Download Simple examples of Lie groups and Lie algebras

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New terms: Lie group, group manifold, generator, Lie algebra, structure constant,
Cartan metric, adjoint representation, compact group, semi-simple group, representation
1. The simple example of the compact Lie group SO(2)
A Lie group is a group whose group elements are functions of some continuously
varying parameters. The parameters which specify the group elements form a smooth
space, a differentiable manifold, called the group manifold with the property that the
group operations are compatible with the smooth structure of the space. One of the
key ideas which will allow us to make significant progress in understanding the theory
of Lie groups is to replace the global object, the group, with its local or linearized
version, its Lie algebra. Most of the following chapters will be about Lie algebras.
However, before we study Lie algebras, we will introduce an example of a Lie group
which is simple enough that we can understand its structure as a group at the same
level that we understood finite discrete groups.
One of the simplest examples of a Lie group is the group of proper rotations in
two-dimensional space, SO(2). These can be defined as the set of 2 × 2 orthogonal
matrices with determinant one. It is very easy to write down a generic form of such
a matrix. We simply find the matrix which implements the rotation of a vector by
an angle φ in the x-y plane. Since the z coordinate always remains unchanged, we do
not need to worry about it, that is, our matrix needs to operate on two-dimensional
vectors only and is therefore a 2 × 2 matrix. The matrix which implements a rotation
by angle φ in the x-y plane is
cos φ − sin φ
(1)
R(φ) =
sin φ cos φ
We see that a rotation is specified by giving the value of one parameter, the angle
φ. This value can be in the range φ ∈ [0, 2π). This is a continuously infinite set of
parameters. There are an infinite number of group elements.
The set of matrices R(φ) defined in equation (1) form a group. The identity is
the identity matrix R(0) = I. It is easy to see that the matrices in (1) have the
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properties
R(φ1 )R(φ2 ) = R(φ1 +φ2 ) , R(φ+2π) = R(φ) , R(−φ) = R(2π−φ) = Rt (φ) = R−1 (φ)
This implies, in particular, the set of matrices R(φ) is closed under multiplication,
and it contains the inverses of all of its elements. It therefore forms a group. Moreover, since R(φ1 )R(φ2 ) = R(φ1 + φ2 ) = R(φ2 )R(φ1 ), it is an abelian group.
Since R−1 (φ) = Rt (φ), R(φ)Rt (φ) = I, R(φ) is said to be an orthogonal matrix.
The set of all 2 × 2 orthogonal matrices forms the group O(2). Since, R(φ) has
det R(φ) = cos2 φ + sin2 φ = 1, R(φ) is a special orthogonal matrix. The subgroup
of O(2) which are the orthogonal matrices with determinant one form the special
orthogonal group SO(2).
SO(2) is a Lie group. A Lie group is a group whose elements depend on continuously varying parameters. Here, we see that R(φ) is a matrix whose matrix elements
are smooth functions of the parameter φ. In fact, SO(2) is a particularly nicely
behaved type of Lie group called a “compact Lie group”. Roughly speaking, the
latter designation is due to the fact that the parameter on which the group elements
depend, φ, itself occupies a compact domain, the interval [0, 2π) of the real number
line, with 2π and 0 identified. One can think of it as parameterizing the points on
a circle, S 1 . The space on which the parameters live is called the “group manifold”.
We would say that the group manifold of SO(3) is the unit circle, or one-dimensional
unit sphere, S 1 .
Generally, we are interested in the question of finding and classifying the representations of a Lie group. In the present example, a representation D(φ) would be a
set of matrices which have the same multiplication law as the group elements, that
is
(2)
D(φ1 )D(φ2 ) = D(φ1 + φ2 ) , D(φ + 2π) = D(φ)
The second identity above is needed if D(φ) is a homomorphism. A homomorphism
is a mapping which must have the property that every element of the range is mapped
onto a unique element of the domain. The image of a fixed element of the group,
R(φ) is the set of matrices D(φ), D(φ±2π), D(φ±4π), . . . which must all be identical
if the representation is a homomorphism.
As in the case of discrete Abelian groups, we say that two representations are
equivalent if there exists an invertible matrix S such that
D̃(φ) = SD(φ)S −1 , ∀φ ∈ [0, 2π)
A representation is said to be reducible if it is equivalent to one which is block
diagonal for all elements of the group. We are generally interested in finding and
classifying the irreducible representations of a group. The defining representation
of SO(2) in (1) is a reducible representation. It can be diagonalized by a similarity
