The su(n) Lie Algebra
Haonan Liu
December 2, 2022
Contents
1 The SU(n) Lie group
1.1 The SU(n) group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 The SU(n) Lie group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
3
2 The su(n) Lie algebra
2.1 Lie algebra . . . . . . . . . . . . . . . . .
2.2 Basis, structure constants, and generators
2.2.1 Basis and generators of su(n) . . .
2.2.2 Structure constants of su(n) . . . .
2.3 States and operators . . . . . . . . . . . .
2.3.1 General definition . . . . . . . . .
2.3.2 Operators in Lie algebra . . . . . .
2.4 The Lie correspondence . . . . . . . . . .
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3
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3 Representations of su(n)
3.1 Representation theory of Lie algebras . . . . . . . .
3.2 Examples of reps . . . . . . . . . . . . . . . . . . .
3.2.1 Trivial/zero rep . . . . . . . . . . . . . . . .
3.2.2 Defining/natural/standard/tautological rep
3.2.3 Adjoint/regular rep . . . . . . . . . . . . .
3.3 Morphism between reps . . . . . . . . . . . . . . .
3.3.1 Intertwining operator . . . . . . . . . . . .
3.3.2 Equivalent reps . . . . . . . . . . . . . . . .
3.3.3 Reps of su(n) and sl(n, C) . . . . . . . . . .
3.4 Operations on reps . . . . . . . . . . . . . . . . . .
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4 Structure theory of sl(n, C) and its reps
4.1 Simple and semisimple Lie algebras . . .
4.1.1 Ideals and commutativity . . . .
4.1.2 Simplicity and semisimplicity . .
4.1.3 Levi decomposition . . . . . . . .
4.2 Reducible and irreducible reps . . . . . .
4.2.1 Reducibility . . . . . . . . . . . .
4.2.2 Complete reducibility . . . . . .
4.3 Complete reducibility and symmetry . .
4.3.1 Schur’s lemma . . . . . . . . . .
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4.4
4.5
4.6
4.3.2 Hamiltonian & first definition of symmetry . . . .
4.3.3 Unitary rep & the second definition of symmetry .
4.3.4 Multiplet . . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Characters . . . . . . . . . . . . . . . . . . . . . .
Reps of sl(n, C) . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Reps of sl(2, C) . . . . . . . . . . . . . . . . . . . .
4.4.2 Cartan subalgebra . . . . . . . . . . . . . . . . . .
4.4.3 Root decomposition . . . . . . . . . . . . . . . . .
4.4.4 Cartan-Weyl basis . . . . . . . . . . . . . . . . . .
4.4.5 Root system . . . . . . . . . . . . . . . . . . . . .
4.4.6 Classification of semisimple Lie algebras . . . . . .
4.4.7 Weight decomposition . . . . . . . . . . . . . . . .
4.4.8 Reps of sl(n, C) . . . . . . . . . . . . . . . . . . . .
4.4.9 Casimir operators . . . . . . . . . . . . . . . . . .
Direct product to direct sum . . . . . . . . . . . . . . . .
4.5.1 Symmetric and antisymmetric tensors . . . . . . .
4.5.2 Direct product into direct sum . . . . . . . . . . .
4.5.3 The Clebsch-Gordan coefficients (CG coefficients)
Summary of properties of sl(n, C) and su(n) . . . . . . . .
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5 The su(2) Lie Algebra and its reps
5.1 Generators and structure constants . . . . . . . .
5.2 The Lie correspondence between SU(2) and su(2)
5.2.1 From SU(2) to su(2) . . . . . . . . . . . .
5.2.2 From su(2) to SU(2) . . . . . . . . . . . .
5.3 Reps . . . . . . . . . . . . . . . . . . . . . . . . .
5.3.1 Defining rep . . . . . . . . . . . . . . . . .
5.3.2 Adjoint rep . . . . . . . . . . . . . . . . .
5.3.3 Irrep . . . . . . . . . . . . . . . . . . . . .
5.3.4 CG coefficients . . . . . . . . . . . . . . .
5.4 Cartan subalgebra . . . . . . . . . . . . . . . . .
5.4.1 Casimir operators . . . . . . . . . . . . .
5.4.2 Fundamental rep . . . . . . . . . . . . . .
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44
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6 The su(3) Lie algebra and its reps
6.1 Generators and structure constants . . . . . . .
6.2 Reps . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Fundamental and antifundamental reps
6.2.2 Tensor product rep . . . . . . . . . . . .
6.2.3 Multiplets . . . . . . . . . . . . . . . . .
6.2.4 Cartan subalgebra . . . . . . . . . . . .
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7 The su(4) Lie algebra and its reps
51
8 Others
51
9 References
52
2
The SU(n) Lie group
1
The SU(n) group
1.1
1
.
Definition 0 [U(n) group]: A unitary group of degree n, U(n), is the group of complex linear automorphisms
of an n-dimensional complex vector space Cn preserving a positive definite Hermitian inner product H on
Cn .
In Def. 0, if the Hermitian form H is “standard”, meaning it is defined associated with the identity
matrix 1 such that H(v, w) = v † · 1 · w, which is usually the case, then the group U(n) is just the group of
n × n unitary matrices U , i.e., U † = U −1 . It can be proved that |det(U )| = 1 for any U ∈ U(n).
Definition 0 [SU(n) group]: A special unitary group, SU(n), is the subgroup of U(n) with determinant 1.
Easy to see, SU(m) is a subgroup of SU(n) for any m < n. Since SU(1) only contains the 1-dimensional
identity, it is trivial and not interesting. Thus we will always have n > 1 for SU(n) in this work.
The SU(n) Lie group
1.2
The SU(n) group is more than just a group, but a Lie group.
Definition 0 [Lie group]: A Lie group is a set endowed simultaneously with the compatible structures of a
group and a C ∞ (infinitely differentiable) manifold, such that both the multiplication and inverse operations
in the group structure are smooth maps. A morphism between two Lie groups is just a map that is both
differentiable and a group homomorphism.
We have already defined SU(n) as a group. Multiplication is smooth because the matrix entries of the
product U1 U2 are polynomials in the entries of U1 and U2 for U1 , U2 ∈ SU(n). Inversion is smooth by
Cramer’s rule. Therefore SU(n) is a smooth manifold and thus a Lie group.2 One can go further and show
that the SU(n) Lie group is compact, simply connected, and nonabelian.3
The benefit of understanding the Lie group nature of the SU(n) group is immediate once we introduce the
one-to-one correspondence between Lie groups and Lie algebra in the next section. With the correspondence,
most of the properties of the SU(n) group that we will use for our physical problems can be derived using
the su(n) or sl(n, C) Lie algebra structure theory.
Some useful properties of SU(n):
1. The matrix elements of an n × n matrix U ∈ SU(n) are analytic functions of d real parameters
x1 , x2 , · · · , xd , with d = n2 − 1. This directly follows from the fact that SU(n) is smooth and that a
unitary matrix with determinant 1 has n2 − 1 degrees of freedom in real numbers. Therefore we also
call an SU(n) group a real Lie group of dimension n2 − 1.
2. More will be added as we complete this note.
The su(n) Lie algebra
2
2.1
Lie algebra
A Lie algebra4 is defined as the following.
1 All
the ??? notations in the work need to be settled. All the "it can be shown" needs to be shown or cited.
SU(n) ⊆ U(n) ⊆ GL(n, C) are all Lie groups and form smooth embeddings.
3 These properties are somewhat “nice” properties in representation theory.
4 Sometimes called a tangent space in literature, related to the smooth manifold nature of the corresponding Lie group.
2 Generally,
3
Definition 0 [Lie algebra]: A Lie algebra g is a vector space together with a skew-symmetric bilinear map 5
(called the Lie bracket)
[ , ]:g×g→
− g
(1)
satisfying the Jacobi identity
[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0
(2)
with a, b, c ∈ g.
Note that, in this work, whenever we mention a Lie algebra, we will always assume that the Lie
algebra is finite-dimensional over an algebraically closed 6 field F of characteristic 7 0 such as
R or C unless explicitly noted. We can then define the Lie algebra su(n) as follows.
Definition 0 [u(n), sl(n, C), and su(n) Lie algebras]: Let gl(n, C) be the Lie algebra that constitutes all the
n × n complex matrices. The u(n) Lie algebra is a Lie algebra that constitutes all the anti-Hermitian n × n
complex matrices, i.e.,
u(n) = U ∈ gl(n, C) | U † = −U .
The sl(n, C) Lie algebra is a Lie algebra that constitutes all the traceless n × n complex matrices, i.e.,
sl(n, C) = {U ∈ gl(n, C) | Tr[U ] = 0}.
The su(n) Lie algebra is a Lie algebra that constitutes all the traceless anti-Hermitian n × n matrices, i.e.,
su(n) = U ∈ gl(n, C) | Tr[U ] = 0 and U † = −U .
For all these algebras, the skew-symmetric bilinear map is the commutator.8
Some comments on these algebras:
1. Notation. In this work, we write
gl(n, C) = gl(Cn ).
(3)
More generally, we write gl(V ) to represent the set of automorphisms (matrices) from vector space V
to V .
2. Both u(n) and su(n) are real Lie algebras. “Real” means that the algebra is a vector space over the
field R, although the matrices themselves can contain complex numbers as entries. On the other hand,
sl(n, C) is a complex Lie algebra.
3. Lie algebra su(n) is a subalgebra of u(n). Here, a subalgebra g0 of Lie algebra g is a subset of g of which
the elements form a Lie algebra with the same commutator over the same field as those of g.
4. Complexification of a real Lie algebra. A real Lie algebra can be complexified if its basis elements stay
linearly independent and the field R the Lie algebra is defined over is replaced by C. Both u(n) and
su(n) satisfy this condition and can be complexified. Specifically, we define
suC (n) ≡ su(n) ⊗ C = su(n) ⊕ isu(n),
5 “Skew-symmetric”,
(4)
or antisymmetric, means [X, Y ] = −[Y, X].
field F is algebraically closed if every polynomial with coefficients in F has a root in F
7 A field has characteristic 0 if the sum of its multiplicative identity never reaches its additive identity.
8 Definition (0) can also be phrased another way using the notion of exponential maps. The anti-Hermitian and traceless
nature of matrices can be derived following rules of exponential maps and the definition of SU(n) Lie groups. However, following
our logic in this work, we will define SU(n) and su(n) separately first, and introduce the Lie correspondence in the end together
with the exponential map.
6A
4
Realizing that the Lie algebras su(n) and isu(n) are isomorphic and that they combined contain all
the traceless n × n complex matrices, we have
suC (n) = sl(n, C),
(5)
i.e., sl(n, C) is the complexification of su(n).
It turns out that sl(n, C) will be very important in the theory. In fact, many of the properties of su(n)
will be studied as a result of sl(n, C).
5. We mention Ado’s theorem here.
Theorem 1 [Ado]: Any finite-dimensional Lie algebra over F is isomorphic to a Lie subalgebra of
gl(n, F).
Ado’s theorem tells us that without loss of generality Lie algebras can be studied with their elements
defined as matrices.
The su(n) Lie algebra is finite-dimensional. Another example of a finite-dimensional Lie algebra is the
Heisenberg algebra which is spanned by the coordinate functions over R, i.e., {1, q, p}, with the skewsymmetric bilinear map defined as the Lie bracket. The Heisenberg algebra is useful in physics. See Miller
or Woit for more.
2.2
Basis, structure constants, and generators
We first discuss the properties of general Lie algebras. For a Lie algebra of dimension d over field F we
can always find a basis of dimension d that we label as ej , j ∈ N. The Lie algebra is then denoted as
spanF {e1 , e2 , · · · , ed } ≡ {e1 , e2 , · · · , ed }F .
(6)
Given a nonsingular matrix R, a new basis can be built using e0k = Rkj ej . Here, for simplicity, we have
assumed the Einstein’s summation convention, and we do not care about the upper or lower indices of a
tensor unless otherwise noted. From now on, we will also use the basis notation (6) to label a Lie algebra.
If the field F is clear from context [R for su(n), C for sl(n, C), etc.] we will discard it as well.
Applying the skew-symmetric bilinear map to basis elements yields another element of the Lie algebra
that can be expanded in the basis, i.e.,
[ej , ek ] = fjkl el .
(7)
Here, the coefficients fjkl are called structure constants relative to the chosen basis. Several properties of
the structure constants are:
1. Given a basis, the structure constants can describe the Lie algebra completely.
2. If the structure constants are all real, then the Lie algebra is real.
3. The structure constants transform between basis in the following way fj0 0 k0 l0 = Rj 0 j Rk0 k fjkl R−1
ll0
4. The structure constants are antisymmetric in the first two indices as a result of the Jacobi identity.
We next focus on the su(n) algebra.
5
.
2.2.1
Basis and generators of su(n)
By definition, the su(n) algebra is spanned by all the traceless anti-Hermitian matrices of dimension
n × n. It follows by counting the real degrees of freedom that su(n), as a vector space, has a finite dimension
d = n2 − 1.
(8)
Therefore, we can find a basis of dimension n2 − 1 for su(n), n > 2. The most common choice of basis in
literature can be constructed in the following way.
We first define a set {λi } as the set of the following traceless Hermitian matrices






 1 0
1 0
0
1 0
0
0






0 1
s



 1 0 1

 0 −1
2






(9)
{λi } = 
−2
.
.
,··· ,
,
, √ 







n(n − 1) 
3





. 
1
. 
0
0
0
−(n − 1)
0
0

 



0 1
0
0 0 1
0
0 0
0

0 0

1 0
 0 0

 




 



(10)
.
.
.

 , 1
,··· ,
,

 




 


.
.
. 1
0
0
0
0
0
1 0
 




0 0 −i
0
0 0
0 
0 −i
0
 0 0

0 0

i 0

 




 




(11)
.
.
.
 , i
,··· ,
 .

 







. 
. −i 
.  
0
0
0
i 0
0
0
In more concise expressions, the set {λi } can be constructed from the following three classes of n×n matrices:


s
for µ < k

1
i
h
2
(1)
=
λk
(12)
δµν × −k for µ = k, k = 1, 2, · · · , n − 1 ;

