Stone M., Goldbart P. Mathematics for Physics: A Guided Tour for Graduate Students

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15.3 Lie algebras 555

and we get the commutator of the representation matrices in the “correct” order only if

we multiply the infinitesimal elements successively from the right.

There appears to be no escape from this sign problem. Many texts simply ignore it,

a few define the Lie bracket of vector fields with the opposite sign, and a few simply

point out the inconvenience and get on with the job. We will follow the last route.

15.3 Lie algebras

A Lie algebra g is a (real or complex) finite-dimensional vector space with a non-

associative binary operation g × g → g that assigns to each ordered pair of elements,

, X

, a third element called the Lie bracket, [X

, X

]. The bracket is:

(a) Skew symmetric: [X , Y ]=−[Y , X ];

(b) Linear: [λX + µY , Z]=λ[X , Z]+µ[Y , Z];

and in place of associativity, obeys

Example: Let M (n) denote the algebra of real n-by-n matrices. As a vector space over

R, this algebra is n

-dimensional. Setting [A, B]=AB − BA makes M (n) into a Lie

algebra.

Example: Let b

denote the subset of M (n) consisting of upper triangular matrices with

any number (including zero) allowed on the diagonal. Then b

with the above bracket

is a Lie algebra. (The “b” stands for the French mathematician and statesman Émile

Borel.)

Example: Let n

denote the subset of b

consisting of strictly upper triangular matrices

– those with zero on the diagonal. Then n

with the above bracket is a Lie algebra. (The

“n” stands for nilpotent.)

Example: Let G be a Lie group, and L

the left-invariant vector fields. We know that

, L

]=f

(15.94)

where [ , ] is the Lie bracket of vector fields. The resulting Lie algebra, g = Lie G is

the Lie algebra of the group.

Example: The set N

of upper triangular matrices with 1’s on the diagonal forms a Lie

group and has n

as its Lie algebra. Similarly, the set B

consisting of upper triangular

matrices, with any non-zero number allowed on the diagonal, is also a Lie group, and

has b

as its Lie algebra.

Ideals and quotient algebras

As we saw in the examples, we can define subalgebras of a Lie algebra. If we want to

define quotient algebras by analogy to quotient groups, we need a concept analogous

556 15 Lie groups

to that of invariant subgroups. This is provided by the notion of an ideal. A ideal is a

subalgebra i ⊆ g with the property that

[i, g]⊆i. (15.95)

In other words, taking the bracket of any element of g with any element of i gives an

element in i. With this definition we can form g − i by identifying X ∼ X + I for any

I ∈ i. Then

[X + i, Y + i]=[X , Y ]+i, (15.96)

and the bracket of two equivalence classes is insensitive to the choice of representatives.

If a Lie group G has an invariant subgroup H that is also a Lie group, then the Lie

algebra h of the subgroup is an ideal in g = Lie G, and the Lie algebra of the quotient

group G/H is the quotient algebra g − h.

If the Lie algebra has no non-trivial ideals, then it is said to be simple. The Lie algebra

of a simple Lie group will be simple.

Exercise 15.17: Let i

and i

be ideals in g. Show that i

∩ i

is also an ideal in g.

15.3.1 Adjoint representation

Given an element X ∈ g, let it act on the Lie algebra, considered as a vector space, by a

linear map ad (x) defined by

ad (X )Y =[X , Y ]. (15.97)

The Jacobi identity is then equivalent to the statement:

(

ad (X )ad (Y ) − ad (Y )ad (X )

)

Z = ad ([X , Y ])Z. (15.98)

Thus

(

ad (X )ad (Y ) − ad (Y )ad (X )

)

= ad ([X , Y ]), (15.99)

[ad (X ),ad(Y )]=ad ([X , Y ]),

(15.100)

and the map X → ad (X ) is a representation of the algebra called the adjoint

representation.

