Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists

222 Chapter

15.

Complete Classification of

All

CSLA Simple Root Systems

Therefore

the

no

branch

point-one double line case

has

given us USp(2l)

for

l

;:::

2,

SO(2l

+

1) for

l

;:::

3,

and

F

4

.

(z)

One branch point, no double line:

Leaving

out

the

circle

at

the

vertex

let

the

three

arms

have

h,

l2,

b

circles respectively,

with

So

to

begin we have

the

graph

Our

experience

in

situation

(y) previously discussed suggests

that

if

the

arms

are

"excessively long",

property

(A)

of

linear independence

of

simple

roots

may

fail. Indeed

this

is so.

It

happens

that:

L e

mma

r(h

;:::

5,

l2

;:::

2,

l3)

~

r'(5,

2,1)

=

Lemm

a

r(ll

;:::

3,

l2

;:::

3,

l3)

~

r'(3

,

3,1)

=

Lemma

r(h;:::

2,l2;::: 2,l3;:::

2)

~

r'(2,2,2)

=

15.2.

The

allowed

Graphs

r

223

After

this

large scale elimination

of

possibilities,

the

residue is

the

following:

h

l2 l3

1f

-system

for

2:5

1 1

l

=

h

+

3:

80(2l)

=

Dl

for

l

2:

8

4

2 1

l

=

8:

Exceptional

group

Es

(see below)

4 1

1

l

=

7:

80(14)=

D7

3

2 1

l

=

7:

Exceptional

group

E7

(see below)

3

1

l

=

6:

80(12)

=

D6

2

2 1

l

=

6:

Exceptional

group

E6

(see below)

2

1 1

l

=

5:

80(10)

=

D5

1 1

1

l

=

4:

80(8)

=

D4

Thus

in this category

we

have

obtained,

at

first

in

a

torrent

but

then

in a

trickle,

the

1f-systems of

80(2l)

=

Dl

for

l

2:

4;

and

th

en

the

three

exceptional

groups

E6, E

7

, Es·

We

can

collect

and

display all allowed Dynkin diagrams, hence also all

possible

C8LA

's, in one

master

table.

In

the

first

three

columns we list

th

e

number

of

branch

points

, double lines, triple lines,

and

in

the

last

column give

th

e

diagrams

and

the

names

of

the

groups.

224

Chapter

15.

Complete Classification of All CSLA Simple Root Systems

Branch

Double

Triple

Name

and

Diagram

Points

Line Line

o o

1

o

1

o

1

o o

o o o

Bl

=

SO(2l

+

1),

l

:::::

3

C

l

=

USp(2l),

l

:::::

2

Dl

=

SO(2l),

l

:::::

4

El,l

=

6,7,8

Al

=

SU(l

+

1),

l

:::::

1

15.3

The

Exceptional

Groups

c:E=O

v'2v'21

0···

0 -

0=0

1 1

v'2

0···

0 -

0=0

0-0=0

- 0

/0

···0-0

"'0

0-0···0

- 0

In

comparison

with

the

results known

to

us from

Chapter

14,

the

new possible

CSLA's found now

are

the

groups

G

2

,

F

4

,

E

6

,

E

7

,

Es -

these

are

in a sense

the

surprises. We will

not

enter

into a detailed discussion of

any

of

them

,

but

only

gather

some information,

at

first in

tabular

form,

then

say something

about

the

root

system for each.

Group

Rank

2

4

6

7

8

Order

Dimension

of

smallest

VIR

14

7

52

26

78 27,

27'

133

56

248 248

Close

relative

of

B3

=

SO(7)

B4

=

SO(9)

A5

=

SU(6)

A7

=

SU(8)

D8

=

SO(16)

Their

root

systems

happen

to

be

the

following (where

~'s

are

unit

vectors

in Euclidean spaces):

15.3.

The Exceptional Groups

225

on

G

2

:

Start

with

the

6

roots

of SU(3):

±f.l'

(1/2)(±f.l

± v3f.2);

to

these,

add

±V3f.2;

V;

(±V3f.l

± f.2), making

12

in all.

F

4

:

Start

with

the

32

roots

of

SO(9):

±f.

a

,

±f.a

±

f.b'

a,

b

=

1,

...

,4:

to

these,

add

on

1

2"(±f.

1

±

f.2

±

f.:3

±

f.4),

making

48 in all.

E6:

Start

with

the

30

roots

of

SU(6)

in

the

(Racah) form

f.a - f.b'

a, b

=

1

...

6, using

unit

vectors

in

R6;

extend

R6

to

R7

by including

f.7;

then

add

on

the

roots

~

f.7

1

±V

2

f.7'±

J2

+

2"

(±f.l

±f.2

±f.3 ±f.4 ±f.5

±f.6)

with

three

plus

and

three

minus signs; get

72

in all.

E7:

Start

with

the

56

roots

of

SU(8)

in

the

(Racah) form

f.a - f.b'

a,

b

=

1

...

8, using

unit

vectors in

R

S

;

then

add

on

~(±f.l

±

f.2

...

± f.s),

with

four

+

and

four - signs; get 126

in

all.