k(k + 1)
µν

0
for µ > k
h
i
(2)
λjk
= δkµ δjν + δkν δjµ ,
j, k = 1, 2, · · · , n, and j < k;
(13)
µν
h
i
(3)
λjk
= −i(δjµ δkν − δjν δkµ ),
j, k = 1, 2, · · · , n, and j < k.
(14)
µν
Here the superscripts (1), (2), and (3) represent the three classes. We can check that the total number of λi
n(n − 1) n(n − 1)
+
= n2 − 1 = d.
matrices is (n − 1) +
2
2
The matrices λi are called the generators of su(n).9 A basis of the su(n) is then just given by the set of
anti-Hermitian traceless matrices {ei } defined from {λi }
1
ej = − iλj ,
2
j = 1, 2, · · · , d.
(15)
The multiplication by i creates anti-Hermitian basis elements from the Hermitian generators. The coefficient
1
− is chosen for historical reasons. As mentioned above, our choice of generators and basis is not unique,
2
but has been shown convenient from literature.
9 Notice that the choice of generators is not unique, as is true for basis. Our choice is only one of the fundamental
representations??? of su(n).
6
2.2.2
Structure constants of su(n)
By the definition of structure constants in Eq. (7), using Eq. (15), we have
[λj , λk ] = 2ifjkl λl .
(16)
With the choice of generators given by Eqs. (9)–(11) or Eqs. (12)–(14), we get a nice normalization condition
for the generators λi , i.e., 10
Tr[λj λk ] = 2δjk
(17)
Now, using Eqs. (16) and (17) together, we get
Tr [λj , λk ], λl = 2ifjkm Tr[λm λl ] = 2ifjkm 2δlm = 4ifjkl ,
(18)
which gives the structure constants as
fjkl =
1
Tr [λj , λk ]λl .
4i
(19)
On the other hand, using Tr[AB] = Tr[BA] in Eq. (18), we get
4ifjkl = Tr [λj , λk ]λl = Tr[λj λk λl − λk λj λl ] = − Tr[λj λl λk − λl λj λk ] = − Tr [λj , λl ]λk = −4ifjlk . (20)
Similarly, we can prove that every odd permutation of the indices of fjkl changes its sign.11 In other words,
fikl is completely antisymmetric in all indices. Therefore all structure constants
2 with
the same indices vanish.
n −1
To find all the nonvanishing terms of structure constants, we only need
independent calculations,
3
given the completely antisymmetric property.
For future use, we can also define the anticommutation relations for the generators λi . Without proof,
for su(n), we have
4
+
λl ,
(21)
[λj , λk ]+ = δjk 1n×n + 2fjkl
n
where
1 +
(22)
fjkl
= Tr [λj , λk ]+ λl .
4
2.3
States and operators
2.3.1
General definition
We first discuss the concepts of states and linear operators in general. For an m-dimensional vector space
over field F, we sometimes refer to the vectors as states or kets. The basis vectors 12 {ψ1 , ψ2 , · · · , ψm } can
also be denoted as {|ψ1 i , |ψ2 i , · · · , |ψm i}. A linear operator is a morphism 13 that maps states from a vector
space to another. For an operator Ω̂ : V → U , with {vj } and {uj } being bases of V and U , we can define
the operator as
k
Ω̂vj = (Ωvj ) uk ,
(23)
k
where we have used the ˆ· notation to label operators. We call Ωkj = (Ωvj ) the matrix of the operator Ω̂.
Ignoring the upper and lower indices (assuming Euclidean metrics for v and u), we get
Ω̂vj = Ωkj uk ,
(24)
where we have chosen the order of the lower indices on purpose. The reason will be explained shortly.
10 This
actually why we chose this basis in the beginning.
arbitrary Lie algebra, fjkl is only antisymmetric in the first two indices. For compact and semisimple Lie algebras
such as su(n), fikl is completely antisymmetric, which is nice.
12 In this work we follow the notation in Eq. (6) such that a set of basis vectors can mean either the basis or the vector space
spanned by the basis. The exact meanings should be clear from context.
13 A morphism is a structure-preserving map used in category theory.
11 For
7
2.3.2
Operators in Lie algebra
Now we go back to the discussion on Lie algebra. So far we have seen that the Lie algebra elements of
su(2) are matrices. Also we know that matrices are morphisms acting on certain vector spaces. Therefore,
we can think of a Lie algebra g as a vector space consisting of linear operators that act on an m-dimensional
vector space V . We assert that the operators that form the Lie algebra are morphisms from V to V . Let
{v1 , v2 , · · · , vm } be a orthonormal basis of V , then we have, for Ω̂ ∈ g, using Eq. (24)
Ω̂vj = Ωkj vk .
(25)
Furthermore, we demand the vector space V to be an inner product space. This allows us to define the inner
product hvj |vk i, where hvj |’s are called bras that are a linear functional of kets living in the dual vector space
V ∗ of V and satisfy
†
hvj | = |vj i .
(26)
In an inner product space, it is always possible to find orthonormal basis for the kets such that
hvj |vk i = δjk ,
j, k ∈ {1, 2, · · · , m}.
(27)
Therefore we will always assume our basis of V and V ∗ to be orthonormal from now on.
With orthonormal basis chosen, we can immediately appreciate our choice of the order of indices in
Eq. (24). Realizing that different orders of indices are equivalent to matrix transpositions, our choice ensures
the matrix Ωjk to have the nice property such that
E
D
(28)
hvk |Ω̂|vj i ≡ vk Ω̂vj = Ωlj hvk |vl i = Ωkj ,
where we have defined a left action. In other words, the elements of the matrix Ωkj are just the operator Ω̂
acted by bra hvk | on the left and ket |vj i on the right following the left action convention. Another nice
property is that
D
E D
ED
E
(ΩΩ0 )jk = hvj |Ω̂Ω̂0 |vk i = vj Ω̂Ω̂0 vk = vj Ω̂vl vl Ω̂0 vk = Ωjl Ω0lk ,
(29)
i.e., the matrices of operators indeed follow our usual convention of matrix multiplications, which assumes
left action.
From now on, we will always think of elements of Lie algebra g as linear operators that act on some inner
vector space V .
2.4
The Lie correspondence
It is no coincidence that the SU(n) Lie group and su(n) Lie algebra have the same dimension d = n2 − 1.
It can be proved that the linear SU(n) Lie group corresponds to the real su(n) Lie algebra, meaning that for
every SU(n) Lie group, there exists a corresponding su(n) Lie algebra, and vice versa. This in general is called
the Lie correspondence given by three fundamental theorems of Lie theory, and the one-to-one correspondence
is only true for simply connected Lie groups, such as SU(n). Mathematically, the correspondence can be
described using the following theorem.
Theorem 2 [Lie correspondence]: The categories of finite-dimensional Lie algebras and connected, simply
connected Lie groups are equivalent. 14
Since SU(n) is connected and simply connected, the categories of SU(n) and su(n) are equivalent, i.e.,
they correspond to each other.
The Lie correspondence between SU(n) and su(n) have the following consequences:
14 I
do not fully understand the equivalence of categories and will not go into details, but this is the correct statement.
8
1. Given a matrix U ∈ SU(n) parameterized by a set of real parameters {xj }, j ∈ {1, 2, · · · , d}, a basis
of the corresponding su(n) Lie algebra can be found by
(el )jk =
∂
Ujk (x1 , x2 , · · · , xd )|x1 =x2 =···=xd =0 .
∂xl
(30)
This relation is local in the parameter space. It directly relates the d = n2 − 1 degrees of freedom of
SU(n) and su(n). From geometry point of view, su(n) is the tangent space to the Lie group SU(n) at
the identity.
2. Given a n × n-dimensional basis {ej } of the su(n) Lie algebra, j ∈ {1, 2, · · · , d}, any matrix U ∈ SU(n)
can be found with parameterization {xj }, j ∈ {1, 2, · · · , d}, by 15
i
U (x) = exp(xj ej ) = exp − xj λj .
(31)
2
This relation is nonlocal since it relies on the exponential map. Following Eq. (31), we can use the basis
(or generators) of su(n) to reconstruct the group elements of SU(n). In other words, the generators of
the su(n) Lie algebra are the infinitesimal generators of the SU(n) Lie group.
In the next section we introduce the representation theory of Lie algebras. Due to the Lie correspondence,
we will mainly focus on the su(n) Lie algebra. However, we will also introduce the representation theory of
Lie groups in times of need.
Representations of su(n)
3
3.1
Representation theory of Lie algebras
Definition 0 [representation of Lie algebras]: A representation (or a rep) of a Lie algebra g over F on a
vector space V (of dimension m) is a homomorphism ρ : g → gl(V ) = End(V ).
Definition. 0 has the following consequences:
1. Notation. In many books the reps are defined to be the vector spaces they act on directly. In our
notation, we will refer to ρ as the rep, and the vector space V as the g-module associated with the rep
ρ of Lie algebra g.
2. Rep ρ is linear (by homomorphism), i.e., for α, β ∈ F and a, b ∈ g
ρ(αa + βb) = αρ(a) + βρ(b).
(32)
3. Rep ρ preserves the Lie brackets (by homomorphism), i.e.,
ρ([a, b]) = [ρ(a), ρ(b)].
(33)
4. Rep ρ can be written as m×m matrices if we specify the elements of the vector space V as m-dimensional
vectors. Then we say rep ρ is m-dimensional.
5. Due to linearity, for a Lie algebra, it suffices to find the corresponding matrices of the basis to define
a specific rep.
15 Surjective,
proof not shown.
9
6. The operators we constructed in Sec. 2.3.2 are just another way to define reps. We will unify the
concepts of operators and reps from now on. A Lie algebra g contains elements that satisfy the
Jacobi identity. Once we say these elements are also linear operators acting on a vector space V , that
automatically means that there is a homomorphism that maps these elements to the endomorphisms
on V .
As an add-on to our last point above, here
n we show
o that the set of operator matrices defined in Sec. 2.3.2
(Ωjk , etc.) is a rep of the Lie algebra g = Ω̂, · · · .
Proof. The linearity is trivial to show. The commutation-preserving property can be shown as follows. Let
Â, B̂ ∈ g be two operators of the Lie algebra. Then their operator matrices are just Ajk and Blm . Thus
([A, B])jk = (AB)jk − (BA)jk = Ajl Blk − Bjl Alk .
What we have done above is more of a consistency check. We specify that for a Lie algebra composed
of operators that act on the inner product space V , the corresponding operator matrices always form a rep
of the Lie algebra. With the different inner product spaces chosen, the operator matrices have different
matrix forms, i.e., the Lie algebra has different reps. From now on, for simplicity, we abandon the ˆ· operator
notation. All the Lie algebra elements we talk about in this work will be taken as linear operators acting
on the same vector space V , which is the g-module of the corresponding rep that maps these Lie algebra
elements to the operators.
Some other useful definitions regarding reps of Lie algebras.
1. Consider a rep ρ : g → gl(V ) of a Lie algebra g. A subspace U ⊂ V is invariant or stable if ∀u ∈ U
and ∀a ∈ g, ρ(a)u ∈ U . Then we say ρ : g → gl(U ) is a subrepresentation (subrep).
2. A rep is faithful if it is injective.
The rep of a Lie group is defined as below.
Definition 0 [representation of Lie groups]: A rep of a Lie group G over F on a vector space V (of dimension
m) is a homomorphism ρG : G → GL(V ) = Aut(V ).
Here GL(V ) is the group of all m × m invertible matrices on the m-dimensional vector space V .
In general, representations allow us to construct elements of the algebra or group as matrices acting
on different spaces. In the cases of SU(n) and su(n), the one-to-one Lie correspondence between them is
naturally inhibited by their reps. Therefore we will still focus on the reps of su(n) with comments on SU(n)
reps when necessary.
We next look at some examples of reps. 16
3.2
3.2.1
Examples of reps
Trivial/zero rep
The trivial/zero rep of a Lie algebra g is the rep that takes all elements of g to the zero linear map, i.e.,
ρ : g → gl(V )
(34)
ρ(a) = 0.
(35)
such that for arbitrary V and all a ∈ g,
Similarly, the trivial rep of a Lie group sends group elements to the identity matrix, so is also called the
identity rep.
16 I was very overwhelmed by all these different concepts so I make sure to include all the names I can find. These concepts
are not mutually exclusive.
10
3.2.2
Defining/natural/standard/tautological rep
The defining/natural/standard/tautological rep is the rep for which the Lie algebra is naturally defined.
For example, we have defined the su(n) Lie algebra as in Def. 0. Therefore, the defining rep is
ρ : su(n) → gl(Cn )
(36)
ρ(ej ) = (ej )kl
(37)
such that
for j ∈ {1, 2, · · · , d} and k, l ∈ {1, 2, · · · , n}, where ej ’s are the basis elements of su(n) and (ej )kl ’s are their
matrix forms on Cn given by Eqs. (12)–(14) and (15).
The defining/natural/standard/tautological rep of a Lie group is defined likewise. For finite groups like
S3 , the defining rep also means that there is a natural geometry interpretation.
3.2.3
Adjoint/regular rep
The adjoint/regular rep of a Lie algebra g of dimension d is the rep for which the g-module is g itself, i.e.,
ad : g → gl(g)
(38)
ad(a)b = [a, b].
(39)
such that for all a, b ∈ g,
Clearly, the effect of the adjoint rep is to take the Lie bracket.
We can define the adjoint operator of a ∈ g as ad(a) by
ad(a)b = [a, b] = ad(a)jk bj ,
(40)
where j, k ∈ {1, 2, · · · , d} since by definition the dimension of the adjoint matrices is d×d. Comparing Eqs. (7)
and (40), we find that the adjoint matrices of basis elements are connected to the structure constants by
ad(ej )lk el = fjkl el .
(41)
This is also the explicit form of the adjoint rep in basis {el }.
The fact that the adjoint rep is a rep is equivalent to the Jacobi identity (2).
Proof. Consider
ad([a, b]) = ad(a)ad(b) − ad(b)ad(a)
(42)
ad([a, b])c = ad(a)ad(b)c − ad(b)ad(a)c
[[a, b], c] = [a, [b, c]] − [b, [a, c]]
[a, [b, c]] + [b, [c, a]] + [c, [a, b]] = 0.
For a compact Lie group G and its corresponding Lie algebra g, the adjoint rep of the Lie group G is the
homomorphism Ad : G → GL(g) such that for b ∈ g and A ∈ G
Ad(A)b = AbA−1 ∈ g,
(43)
i.e., the adjoint operation is given by the conjugation of b by A. Let A = exa for x ∈ F and a ∈ g, then we
have by BCH
AbA−1 = exa be−xa = b + [a, b]x + O x2 .
(44)
11
Therefore, using the Lie correspondence we have
ad(a)b = [a, b]
(45)
which is exactly Eq. (39).
The adjoint rep is useful in physics since it is closely related to invariant theory, which can be seen from
its form in Eq. (43). 17 In Sec. 4.3.2 we will see how we can relate adjoint reps to symmetry through this
reasoning.
3.2.3.1
Killing form
Another useful tool constructed from the adjoint rep is the Killing form K(a, b).
Given a rep ρ of g on V , a bilinear form 18 B on V is invariant under the action of g if
B(ρ(a)v, w) + B(v, ρ(a)w) = 0
(46)
for v, w ∈ V and a ∈ g. Now let ρ be the adjoint rep with V = g. Then we say a bilinear form B on g is
invariant if
B(ad(a)b, c) + B(b, ad(a)c) = 0
(47)
B([a, b], c) + B(b, [a, c]) = 0
(48)
or
for a, b, c ∈ g.
One important example of such invariant bilinear forms on g is the Killing form K(a, b), defined as
K(a, b) = Tr [ad(a)ad(b)]
(49)
for a, b ∈ g. One can check that K(a, b) is also symmetric. For this reason, if the Killing form is also
invertible, it can be interpreted as a metric tensor. This property will be used in Sec. 4.4.
The Killing form of two basis elements of g, following Eq. (41), can be given by the structure constants,
i.e.,
K(ej , ek ) = fjlm fkml .
(50)
Specifically for the su(n) Lie algebra, the Killing form can be further simplified and yields
K(ej , ek ) = fjlm fkml = −fjlm fklm .
(51)
K(ej , ej ) = −fjlm fjlm ,
(52)
Therefore the diagonal elements are
which means Killing form on su(n) is negative definite.
We say a Lie algebra is compact if its Killing form is negative definite. 19 A compact Lie algebra can be
seen as the smallest real form of a complexification. In our case, su(n) is the real form of sl(n, C).
We can also directly calculate the Killing form of su(n) using their defining reps. For x, y ∈ su(n), the
Killing form is given by