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transform and, it is equivalent to
iφ
e
0
D(φ) =
0 e−iφ
(3)
We can easily extend the proof that we found from discrete Abelian groups that
any representation of SO(2) is equivalent to a unitary representation. Essential to
being able to extend the proof it the construction of the matrix
Z 2π
1
(4)
H=
dφD(φ)D† (φ)
2π 0
In this formula, the “volume” of the group manifold, 2π, takes the place of the order
of the finite discrete group |G|. The integral over the parameter φ takes the place of
the summation over group elements. H has the property D(φ)HD† (φ) = H. This
property comes from the translation invariance of the integration measure in (4),
(5)
1
D(φ)HD (φ) = D(φ)
2π
†
Z
0
2π
1
dφ̃D(φ̃)D (φ̃) D (φ) =
2π
†
†
Z
2π
dφ̃D(φ̃ + φ)D† (φ̃ + φ) = H
0
We note that, if the representation is unitary, H = I. If not, since H is a positive
1
Hermitian matrix, we can define H ± 2 and it can be used to conjugate the represen1
1
tation as D̃(φ) = H − 2 D(φ)H 2 . Then D̃(φ) is a unitary matrix, D̃(φ)D̃† (φ) = I.
Thus, we can assume that a representation is unitary. SO(2) is an abelian group.
Any two elements commute R(φ1 )R(φ2 ) = R(φ2 )R(φ1 ) and, therefore, for any representation, D(φ1 )D(φ2 ) = D(φ2 )D(φ1 ). The matrix corresponding to any group
element commute with the matrix corresponding to any other group element. If they
are unitary matrices, there is, then, always an equivalent representation where the
matrices are simultaneously diagonal. This means that any representation can be
reduced to a direct sum of one-dimensional representations. Thus, we see that all of
the irreducible representations of SO(2) are one-dimensional.
Consider a one-dimensional representation D(m) (φ). We know that, being a representation, D(m) (φ1 )D(m) (φ2 ) = D(m) (φ1 + φ2 ). and D(m) (φ + 2π) = D(m) (φ). For
one-dimensional matrices, these are only possible if
(6)
D(m) (φ) = eimφ , m = 0, ±1, ±2, . . .
Thus, we find an infinite array of one-dimensional irreducible representations. Of
these, only D(1) (φ) and D(−1) (φ) are isomorphisms. The other representations, with
index m 6= ±1 are homomorphisms with kernels.
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Using the explicit list of irreducible representations (6), we can easily find an
analog of the orthogonality theorem,
Z 2π
1
(7)
dφD(m) (φ)D(n)† (φ) = δmn
2π 0
Moreover, the characters of one-dimensional matrices are just the matrices themselves, so we see that the orthogonality theorem for characters also generalizes,
∞
∞
X
X
(m)
(m)†
(8)
D (φ1 )D
(φ2 ) = 2πδ(φ1 − φ2 ) =
eimφ1 e−imφ2
m=−∞
m=−∞
The defining representation in (1) is reducible and it contains the m = 1 and the
m = −1 irreducible representations.
The decomposition of direct products of irreducible representations into direct
sums of irreducible representations is really easy in this case,
(9)
D(m) (φ)D(n) (φ) = D(m+n) (φ)
Finally, we note that we will also be interested in unitary groups. The oneparameter unitary group is the group U (1) of one-dimensional unitary matrices. An
element of U (1) is a one-by-one matrix, that is, a number whose complex conjugate
is its inverse. That is, it is a phase,
U (1) = {u| u = eiφ , 0 ≤ φ < 2π}
It is easy to confirm that this set forms a group. In fact, it is isomorphic to
SO(2), being identical to one of the two representations of SO(2) which is in fact
an isomorphism, as well as being a homomorphism. (The other “faithful” representation of SO(2) is D(φ) = e−iφ . Clearly D(0) (φ) = 1 is not an isomorphism. The other representations, D(m) (φ) = eimφ with |m| > 1 have a kernel
φ = 0, 2π/|m|, 4π/|m|, ...2π − 2π/|m| and are therefore not isomorphisms. )
2. The Lie groups SO(3) and SU (2)
SO(3) is the group of 3 × 3 orthogonal matrices with determinant one. Orthogonal
−1
t
matrices have the property RRt = I, or Rij
= Rij
= Rji . This implies that
the columns of R form a set of three ortho-normal vectors. Special matrices have
determinant one. SO(3) is the group of special orthogonal matrices.
SO(3) has a continuously infinite number of elements and it is another example
of a Lie group. These elements correspond to the set of all possible proper rotations
in three-dimensional space. There are various ways of denoting the rotations using
parameters.
There are other choices of the three parameters which are useful for various purposes. For example, the Euler angles are used to describe the orientation of rigid
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Figure 1. The orientation of an airplane is described by the roll,
pitch and yaw angles by which it rotates from a reference position.
bodies in classical mechanics. They stem from the observation that any rotation
can be implemented by first making a rotation by angle α about the x-axis, then a
rotation by β about the z-axis and finally a rotation by angle γ about the x-axis. As
a matrix, the rotation is