15.3 Lie algebras 557

The linear map “ad (X )” exponentiates to give a map exp[ad (tX )] defined by

exp[ad (tX )]Y = Y + t[X , Y ]+

[X , [X , Y ]]+···. (15.101)

You probably know the matrix identity

−tA

= B + t[A, B]+

[A, [A, B]]+···. (15.102)

Now, earlier in the chapter, we defined the adjoint representation “Ad ” of the group on

the vector space of the Lie algebra. We did this setting gXg

−1

= Ad (g)X . Comparing

the two previous equations we see that

Ad (Exp Y ) = exp(ad (Y )). (15.103)

15.3.2 The Killing form

Using “ad ” we can define an inner product  ,  on a real Lie algebra by setting

X , Y =tr (ad (X )ad (Y )). (15.104)

This inner product is called the Killing form, after Wilhelm Killing. Using the Jacobi

identity and the cyclic property of the trace, we find that

ad (X )Y , Z+Y ,ad(X )Z=0, (15.105)

or, equivalently,

[X , Y ], Z+Y , [X , Z] = 0. (15.106)

From this we deduce (by differentiating with respect to t) that

exp(ad (tX ))Y , exp(ad (tX ))Z=Y , Z,

(15.107)

so the Killing form is invariant under the action of the adjoint representation of the

group on the algebra. When our group is simple, any other invariant inner product will

be proportional to this Killing-form product.

In case you do not, it is easily proved by setting F(t) = e

−tA

, noting that

F(t) =[A, F(t)], and

observing that the RHS is the unique series solution to this equation satisfying the boundary condition

F(0) = B.

558 15 Lie groups

Exercise 15.18: Let i be an ideal in g. Show that for I

, I

∈ i

I

, I



=I

, I



where  , 

is the Killing form on i considered as a Lie algebra in its own right. (This

equality of inner products is not true for subalgebras that are not ideals.)

Semisimplicity

Recall that a Lie algebra containing no non-trivial ideals is said to be simple. When the

Killing form is non-degenerate, the Lie algebra is said to be semisimple. The reason for

this name is that a semisimple algebra is almost simple, in that it can be decomposed

into a direct sum of decoupled simple algebras:

g = s

⊕ s

⊕···⊕s

. (15.108)

By “decoupled” we mean that the direct sum symbol “⊕” implies not only a direct sum

of vector spaces but also that [s

, s

]=0 for i = j.

The Lie algebra of all the matrix groups O(n),Sp(n),SU(n), etc. are semisimple

(indeed they are usually simple) but this is not true of the algebras n

and b

Cartan showed that our Killing-form definition of semisimplicity is equivalent to his

original definition of a Lie algebra being semisimple if the algebra contains no non-zero

abelian ideal – i.e. no ideal with [I

, I

]=0 for all I

∈ i. The following exercises

establish the direct sum decomposition, and, en passant, the easy half of Cartan’s result.

Exercise 15.19: Use the identity (15.106) to show that if i ⊂ g is an ideal, then i

⊥

, the

set of elements orthogonal to i with respect to the Killing form, is also an ideal.

Exercise 15.20: Show that if a is an abelian ideal, then every element of a is Killing

perpendicular to the entire Lie algebra. (Thus, non-degeneracy implies no non-trivial

abelian ideal. The null space of the Killing form is not necessarily an abelian ideal,

though, so establishing the converse is harder.)

Exercise 15.21: Let g be a semisimple Lie algebra and i ⊂ g an ideal. We know from

Exercise 15.17 that i ∩ i

⊥

is an ideal. Use (15.106), coupled with the non-degeneracy

of the Killing form, to show that it is an abelian ideal. Use the previous exercise to

conclude that i ∩ i

⊥

={0}, and from this that [i, i

⊥

]=0.

Exercise 15.22: Let  ,  be a non-degenerate inner product on a vector space V . Let

W ⊆ V be a subspace. Show that

dim W + dim W

⊥

= dim V .

(This is not as obvious as it looks. For a non-positive-definite inner product, W and W

⊥

can have a non-trivial intersection. Consider two-dimensional Minkowski space. If W

15.3 Lie algebras 559

is the space of right-going, light-like, vectors then W ≡ W

⊥

, but dim W +dim W

⊥

still

equals two.)

Exercise 15.23: Put the two preceding exercises together to show that

g = i ⊕ i

⊥

Show that i and i

⊥

are semisimple in their own right as Lie algebras. We can therefore

continue to break up i and i

⊥

until we end with g decomposed into a direct sum of simple

algebras.

Compactness

If the Killing form is negative definite, a real Lie algebra is said to be compact, and

is the Lie algebra of a compact group. With the physicist’s habit of writing iX

for the

generators of the Lie algebra, a compact group has Killing metric tensor

def

= tr {ad (X

)ad (X

)} (15.109)

that is a positive-definite matrix. In a basis where g

= δ

, the exp(ad X ) matrices of the

adjoint representations of a compact group G form a subgroup of the orthogonal group

O(N ), where N is the dimension of G.