K(x, y) = 2n Tr[xy].
(53)
Equation (53) also works for sl(n, C) although technically we have not given its defining rep (see Sec. 4.4.8).
Killing forms are useful in the structure theory of Lie algebras in Sec. 4.
17 Is
this correct???
bilinear form B(x, y) can have a coordinate representation as B(x, y) = xT Ay where A is the matrix of the bilinear
form. This footnote is a self reminder and has nothing to do with the context.
19 There is some ambiguity here regarding whether the definition of a compact Lie algebra should correspond to a Killing
form being negative definite or negative semidefinite. But su(n)’s Killing form is negative definite anyway so we do not care.
18 A
12
3.3
Morphism between reps
In this section we discuss morphisms between reps.
3.3.1
Intertwining operator
Definition 0 [intertwining operator]: An intertwining operator between two reps ρV and ρW of g (respectively, G) is a morphism f : V → W that commutes with the action of g (respectively, G), i.e.,
f ρV (a)v = ρW (a)f (v)
(54)
for a ∈ g (respectively, G) and v ∈ V .
Notice that here we follow the convention that intertwining operators map one g-module to another
g-module, instead of from rep ρV to rep ρW . The space of all g-morhisms between V and W is denoted as
Hom(V, W ).
3.3.2
(55)
Equivalent reps
If f is bijective (isomorphism), then V and W are equivalent. The two reps ρV and ρW are then called
equivalent reps. When two reps are equivalent reps, we say they are the same rep up to isomorphism, denoted
by
∼ ρW .
ρV =
Equivalent reps can be transformed to one another using similarity transformations.
3.3.3
(56)
20
Reps of su(n) and sl(n, C)
Theorem 3: Let g and gC be a real Lie algebra and its complexification. Then categories of complex reps
of g and gC are equivalent.
Theorem 3 means that any complex rep of g has a unique corresponding rep of gC , and Hom(V, W )g =
Hom(V, W )gC . In other words, if we classify the reps of gC , the reps of g are also classified.
In our case, Thm. 3 tells us that categories of complex reps of su(n) and sl(n, C) are equivalent. In fact,
considering Thm. 2, the categories of finite-dimensional reps of SL(2, C), SU(2), sl(2, C), and su(2) are all
equivalent. 21
3.4
Operations on reps
Theorem 4: Let V and W be g-modules associated with reps ρV and ρW , respectively. Then there is a
canonical structure 22 of a rep on V ⊕ W , V ⊗ W , and V ∗ . Specifically, the actions of g on these spaces are:
1. Action of g on V ⊕ W is given by ρV ⊕W : g → gl(V ⊕ W ) such that for a ∈ g, v ∈ V , and w ∈ W ,
ρV ⊕W (a)(v ⊕ w) = ρV (a)v ⊕ ρW (a)w.
(57)
2. Action of g on V ⊗ W is given by ρV ⊗W : g → gl(V ⊗ W ) such that for a ∈ g, v ∈ V , and w ∈ W ,
ρV ⊗W (a)(v ⊗ w) = ρV (a)v ⊗ w + v ⊗ ρW (a)w.
20 A
similarity transformation is diagonalization if the transformed matrix becomes diagonal.
is true for n right???
22 I think this means we can define a rep following the definition???
21 This
13
(58)
3. Action of g on V ∗ is given by ρV ∗ : g → gl(V ∗ ) such that for a ∈ g, v ∈ V , and v ∗ ∈ V ∗ ,
ρV ∗ (a)v ∗ = −ρ∗V (a)v ∗ ,
(59)
where ρ∗V (a) is the adjoint operator of ρV (a).
Comments on Thm. 4:
1. The corresponding Lie group actions on V ⊕ W , V ⊗ W , and V ∗ for A ∈ G are given by
ρV ⊕W (A)(v ⊕ w) = ρV (A)v ⊕ ρW (A)w,
(60)
ρV ⊗W (A)(v ⊗ w) = ρV (A)v ⊗ ρW (A)w,
(61)
ρV ∗ (A)v =
∗
ρ∗V
(A
−1
)v .
∗
(62)
2. The definition of ρV ⊗W is a natural result of linearity and the Lie correspondence.
3. The definition of ρV ∗ can be understood as follows. Consider the bilinear form h·, ·i which is given by
h·, ·i : V × V ∗ → C,
(63)
where V ∗ is the dual space of V . Then following from the fact that the natural pairing V × V ∗ and the
tensor product V ⊗ V ∗ have a universal property 23 , we can take V × V ∗ as the vector space V ⊗ V ∗
and understand the bilinear form h·, ·i as an intertwining operator between V ⊗ V and C (trivial rep).
Using the definition of the intertwining operator and Eq. (58), we get
hρV (a)v, v ∗ i + hv, ρV ∗ (a)v ∗ i = 0.
(64)
Equation (59) thus follows. Here the adjoint operator is the Hermitian adjoint or Hermitian conjugate
operator in physics (usually labeled with “†”). In Dirac notation, Eq. (64) is just saying
hv|ρV ∗ (a)ui = − hρV (a)v|ui
E
D
hv|ρV ∗ (a)ui = − v ρ†V (a)u .
(65)
Thus we get Eq. (59).
4. Observing Eqs. (78) and (59), we see that ρV is a unitary rep if ρV = ρV ∗ .
5. As a direct result of Thm. 4, the following vector spaces are all g-modules if V and W are g-modules:
(a) any tensor space V ⊗j ⊗ (V ∗ )
⊗k
for j, k ∈ N.
(b) Hom(V, W ), with the action given by
ρ(a)h = ρW (a)h − hρV (a)
(66)
for h ∈ Hom(V, W ).
Specifically, when W = V , Hom(V, V ) = End(V ) ' V ⊗ V ∗ , i.e., we now have
ρ : g → End(End(V )) = gl(gl(V )) = gl(V ⊗ V ∗ )
(67)
ρ(a)h = ρV (a)h − hρV (a)
(68)
with
where in Dirac notation h is just |vihu| for |vi ∈ V and hu| ∈ V ∗ . But Eq. (68) is just Eq. (39)!
So we have recovered the adjoint rep.
23 See
category theory.
14
Structure theory of sl(n, C) and its reps
4
We study the structure theory of Lie algebras with sl(n, C) as an example. With sl(n, C) being the
complexification of su(n), all of the useful results regarding su(n) can be derived from those regarding
sl(n, C). Mathematically, as we have mentioned, this means that categories of complex finite-dimensional
reps of su(n) and sl(n, C) are equivalent. Therefore, with su(n) as our target, we investigate the classification
theory of sl(n, C).
We start from formalizing the simplicity of Lie algebra.
4.1
4.1.1
Simple and semisimple Lie algebras
Ideals and commutativity
The first property of the structure of a Lie algebra we discuss is its commutativity (or abelianity), i.e.,
how close a Lie algebra is to an abelian Lie algebra. The tool we use to study commutativity is called the
ideal of a Lie algebra. A subalgebra g0 is invariant if [a, b] ∈ g0 , ∀a ∈ g0 and ∀b ∈ g. Then we define ideals
as follows.
Definition 0 [ideal]: An ideal i of a Lie algebra g is its invariant subalgebra, i.e., ∀g ∈ g, a ∈ i, [a, g] ∈ i.
24
Obviously, {0} and g are both ideals of g. Also, given i1 and i2 as ideals of g, i1 ∩ i2 , i1 + i2 , and [i1 , i2 ]
are all ideals, where
[i1 , i2 ] = {[a, b]}
(69)
for a ∈ i1 and b ∈ i2 .
With the ideal defined, we now examine the commutativity of a Lie algebra. Specifically, we make the
following definitions given a Lie algebra g:
1. The commutant of g is the ideal [g, g]. The smaller the commutant is, the closer g is to being abelian.
We say that a Lie algebra g is abelian if [a, b] = 0, ∀a, b ∈ g. Then g is abelian when [g, g] = 0.
2. Lie algebra g is solvable if Dn g = 0 for some n ∈ N, where
( j+1
D g = Dj g, Dj g
D0 g = [g, g]
.
(70)
3. A radical rad(g) is the unique solvable ideal that contains any other solvable ideal of g.
4. Lie algebra g is nilpotent if Dn g = 0 for some n ∈ N, where
(
Dj+1 g = [g, Dj g]
D0 g = g
.
(71)
Both nilpotent and solvable Lie algebras are “almost abelian” algebras, meaning that an abelian algebra can
be made by successive extensions of abelian algebras. As an example, let b ⊂ gl(n, F) be the subalgebra of
upper triangular matrices and n ⊂ gl(n, F) be the subalgebra of all strictly upper triangular matrices. Then
b is solvable, and n is nilpotent. We mention here there is a Cartan criterion for solvability, i.e., g is solvable
iff K([g, g], g) = 0.
24 Since
the commutator is skew-symmetric, the left ideal and right ideal are the same.
15
4.1.2
Simplicity and semisimplicity
We next introduce semisimplicity and simplicity to describe how far a Lie algebra is from being abelian.
Definition 0 [semisimplicity of Lie algebras]: A Lie algebra g is semisimple if any of the following equivalent
statements is true:
1. Lie algebra g does not have a nonzero abelian ideal.
2. Lie algebra g does not have a nonzero solvable ideal.
3. The radical rad(g) = 0.
4. (Cartan criterion) The Killing form K(a, b) for a, b ∈ g is nondegenerate, i.e., det [K(a, b)] 6= 0.
25
Definition 0 [simplicity of Lie algebras]: A Lie algebra g is simple if it is not abelian and does not possess
an ideal other than {0} and g.
Here we give several useful criteria for semisimplicity of a Lie algebra:
1. A Lie algebra is semisimple iff it is a direct sum of simple Lie algebras.
2. Any compact Lie algebra is semisimple.
3. If a Lie algebra is simple, then it is semisimple.
4. gC is semisimple iff g is semisimple, where g and gC are a Lie algebra and its complexification.
We will see in Sec. 4.2 that semisimple Lie algebras are completely reducible, which is a very desired property
in physics. Semisimplicity means far away from commutativity. If a Lie algebra g is semisimple, then its
commutant [g, g] = g.
All of the classical Lie algebras (also the most useful Lie algebras in physics) [see Sec. 4.4.6] except for D1
and D2 are simple. 26 Since sl(n, C) is a classical Lie algebra, it is simple, thus semisimple. Therefore su(n)
is semisimple. One counterexample is the Heisenberg algebra {X, P, 1} that is neither simple nor semisimple.
4.1.3
Levi decomposition
Given the tools to describe a Lie algebra’s commutativity, we have the Levi theorem.
Theorem 5 [Levi decomposition]: Any Lie algebra g can be written as a direct sum
g = rad(g) ⊕ gss
(72)
where gss is a semisimple Lie algebra (not necessarily an ideal).
The Levi theorem states that any Lie algebra can be written as a direct sum of an abelian part and a
semisimple part.
We can add more structure to the abelian part. If rad(g) = z(g), where z(g) = {a ∈ g | [a, b] = 0, ∀b ∈ g}
is the center of g, then we say g is reductive. 27 Any simple Lie algebra is reductive.
25 Recall that a symmetric bilinear form on a finite-dimensional vector space V is nondegenerate/invertible if for all a 6= 0
there exists b such that K(a, b) 6= 0.
26 What is D ???
1
27 Reductivity is a nice property for Lie algebras since it ensures [g, rad(g)] = 0, i.e., elements in an irrep will act as zero only
when they are zeros.
16
4.2
4.2.1
Reducible and irreducible reps
Reducibility
We now turn from Lie algebras to their reps. We make the following definitions.
Definition 0 [reducibility of reps]: A nontrivial rep ρ : g → gl(V ) of a Lie algebra g is irreducible if it has
no subreps other than the trivial rep or itself. A rep is reducible if it is not irreducible.
For convenience, we say the V is an irreducible sl(2, C)-module, or V is irreducible, if rep ρ : g → gl(V )
is irreducible.
As an example, we have the following theorem.
Theorem 6: The rep ρ : sl(n, C) → gl(Cn ) is irreducible.
Theorem. 6 also means the defining reps of sl(n, C) are irreducible. Following Thm. (6), we clarify the
difference between the direct sum of vector spaces Cl and Cm and the direct of sum of g-modules Cl and
Cm . For arbitrary vector spaces Cl and Cm , we have Cl+m = Cl ⊕ Cm . However, for vector spaces Cl and
Cm acted upon by reps of g, we have
Cl+m 6= Cl ⊕ Cm (g-modules).
(73)
u
with a ∈ g, u ∈ Cl , and
This is because for rep ρCl+m : g → gl Cl+m , we require action ρCl+m (a)
v
v ∈ Cm . However for reps ρCl : g → gl Cl and ρCm : g → gl(Cm ), following Eq. (57), the rep of their direct
u
0
u
+ ρCm (a)
. Clearly, by definition ρCl ⊕Cm is reducible, while ρCl+m
= ρCl (a)
sum acts as ρCl ⊕Cm (a)
0
v
v
can be irreducible. In this sense, we have Eq. (73). Therefore Thm. 6 tells us that any rep of sl(n, C) on Cn
is irreducible, n ∈ Z and n ≥ 2. Nevertheless any rep of sl(n, C) on Cm ⊕ Cn−m is obviously reducible, for
m, n − m ∈ Z+ .
4.2.2
Complete reducibility
Irreducible reps, or irreps, of semisimple Lie algebras have nice properties that they can be classified
completely as we will see later. In fact, one of the most commonly used technique in physics is to decompose
a direct product of irreps as a direct sum of irreps, which will then be useful due to their block diagonal
nature. We formalize this process by introducing the complete reducibility of reps.
Definition 0 [complete reducibility of reps]: A rep of g acting on V is completely reducible if it is isomorphic
to a direct sum of irreps, i.e., V ' ⊕nj Vj for nj ∈ N. Here nj ’s are called multiplicities, and Vj ’s are gmodules of pairwise nonisomorphic reps.
For convenience, we say V is a completely reducible g-module, or V is completely reducible, if the rep
on V is completely reducible.
Not every rep of a Lie algebra is completely reducible. Mathematically, we want to answer the following
three questions:
1. For what Lie algebras are all the reps completely reducible?
2. For a given completely reducible rep of a Lie algebra, what is the decomposition into irreps?
3. For a given Lie algebra, classify all irreps.
In this section, we answer the first question. The second question will be answered theoretically use characters
in Sec. 4.3.5, but a practical method using weight decomposition will be given in Sec. 4.4. The third question
will be briefly answered in Sec. 4.4.6.
The answer to the first question is the following theorem.
17
Theorem 7: Any complex finite-dimensional rep of a semisimple Lie algebra is completely reducible.
Since sl(n, C) is semisimple, any complex finite-dimensional rep of sl(n, C) is completely reducible, and
so is that of su(n). In Sec. (4.4), we will focus on the decomposition of reps of semisimple Lie algebras, with
sl(n, C) as an example.
In the next section, we examine other theorems that are related to complete reducibility. This is a
deviation from purely Lie algebra structure theory since we discuss compact Lie groups and their associated
real Lie algebras such as SU(n) and su(n). However, the contents are very important in physics.
4.3
4.3.1
Complete reducibility and symmetry
Schur’s lemma
One useful tool to decompose completely reducible reps into irreps is the use of intertwining operators.
Specifically, there is Schur’s lemma.
Theorem 8 [Schur’s lemma]: Let ρV and ρW be complex irreps of g (respectively, G).
1. If ρV ∼
= ρW (V = W ), then Hom(V, W ) = Hom(V, V ) = C1.
2. If ρV ρW , then Hom(V, W ) = 0.
Schur’s lemma tells us how inequivalent irreps are fundamentally different from each other. Part 1 says
equivalent irreps only differ by coefficients. Part 2 says inequivalent irreps cannot be intertwined unless
trivially.
The following theorems are direct results of Schur’s lemma.
Theorem 9: If g (respectively, G) is commutative, then any complex irrep of g (respectively, G) is onedimensional.
Theorem 10: Let ρV be a completely reducible rep of g (respectively, G).
L
1. If V =
Vj , ρVj irreducible and pairwise nonisomorphic, then any intertwining operator Φ : V → V
L
is of the form Φ =
λj 1Vj .
L
L nj
2. If V =
nj Vj =
C ⊗ Vj , ρVj irreducible and pairwise nonisomorphic, then any intertwining
L
operator Φ : V → V is of the form Φ =
Aj ⊗ 1Vj , for Aj ∈ End(Cnj ).
Theorem 10 gives us a very effective way to analyze intertwining operators. Specifically, it is extremely
useful in physics as the Hamiltonian can be thought of as an intertwining operator.
4.3.2
Hamiltonian & first definition of symmetry
Consider a Hilbert space V , which is a complex vector space, and a Hamiltonian operator H : V → V .
Then the following statements are equivalent:
1. There is a symmetry described by a Lie group G.
2. There is an action of G on V that leaves H invariant.
3. For all A ∈ G,
AHA−1 = H.
4. H is an intertwining operator that commutes with actions of G.
18
(74)
The above statements can be taken as a first definition of symmetry. Specifically, it predicts a very nice
physics picture as illustrated below.
Given H as an intertwining operator H : V → V , if the Lie group G has a completely reducible rep on
L
V such that V =
nj Vj is the G-module, then immediately from Thm. 10 we see that H has the form