0
0
cos β − sin β 0 1
1
0
0
R(α, β, γ) = 0 cos γ − sin γ   sin β cos β 0 0 cos α − sin α
1
0
0 0 sin α cos α
0 sin γ cos γ
which furnishes a rather explicit parameterization of the SO(3) matrix R. Another
set of angles which are used in navigational systems to specify the orientation of a
ship or an airplane. They are the Tait-Bryan angles angles which compose a rotation
by subsequent rotations about the x, y and z axes,
(10)




1
0
0
cos θ2 0 sin θ2
cos θ3 − sin θ3 0
1
0  0 cos θ1 − sin θ1 
R(θ1 , θ2 , θ3 ) =  sin θ3 cos θ3 0  0
0 sin θ1 cos θ1
0
0
1 − sin θ2 0 cos θ2
In navigation, the angle θ1 is the “roll” , θ2 is the “pitch” and θ3 the “yaw”, as
illustrated in figure 1. It is this parameterization of the space of all possible rotations
that we will use extensively in the following. What distinguishes SO(3) as a Lie group
is the fact that, apart from a few singular points, the group elements themselves are
smooth functions of the parameters.
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Infobox 2.1 The Pauli matrices
Consider the three 2 × 2
0
(11)
σ1 =
1
hermitian matrices
1
0 −i
1 0
, σ1 =
, σ1 =
,
0
i 0
0 −1
They are called the Pauli matrices. They have the following useful properties:
(12)
σi σi + σj σi = 2δij , σi σi − σj σi = 2i
3
X
ijk σk
k=1
σi†
(13)
σi =
(14)
Tr (σi ) = 0 , Tr (σi σj ) = 2δij , Tr (σi σj σk ) = 2iijk
with ijk the totally antisymmetric tensor with 123 = 1.
A way of parameterizing SO(3) which is useful for understanding its group manifold, comes from embedding it in a slightly larger group, SU (2). The Lie group SU (2)
is the group of all 2 × 2 unitary matrices. A unitary matrix, U is one which obeys
the equation U U † = I. A special unitary matrix the additional property det U = 1.
The set of all 2 × 2 unitary matrices forms the Lie group U (2). SU (2) is the subgroup of U (2) consisting special unitary matrices. It is convenient to parameterize
the elements of SU (2) as
U = a0 I + i
3
X
ai σi , (a0 , a1 , a2 , a3 ) = (a∗0 , a∗1 , a∗2 , a∗3 ) , a20 + a21 + a22 + a23 = 1
i=1
where σi are the Pauli matrices whose properties are summarized in Infobox 2.1.
We can think of (a0 , a1 , a2 , a3 ) as being the parameters which specify an element of
SU (2). The group manifold of SU (2) is the set of all values that these four real
numbers can have. They comprise the three-dimensional unit sphere, S 3 embedded
in four dimensional euclidean space, that is, the locus of the equation
a20 + a21 + a22 + a23 = 1
This is a compact space and SU (2) is a compact Lie group.
To see the correspondence between SU(2) and SO(3), we use the Pauli matrices
x1