Totally anti-symmetric structure constants

Given a basis iX

for the Lie-algebra vector space, we define the structure constants f

through

, X

]=if

. (15.110)

In terms of the f

, the skew symmetry of ad (X

), as expressed by Equation (15.105),

becomes

0 =ad (X

, X

+X

,ad(X



≡[X

, X

], X

+X

, [X

, X

]

= i(f

+ g

)

= i(f

kij

+ f

kji

). (15.111)

In the last line we have used the Killing metric to “lower” the index l and so define

the symbol f

ijk

. Thus, f

ijk

is skew symmetric under the interchange of its second pair of

indices. Since the skew symmetry of the Lie bracket ensures that f

ijk

is skew symmetric

under the interchange of the first pair of indices, it follows that f

ijk

is skew symmetric

under the interchange of any pair of its indices.

560 15 Lie groups

By comparing the definition of the structure constants with

, X

]=ad (X

= X

[

ad (X

)

]

, (15.112)

we read off that the matrix representing ad (X

) has entries

[

(ad (X

)

]

= if

. (15.113)

Consequently

= tr {ad (X

)ad (X

)}=−f

. (15.114)

The quadratic Casimir

The only “product” that is defined in the abstract Lie algebra g is the Lie bracket [X , Y ].

Once we have found matrices forming a representation of the Lie algebra, however, we

can form the ordinary matrix product of these. Suppose that we have a Lie algebra g

with basis X

, and have found matrices

with the same commutation relations as the

. Suppose, further, that the algebra is semisimple and so g

, the inverse of the Killing

metric, exists. We can use g

to construct the matrix

= g

. (15.115)

This matrix is called the quadratic Casimir operator, after Hendrik Casimir. Its chief

property is that it commutes with all the

[

]=0. (15.116)

If our representation is irreducible then Shur’s lemma tells us that

= c

I, (15.117)

where the number c

is referred to as the “value” of the quadratic Casimir in that irrep.

Exercise 15.24: Show that [

, X

]=0 is another consequence of the complete skew

symmetry of the f

ijk

Mathematicians do sometimes consider formal products of Lie algebra elements X , Y ∈ g. When they do,

they equip them with the rule that XY −YX −[X , Y ]=0, where XY and YX are formal products, and [X , Y ]

is the Lie algebra product. These formal products are not elements of the Lie algebra, but instead live in an

extended mathematical structure called the Universal enveloping algebra of g, and denoted by U (g). The

quadratic Casimir operator can then be considered to be an element of this larger algebra.

15.3 Lie algebras 561

15.3.3 Roots and weights

We now want to study the representation theory of Lie groups. It is, in fact, easier to

study the representations of the corresponding Lie algebra and then exponentiate these

to find the representations of the group. In other words, given an abstract Lie algebra

with bracket

, X

]=if

, (15.118)

we seek to find all matrices

such that

[

]=if

. (15.119)

(Here, as with the representations of finite groups, we use the superscript J to distinguish

one representation from another.) Then, given a representation

of the Lie algebra,

the matrices

(g(ξ )) = exp



iξ



, (15.120)

where g(ξ ) = Exp {iξ

}, will form a representation of the Lie group. To be more

precise, they will form a representation of the part of the group that is connected to the

identity element. The numbers ξ

serve as coordinates for some neighbourhood of the

identity. For compact groups there will be a restriction on the range of the ξ

, because

there must be ξ

for which exp{iξ

}=I.

The Lie algebra of SU(2)

The quantum mechanical angular momentum algebra consists of the commutation

relation

, J

]=iJ

, (15.121)

together with two similar equations related by cyclic permutations. This, once we set

 = 1, is the Lie algebra su(2) of the group SU(2). The goal of representation theory is

to find all possible sets of matrices that have the same commutation relations as these

operators. Since the group SU(2) is compact, we can use the group-averaging trick from

Section 14.2.2 to define an inner product with respect to which these representations are

unitary, and the matrices J

are hermitian.

Remember how this problem is solved in quantum mechanics courses, where we find a

representation for each spin j =

,1,

, etc. We begin by constructing “ladder” operators

def

= J

+ iJ

, J

−

def

= J

†

= J

− iJ

, (15.122)

which are eigenvectors of ad (J

)

ad (J

=[J

, J

]=±J

. (15.123)

562 15 Lie groups

From (15.123) we see that if |j, m is an eigenstate of J

with eigenvalue m, then J

|j, m

is an eigenstate of J

with eigenvalue m ± 1.