···


c11 1Vj · · · c1nj 1Vj

M



(75)
H=
Aj ⊗ 1Vj = 
···
···
···
.




cnj 1 1Vj · · · cnj nj 1Vj
···
If all the nj ’s are 1, then
H=
M
Ej 1Vj

···
=

(76)
.
Ej 1Vj
···
Since H has a block diagonal form, the following statements are true:
1. If the H commutes with a rep of G, then the energy eigenspaces are irreps. Group actions of G on an
energy eigenstate will not change its energy.
2. A state that lives in the vector space of an irrep will stay there under time evolution.
28
All this seems nice, only if G has a completely reducible rep.
4.3.3
Unitary rep & the second definition of symmetry
It turns out that in our case G does have a completely reducible rep. To see this we first introduce the
notion of unitary reps.
Definition 0 [unitary rep]: A complex rep ρ : g → gl(V ) of a real Lie algebra g is unitary if there is an
inner product that is g-invariant, i.e., for all a ∈ g and v, w ∈ V
(ρ(a)v, w) + (v, ρ(a)w) = 0,
(77)
ρ(a)† = −ρ(a).
(78)
or equivalently, if ρ(a) ∈ u(V ), i.e.,
A complex rep ρG : G → GL(V ) of a real Lie group G is unitary if there is an inner product that is
G-invariant, i.e., for all A ∈ G and v, w ∈ V ,
(ρG (A)v, ρG (A)w) = (v, w),
(79)
ρG (A)† = ρG (A)−1 .
(80)
or equivalently, if ρG (A) ∈ U(V ), i.e.,
Unitary reps are nice because they are completely reducible.
Theorem 11: Every unitary rep is completely reducible.
Therefore if we can show that Lie groups have unitary reps then we are done. This is given by the
following theorem.
28 We
will discuss this vector space, the multiplet, in the next section.
19
Theorem 12 [Peter-Weyl, Part II]: Any finite-dimensional rep of a compact Lie group G (respectively, g)
is unitary and thus completely reducible.
Notice that Thm. 12 discusses a narrower class of Lie algebras than Thm. 7, since compact Lie algebras
[e.g., su(n)] are the compact real forms of the corresponding complex semisimple Lie algebras [e.g., sl(n, C)].
However, Thm. 12 does provide the unitarity property that Thm. 7 does not guarantee.
From Thm. 12 we get the following statement.
Theorem 13: Any finite-dimensional rep of SU(n) and su(n) is unitary and completely reducible.
We can now collect our thoughts. Given a compact Lie group G with its associated real Lie algebra g and
the complexification gC , we know that any finite-dimensional rep of G, g, and gC is completely reducible.
Therefore, given a Hamiltonian operator, which is also an intertwining operator H : V → V , all the results
from Sec. 4.3.2 will stand.
This is nice, but where did the unitarity of reps of G and g play a role?
Recall in Eq. (43) we have introduced the adjoint rep of a Lie group that leaves a Lie algebra element
invariant, i.e., for a ∈ g and A ∈ G,
AaA−1 ∈ g.
(81)
Now that we know any finite-dimensional rep of G or g is unitary from Thm. 12, we can define the unitary
reps ρG : G → U(V ) and ρ : g → u(V ). Then for A ∈ G and a ∈ g we have
ρG (A)ρ(a)ρG (A)−1 ∈ u(V ).
(82)
Let us look at Eq. (82) and think about symmetry again. If states in a Hilbert space transform under a Lie
group action that is unitary, then we say the group is a group of symmetry of the Hilbert space since the
group action preserves the inner product of states. Then observables, as linear operators on the Hilbert space,
are well-defined if their transformation under the group action follows the transformation of the states in
such a way that if the observables and states are transformed together then no information should be able to
get measured. Since states transform with ρG (A), the observables should transform by a conjugation of these
unitary reps, i.e., exactly given by Eq. (82). Considering Hermitianity, a Hermitian operator (observable)
acting on V can thus be constructed as iρ(a), which are the generators of the Lie algebra. From Eq. (82),
after transformation under the group action, the observables remain Hermitian.
In this sense, we give the second definition of symmetry in quantum mechanics.
Definition 0 [symmetry]: We say that there is a symmetry when there is a unitary rep of a Lie group acting
on the Hilbert space.
The Lie algebra generators both generate the unitary Lie group rep, and also transform under it as
observables via conjugation. Normally, observables transform into other generators of the Lie algebra.
Specifically, when the observable is the Hamiltonian H, and the group action leaves H invariant, we retain
the first definition of symmetry in Sec. 4.3.2.
4.3.4
Multiplet
L
In Sec. 4.3.2 we see that for a completely reducible rep, its g-module can be written as V =
nj V j .
A state that lives in one of the g-modules of the irreps Vj remains in Vj when acted by the intertwining
operator H. These g-modules of the irreps are called multiplets in physics.
Definition 0 [multiplet]: The g-module of an irrep ρ of Lie algebra g is called the multiplet of ρ.
The concept of multiplets appears everywhere in quantum physics. Here are some important properties:
1. Stability. Let ρ be a rep. If ρ is irreducible, then by definition states transform in the same multiplet
under Lie algebra elements. If ρ is completely reducible, then due to the block diagonal form, states
20
that are in a certain multiplet will transform within the same multiplet under actions of Lie algebra
elements.
2. State multiplets. We refer to the multiplet expressed in a specific state basis {vj } as the state multiplet
associated with the basis.
3. The su(n) and sl(n, C) Lie algebras have the same multiplets, as a result of Thm. 3.
Next we define the fundamental multiplets of the su(n) algebras.
29
Definition 0 [fundamental multiplet]: If the sl(n, C)-module is Cn , then we call this sl(n, C)-module the
fundamental multiplet of sl(n, C). Similarly for su(n).
By Thm. 6, the fundamental multiplet of of sl(n, C) is well-defined. The fundamental multiplet of sl(n, C)
is the defining multiplet of sl(n, C), as well as the multiplet of lowest dimension corresponding to the irreps
of sl(n, C).
4.3.5
Characters
So far we have seen that a large class of reps are completely reducible. However, we have not talked
about the decomposition of completely reducible reps or the multiplicity yet. In this section we introduce
the concept of characters.
Definition 0 [characters of real compact Lie groups and algebras]: Let G and g be a real compact Lie group
and its associated Lie algebra.
The character XV of a finite-dimensional rep ρG : G → GL(V ) is the function on G defined by
XV (A) = Tr[ρG (A)].
The character χV of a finite-dimensional rep ρ : g → gl(V ) is the function on g defined by
h
i
χV (a) = Tr eρ(a) .
(83)
(84)
Definition .0 has the following consequences:
1. Characters are basis free.
2. The characters are connected by
χV (a) = XV (ea ).
(85)
Since the characters of G and g are defined to be equal for corresponding elements, below we only
discuss XV .
3. If ρG is trivial rep, then XV (A) ≡ 1.
4. For A, B ∈ G,
XV ⊕W = XV + XW ,
(86)
XV ⊗W = XV XW ,
(87)
XV ∗ =
X(ABA
29 I
−1
XV∗ ,
(88)
) = XV (B).
(89)
do not know if this definition is useful at all. See Greiner for more on multiplets.
21
Characters offer a theoretical way to determine the multiplicities through an inner product on C ∞ (G, C)
given by
ˆ
(f1 , f2 ) =
f1 (A)f2∗ (A) dA
(90)
G
where dA is the Haar measure. From Schur’s lemma in Thm. (8), we can prove
Theorem 14: Let ρV and ρW be nonisomorphic complex irreps of a compact Lie group G. Then
1. (XV , XW ) = 0, which means the characters XV and XW are orthogonal with respect to the inner
product.
2. (XV , XV ) = 1.
This implies the following theorem.
Theorem 15: Let ρV be a complex rep of a compact real Lie group G. Then
1. ρV is irrep iff (XV , XV ) = 1.
L
2. V can be uniquely written in the form V =
nj Vj , where Vj ’s are G-modules of pairwise nonisomor
phic irreps and the multiplicities nj are given by nj = XV , XVj .
Theorem. 15 gives a way to calculate nj . In reality, it is usable only for finite groups and special cases of
Lie groups due to the difficulty to calculate the inner product (90). In Sec. 4.4 we will develop a much more
practical method using weight decomposition.
4.4
Reps of sl(n, C)
In this section we answer the second question we asked in Sec. 4.2, i.e., how to find the decomposition
of a completely reducible rep. We will assume g to be a finite-dimensional complex semisimple
Lie algebra for the remainder of Sec. 4. Specifically, we use sl(n, C) as an example, which is also our
primary goal. For simplicity, whenever we specify the g-module, we will write ρ(a)v = av for a ∈ g, i.e., we
treat a not only as the Lie algebra element, but also as an operator. This notation is closer to the one used
in physics.
We start from the simplest case of sl(n, C), the sl(2, C) Lie algebra.
4.4.1
Reps of sl(2, C)
One most commonly used basis of sl(2, C) is {e, f, h} that satisfies
[h, e] = 2e,
[h, f ] = −2f,
[e, f ] = h.
In the defining rep, which is on C2 , these basis elements are commonly chosen as
0 1
0 0
1 0
e = σ+ =
, f = σ− =
, h = σz =
.
0 0
1 0
0 −1
(91)
(92)
Notice that h is diagonal and has eigenvalues ±1.
More generally, in an arbitrary rep, we know by the fundamental theorem of algebra that h can be
diagonalized with complex eigenvalues. This can be formalized as the weight decomposition.
22
Definition 0 [weight, sl(2, C)]: Let V be an sl(2, C)-module. A vector v ∈ V is called a vector of weight λ,
λ ∈ C, if it is an eigenvector of h with eigenvalue λ, i.e.,
hv = λv ⊂ Vλ ,
(93)
where the subspace with eigenvalue λ is denoted as Vλ ⊂ V .
Then the actions of e and f on Vλ can be derived from commutation relations (91), yielding
ev ⊂ Vλ+2 ,
(94)
f v ⊂ Vλ−2 .
(95)
Since e brings a vector from Vλ to Vλ+2 , operator e is usually called the raising operator. Similarly, since f
brings a vector from Vλ to Vλ−2 , operator f is usually called the lowering operator.
We can thus apply the weight decomposition to reps of sl(2, C).
Theorem 16 [weight decomposition, sl(2, C)]: Every finite-dimensional rep of sl(2, C) has a weight decomposition, i.e.,
M
V =
Vλ .
(96)
λ
By Thm. 7, any complex finite-dimensional rep of sl(2, C) is completely reducible. Therefore, it suffices
to discuss only the weight decomposition of irreps. Specifically, we define the highest weight of V .
Definition 0 [highest weight]: Let V be irreducible. A weight λ of V (Vλ 6= 0) is the highest weight if for
every weight λ0 of V we have
Re λ ≥ Re λ0 .
(97)
We can now write down the classification of irreps of sl(2, C).
Theorem 17 [classification, irreps of sl(2, C)]: Let {h, e, f } be the basis of sl(2, C).
1. For any n ∈ N, let Vn be a (n + 1)-dimensional vector space with basis {v0 , v1 , · · · , vn }. Define the
action of sl(2, C) by
hvk = (n − 2k)vk ,
(
f vk = (k + 1)vk+1 ,
(98)
k < n;
f vn = 0,
(
evk = (n + 1 − k)vk−1 ,
k > 0;
ev0 = 0,
(99)
(100)
Then ρVn : sl(2, C) → gl(Vn ) is an irrep. We call this irrep the irrep with highest weight n.
2. For n 6= m, reps ρVn and ρVm are nonisomorphic.
3. Every finite-dimensional irrep of sl(2, C) is isomorphic to one of the reps ρVn .
See Fig. 1 for a figure presentation of Thm. 17.
The classification of irreps of sl(2, C) provides the fundamental intuition for the classification of irreps
of semisimple Lie algebras. To generalize this weight decomposition to sl(n, C) and other semisimple Lie
algebras, we require the concept of the Cartan subalgebra.
23
1
vn
n−2k
−n+2
−n
n
2
v n−1
n+1−k
n−k
vk
···
n−1
n−2
k+1
n
n
n−1
v1
···
2
k
v0
1
Figure 1: Action of sl(2, C) on Vn in the irrep ρVn . Operator h sends a vector vk back to itself via the self
loop arrows. Operators e and f act as the top and bottom arrows, respectively.
4.4.2
Cartan subalgebra
For a finite-dimensional complex semisimple Lie algebra, the Cartan subalgebra can be defined as follows.
Definition 0 [Cartan subalgebra of complex semisimple Lie algebras]: A Cartan subalgebra h is the maximal
abelian subalgebra of Lie algebra g consisting semisimple elements. Semisimple elements are elements x ∈ g
such that the adjoint operator ad(x) : g → g is semisimple (diagonalizable).
In short, the Cartan subalgebra is the maximal abelian subalgebra with diagonalizable elements in the
adjoint rep. In every complex semisimple Lie algebra, there exists a unique Cartan subalgebra up to isomorphism. The dimension l of this unique Cartan subalgebra h is called the rank of a Lie algebra.
Definition 0 [rank]: The rank l of g is the dimension of the Cartan subalgebra h, i.e.,
l = dim h.
(101)
For sl(n, C),
h=


diag{c1 , c2 , · · · , cn } | cj ∈ C,

X
j


cj = 0

(102)
is a Cartan subalgebra. This will be the most important Cartan subalgebra we use in this work.
The rank of g is the largest number of commuting basis elements (generators) if they are all diagonalizable
in the adjoint rep. For example, the rank of sl(n, C) is l = n − 1, following from Eq. (102). Similarly, the
rank of su(n) is also l = n − 1.
The above definitions are enough for our use of Cartan subalgebras in physics. The more general definition
of a Cartan subalgebra is rather subtle. Below we add a brief compendium of the full construction of the
Cartan subalgebra. We first introduce the following concepts:
1. Normalizer. The normalizer of a subset s in a Lie algebra g is
Ng (s) = {x ∈ g | [x, s] ∈ s, ∀s ∈ s}.
(103)
A subalgebra h of a Lie algebra g is self-normalizing if Ng (h) = h.
2. Centralizer. The centralizer of a subset s in a Lie algebra g is
Cg (s) = {x ∈ g | [x, s] = 0, ∀s ∈ s}.
(104)
An element a ∈ g is regular if the dimension of its centralizer (if we take the element as the subset) is
minimal among all centralizers of elements of g.
Consider a finite-dimensional Lie algebra g over a field F. Then we have the following definition.
Definition 0 [Cartan subalgebra, general]: A Cartan subalgebra is a nilpotent subalgebra h of a Lie algebra
g that is self-normalizing. 30
30 In Kirillov there is another way to define Cartan subalgebra from its toral property. In Greiner and Muller P440 there is
yet another way using simple linear algebras.
24
Specifically,
1. if F is infinite, then g has Cartan subalgebras;
2. if F is of characteristic zero, then there is a Cartan subalgebra for every regular element of g such that
each regular element belongs to one and only one Cartan subalgebra and all Cartan subalgebras have
the same dimension l;
3. if F is algebraically closed, then all Cartan subalgebras are isomorphic;
4. if g is semisimple, then all Cartan subalgebras are Abelian.
Therefore, for semisimple Lie algebras over R or C, Def. 0 follows from Def. 0 and the properties above.
For sl(2, C), the Cartan subalgebra is just Ch, which is one-dimensional. Theorem. 17 gives a full
classification of all the irreps of sl(2, C) using the Cartan subalgebra and its corresponding raising and
lowering operators. For a more complicated semisimple Lie algebra g, its turns out that we can always
consider g as a module over sl(2, C) and use Thm. 17 to classify its reps. However, to understand this result
we need to work with a special decomposition of g, the root decomposition, which is the weight decomposition
in the adjoint rep.
4.4.3
Root decomposition
Consider a semisimple Lie algebra g [such as sl(n, C)] and its Cartan subalgebra h as defined in Def. 0
which is unique up to isomorphism.
Let {Hj }, j ∈ {1, 2, · · · , l}, be a basis of h ⊂ g. By definition, all Hj ’s commute with each other and can
be simultaneously diagonalized in the adjoint rep. This means that the simultaneous eigenvectors of Hj ’s
must span g, thus forming a basis. The eigenvectors with zero eigenvalues span h. Therefore, without loss
of generality, we can choose these eigenvectors as Hj themselves.
The eigenvectors with nonzero eigenvalues span g/h. For each of these eigenvectors, the set of l nonzero
eigenvalues associated with the Hj ’s constitutes another vector of dimension l, called a root. Denoting such
an eigenvector as Eα and its associated root as α, we can express the eigendecomposition of Hj by
[Hj , Eα ] = αj Eα .
(105)
The form of Eq. (105) implies duality. We can take the root α as a linear functional on h by considering
some bilinear form h·, ·i. In other words, all the roots together with zero span h∗ , the dual space of h. For
convenience we define E0 ∈ h. However, when we refer to roots we always assume they are nonzero. We
thus have the root decomposition theorem.
Theorem 18 [root decomposition]: The root decomposition of g is
!
M
g=h⊕
gα
(106)
α∈R
where α’s are called the roots, the subspaces
gα = {x ∈ g | [h, x] = hα, hi x, ∀h ∈ h}
(107)
are called the root subspaces, and the set
R = {α ∈ h∗ \{0} | gα 6= 0}
(108)
g0 = h.
(109)
is called the root system.
Specifically, we define
25
The root decomposition is the weight decomposition in the adjoint rep. As an example, consider g =
sl(n, C) with h defined in Eq. (102). Define ej : h → C as the functional