to embed a three dimensional vector ~x = x2  in a 2 × 2 traceless Hermitian matrix
x3
as
3
X
x3
x1 − ix2
(15)
x=
σi xi = ~x · σ =
x1 + ix2
−x3
i=1
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This gives us a one-to-one correspondence between 2 × 2 Hermitian matrices and
real-valued three-dimensional vectors. If we have a three-dimensional vector, we can
form a traceless hermitian matrix as in (15). If we have a traceless hermitian matrix,
x, we can find the three dimensional vector to which it corresponds using the identity
1
(16)
xi = Trσi x
2
Now that we have embedded three-dimensional vectors in 2 × 2 matrices, you
might ask how a rotation might act. In the vector notation, of course, a rotation
is xi → Rij xj where Rij is an element of SO(3). This is a mapping of xi onto the
coordinates in a rotated reference frame. The new coordinates are linear functions of
the old coordinates. We need a mapping of the matrix x such that the components
of the new matrix are linearP
functions of the old coordinates. Another observation
that helps is that det x = − 3i=1 x2i = −~x2 . The transformation of x should be one
which preserves its determinant, as well as the properties that it is hermitian and
traceless. The most general such transformation is
x → x̃ = U xU †
where U is an SU (2) matrix.
Such a mapping must be a rotation. In fact, it is easy to find the corresponding
SO(3) matrix,
1
Rij = Tr σj U σi U †
(17)
2
Thus, for every matrix U there is a rotation. One might ask whether the converse is
true, is there a U for every rotation R? The answer is yes, in fact there are two U ’s
for each R. It is clear that, for a given R, if U satisfies (17) then −U also satisfies
(17). Therefore, for each R there are at least two U ’s. In fact, it turns out that this
is the only multiplicity. SU (2) has exactly two elements for every element of SO(3).
The group manifold of SO(3) is then gotten from the group manifold of SU (2) by an
identification which identifies U and −U . In terms of the parameters (a0 , a1 , a2 , a3 )
it is the three-sphere S 3 , the locus of
a20 + a21 + a22 + a23 = 1
with an identification (a0 , a1 , a2 , a3 ) ≡ (−a0 , −a1 , −a2 , −a3 ). This is the real projective space called RP 3 .
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Infobox 2.2 Lie algebra
A Lie algebra, L, is a linear vector space consisting of “vectors” M1 , M3 , M3 , . . .
which is equipped with a binary operation (M1 , M2 ). First of all, L has the
properties of a vector space: for all M1 , M2 , M3 ∈ L,
(1) L is closed under addition:
M1 + M2 ∈ L
(2) Addition is associative:
M1 + (M2 + M3 ) = (M1 + M2 ) + M3
(3) Addition is commutative:
M1 + M2 = M2 + M1
(4) L contains the zero vector 0 with the property 0 + M1 = M1 .
Any vector space is associated with a field whose elements are used to
do “scalar multiplication” of the vectors by scalars. In the following, we
shall use the field of the real numbers exclusively.
If a, b, c, ... are real numbers,
(5) Closure of scalar multiplication
aM1 ∈ L
(6) Scalar multiplication distributes over vector addition:
a(M1 + M2 ) = aM1 + aM2
(7) Another distributive law:
(a + b)M1 = aM1 + bM2
In addition, the binary operation has the properties
(1) Closure,
(M1 , M2 ) ∈ L
(2) Antisymmetric,
(M1 , M2 ) = −(M2 , M1 )
(3) Linear:
(aM1 + bM2 , M3 ) = a(M1 , M3 ) + b(M2 , M3 )
(4) Jacobi identity
((M1 , M2 ), M3 ) + ((M2 , M3 ), M1 ) + ((M3 , M1 ), M2 ) = 0
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In practically all of our uses of a Lie algebra, the elements of the algebra will be
square matrices, the scalars will be real numbers and the binary operation will
be proportional to the commutator of the matrices, (M1 , M2 ) = −i[M1 , M2 ].
We are generally going to be interested in problems which do not make use of the
full space of parameters of the rotations. Generally, we are interested in finding
