

iii.
(c: complex or real number)

iv.

v. , if and only if = 0.

Note that we obtain
from i and iii. The state
is named adjoint
of
.

In an inner product vector space with
linearly independent basis functions
it is always possible to construct an orthogonal
basis
, which satisfies

*i*, *k* = 1, 2, ... ,*d*. (1.3.5)

In the
following, we will presuppose that the basis functions are orthonormalized,
which is not a restriction. Assuming this property we form the inner product of
both sides of eq. (1.3.2) with the basis function

. (1.3.6)

In section
1.4 we will refer to the quadratic matrix
, which contains the matrix elements
.

**Further
properties**

We look
into further properties of the operators constituting a Lie algebra and of the
affiliated basic functions. In analogy to (1.3.6) we make up the inner product
of both sides of eq. (1.3.1) with the basis function
as follows

, (1.3.7)

which we
insert again in eq. (1.3.1) like this

. (1.3.8)

The
functions
and
are ket states and can be marked by the ket
symbol
this way

. (1.3.9)

Formally
we can regard the expression
as an operator which restores the state
like the identity operator
.** **That is the *completeness**
relation*

. (1.3.10)

We apply
it in order to formulate a **matrix element of a sequence of two operators**,
say ** x** and

**, like this**

*y*
. (1.3.11)

Making use
of (1.3.6) we have the result

(1.3.12)

I.e., the
matrices
multiply in analogy with the corresponding
operators.

Now, we
investigate the matrix
containing the matrix elements
, the adjoint of this matrix and adjoint operators. By
definition, the adjoint matrix of
,
, is transposed and conjugate complex, i.e.,

. (1.3.13)

condition
i of the inner product definition yields

. (1.3.14)

We define
the adjoint operator
by

. (1.3.15)

Using this
relation we write the matrix element of a sequence of two operators this way
or
. Thus we have found

(1.3.16)

in
accordance with the matrix relation (2.2.3).

### 1.4 Representation of a Lie algebra

**Definition**

Suppose that
and
are elements of a Lie algebra
and that to every
there exists a
-matrix
such that

and (1.4.1)

. (1.4.2)

Then these
matrices are said to form a
-dimensional representation of
.

Clearly, the set of matrices
forms a Lie algebra over the same field (with
real or complex coefficients
) as
.

In
(1.1.21) the **adjoint** or
** matrices** were introduced. Starting from
(1.1.22) and (1.1.23) for simple Lie algebras it can be proved that the matrices
constitute a representation of the algebra
. This is the* regular or adjoint representation.*

We come
back to the elements
, (1.3.2) and (1.3.8), which make up the matrix
. We now maintain that the *matrices*
,
, *... constitute a representation of the Lie algebra
, ...* . As in section 1.3 we presuppose that the
set of basis functions
is associated to the Lie algebra and the
relation (1.3.2) holds as
with
, see (1.3.6). According to (1.3..5) the states
can be regarded as orthonormalized, i.e.
.

We prove
our assertion. First we deal with the linearity property of the matrices
. For the moment we handle with the matrix element (*lk*):

(1.4.3)

Of course,
the same relation holds for the whole matrices:

. (1.4.4)

Next we
treat the commutator relation, which will be similar to (1.4.2). For the
commutator
we don't take the abstract form (1.1.2) but we
choose

. (1.4.5)

Following
(1.4.4) we write

. (1.4.6)

With eq.
(1.3.12) we obtain

. (1.4.7)

Therefore,
the equations (1.4.4) and (1.4.7) show that the
represent the Lie algebra
.

### 1.5 Reducible and irreducible representations, multiplets.

Let's
suppose that the vector space of functions of a Lie algebra is the direct sum of
two subspaces
and
, i.e. that the basis functions are split into two sets
and
. Furthermore, we assume that the subspaces are *invariant*,
i.e., for every operator
of the Lie algebra the relation (1.3.2) is
modified like this

(1.5.1)

That is,
the transformation of the functions takes place only in the vector subspace of
the original basis function. This means that

(1.5.2)

Consequently only those coefficients
are non-zero which meet the conditions (1.5.2).
Therefore the matrix
has the following form

(1.5.3)

If in a
Lie algebra all matrices
are divided in this way in two or more squares
along the diagonal, the representation is named *reducible*
. some authors call it "*completely reducible*".

On the
other hand, if the operators constituting the algebra transform every function
among functions of the entire vector space, there is *no nontrivial invariant
subspace* (as defined at the beginning of this section). In this case, the
matrices
cannot be decomposed in two or more square
matrices along the diagonal, and the representation is named *irreducible*.
The vector space of the functions of such a representation is called a *
multiplet*. For particle physics, multiplets are very important.

We go back
to the representation (1.5.3). The squares in the matrix are irreducible
representations, i.e. the reducible representation is decomposed into the*
direct sum of* *irreducible representations*.

If we
interchange the basis functions or if we transform the basis more generally, it
happens that we lose the structure of (1.5.3) i.e. the non-zero matrix elements
can be spread over the entire matrix and the representation seems not to be
reducible. However, one can reduce such a representation to the form (1.5.3) by
a similarity transformation, where every matrix of the representation is
transformed in the same way as set out below. H. Weyl proved that *every
finite dimensional representation of a semi-simple algebra decomposes into the
direct sum of irreducible *representations. Since the algebras
are simple, this theorem holds also for them.

We
investigate the **transformations of representations**. If
is a non-singular
matrix, starting from a representation of the Lie algebra with
the *d*-dimensional matrices
, we are able to construct a new representation constituted by
the elements

. (1.5.4)

The matrix
is independent of
.

First, we
have to show that the matrices
meet the linearity relation (1.4.4), namely

(1.5.5)

Then, the
commutator relation is treated like this

(1.5.6)

Consequently the set of matrices
also forms a *d*-dimensional representation
.
and
are said to be *equivalent*
representations.

A given
representation
is reducible, if one can find the matrix
in (1.5.4) so that all representation matrices
of the equivalent representation have the same
block-diagonal form (1.5.3) with two or more squares. If the matrices of the
basis elements,
have obtained this form, obviously every matrix
of the equivalent representation shows the same form (Note: the commutator of
two matrices with the same block diagram structure generates a matrix which has
the same structure).

Assume
that the basis
functions
are assigned to the representation
. What can be said about the **basis functions**
**which belong to the equivalent representation**
We claim that the basis functions

(with
from (1.5.4)) (1.5.7)

meet the
relation
. We let the linear operator
act on equation (1.5.7). Making use of (1.3.2)
we get

.

Because
is non-singular the equation (1.5.7) can be
inverted like this
, which we insert using (1.5.4) like this

(1.5.8)

If the
matrices
of the equivalent representation have the form
(1.5.3) with two or more squares on the diagonal, the vector space of the
functions
is divided in invariant subspaces. Due to
(1.5.8) the operator
(and all the operators of the Lie algebra)
transform the states of a subspace among themselves as described in the example
at the beginning of this section. The structure of the basis functions is given
in (1.5.7).

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