h1


..
ej : 
(110)
 → hj .
.
hn
P
Since j ej = 0, we have

,

M
X
h∗ = 
Cej  C
ej .
(111)
j
j
Notice that matrix units Ejk are eigenvectors of ad(h) for h ∈ h since
ad(h)Ejk = [h, Ejk ] = (hj − hk )Ejk = hej − ek , hi Ejk .
(112)
Thus the root decomposition of sl(n, C) is given by
R = {ej − ek | j 6= k} ⊂ h∗ ,
gej −ek = CEjk .
(113)
(114)
With the root decomposition, we can show the following properties:
1. For root subspaces,
[gα , gβ ] ⊂ gα+β .
(115)
2. If α + β 6= 0 then gα and gβ are orthogonal with respect to the Killing form K.
3. For any α, the Killing form K gives a non-degenerate pairing
gα ⊗ g−α → C.
(116)
In particular, restriction of K to h is non-degenerate.
So far we have not specified the natural pairing h·, ·i introduced in Eq. (107). One standard procedure to
define this natural pairing is to consider a nondegenerate symmetric bilinear form (·, ·), i.e., a metric, defined
on g. Such a metric exists. For example, by the Cartan criterion in Def. 0 and the last property above, the
Killing form is one such metric for semisimple Lie algebras. In fact, if we take the metric as the Killing form,
as we will see later, we can define the so-called Cartan-Weyl basis for a semisimple Lie algebra. However, for
now, we just assume there exists such a metric (·, ·). Since the restriction of (·, ·) to the Cartan subalgebra h
is nondegenerate, we can also define a metric on its dual h∗ , which we also denote as (·, ·). Finally, denoting
the dual element of root α ∈ h∗ as Hα ∈ h, we make the following definition
hα, Hβ i = (Hα , Hβ ) = (α, β),
(117)
for α, β ∈ h∗ . Immediately we can show that
1. Let e ∈ gα , f ∈ g−α . Then
[e, f ] = (e, f )Hα .
(118)
(α, α) = (Hα , Hα ) 6= 0.
(119)
2. For α ∈ R,
26
3. Let e ∈ gα , f ∈ g−α be such that
2
,
(α, α)
(e, f ) =
(120)
and let
hα =
2Hα
.
(α, α)
(121)
Then hhα , αi = 2 and the elements e, f, hα satisfy the commutation relations (91) of sl(2, C). We
denote such a subalgebra by sl(2, C)α ⊂ g. Moreover, the so defined hα is independent of the choice of
the metric (·, ·).
4. The vector space
M
V = Chα ⊕
gkα ⊂ g
(122)
k∈Z,k6=0
is an sl(2, C)α -module.
From the last statement (lemma) we see that the root decomposition offers a way to study the structure
of any semisimple Lie algebra g by decomposing it into sl(2, C)α ’s and then use the theory of reps of sl(2, C)
given in Sec. 4.4.1 to study it. The main theorem is given below.
Theorem 19 [structure of semisimple Lie algebras]: Let g be a complex semisimple Lie algebra with Cartan
subalgebra h and root decomposition given in Eq. (106). Let (·, ·) be a nondegenerate symmetric invariant
bilinear form on g.
1. The root system R spans h∗ as a vector space. The elements hα , α ∈ R, defined by
hα =
2Hα
,
(α, α)
(123)
span h as a vector space.
2. For any α ∈ R, the root subspace gα is one-dimensional.
3. For any α, β ∈ R, the number
hhα , βi =
2(α, β)
(α, α)
(124)
is an integer.
4. For any α, β ∈ R, the reflection operator sα : h∗ → h∗ , defined by
sα (β) = β − hhα , βi α = β −
2(α, β)
α
(α, α)
(125)
is also a root. In particular, sα (α) = −α ∈ R.
5. For any α ∈ R, the only multiples of α that are also roots are ±α.
6. For any α, β ∈ R such that β 6= ±α, the subspace
M
V =
gβ+kα
(126)
k∈Z
is an sl(2, C)α -module.
7. For any α, β ∈ R such that α + β ∈ R, we have [gα , gβ ] = gα+β .
We will not give the proof here. Rather we will try to understand the construction by studying the
abstract root space Sec. 4.4.5. However, before leaving the Lie algebra and dive into the dual space, we
derive the Cartan-Weyl basis which replies on taking the Killing form to be the metric (·, ·).
27
4.4.4
Cartan-Weyl basis
In this section we derive the Cartan-Weyl basis. While the construction is not more insightful than what
we have done in the last section, it is useful in physics by explicitly defining a usable basis for a semisimple
Lie algebra. We would like to use the Hj ’s and Eα ’s defined in Eq. (105) as a basis of the Lie algebra g. To
do this, it remains to uniquely define each eigenvector Eα and the corresponding root α given a basis {Hj }
of h. With our construction this still requires two constraints:
1. a rescaling factor for the Hj and the corresponding root components αj ,
2. a normalization condition for Eα .
Both of these can be chosen by explicitly defining the bilinear form (the natural pairing in this case) on h
and h∗ . Below we define such a natural pairing associated with the the Killing form K(a, b), a, b ∈ g.
It is natural to consider the Killing form since we work in the adjoint rep. By Eq. (49), the Killing form
is a symmetric bilinear invariant form. By Def. 0, the Killing form is nondegenerate given a semisimple Lie
algebra g. Therefore we can identify it as a metric tensor on g, denoted as
gab = K(a, b) ≡ (a, b).
(127)
Using the definition of the structure constant in Eq. (7) and the relation between the Killing form and the
structure constants in Eq. (50), we can show the following results:
1. If α is a root, then −α is a root.
2. We can write the nondegenerate Killing form in a block diagonal. Denoting
gjk = (Hj , Hk )
(128)
gα,−α = (Eα , E−α ),
(129)
and
we have
gab

gjk




=





0
gα,−α
gα,−α
0
0
gβ,−β
gβ,−β
0
..
.




,




(130)
where j, k ∈ {1, 2, · · · , l} and α, β label different roots. Notice here gjk is an l × l dimensional matrix.
Nondegeneracy requires that we always have det [gjk ] 6= 0 and gα,−α =
6 0 for any root α.
Both of these results have been mentioned in the previous section, but we show them here as a result of
choosing the Killing form as the metric. The fact that gjk itself is a metric tensor on h allows us to use it as
a prescription to uniquely define the bilinear form h·, ·i on h and h∗ . Below I give the standard procedure to
construct the corresponding inner products in and between a vector space and its dual given a metric. This
logic seems not intuitional to physicists.
Consider an element in h as
X
Hλ =
λ0j Hj ,
(131)
j
28
where we write out the summation symbol on purpose, and λ0j ’s are some coefficients. With the dual space
structure and the metric tensor, the element Hλ can also be expressed as
Hλ = λj g jk Hk = λj Hj .
(132)
Here we have used the Einstein notation and asserted
j
g jk gkm = δm
,
(133)
j
where δm
in matrix form is the identity matrix, and g jk is the inverse of the metric tensor gjk . We see that
by defining
λj = λ0j
(134)
we have created a bijective mapping between Hλ ∈ h and λ ∈ h∗ , defined uniquely by the metric tensor. We
denote Eq. (132) by the bilinear form (inner product) hλ, Hi such that
Hλ = hλ, Hi = λj g jk Hk = λj Hj .
(135)
Notice that Hλ is both the element in h corresponding to root λ but also the result of the bilinear form. The
Killing form of two such elements Hα and Hβ in the Cartan subalgebra h is
(Hα , Hβ ) = αj (Hj , Hk )β k = αj gjk β k = αj g jk βk .
(136)
For convenience, we denote the dual of the Killing form in h∗ also by the inner product
αj g jk βk = (α, β).
(137)
Immediately, we see that our definition above satisfies
(Hα , Hβ ) = (α, β) = hα, Hβ i ,
(138)
which is consistent with Eq. (117).
So far we have fixed the rescaling factor between Hα and α by connecting them with a metric. Now we
determine the eigenvector Eα given its root α. Using the Jacobi identity (2) and Eq. (105), we can show
[Hj , [Eα , Eβ ]] = (αj + βj )[Eα , Eβ ].
(139)
1. If αj + βj = 0 for some j, then [Eα , Eβ ] ∈ h, so α + β = 0. We then calculate
([Eα , E−α ], Hβ ) = (Eα , [E−α , Hβ ]) = hα, βi (Eα , E−α ) = hHα , Hβ i (Eα , E−α ),
(140)
where we have used the fact that the Killing form is invariant [this also proves Eq. (118)]. Thus we
have
[Eα , E−α ] = (Eα , E−α )Hα .
(141)
2. If αj + βj 6= 0 for some j, then [Eα , Eβ ] ∈ g/h, so α + β 6= 0. Then [Eα , Eβ ] is an eigenvector of Hj
with eigenvalue (αj + βj ). We write
[Eα , Eβ ] = Nαβ Eα+β ,
where the matrix Nαβ is defined with the following properties:
(a) The matrix Nαβ depends on the normalization of Eα ’s.
(b) If α + β is not a root then Nα+β = 0.
29
(142)
Choose the normalization (Eα , E−α ) = 1, we have the Cartan-Weyl basis.
Definition 0 [Cartan-Weyl basis 31 ]: The Cartan-Weyl basis of a semisimple Lie algebra g is a basis of g
that consists of {Hj } and {Eα }, where Hj ’s are a basis of the Cartan subalgebra h of g and Eα ’s are a basis
of g/h. The defining Lie products are
[Hj , Hk ] = 0,
(143)
[Hj , Eα ] = αj Eα ,
(144)
[Eα , E−α ] = hα, Hi ,
(145)
[Eα , Eβ ] = Nαβ Eα+β
(α + β 6= 0).
(146)
Notice that although both semisimple Lie algebras over R [such as su(n)] and over C [such as sl(n, C)]
have Cartan subalgebras, the Cartan-Weyl approach automatically introduces the complex extensions of the
real Lie algebras since diagonalization generally results in complex eigenvalues and complex coefficients of
basis vectors in the expression of eigenvectors by the fundamental theorem of algebra. In the adjoint rep,
the latter means the complexification of real Lie algebras. As an example, consider su(2), where we normally
choose basis {e1 , e2 , e3 } such that
[e1 , e2 ] = e3 ,
[e2 , e3 ] = e1 ,
[e3 , e1 ] = e2 .
(147)
We choose {e3 } as the Cartan subalgebra without loss of generality and find its eigendecomposition in the
adjoint rep, i.e.,
[e3 , e1 + ie2 ] = −i(e1 + ie2 ).
(148)
Therefore, the diagonalization of the Cartan subalgebra of the real su(2) algebra automatically leads to an
expression that is only defined in its complex extension sl(n, C). This is also why we study sl(n, C) instead
of su(n) in the first place.
4.4.5
Root system
4.4.5.1
Definition
In Secs. 4.4.3–4.4.4 we see how the root system can tell us about the structure of a semisimple Lie algebra.
In this section we examine the abstract root system as an independent object, which will help us understand
Thm. 19. It can be proved that the dual space generated 32 by α ∈ R is completely determined by the real
vector space generate by α ∈ R, denoted as h∗R , or in mathematical form h∗ = h∗R ⊕ ih∗R . Therefore below
we only need to define the real vector space.
Definition 0 [abstract root system]: An abstract root system is a finite set of elements R ⊂ E\{0}, where
E is a Euclidean vector space 33 such that the following properties hold:
1. R generates E as a vector space. The number r = dim E is called the rank of R.
2. For any two roots α and β, the number
nαβ =
2(α, β)
(β, β)
(149)
is an integer.
31 The Cartan-Weyl basis is not unique. Using h , defined in Eq. (124), instead of H , we get the Chevalley basis where
α
α
Nαβ become all integers.
32 Here a vector space “generated” by v means the minimum set of elements that contains v and has the structure required
by a vector space (zero, linearity, closeness, etc.). Similarly for a Lie algebra generated by some element a.
33 A real vector space with an inner product.
30
3. Let sα : E → E be defined by
sα (λ) = λ −
2(α, λ)
α.
(α, α)
(150)
Then for any roots α and β, sα (β) ∈ R.
4. If, in addition, R satisfies the property that if α and cα are both roots then c = ±1, then R is called a
reduced root system. 34
Definition 0 defines the abstract root system as an independent mathematical construction, but from
Thm. 19 we immediately realize the following theorem.
Theorem 20 [semisimple Lie algebra and reduced root system]: Let g be a semisimple complex Lie algebra,
with root decomposition given in Eq. (106). Then the set of roots R ⊂ h∗R \{0} is a reduced root system.
Therefore we have turned the classification of all semisimple Lie algebras into the classification of all
possible reduced root systems, although strictly speaking the one-to-one mapping is not yet established. To
better understand the root system, we can give Def. 0 very specific geometric meanings.
1. The number nαβ is twice of the projection of α onto β, which is an integer.
2. The operator sα (λ) is the reflection of λ around the hyperplane
Lα = {λ ∈ E | (α, λ) = 0}.
(151)
If (α, λ) = 0 then sα (λ) = λ. If λ = α then sα (α) = −α.
We also define the coroot α∨ ∈ E ∗ such that
hα∨ , λi =
2(α, λ)
.
(α, α)
(152)
Clearly, for the root system of of a semisimple Lie algebra, we have
α∨ = hα .
(153)
Then from Eqs. (124)–(125) we have
hα∨ , αi = 2,
(154)
nαβ = hα, β i ,
(155)
∨
sα (λ) = λ − hλ, α i α.
∨
(156)
As an example, let {ej } be a basis of Rn , with the inner product being the dot product (ej , ek ) = δjk .
Let
n
o
X
E = (λ1 , λ2 , · · · , λn ) ∈ Rn |
λj = 0
(157)
R = {ej − ek | 1 ≤ j, k ≤ n, j 6= k} ⊂ E.
(158)
and
Then R is a reduced root system. The reflection sα is just the transposition of j, k entries for α = ej − ek ,
denoted as sjk . This root system is of rank n − 1, and historically called the root system of type An−1 .
Clearly, this reduced root system is the root system of sl(n, C) given in Eq. (113).
We next consider the possible geometric configurations a reduced root system can have. From conditions
2 and 3 of Def. 0, we realized that there is only a finite set of configurations (relative angles) two noncolinear
roots α, β can have, which we will not shown here. For each configuration, its nαβ , nβα values can be
determined correspondingly. A special class of automorphisms of a root system that keep nαβ invariant is
the Weyl group.
34 In
this work we only consider reduced root systems.
31
Definition 0 [Weyl group]: The Weyl group W of a root system R is the subgroup of GL(E) generated by
reflections sα , α ∈ R.
The Weyl group W is a finite subgroup in the orthogonal group O(E), and the root system R is invariant
under the action of W . For any w ∈ W , α ∈ R, and we have sw(α) = wsα w−1 . Specifically, for R being
the root system of type An−1 , W is the group generated by transpositions sjk , which is none other than the
symmetric group Sn . This is why the Lie algebra sl(n, C) is closely connected to the symmetric group Sn :
because the Weyl group of the reduced root system of sl(n, C) is Sn !
4.4.5.2
Positive roots and simple roots
It is possible to find for each root system some small set of “generating roots”. To do so requires some
new concepts. Let t ∈ E be such that for any root α, (t, α) 6= 0. Then t is called a regular element of E.
Then we decompose R as R = R+ t R− , with
R+ = {α ∈ R | (α, t) > 0},
(159)
and R− = {α ∈ R | (α, t) < 0}. Such a decomposition is called a polarization of R, which depends on our
choice of t. The roots α ∈ R+ will be called positive roots, and the roots α ∈ R− will be called negative
roots.
Definition 0 [simple roots]: A positive root α ∈ R+ is a simple root if it can not be written as a sum of two
positive roots.
We denote the set of simple roots by Π ⊂ R+ . Since (α, t) can take only finitely many values, we can show
that any positive root can be written as a sum of simple roots. Therefore we have the following theorem.
Theorem 21 [simple roots are a basis]: Let R = R+ t R− ⊂ E be a root system. Then the simple roots
form a basis of the vector space E.
Another useful property of simple roots is that if α, β ∈ R+ are simple then (α, β) ≤ 0.
From the geometry we can prove that every α ∈ R can be uniquely written as a linear combination of
simple roots with integer coefficients, i.e.,
α=
r
X
(160)
nj αj
j=1
where nj ∈ Z and {α1 , · · · , αr } = Π is the set of simple roots. For α ∈ R+ , all nj ≥ 0. For α ∈ R− , all
nj ≤ 0. We define the height of a positive root α ∈ R+ by