and classifying the irreducible representations of SO(3). To find the possibilities for
possible representations, it is very useful to study those group elements which are
infinitesimally close to the identity. These group elements are of the form
Rij = δij + ωij
where ωij are infinitesimal constants. RRt = I if and only if ω = −ω t or, with
indices, ωij = −ωji . That is, ω should be an anti-symmetric matrix. Thus we
see that any SO(3) group element which is close to the identity identifies an antisymmetric matrix. Any anti-symmetric matrix inserted there with an infinitesimal
coefficient should do the job, it should present an SO(3) group element. We can find
a basis for anti-symmetric matrices (multiplied by a factor of i so that they are also
hermitian)
(18)
(Ti )jk = −iijk
These basis matrices obey the commutator algebra
(19)
[Ti , Tj ] = iijk Tk
This commutator algebra is the binary operation for the so(3) Lie algebra. Some of
the details are summarized in infobox 2.3.
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Infobox 2.3 SO(3) and the Lie algebra so(3)
The formal definition of Lie algebra is reviewed in info box 2.2. The Lie algebra so(3) corresponding to the Lie group SO(3) is the set of all imaginary
anti-symmetric matrices. This set clearly forms a vector space where scalar
multiplication uses real numbers. The binary operation is proportional to the
commutator, (M1 , M2 ) = −i[M1 , M2 ]. It is easy to see that the commutator of
any two objects in this set is automatically a member of the set. If M1 and M2
both obey Mi∗ = −Mi and Mit = −Mit , then
(−i[M1 , M2 ])∗ = i[M1∗ , M2∗ ] = i[−M1 , −M2 ] = −(−i[M1 , M2 ])
(−i[M1 , M2 ])t = −i[M2t , M1t ] = −i[−M2 , −M1 ] = −i[M2 , M1 ] = −(−i[M1 , M2 ])
We will choose a basis for the set of all imaginary anti-symmetric matrices
which is made out of i times the totally anti-symmetric tensor ijk . We remind
the reader that the anti-symmetric tensor is defined as having the value 123 = 1
and as being antisymmetric under the exchange of any two neighbouring indices.
Its non-zero values are then
123 = 231 = 312 = 1
213 = 132 = 321 = −1
Any 3 × 3 imaginary anti-symmetric matrix can by expressed as
Mab =
3
X
θj (Tj )ab =
j=1
3
X
θj (−ijab ) , (Tj )ab = −ijab
j=1
for some choice of the three parameters θj . In fact, we observe that, the only
non-zero components of M are M12 = −M21 , M23 = −M32 and M31 = −M13
and then, explicitly, θ1 = iM23 , θ2 = iM31 and θ3 = iM12 .
A very useful identity, which can be proven by just comparing non-zero elements of both sides, is
ijk klm ≡
3
X
ijk klm = δil δjm − δim δjk
k=1
On the left-hand-side of the above equation, we are using the summation convention where repeated indices are summed over.
The generators which we have chosen are
(Ti )ab = −iiab
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where we have been careful to distinguish the indices of the matrix from the label
of the matrix, although in this case all three run over the same domain. The
matrix is anti-symmetric. It is also purely imaginary. It is therefore hermitian.
We can use the identities which we discussed above to compute the commutator of two generators,
X
[Ti , Tj ]ac =
[(−iiab )(−ijbc )−(−ijab )(−iibc )] = −δaj δic +δac δij +δai δjc −δac δij
b
= −δaj δic + δai δjc = iijk (−ikac ) = iijk (Tk )ac
In summary, the basis for the so(3) Lie algebra which we have chosen has generators Ta which obey the commutation relations
[Ti , Tj ] = iijk Tk
The corresponding development for SU (2) studies elements in the vicinity of the
identity matrix,
U ≈ I + iH
where H is a matrix of infinitesimal magnitude. If we impose the condition that U
is unitary, to leading order in H,
(I + iH)(I − iH † ) ≈ I + iH − iH † = I → H = H †
and H must be a hermitian matrix. Also, if the determinant is unity,
det(I + iH) ≈ 1 + iTr(H) = 1 → TrH = 0
and H must be a traceless matrix. Then, moreover, consider the combination of