Now, in any finite-dimensional representation there must be a highest weight state,

|j, j, such that J

|j, j=j|j, j for some real number j, and such that J

|j, j=0. From

|j, j we work down by successive applications of J

−

to find |j, j − 1, |j, j −2, ...We

can find the normalization factors of the states |j, m∝(J

−

)

j−m

|j, j by repeated use of

the identities

−

= (J

+ J

) − (J

− J

−

= (J

+ J

) − (J

+ J

). (15.124)

The combination J

≡ J

is the quadratic Casimir of su(2), and hence in any

irrep is proportional to the identity matrix: J

= c

I. Because

0 =J

|j, j

=j, j|J

†

|j, j

=j, j|J

−

|j, j

=j, j|



− J

+ 1)

|j, j

=[c

− j(j + 1)]j , j|j, j, (15.125)

and j, j|j, j≡|j, j

is not zero, we must have c

= j(j +1).

We now compute

J

−

|j, m

=j, m|J

†

−

|j, m

=j, m|J

−

|j, m

=j, m|



− J

− 1)

|j, m

=[j(j + 1) − m(m − 1)]j, m|j, m, (15.126)

and deduce that the resulting set of normalized states |j, m can be chosen to obey

|j, m=m|j, m,

−

|j, m=



j(j + 1) − m(m − 1)|j, m − 1,

|j, m=



j(j + 1) − m(m + 1)|j, m + 1. (15.127)

If we take j to be an integer or a half-integer, we will find that J

−

|j, −j=0. In this

case we are able to construct a total of 2j +1 states, one for each integer-spaced m in the

range −j ≤ m ≤ j. If we select some other fractional value for j, then the set of states

15.3 Lie algebras 563

will not terminate, and we will find an infinity of states with m < −j. These will have

J

−

|j, m

< 0, so the resultant representation cannot be unitary.

SU(3)

The strategy of finding ladder operators works for any semisimple Lie algebra. Consider,

for example, su(3) = Lie(SU(3)). The matrix Lie algebra su(3) is spanned by the

Gell-Mann λ-matrices

⎛

⎝

010

100

000

⎞

⎠

⎛

⎝

0 −i 0

i 00

000

⎞

⎠

⎛

⎝

100

0 −10

000

⎞

⎠

⎛

⎝

001

000

100

⎞

⎠

⎛

⎝

00−i

00 0

i 00

⎞

⎠

⎛

⎝

000

001

010

⎞

⎠

⎛

⎝

00 0

00−i

0 i 0

⎞

⎠

√

⎛

⎝

10 0

01 0

00−2

⎞

⎠

, (15.128)

which form a basis for the real vector space of 3-by-3 traceless, hermitian matrices. They

have been chosen and normalized so that

tr (

) = 2δ

, (15.129)

by analogy with the properties of the Pauli matrices. Notice that

and

commute

with each other, and that this will be true in any representation.

The matrices

(

± i

(

± i

(

± i

) (15.130)

have unit entries, rather like the step-up and step-down matrices =σ

(=σ

± i=σ

Let us define 

to be abstract operators with the same commutation relations as

and define

(

± i

(

± i

(

± i

). (15.131)

564 15 Lie groups

These are simultaneous eigenvectors of the commuting pair of operators ad (

) and

ad (

=[

, T

]=±2T

ad (

=[

, V

]=±V

ad (

=[

, U

]=∓U

ad (

=[

, T

]=0,

ad (

=[

, V

]=±

√

ad (

=[

, U

]=±

√

. (15.132)

Thus, in any representation, the T

, U

, V

act as ladder operators, changing the

simultaneous eigenvalues of the commuting pair 

, 

. Their eigenvalues, λ

, λ

, are

called the weights, and there will be a set of such weights for each possible representation.

By using the ladder operators one can go from any weight in a representation to any other,

but one cannot get outside this set. The amount by which the ladder operators change

the weights are called the roots or root vectors, and the root diagram characterizes the

Lie algebra (Figure 15.2).

In a finite-dimensional representation there must be a highest-weight state |λ

, λ



that is killed by all three of U

, T

and V

. We can then obtain all other states in the

representation by repeatedly acting on the highest-weight state with U

−

, T

−

or V

−

and

their products. Since there is usually more than one route by which we can step down from

the highest weight to another weight, the weight spaces may be degenerate – i.e. there

may be more than one linearly independent state with the same eigenvalues of 

and



. Exactly what states are obtained, and with what multiplicity, is not immediately

–

2–2

–



Figure 15.2 The root vectors of su(3).