r
r
X
X
ht(α) = ht
nj αj  =
nj ∈ N.
(161)
j=1
j=1
For the root system R of type An−1 , suppose we choose the polarization as
R+ = {ej − ek | j < k}.
(162)
αj = ej − ej+1 ,
(163)
Then the simple roots are
with j ∈ {1, · · · , n − 1}. The height of any positive root is given by ht(ej − ek ) = k − j.
32
4.4.5.3
Root and weight lattices
Due to the fact that every root can be written as a linear combination of simple roots with integer
coefficients, it is natural to think about the roots as a root lattice. A lattice in a real vector space E is an
abelian group generated by a basis in E. Even more importantly we can also construct a weight lattice for
a root system. We make the following definitions.
Definition 0 [root and weight lattices]: A root lattice is the abelian group Q ⊂ E generated by α ∈ R. A
coroot lattice is the abelian group Q∨ ⊂ E ∗ generated by α∨ . A weight lattice P ⊂ E is the dual lattice of
Q∨ , defined by
P = {λ ∈ E | hλ, α∨ i ∈ Z, ∀α ∈ R} = {λ ∈ E | hλ, α∨ i ∈ Z, ∀α∨ ∈ Q∨ }.
(164)
Elements of P are called integral weights.
Since we can identify the Cartan elements hα as the coroots α∨ , the elements of the weight lattice are
indeed the generalization of the weights as seen in the sl(2, C) case in Sec. 4.4.1. This actually implies a
weight decomposition for general semisimple Lie algebras, but we will discuss this later.
Given a polarization of R and a set of simple roots Π, the root lattice is just
M
Q=
Zαj
(165)
j
for αj ∈ Π. The weight lattice is then
P = λ ∈ E | λ, αj∨ ∈ Z, ∀αj ∈ Π .
(166)
Clearly, the root lattice is sublattice of the weight lattice, i.e., Q ⊂ P .
We can define a basis of P by introducing the fundamental weights.
Definition 0 [fundamental weights]: The fundamental weights ωj ∈ E are weights that satisfy
hωj , αk∨ i = δjk .
(167)
Then we see that {ωj } is a basis of E and that
P =
M
(168)
Zωj .
j
Hence we have constructed two sets of basis of E, i.e., the simple roots and fundamental weights.
4.4.5.4
Weyl chambers
We have seen that the choice of polarization depends not on the specific element t ∈ E but on the sign of
(t, α). Therefore a polarization is invariant if we change t without crossing any hyperplane Lα . We call the
connected component of the complement of the hyperplanes Lα as a Weyl chambers, denoted by C. In other
words, a Weyl chamber C is a region of E bounded by hyperplanes Lα that uniquely defines a polarization
by
R+ = {α ∈ R | (α, t) > 0},
t ∈ C.
(169)
Conversely, given a polarization we can always define a corresponding positive Weyl chamber C+ by
C+ = {λ ∈ E | (λ, α) > 0, ∀α ∈ R+ } = {λ ∈ E | (λ, αj ) > 0, ∀αj ∈ Π}.
(170)
Therefore there is a bijection between all polarizations and all Weyl chambers. For different Weyl chambers,
the Weyl group acts transitively on the set of Weyl chambers, meaning that
Cl = sβl (Cl−1 )
33
(171)
for two adjacent Weyl chambers separated by the hyperplane Lβl . Given a polarization (Weyl chamber) and
a set of simple roots Π, it is then straightforward to see that we can recover the complete root space from
simple reflections defined be Weyl group elements sj ≡ sαj acting on simple roots αj ’s.
For the root system An−1 , the Weyl group is Sn , and the simple reflections sj ’s are transpositions of the
jth and j + 1th entries. Then we can define the positive Weyl chamber to be
C+ = {(λ1 , · · · , λn ) ∈ E | λ1 ≥ · · · ≥ λn }.
All the other Weyl chambers are obtained by applying to C+ permutations σ ∈ Sn , i.e.,
Cσ = (λ1 , · · · , λn ) ∈ E | λσ(1) ≥ · · · ≥ λσ(n) .
(172)
(173)
We will discuss the relation to Sn in Sec. ??.
4.4.5.5
Dynkin diagrams
We have seen that given a set of simple roots we can recover a root system. Therefore classifying root
systems is equivalent to classifying possible sets of simple roots. We first define reducible and irreducible root
systems.
Definition 0 [reducibility of root systems]: A root system R is reducible if it can be written in the form
R = R1 t R2 with R1 ⊥ R2 . Otherwise R is called irreducible.
It can be shown that every reducible root system can be uniquely written in the form R1 t · · · Rn
where Rj ’s are mutually orthogonal irreducible root systems. Therefore it suffices to classify irreducible root
systems. Suppose R is an irreducible reduced root system and we have chosen a set of simple roots Π. To
uniquely classify the simple roots, we define the Cartan matrices.
Definition 0 [Cartan matrices]: The Cartan matrix A of a set of simple roots Π ⊂ R is the r × r matrix
with entries
ajk = nαj αk = αj∨ , αk =
2(αj , αk )
.
(αk , αk )
(174)
Immediately we have the following properties of the Cartan matrix:
1. For any j, ajj = 2.
2. For any j 6= k, ajk is a nonpositive integer.
3. For any j 6= k, ajk akj = 4 cos2 ϕ, where ϕ is the angle between αj and αk . If ϕ 6= π/2, then
|αj |
2
|αk |
2
=
akj
.
ajk
For An−1 , r = n − 1, and the Cartan matrix is of dimension (n − 1) × (n − 1) that has the form


2 −1
−1 2 −1





−1 2 −1


.
A=
..
..
..


.
.
.





−1 2 −1
−1 2
(175)
(176)
The information contained in the Cartan matrix can be presented in a graphical way, called the Dynkin
diagram. 35
35 There are other kinds of diagrams in the study of semisimple Lie algebras. For example, the Satake diagrams classify
simple Lie algebras over R.
34
Definition 0 [Dynkin diagram]: Let Π be a set of simple roots of a root system R. The Dynkin diagram of
Π is the graph constructed in the following manner.
1. For each simple root αj , we construct a vertex vj of the Dynkin diagram.
2. For each pair of simple roots αj 6= αk , we connect the corresponding vertices by n edges, where n
depends on the angel ϕ between αj and αk :
π
, n = 0.
2
2π
, n = 1.
(b) For ϕ =
3
3π
(c) For ϕ =
, n = 2.
4
5π
, n = 3.
(d) For ϕ =
6
(a) For ϕ =
3. For every pair of distinct simple roots αj 6= αk , if |αj | 6= |αk | and they are not orthogonal, we orient
the corresponding (multiple) edge by putting on it an arrow pointing from the longer root to the shorter
root.
The Dynkin diagram has the following properties.
1. The Dynkin diagram of Π is connected iff R is irreducible.
2. The Dynkin diagram determines the Cartan matrix A.
3. R is determined by the Dynkin diagram uniquely up to an isomorphism.
Therefore the classification of all irreducible root systems reduces to finding all the possible Dynkin diagrams
of irreducible root systems.
Theorem 22 [classification of Dynkin diagrams]: Let R be a reduced irreducible root system. Then its
Dynkin diagram is isomorphic to one of the diagrams below 36 .
• An (n ≥ 1):
• Bn (n ≥ 2):
• Cn (n ≥ 2):
• Dn (n ≥ 4):
• E6 :
• E7 :
• E8 :
• F4 :
36 In
each diagram the subscript is equal to the number of vertices.
35
• G2 :
In our work, we only focus on the simply-laced An type 37 , which corresponds to the sl(n, C) Lie algebra.
For completeness, below we give the special cases when n is smaller than required in the diagrams above.
1. For n = 1, A1 = B1 = C1 , which corresponds to the Lie algebra isomorphisms sl(2, C) ' so(3, C) '
sp(1, C).
2. For n = 2, B2 = C2 , which corresponds to the Lie algebra isomorphism so(5, C) ' sp(2, C).
3. For n = 2, D2 = A1 ∪ A1 , which corresponds to so(4, C) ' sl(2, C) ⊕ sl(2, C).
4. For n = 3, D3 = A3 ∪ A3 , which corresponds to so(6, C) ' sl(4, C).
Other than these special cases, all root systems listed in Thm. 22 are distinct.
4.4.6
Classification of semisimple Lie algebras
With the discussion on root systems classification of Dynkin diagrams, we can now go back to the
classification of semisimple Lie algebras. We have shown that every semisimple Lie algebra defines a reduced
root system. In fact, we can also recover a semisimple Lie algebra from a reduced root system. This is shown
by the following theorems. We first have the triangular decomposition of semisimple Lie algebras.
Theorem 23 [triangular decomposition]: Let g be a semisimple Lie algebra with root system R ∈ h∗ , and
(·, ·) be a nondegenerate invariant symmetric bilinear form on g. Let R = R+ t R− be a polarization of R
and Π = {α1 , · · · , αr } be the corresponding system of simple roots. Then the subspaces
M
n± =
gα
(177)
α∈R±
are subalgebras in g, and
g = n− ⊕ h ⊕ n+ .
(178)
This comes directly from the root decomposition theorem and the fact that the sum of positive roots is
positive. Given the triangular decomposition, we have the Serre relations 38 .
Theorem 24 [Serre relations]: Given the definitions in Thm. 23, let ej ∈ gαj , fj ∈ g−αj be chosen such
that
2
(ej , fj ) =
,
(179)
(αj , αj )
and let hj = hαj ∈ h be defined by Eq. (123), i.e.,
hj =
2Hαj
.
(αj , αj )
(180)
Then for j ∈ {1, · · · , r}, {ej } generate n+ , {fj } generate n− , and {hj } form a basis of h. In particular
{ej , fj , hj } generate g.
Moreover, the elements ej , fj , hj satisfy the following Serre relations:
[hj , hk ] = 0,
(181)
[hj , ej ] = ajk ek ,
(182)
[hj , fj ] = −ajk ek ,
(183)
[ej , fk ] = δjk hj ,
(184)
1−ajk
ek = 0,
(185)
1−ajk
fk = 0,
(186)
[ad(ej )]
[ad(fj )]
37 A
Dynkin diagram with no multiple edges is called a simply-laced diagram (ADE types).
Chevalley-Serre relations.
38 Or
36
where
ajk = nαj αk = αj∨ , αk
(187)
are the entries of the Cartan matrix.
Each Cartan matrix ajk determines a unique semisimple complex Lie algebra via the Serre relations. In
fact, one can show the following theorem.
Theorem 25 [bijection between root systems and semisimple Lie algebras]: There is a natural bijection
between the set of isomorphism classes of reduced root systems and the set of isomorphism classes of finitedimensional complex semisimple Lie algebras. Moreover, the Lie algebra is simple iff the reduced root system
is irreducible.
Combining Thms. 22 and 25, we have the following result.
Theorem 26 [classification of simple Lie algebras]: Simple finite-dimensional complex Lie algebras are
classified by Dynkin diagrams An , Bn , Cn , Dn , E6 , E7 , E8 , F4 , and G2 listed in Thm. 22.
4.4.7
Weight decomposition
We can now classify the reps of complex semisimple Lie algebras, which reduces to the classification of
irreps and finding a way to determine the multiplicities. To do this, we consider the weight decomposition
of a finite-dimensional rep.
Definition 0 [weight, general]: Let V be a g-module. A vector v ∈ V is called a vector of weight λ ∈ h∗ if
for any h ∈ h we have
hv = hλ, hi v.
(188)
The space of all vectors of weight λ is called the weight space and denoted as V [λ], i.e.,
V [λ] = {v ∈ V | hv = hλ, hi v, ∀h ∈ h}.
(189)
If V [λ] 6= {0} then λ is called a weight of V .
Definition 0 is the generalization of Def. 0 for the sl(2, C) case. The set of all weights of V is denoted by
P (V ), i.e.,
P (V ) = {λ ∈ h∗ | V [λ] 6= 0}.
(190)
Since vectors of different weights are linearly independent, P (V ) is finite for finite-dimensional reps.
Theorem 27 [weight decomposition, general]: Every finite-dimensional rep of g admits a weight decomposition given by
M
V =
V [λ].
(191)
λ∈P (V )
Moreover, all weights of V are integral, meaning that P (V ) ⊂ P , where P is the weight lattice defined
in Def. 0.
This comes from Thms. 17 and 19. As in the sl(2, C) case this weight decomposition agrees with the root
decomposition of g, meaning that if x ∈ gα , then x.V [λ] ⊂ V [λ + α].
We next study the dimensions of the weight subspaces V [λ]. This is done by defining the formal generating
series for these dimensions, i.e., the character of V .
37
Definition 0 [character, weight decomposition]: Let V be a finite-dimensional g-module. Then its character
ch(V ) is defined by
X
ch(V ) =
(dimV [λ])eλ .
(192)
λ
for λ ∈ P (V ).
Here ch(V ) ∈ C[P ] where C[P ] is the algebra generated by formal expressions eλ , λ ∈ P , subject to
relations eλ eµ = eλ+µ and e0 = 1. It can be shown that algebra C[P ] is isomorphic to the algebra of Laurent
polynomials in r variables, where r is the rank of g. Therefore, we can understand the character as a series
α
of polynomials. As an example, for sl(2, C), we have P = Z . So C[P ] is generated by enα/2 , n ∈ Z. 39
2
Denoting eα/2 = x, we have C[P ] = C[x, x−1 ]. Thus by Thm. 17 the character of each irreducible Vn is given
by
ch(Vn ) = xn + xn−2 + · · · + x−n =
xn+1 − x−(n+1)
.
x − x−1
(193)
We have already seen the definition of characters for real compact Lie groups and Lie algebras in Sec. 4.3.5.
Although both characters can be used to calculate the multiplicities of irreps, in Sec. 4.3.5 the characters are
calculated from a group approach while here the characters are calculated using the weight decomposition
of Lie algebras. 40 The characters have the following properties.
1. ch(C) = 1.
2. ch(V1 ⊕ V2 ) = ch(V1 ) + ch(V2 ).
3. ch(V1 ⊗ V2 ) = ch(V1 )ch(V2 ).
4. ch(V ∗ ) = ch(V ), where · is defined by eλ = e−λ .
In the sl(2, C) case we see that the characters are symmetric with respect to Weyl group actions (x → x−1 )
in this case. More generally, we have the following theorem.
Theorem 28 [dimension of weight spaces are Weyl invariant]: If V is a finite-dimensional g-module, then
the set of weights and dimensions of weight subspaces are Weyl group invariant, meaning that for any w ∈ W ,
dim V [λ] = dim V [w(λ)].
(194)
w.ch(V ) = ch(V ),
(195)
Equivalently, we can write
where the action of W on C[P ] is defined by w.eλ = ew(λ) .
We now consider irreps and their properties.
Definition 0 [highest weight rep]: A nonzero rep ρV of g is called a highest weight rep if V (not necessarily
finite-dimensional) is generated by a vector v ∈ V [λ] such that x.v = 0 for all x ∈ n+ , where n+ is defined
by the triangular decomposition of g in Thm. 23. Then v is called the highest weight vector, and λ is the
highest weight of V .
The highest weight rep is important because of the following theorem.
Theorem 29 [every finite-dimensional irrep is highest weight rep]: Every finite-dimensional irrep of g is a
highest weight rep.
39 ???
40 I am not perfectly clear of the differences and relations between these two definitions. It seems like they are self-consistent.
Anyway I will leave some questions marks here???
38
In any highest weight rep there is a unique highest weight and unique-up-to-a-scalar highest weight vector.
In any highest weight rep with highest weight vector vλ ∈ V [λ], the following conditions hold:
hvλ = hh, λi vλ ,
xvλ = 0,
∀h ∈ h,
∀x ∈ n+ .
(196)
(197)
We can define a module Mλ as the g-module generated by a vector vλ satisfying only Eqs. (196)–(197).
Formally, M λ is called the Verma module, but we will not dig into the details at this time. The point of
introducing the Verma module is...
here put comments from youtube video
By Thm. 29, to classify all finite-dimensional irreps is to classify all highest weight reps which are finitedimensional and irreducible.
Theorem 30 [existence of an irreducible highest weight rep]: For any λ ∈ h∗ , there exists a unique up to
isomorphism irreducible highest weight rep with highest weight λ. This g-module of this rep is denoted Lλ .
For sl(2, C), if λ ∈ N, then Lλ = Vλ is the finite-dimensional irreducible module of dimension λ + 1. If
λ∈
/ N, then Lλ = Mλ . More generally, for any Lie algebra, for “generic” λ, the Verma module is irreducible
so Mλ = Lλ . Also by Thm. 29, every irreducible finite-dimensional V must be isomorphic to one of Lλ .
Thus to classify all finite-dimensional irreps of g, we need to find out which of Lλ are finite-dimensional. We
make the following definitions.
Definition 0 [dominant weight]: A weight λ ∈ h∗ is called a dominant integral weight or a dominant weight
if for all α ∈ R+ ,
hλ, α∨ i ∈ N.
(198)
The set of all dominant integral weights is denoted P+ .
The point of introducing dominant integral weights is the following theorem.
Theorem 31: For every λ ∈ P+ , Lλ is irreducible and finite-dimensional. These reps are pairwise nonisomorphic, and every irreducible finite-dimensional rep is isomorphic to one of them.
To study the structure of the reps, one requires the BGG resolution and Weyl character formula, which
are beyond our scope. At this point, we know we can write any g-module as
M
V =
nλ Lλ .
(199)
λ∈P+
The multiplicities nλ can be found by writing the character ch(V ) in the basis ch(Lλ ). We skip the derivations
and only give the method to calculate multiplicities for sl(n, C) in the next section.
4.4.8
Reps of sl(n, C)
In this section, we classify all of the irreducible reps of sl(n, C) both as an example and as our purpose
in this work. 41
Recall the root system of sl(n, C) is given by
R = {ej − ek | j 6= k} ⊂ h∗ = Cn /C(1, · · · , 1).
(200)
The set of positive roots is given by
R+ = {ej − ek | j < k}.
(201)
41 I do not know if in my life I will need to use any of the knowledge above on any other Lie algebras, but at least knowing
these stuff is fun.
39
The weight lattice and the set of dominant roots are
P = {(λ1 , · · · , λn ) ∈ h∗ | λj − λk ∈ Z},
(202)
P+ = {(λ1 , · · · , λn ) ∈ h | λj − λj+1 ∈ N}.
(203)
∗
Since adding a multiple of (1, · · · , 1) does not change the weight lattice and the dominant roots (changing
P
the basis of h∗ by moving along the line
ej = 0), we can represent P and P+ as
P = {(λ1 , · · · , λn−1 , 0) | λj ∈ Z},
(204)
P+ = {(λ1 , · · · , λn−1 , 0) | λj ∈ N, λ1 ≥ · · · ≥ λn−1 ≥ 0}.
(205)
Equation (205) means that the set of dominant integer weights for sl(n, C) can be identified with the set
of partitions with n − 1 parts, which can then be represented graphically by the Young diagrams. We will
introduce the Young diagrams and Young tableaux in Sec. ??.
eg 8.42
eg8.43
eg 8.44
ex8.4
ex8.5
ex8.7
ex8.9
P
Any irreducible finite-dimensional Lλ as an sl(n, C)-module appears as a subspace in (Cn )N , N =
λj ,
determined by suitable symmetry properties, i.e. transforming ina certain way under the action of symmetric
group Sn .
We can calculate the characters of irreps of sl(n, C).
Theorem 32 [Weyl character formula, sl(n, C)]: Let λ = (λ1 , · · · , λn ) ∈ P+ be a dominant weight for
sl(n, C), λj ∈ N, λ1 ≥ λ2 ≥ · · · ≥ λn . Then the character of the corresponding irrep of sl(n, C) is given by
ch(Lλ ) =
Aλ1 +n−1,λ2 +n−2,··· ,λn
Aλ1 +n−1,λ2 +n−2,··· ,λn
Q
=
,
An−1,n−2,··· ,0
j<k (xj − xk )
(206)
where
Aµ1 ,µ2 ,··· ,µn = det xµj k
1≤j,k≤n
=
X
1
n
sgn(s)xµs(1)
· · · xµs(n)
.
(207)
s∈Sn
One may recognize that in Eq. (206), the nominator is the general alternating polynomial, the denominator
is the Vandermonde polynomial, and the RHS itself is the Schur polynomial.
4.4.9
Casimir operators
The adjoint representation is irreducible iff the Lie algebra is simple, since subrepresentations correspond
to ideals. have I included this???
casimir elements 6.15 for quadratic, others? given in Greiner p517
Can find casimir operaters this way
greiner 115, 137
greiner chap 12
pf 112-117
simplicity pf p6
lilian notes
40
For a Lie algebra, casimir operators are defined such that they commute with every generator of the Lie
algebra, and can therefore be used to label irreducible representations in the Cartan-Weyl basis42 . We can
now define the rank of a Lie group.
There are no general method to get all the Casimir operators for an arbitrary Lie algebra. However, for
SU(n), there are methods available43 . In general there is no method to construct the casimir operators for
arbitrary semisimple groups. Casimir operators are not unique. quadratic casimir operators are conserved
quantity for 2 dim phase space Miller 50
Racah theorem: Greiner 109
Each multiplet of semisimple Lie group can be uniquely characteried by the eigenvalues C1 C2 Cl of the
l Casimir operators C1, C2, Cl
One of the Casimir operators is always given by the quadratic form Griener 116
Harish-Chandra isomorphism, 8.8 Kirillov
4.5
4.5.1
Direct product to direct sum
Symmetric and antisymmetric tensors
bilinear form
decomposition into symmetric and anti-symmetric
invariant bilinear form in rep theory; G-invariant
Define the averaging map Av : V ⊗n → V ⊗n such that
n
n
O
1 X O
Av(
vj ) =
vσ(1)
n!
j=1
j=1
(208)
σ∈Sn
Then the symmetric map is defined as Symn V = Image(Av : V ⊗n → V ⊗n ). If V is a rep then
1. Av is a morphism of reps V ⊗n → V ⊗n (interwining map)
2. Image of a morphism is a subrep.
Symn V is a homogeneous polynomials of degree n in a basis of V . Symn C2 is (n + 1)-dimensional.
Symn C2 : su(2) → gl(n + 1, C). This gives the state multiplet.
Symmetric tensors and skey-symmetric/antisymmetric tensors Λ2 C2 the exterior square, antisymmetric
tensors
Sym2 Sym2 C2 ' Sym4 C2 ⊕ C. This looks like quadratic forms.
invariant theory
4.5.2
Direct product into direct sum
Direct sum is reducible. Direct product may be irreducible. Symmetric powers may be irreducible. But
symmetric powers have a subrep that is irrep, the symmetric tensors.
Sym2 C2 = 12 {e1 ⊗ e1 , e2 ⊗ e2 , e1 ⊗ e2 + e2 ⊗ e1 }
direct product and sum of semisimple Lie algebras are both semisimple Lie algebras
4.5.3
The Clebsch-Gordan coefficients (CG coefficients)
For physicists, one of the most useful properties
theorem of complete reducibility.
44
42 I
regarding reducibility and simplicity is the following
will add more to the Cartan-Weyl basis, roots, and multiplets.
Cvitanović in Reference
44 One can also look at the concept of reductive algebras. The su(n) Lie algebra is semisimple and reductive. The rep of
su(n) is reducible and decomposable.
43 See
41
Theorem 33 [Weyl’s theorem on complete reducibility 45 ]: Every finite-dimensional rep of a semisimple
algebra decomposes into the direct sum of irreducible reps (or irreps).
One should also note the difference between reducibility and decomposability. Without proof, we state
here that a rep is reducible if under similarity transformation it can be written in a nontrivial upper triangular
block form (Lie’s theorem). A rep is decomposable if under similarity transformation it can be written in a
nontrivial block diagonal form. Decomposability implies reducibility, but not conversely.
We can define the direct product of two reps. For example, the direct product of su(n) reps is used
across quantum physics. By Thm. 33, The CG coefficients are used to perform the explicit direct sum
decomposition of the tensor product of irreducible representations in cases where the numbers and types
of irreducible components are already known abstractly. The detailed constructions and derivations will
be shown as an example for su(2) in Sec. 5.3.4. The CG coefficients for a general Lie algebra is unknown.
However, the algorithms to produce CG coefficients for su(n) are known.
weights are connected here
Summary of properties of sl(n, C) and su(n)
4.6
1. Lie algebra.
Let g = sl(n, C) and h = {diag{c1 , c2 , · · · , cn } | cj ∈ C,
P
cj = 1}.
Define ej ∈ h as the functional
∗