quantities
U1 U2 U1−1 U2−1 ≈ (I + iH1 )(I + iH2 )(I − iH1 )(I − iH2 ) ≈ I + [H1 , H2 ] + . . .
where we have kept terms to second order in the deviation from the identity. (It
is easy to check that, if we keep second order terms for each u, they cancel.) This
product of group elements must still be a group element. Therefore −i[H1 , H2 ] must
be a hermitian matrix. Indeed, it is hermitian when H1 and H2 are hermitian.
We can define a basis {T1 , T2 , ...} for the set of traceless hermitian 2×2 matrices as
a complete set of matrices in which any traceless hermitian matrix can be expanded,
X
H=
Ta θa , Ta = Ta† , TrTa = 0
a
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The number of matrices which are needed is equal to the dimension of the group;
which is also the dimension of the group manifold. The basis matrices Ta are called
generators.
A convenient basis for SU (2), which has dimension three, uses the Pauli matrices
1
Ta = σa
2
These matrices have the commutation relation
(20)
[Ta , Tb ] = iabc Tc
where the summation over c on the right-hand-side is understood.
We see that the Lie algebra su(2) has an identical number of generators and the
same structure constants as the Lie algebra of so(3).
2.1. Representations. A representation of a Lie algebra is a set of matrices which
has the same commutation relations as are dictated by the binary operation of the Lie
algebra. The dimension of the matrices is not necessarily the same as the dimension
of the matrices behind the definition of the Lie algebra. For example, a set of three
k × k matrices, Ta which satisfy the commutation relation (20) is a representation of
either su(2) or so(3).
If Ta is a representation of the algebra, a similarity transformation of the generators, STa S −1 is also a representation. Two representations are said to be equivalent
if they are related by a similarity transformation.
As in the case of groups, we can define a reducible representation of a Lie algebra
as one where all of the basis elements Ta can simultaneously be made block diagonal
by a similarity transformation. A block diagonal representation is said to be a direct
sum of smaller representations.
A representation of a Lie algebra is said to be irreducible if it is not reducible.
Finding the irreducible representations of a Lie algebra is equivalent to finding the
irreducible representations of the corresponding Lie group. The representation of the
Lie algebra gives us an explicit representation of those group elements in an infinitesimal neighbourhood of the identity. It might be difficult to construct generic group
elements which are not near the identity in a given representation. However, in applications to quantum mechanics, they are generally not needed. What is important
is the dimension of irreducible representations and the decomposition of reducible
representations to direct sums of irreducible representations. This information can
be extracted from the representations of the Lie algebra.
2.1.1. The irreducible representations of su(2). Now, let us consider the su(2) Lie
algebra. Let us assume that Ta are a set of d × d hermitian matrices which satisfy the
algebra in equation (20). We will ask the question as to what are the dimensions, d,
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of the irreducible representations. Then, we will go as far as we can in constructing
the matrices in each representation.
To answer these questions, we will change the basis of the Lie algebra. This change
of basis should be familiar to students who have taken a quantum mechanics course.
We have three hermitian matrices which do not commute with each other. Since
they do not commute, they cannot be simultaneously diagonalized. In fact, only one
of them can be diagonalized at a time. In the representation theory of Lie algebras,