h1

ej : 
..

 → hj .
.
(209)
hn
Then
h∗ =
M
Cei /C
X
ej
(210)
and
E = h∗R =
M
Rej /R
X
ej .
(211)
The inner product is defined by
(λ, µ) = λj µj
if λ and µ are chosen such that
P
λj =
P
(212)
µj = 0.
2. Root system.
The root system is
R = {ej − ek | j 6= k}.
(213)
The root subspace corresponding to α = ej − ek is
gα = CEjk
(214)
hα = Ejj − Ekk .
(215)
and the corresponding coroot hα = α∨ ∈ h is
45 See
Kirillov and Fulton for Weyl’s construction with radicals and nilpotents.
42
3. Positive and simple roots.
We choose the set of positive roots as
R+ = {ej − ek | j < k}
(216)
with
|R+ | =
n(n − 1)
.
2
(217)
The set of simple roots is
Π = {α1 , · · · , αn−1 },
(218)
αj = ej − ej+1 .
(219)
4. Dynkin diagram.
5. Cartan matrix.
2
−1



A=





−1
2
−1

−1
2
..
.
−1
..
.
−1
..
.
2
−1




.



−1
2
(220)
6. Weyl group.
The Weyl group is
W = Sn
(221)
sj =??
(222)
acting on E by permutations.
Simple reflections are
7. Weight and root lattices.
The weight lattice is
P = {(λ1 , · · · , λn ) | λj − λk ∈ Z}/R(1, · · · , 1)
= {(λ1 , · · · , λn−1 , 0) | λj ∈ Z}.
(223)
n
o
X
Q = (λ1 , · · · , λn ) | λj ∈ Z,
λj = 0 .
(224)
∼ Z/nZ.
P/Q =
(225)
The root lattice is
43
8. Dominant weights and Weyl chamber.
C+ = {(λ1 , · · · , λn ) | λ1 > · · · > λn }/R(1, · · · , 1)
= {(λ1 , · · · , λn−1 , 0) | λ1 > · · · > λn−1 > 0}.
(226)
P+ = {(λ1 , · · · , λn ) | λj − λj+1 ∈ N}/R(1, · · · , 1)
= {(λ1 , · · · , λn−1 , 0) | λj ∈ Z, λ1 ≥ · · · ≥ λn−1 ≥ 0}.
(227)
As algebras,
Then for simple Lie algebras, the kernel is {0}, which means that all the nontrivial reps of simple Lie
algebras are faithful.We can prove that the kernel of a rep of Lie algebra g is the ideal of g.
Reps of su(n).
Here we list the properties of su(n) regarding reducibility and multiplets:
1. Since sl(n, C) is semisimple, any complex finite-dimensional rep of su(n) is completely reducible.
2. All defining reps of su(n) are irreps.
3. All nontrivial reps of su(n) are faithful.
4. For each m ∈ {2, 3, · · · }, su(n) has exactly one irrep on Cm up to isomorphism.
5. The fundamental multiplet of su(n) is the same as the vector space associated to its natural/defining/standard
rep, both being Cn .
character of a rep???
The su(2) Lie Algebra and its reps
5
In the next three sections we discuss in detail the su(2), su(3), and su(4) Lie algebra.
5.1
Generators and structure constants
By Eq. (8), the su(2) Lie algebra is of dimension 3. From Eqs. (12)–(13), the 3 generators of su(2) are
given by the Pauli matrices, i.e.,
0 1
0 −i
1 0
x
y
z
λ1 = σ =
,
λ2 = σ =
, λ3 = σ =
.
(228)
1 0
i 0
0 −1
The corresponding basis, according to Eq. (15), are
i x
i 0 1
i y
i 0
e1 = − σ = −
,
e2 = − σ = −
2
2 1 0
2
2 i
−i
,
0
i z
i 1
e3 = − σ = −
2
2 0
0
.
−1
(229)
The structure constants are found by Eq. (19). There is only one nonzero independent structure constant
up to the completely antisymmetric nature, i.e.,
f123 =
1
Tr [[σ x , σ y ]σ z ] = 1.
4i
This agrees with our knowledge of the Pauli matrices.
44
(230)
5.2
5.2.1
The Lie correspondence between SU(2) and su(2)
From SU(2) to su(2)
Let U ∈ SU(2) be a 2 × 2 unitary matrix with unity determinant. One way to parameterize U with real
parameters xj , j ∈ {1, 2, 3}, is
 q

− 1 − 14 xj xj − 2i x3
− 21 x2 − 2i x1
.
q
U =
(231)
1
i
1
i
x
−
x
−
1
−
x
x
+
x
2 2
2 1
4 j j
2 3
Using Eq. (30), we can check that this choice of parameterization matches our basis choice in Eq. (229), i.e.,
(el )jk =
5.2.2
∂
Ujk (x1 , x2 , x3 )|x1 =x2 =x3 =0 .
∂xl
(232)
From su(2) to SU(2)
By Eq. (31), given a basis in Eq. (229), any matrix U ∈ SU(2) can be parameterized by
i
U (x1 , x2 , x3 ) = exp (xj ej ) = exp − (x1 σ x + x2 σ y + x3 σ z ) .
2
(233)
One can check for self-consistency by combining Eqs. (232) and (233).
5.3
5.3.1
Reps
Defining rep
By Def. 0, the su(2) Lie algebra over R constitutes all the anti-Hermitian 2 × 2 matrices
with vanishing
n
o
traces. Thus the defining rep is none other than the 2-dimensional rep that we label as (e1 )jk , (e2 )jk , (e3 )jk ,
where the basis operator matrices {ej } are given in Eq. (229). The vector space of the rep is C2 .
5.3.2
Adjoint rep
The adjoint rep of su(2) is of dimension 3. The adjoint matrices of the three basis elements are found by
Eq. (41), i.e.,
k=1
j=1
ad(e1 )jk = f1kj =
j=2
j=3
k=2
0
 0
0

0
0
1
k=3

0
−1 ,
0

0 0 1
=  0 0 0,
−1 0 0


0 −1 0
= 1 0 0,
0 0 0
(234)

ad(e2 )jk = f2kj
ad(e3 )jk = f3kj
(235)
(236)
where
n we have detailed the labels
o for ad(e1 ). Thus the adjoint rep is the 3-dimensional
rep ad(e1 )jk , ad(e2 )jk , ad(e3 )jk . The vector space of the rep is the su(2) Lie algebra itself.
The Killing form of su(2) can be found following Eq. (51), i.e.,