the number of generators that can be simultaneously diagonal is called the rank of
the Lie algebra. su(2) has rank one. Let us choose T3 , to be diagonal. Then, it is
convenient to organize T1 and T2 into two complex matrices
T+ = T1 + iT2 , T− = T1 − iT2 , T+ = T−†
(21)
With these matrices, the commutation relations for the Lie algebra elements are
[T3 , T± ] = ±T± , [T+ , T− ] = 2T3
(22)
Since T3 is a hermitian matrix, it has real eigenvalues. Assume that |m > is a
normalized eigenvector of T3 with eigenvalue m, so that
T3 |m > = m |m > , < m|m >= 1
In the language of representation theory , m, an eigenvalue of the diagonalized generator, is called a weight. It will turn out to be either a half-integer. We will learn
that by studying the algebra.
Using the algebra, we see that T+ |m > is also an eigenstate of T3 with weight
m + 1 and T− |m > is an eigenstate of T3 with weight, m − 1. Explicitly
±T± |m > = T3 T± |m > −T± T3 |m >→ T3 (T± |m >) = (m ± 1)(T± |m >)
that is
T+ |m >= cm |m + 1 >
and
T− |m >= c∗m−1 |m − 1 >
where cm is a constant (generally a complex number). It is given by computing the
matrix element
cm =< m + 1|T+ |m >=< m|T− |m + 1 >∗
In words, T+ is a raising operator and T− is a lowering operator and they raise and
lower the weight by one unit. The unit by which an eigenvalue is raised in this way,
in this case one, is called a “root”. The terminology which uses roots and weights
will become more important when we study higher rank Lie algebras.
We will assume that T3 is a finite dimensional matrix and that its eigenvalues are
finite real numbers. Then, there must be an eigenvalue which is the largest. We
will assume for the moment that there is only one such eigenvalue, that is, that the
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largest eigenvalue is non-degenerate. Let us call that eigenvalue j and the eigenvector
|j >. j is called the highest weight of the representation. We have
T3 |j > = j |j >
Since |j > is the highest weight, we must have
T+ |j >= 0 or cj = 0
We can obtain other states in the representation by operating lowering operators on
the highest weight state,
(T− )p |j >∼ |j − p >
Each action of T− lowers the eigenvalue by one. We will make use of the fact that the
dimension of the representation is finite. This means that there must be an integer
pmax beyond which we cannot lower, that is,
(23)
(T− )pmax +1 |j >= 0
Consider the matrix element of the commutator,
(24)
< m| [T+ , T− ] |m >= |T− |m > |2 − |T+ |m > |2 = |cm−1 |2 − |cm |2 = 2m
This identity for the weights j, j − 1, .... gives the sequence of equations
|cj−1 |2 − |cj |2 = 2j
|cj−2 |2 − |cj−1 |2 = 2(j − 1)
|cj−3 |2 − |cj−2 |2 = 2(j − 2)
......
|cj−p+1 |2 − |cj−p+2 |2 = 2(j − p + 2)
(25)
|cj−p |2 − |cj−p+1 |2 = 2(j − p + 1)
If remember cj = 0, and we add together the left-hand-sides of the above equations
we arrive at
p−1
X
|cj−p |2 =
2(j − k) = 2jp − p(p − 1)
k=0
P
P
where we have used the summation formulae 0p−1 1 = p and p−1
k=0 k = p(p − 1)/2.
The result is
p
(26)
cj−p = p(2j + 1 − p)
where we have chosen the phase convention that these coefficients are real.
Since the fact that the number of states must be finite implies that there is a
lowest weight, which is obtained by lowering the highest weight by mmax operations
of T− , we need
|cj−pmax −1 |2 = 0 = (pmax + 1)(2j − pmax )
GROUP THEORY PRIMER
15
This has the solution
(27)
pmax = 2j
Thus, the highest weight in the sequence must be j and the lowest weight is j − 2j =
−j. What is more, since the weight is lowered by one unit each time, the highest and
lowest weight states must have weights differing by an integer. This is only possible
when 2j =integer and the eigenvalues of T3 are j, j − 1, ..., −j + 1, −j. j is either an
integer or a half-odd-integer. If it is na integer, all of the weights are integers. If it
is a half-odd-integer, all of the weights are half-odd-integers. The explicit matrix is