−2 0
0
gjk = K(ej , ek ) =  0 −2 0  = −213×3 .
0
0 −2
45
(237)
5.3.3
Irrep
fast finish!!!!
this is isomorphism !!! Kirillov p47
Theorem 34 [classification of su(2)]: Any irrep of su(2) is isomorphic to Symn C2 for some n ∈ N.
has weight diagram
−n −(n − 2) n − 2
n
46
This
(238)
For any n ∈ N+ there is one and only one irrep of dimension n for su(2) up to isomorphism.
1
H=
0
0
−1
0
E=
0
1
0
0
F =
1
0
.
0
(239)
[H, E] = 2E,
(240)
[H, F ] = −2F,
(241)
[E, F ] = H.
(242)
The su(2) Lie algebra has irreps. Consider basis operators
specifying the dimension of
h {eµi } without
h
i h
i
1
(j)
(j)
(j)
e1
, e2
, e3
, for j ∈ N =
its state space. A (2j + 1)-dimensional irrep is found by
2
µν
µν
µν
1
3
47
0, , 1, , · · · , where the basis operators are defined as
2
2
e1 = −iJ x ,
(243)
e2 = −iJ ,
(244)
e3 = −iJ .
(245)
y
z
Here physicists will recognize the J µ operators as the angular momentum operators. With the definitions
J ± = J x ± iJ y ,
2
(246)
2
2
2
J 2 = (J x ) + (J y ) + (J z ) = (J z ) + J z + J − J + ,
the angular momentum operators satisfy
(247)
48
J 2 |jmi = j(j + 1) |jmi ,
(248)
J |jmi = m |jmi ,
p
J ± |jmi = (j ∓ m)(j ± m + 1) |j(m ± 1)i ,
(249)
z
(250)
where we have defined the orthomormal basis states with the azimuthal quantum number j and the magnetic
1
quantum number m such that given j ∈ N, m ∈ {j, j − 1, · · · , −j + 1, −j}.
2
Here are some properties of the irreps of su(2):
1. The fundamental multiplet.
1
The fundamental multiplet is found when 2j + 1 = 2, i.e., j = . We can write the fundamental
2
1 1
1 1
multiplet as the state multiplet
,
, ,−
. When the other j values are irrelevant, the
2 2
2 2
n = 0, Sym0 C2 = C
case j = 0 is the trivial rep.
48 Derivations of these relations can be found in any quantum mechanics textbook.
46 For
47 The
46
fundamental multiplet can also be labelled as {|1i , |0i}, {|↑i , |↓i}, or {|ei , |gi} within different contexts.
This agrees with our discussion in Sec. 4.3.4 that the fundamental multiplet for su(2) is C2 . The fact
that the state space of the defining rep is also C2 means that in physics, we define su(2) with its
fundamental rep conventionally.
1
For j = , we have
2

 1
1
i
1 x
( )
x (2)




e1 2 = − σ x
=
σ
(J
)



1


2
2


 J ± ( 2 ) = σ±

 1
1
1 y
i
(2)
y (2)
(251)
,
= σ ,
(J )
e2 = − σ y ,
1
3
)
(
2



2
2
2


=
1
J
2×2


1

 (2)
1
4
i


 (J z )( 2 ) = 1 σ z
 e
= − σz
3
2
2
which agrees with Eq. (229).
2. Higher-dimensional multiplets and explicit operator forms.
1
For arbitrary j ∈ N, the corresponding (2j+1)-dimensional multiplet is the state multiplet {|j, m = ji , |j, m = j − 1i , ·
2
For j = 1, we have

√



2 0
0

√
√
 √



1
 (J x )(1) =  2 0

0
2 0
2


√


√
(1)
2




J+
= 0 0
2 


0
2 0




(1)
(1)
√







0 0
0
e1 = −i(J x )



0
0 − 2



√


i √
(1)
(1)
(1)
0
0 0 ,
. (252)
(J y ) =  2
0
− 2 ,
e2 = −i(J y )
√
√
(1)



2


=  2 0 0 

 J−

(1)
(1)
2
0
0
z


√
e3 = −i(J )








2 0
0


1
0
0






(1)

2 (1)

(J z ) = 0 0 0 
J
= 213×3



0 0 −1
Notice that the multiplet with j = 1 (the triplet rep) is of the same dimension as the state space of
the adjoint rep. It turns out that the adjoint rep of su(2) given in Eqs. (234)–(236) is an equivalent
rep of the triplet rep given in Eq. (252). In fact, su(2) has a unique (2j + 1)-dimensional irrep up to
equivalent classes.
3. The sl(2, C) Lie algebra. We notice that {J + , J − , J z } over C is also a Lie algebra given the commutations
+ −
J , J = 2J z ,
, J −, J z = J −,
, [J z , J x ] = J + .
(253)
In fact, the Lie algebra {J + , J − , J z } over C is the complexified Lie algebra [i.e., sl(2, C)] of the Lie
algebra {e1 , e2 , e3 } [i.e., su(2)]. Their transformations are given by
   i
 + 
e1
− 2 − 2i
J
0
1
e2  = − 1
J − ,
0
2
2
e3
0
0 −i J z
 + 
 
i −1 0 e1
J
J −  =  i 1 0e2 .
Jz
0 0 i e3
47
(254)
(255)
The transformations between the generators {λ1 , λ2 , λ3 } and {J + , J − , J z } are
49
  
 
λ1
1 1 0 J+
λ2  = −i i 0J − ,
λ3
0 0 2 Jz
 +  1
 
i
J
0 λ1
2
2
J −  =  1 − i 0λ2 .
2
2
Jz
0
0
i λ3
(256)
(257)
The sl(2, C) and su(2) have the same set of multiplets.
4. Notation. For future use, we use the notation [j] to represent the (2j + 1)-multiplet.
5.3.4
CG coefficients
We introduce the decomposition of direct products of sl(2, C) irreps in this section. Since sl(2, C) and
su(2) have the same set of multiplets, the direct sum of multiplets of sl(2, C) resulted from the decomposition
is also that of su(2).
5.3.4.1
Direct product of reps
Consider the direct product of two sl(2, C) irreps J1± , J1z and J1± , J1z with state multiplets [j1 ] ≡
{|j1 m1 i} and [j2 ] ≡ {|j2 m2 i}. Without loss of generality, we let j1 ≥ j2 . The resulting rep has operators
that act on the vector space given by [j1 ] ⊗ [j2 ] ≡ {|j1 m1 i ⊗ |j2 m2 i}, where the basis |j1 m1 i ⊗ |j2 m2 i ≡
|j1 m1 , j2 m2 i is named the product basis.
We introduce the total operators
J µ = Jµ ⊗ 1 + 1 ⊗ Jµ
(258)
for µ ∈ {x, y, z, +, −}. Then it is straightforward to check that {J x , J y , J z } is the su(2) Lie algebra and
{J ± , J z } is the sl(2, C) algebra. It can be shown that the rep of {J ± , J z } acting on the vector space [j1 ]⊗[j2 ]
is reducible 50 . The state basis in which the rep is in block diagonal form (manifestly reducible) is called the
total basis labelled as [J] ≡ |JM i, where J ∈ {j1 + j2 , j1 + j2 − 1, · · · , j1 − j2 }, M ∈ {J, J − 1, · · · , −J}.
Indeed, the dimensions of the vector spaces {|JM i} and {|j1 m1 , j2 m2 i} are both (2j1 + 1)(2j2 + 1).
5.3.4.2
CG coefficients
Completeness of basis states allows us to write
X
|JM i =
|j1 m1 , j2 m2 i hj1 m1 , j2 m2 |JM i .
(259)
m1 ,m2
Here, the coefficient hj1 m1 , j2 m2 |JM i is the CG coefficient we are looking for. It has the following properties:
1. Triangle inequality. hj1 m1 , j2 m2 |JM i =
6 0 only if j1 − j2 ≤ J ≤ j1 + j2 .
2. hj1 m1 , j2 m2 |JM i =
6 0 only if m1 + m2 = m.
3. By the Condon-Shortley convention,
(a) The CG coefficients are chosen to be real.
(b) hj1 j1 , j2 (J − j1 )|JJi is positive.
49 This
50 See
is consistent with the choice in my thesis which is nice.
Chap. 3.5, Pfeifer.
48
4. hj1 m1 , j2 m2 |JM i = (−1)
j1 +j2 −J
hj1 (−m1 ), j2 (−m2 )|J(−M )i.
5. Since CG coefficients relate one orthonormal basis to another, the CG matrix is unitary. In the
Condon-Shortley convention, it is orthogonal, i.e., hj1 m1 , j2 m2 |JM i = hJM |j1 m1 , j2 m2 i.
6. The algorithm to find CG coefficients:
(a) Start from |j1 + j2 , j1 + j2 i = |j1 j1 , j2 j2 i.
(b) Use the lowering operator to get all |j1 + j2 , M i for M ≥ 0.
(c) Find |j1 + j2 − 1, j1 + j2 − 1i using properties 1–3.
(d) Using the lowering operators to get all |j1 + j2 − 1, M i for M ≥ 0.
(e) Loop until the end.
(f) Using property 4 to get the M < 0 entries.
Given the CG coefficients, we can transform the reps of {J ± , J z } between the product basis and the total
basis by
X
hJM 0 |J µ |JM i =
hJM 0 |j1 m01 , j2 m02 i hj1 m01 , j2 m02 |J µ |j1 m1 , j2 m2 i hj1 m1 , j2 m2 |JM i .
(260)
m1 m2 m01 m02
for µ ∈ {±, z}. By definition the J µ operators are in block diagonal forms in the total basis.
Other notations of the CG coefficients in the literature include
√
j1
j2
J
jj 0
j1 −j2 +M
2J + 1
.
hj1 m1 , j2 m2 |JM i = Smm0 ,JM = (−1)
m1 m2 −M
5.3.4.3
(261)
Direct sum of multiplets
We have decomposed the rep that comes from a direct product of irreps into a direct sum of irreps. In
literature, this direct sum of irreps is called the Clebsch-Gordan series (CG series). Since irreps can be
labelled by their multiplets, we can use the notation for multiplets such as [j] to label the irreps. For two
su(2) multiplets [j1 ] and [j2 ] with j1 ≥ j2 , the CG series read
[j1 ] ⊗ [j2 ] = [j1 + j2 ] ⊕ [j1 + j2 − 1] ⊕ · · · ⊕ [j1 − j2 ]
For example, we have for the fundamental multiplet of su(2)
1
1
⊗
= [1] ⊕ [0],
2
2
1
1
1
3
1
⊗
⊗
=
⊕2
.
2
2
2
2
2
(262)
(263)
(264)
A drawing technique for general reduction rules can be found in Pfeifer.
5.4
Cartan subalgebra
The Cartan subalgebra is one-dimensional in σ3 .
5.4.1
Casimir operators
There is only 1 Casimir operator, defined by C2 = σi σi ∝ J 2 , where Ji = 21 σi are the angular momentum
operators.
The su(2) subalgebra
49
5.4.2
Fundamental rep
su(2) in 2D, su(3) in 3D, etc; for su(2), only 1 fundamental rep
dual rep of su(2) has the same weight diagram, thus ρ ' ρ∗ for su(2). for su(3) not the case. antiquarks
live in the dual space.
p326, Greiner
The su(3) Lie algebra and its reps
6
greiner chapter 7
pf p50-86
6.1
Generators and structure constants
Gell-Mann’s matrices
6.2
6.2.1
Reps
Fundamental and antifundamental reps
fundamental rep (3)
antifundamental rep 3̄ is the complex conjugate of the fundamental rep.
6.2.2
Tensor product rep
9-dimensional, reducible, can be decompose into 6 + 3̄
cartan-weyl basis
weight = m, j = heighest weight
weight lattice vs root lattice
killing form
simple roots
Dynkin diagram
representation = collection of weights Pi
casimir operator
6.2.3
Multiplets
The multiplet states are defined by |jmi states in the usual angular momentum language. in su3, 3 of 8
generators form an su2 subalgebra, other 5 form two doublets and a singlet.
weyl group
6.2.4
Cartan subalgebra
cartan subalbegra is the maximal abelian subalgebra. λ3 and λ8 generate a Cartan subalgebra.
rank = 3.
For any reps of su(3), the basis vector can be chosen to be the eigenvectors of (H1 , H2 ) = (T3 , T8 ).
Ha |~
µi = µa |~
µi, a = 1, 2.
The other 6 operators are ladder operators. Let us define
E±~α1 ≡ T1 ± iT2 ,
(265)
E±~α2 ≡ T4 ± iT5 ,
(266)
E±~α3 ≡ T6 ± iT7 .
(267)
50
Then
[Hα , E±~α ] = α
~ α E±~α .
(268)
2π
α
~ 1 = (1, 0), α
~ 2 = cos π3 , sin π3 , α
~ 3 = cos 2π
3 , sin 3 .
The ladder operators shift eigenvalues of Ha by the corresponding α
~ vector.
rank is dim of Cartan subalgebra (2 for SU(3))
p166 figure 8.1 Kirillov
The su(4) Lie algebra and its reps
7
chapter 11, Greiner
su4, 60, 60, 90 degress!!
pf p87-106
The generators of SU(4) can be written down from Eqs. (9)–(11). By convention, they are labeled in the
following way

0
1
λ1 = 
0
0

0
0
λ5 = 
i
0

0
0
λ9 = 
0
1

0
0
λ13 = 
0
0
1
0
0
0
0
0
0
0
0
0
0
0
−i
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1


0
0 −i
i 0
0
 , λ2 = 
0 0
0
0
0 0


0
0 0
0 0
0
 , λ6 = 
0 1
0
0
0 0


1
0 0
0 0
0
 , λ10 = 
0 0
0
0
i 0


0
0 0


0
0 0
, λ14 = 
0 0
1
0
0 0




0 0
1 0 0 0
0 0
0 −1 0 0
0 0
0 0
 , λ3 = 


0 0 0 0 , λ4 = 1 0
0 0
0 0
0 0 0 0
0 0




1
0 0
0 0 0 0
0
0 0 −i 0
1
1 0

 , λ7 = 

0 i 0 0 , λ8 = √3 0
0 0
0
0 0
0 0 0 0




0 −i
0 0 0 0
0 0
0 0 0 1
0 0
0 0
 , λ11 = 


0 0 0 0 , λ12 = 0 0
0 0
0 0
0 1 0 0
0 i



1 0 0 0
0 0


1 0 1 0 0
0 0
.
, λ15 = √ 

0 −i
6 0 0 1 0 
0 0 0 −3
i 0
1
0
0
0

0
0
,
0
0
0 0
1 0
0 −2
0 0
0
0
0
0

0
0
,
0
0
0
−i
,
0
0

(269)
There are 3 Casimir operators, of order 2, 3, and 451 . The quadratic Casimir operator C2 is given by
C2 = λi λi
+
The nonzero structure constants fjkl and coefficients fjkl
are given in figure (2).
+
Figure 2: The nonvanishing fjkl and fjkl
for SU(4).
su(4) has 3 su(2) subalgebras
8
Others
3. griener and muller fill it up, p456 for sl3 and sl4, go back to greiner after everything!!!!!
51 I
will include higher orders of Casimir operators and the Cartan-Weyl basis later.
51
(270)
example 4.22 p59 Kirillov, symmetric tensor, sym2 and gamma2, wiki
su(3)
torus,
weight decomposition is now lattices. since not perfect equilateral triangle, want an angle.
L
Theorem 35: If ρ : su(3) → gl(V ) is a complex rep then V =
Wk,l where Wk,l = {v ∈ V : ρ(D(θ1 , θ2 ))v} =
ei(kθ1 +lθ2 v for k, l ∈ Z.
sl(2, C) is a subalgebra of sl(3, C) from weight decomposition.
sl(3, C) splits as a sum of sl(2, C) reps in 3 ways. Weyle symmetry, Weyl group isomorphic to S3 . Can
use Weyl group and one highest weight to get the whole weight decomposition.
Theorem 36: For every k, l ∈ N, there is an irrep Γk,l of su(3), unique up to isomorphism, whose weight
diagram is the following. Moreover this is a complete list.
classification -reflex using Weyle group -take the convex hull of these points to get the polygon p -take
as weights the points in p of the form λ + r with r in the root lattice
root lattice is a sublattice of the wight lattice, a linear combination of roots
each dot is a weight space, being a subspace of the rep
still need a way to get the dimension of the weight space (multiplicity), there is a algorithm for this.
⊗3
3 quarks = C3
Dual rep: ρ∗ : g → gl(V ∗ ) so that ρ∗ (a)v = v ∗ ρ(a)−1 .
Dual space:
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References
Most useful references.
1. The Lie Algebra su(N) An Introduction, Pfeifer
2. Introduction to Lie Groups and Lie Algebras, Kirillov
3. Quantum Mechanics Symmetries, Greiner and Müller
4. Representation Theory and Quantum Mechanics, Miller
5. Quantum Theory, Groups, and Representations, Woit
6. Introduction to Lie Algebras and Representation Theory, Humpherys
7. Youtube videos, Jonathan Evans
Others.
1. Introduction to Smooth Manifolds, Lee
2. Representation Theory, Fulton and Harris
3. Young Tableaux, Fulton
4. Group Theory in Physics, Tung
5. Group Theory: Birdtracks, Lie’s, and Exceptional Groups, Cvitanović
6. Lecture notes on Hadron Physics, Eichmann
7. Shankar
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