j
0
...
0
0
 0 j − 1 ...
0
0



...
. . .
T3 = . . . . . . . . .

0
0
. . . −j + 1 0 
0
0
...
0
−j
We can also find
T+ =
j
X
cm |m + 1 >< m| , T− =
m=−j
j
X
c∗m−1 |m − 1 >< m|
m=−j
or, more explicitly,

0 cj−1 0
0 ...
0
0
0 cj−2 0 . . .
0
0 
0


0
0 cj−3 . . .
0
0 
0
T+ = 

. . . . . . . . . . . . . . . c−j+1 0 
0
0
0
0 ...
0
c−j 
0
0
0
0 ...
0
0


0
0
0 0 ... 0 0
c∗j−1 0
0 0 . . . 0 0


∗
 0 cj−2 0 0 . . . 0 0


T− = 

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
0


 0
0
0 0 . . . 0 0
0
0
0 0 . . . c∗−j 0
p
cj−p = p(2j + 1 − p)
This is clearly an irreducible representation. To see this, consider the vectors space
on which it acts. To be reducible, there would have to be a sub-space of that vector
space with the property that, when su(N ) generators operate on the vectors in the
subspace, they produce only vectors in the subspace. But, in the above development
it is clear that by operating with T+ or T− some number of times we can produce

16
GROUP THEORY PRIMER
any basis vector from any other basis vector. Thus, there cannot be an invariant
subspace and the representation is irreducible.
We have constructed this representation with the assumption that the highest
weight was non-degenerate. It is clear that, if it were degenerate, we could begin
with two orthogonal states. Then operation of the lowering operators on the two
orthogonal states would simply generate two identical orthogonal copies of the representation that we have found. In that case, it would be reducible. An irreducible
representation must therefore has a unique, non degenerate highest weight state.
We have now answered the question that we asked at the outset. Finite dimensional
irreducible representations of su(2) are characterized by the highest weight j. They
are d = 2j + 1-dimensional and the possible values of j are
1
3
j = 0, , 1, , . . .
2
2
The defining representation of su(2) is the j = 12 representation and the defining
representation of so(3) is the j = 1 representation.
We see here, rather explicitly, that finding the representations of the Lie algebra are
enough to understand the representations of the Lie group. The dimension and some
other aspects of the representations of the Lie group are completely determined by
the representations of the Lie algebra. The representation of the algebra determines
the representation of the group in the vicinity of the identity. How the representation
is extended away from the infinitesimal patch near the identity is somewhat arbitrary.
Different extensions are equivalent to different choices of the coordinates of the group
manifold. One common extension is called the exponential map. It takes an element
of the Lie algebra, X and simple writes the group element as eiX .