Es:

Start

with

the

112

roots

of

SO(16):

±f.a

±

f.b'

a, b

=

1

...

8;

then

add

on

~

(±f.l ±

f.2

...

± f.s), even

number

of

plus signs;

get

240

roots

in all.

The

distribution

of simple

roots

of

various lengths

in

all CSLA

can

be

conveniently

taken

as follows:

Group

Length

1

Length

J2

Length

v3

SU(l

+

1)

=

Al

1

SO(2l

+

1)

=

Bl

1

1 - 1

USp(2l)

=

C

l

1 - 1

1

SO(2l)

=

Dl

1

G

2

1

F4

2 2

El,l

=

6,7,8

I

This page intentionally left blankThis page intentionally left blank

Chapter

16

Representations

of

Compact

Simple

Lie

Algebras

In

earlier

Chapters

we

have

dealt

with

certain

specific

UIR's

of

the

groups

of

interest

to

us,

namely

the

adjoint

representation,

and

for each

of

the

four

classical families a defining

representation.

Now

we

shall

gather

and

describe

some

aspects

of

a generic UIR,

to

be

denoted

D,

of

any

CSLA. We

assume

D

acts

on

a complex linear

vector

space

V

of

some dimension.

The

aim

will

be

to

convey

an

overall

appreciation

of

structure

within

a UIR, as well as

patterns

holding for

the

set

of

all

UIR's

of

CSLA's,

without

entering

into

detailed

proofs

of

all

statements.

16.1

Weights

and

Multiplicities

We have some

compact

simple Lie

group

G

with

Lie

algebra

.c,

spanned

by

the

Cartan-Weyl

basis

H

a

, Ea.

Since

the

Ha

are

commuting

hermitian

operators

on

V,

let us

simultaneously

diagonalise

them

and

denote

a

common

eigenvector

by

It:

...

):

Halt:

..

·)

=

fLalt:

..

. ),

a

=

1,2,

...

, l

(16.1)

We

assume

It:

...

)

is normalised

to

unity. Here

t:

is a

reall-component

vector

with

components

fLa

being

the

eigenvalues of

Ha.

We have

already

introduced

the

term

weight vector

for

fL:

it

is a

vector

in

the

same

l-dimensional

Euclidean

space

in which

roots

lie. We

can

construct

a basis for

V

with

vectors

IfL

...

).

Vectors

It:

...

)

with

different weights

t:

are

clearly linearly

independent.

-

While

the

Ha

do

commute

pairwise,

they

may

not

be

a

complete

commuting

set.

This

may

even

depend

on

the

particular

UIR

D.

So we have

put

dots

after

fL

in

the

ket vectors

IfL

...

)

E

V:

for a given

fL,

there

may

be

several

independent

Simultaneous eigenkets.

This

is

the

proble~

of

multiplicity.

A

nondegenerate

fL,

i.e., a weight for which

there

is

just

one eigenket

IfL),

is

said

to

be

a

simple

weight (of course for

the

UIR

D

under

consideration)'

Thus,

one

of

Cartan's

228

Chapter

16.

Representations

of

Compact

Simple Lie Algebras

theorems

can

be

expressed as saying

that

the

non-zero weights of

the

adjoint

representation

are

all simple.

Given

the

UIR

V

acting on

V,

we will often denote

the

set of all weights

occurring

by

W:

here no

attention

is

paid

to

the

possible multiplicity of eigen-

vectors in

V

for each

I!:

E

W.

16.2

Actions

of

Ea

and

SU(2)(a) -

the

Weyl

Group

How do

the

nonhermitian

generators

Ea

affect a ket

1J.t

...

)?

The

Ha

-

Ea

commutation

rule tells us

that

in general

Ea

changes

the

weight in a definite

way - it shifts it by a definite amount:

HaEall!:"')

=

[Ha,

Ea]II!:"')

+

EaHall!:"')

=

(I!:

+

Q)aEall!:"')

(16.2)

Therefore either

EalJ.t

...

)

vanishes,

or

it

is

another

vector in

V

with

weight

J.t

+

Q.

In

the

latter

case,

the

multiplicities of

J.t

and

J.t

+

Q

could

be

different.

We write symbolically - -

(16.3)

Thus

the

relationship (14.1) is easily understood.

Now

take

a

root

Q

E

9t,

its associated

group

SU(2)(a),

and

analyse a weight

J.t

E

W

with

respect

to

it. Recall

that

the

SU(2)(a) generators are

The

ket

II!:'

..

)

is

an

eigenket of

J

3

:

J

3

11!:

..

·)

=

mil!:"

.),

m

=

Q .

1!:/IQ12

(16.4)

(16.5)

From

our

knowledge of

UIR's

of SU(2), we see

that

m is quantized

and

must

have one of

the

values

0,

±1/2,

±1,

....

More precisely,

Q

E

9t'1!:

E

W

=}

2Q

.

1!:/IQ12

=

0,

±1, ±2,

...

(16.6)

Again from

the

structure

of SU(2) UIR's, we

can

see by applying

E±a

repeatedly

to

II!:'

..

)

that

there

must

be

integers

p, q

~

0

such

that

I!:

+

PQ,

I!:

+

(p

- l)Q,

...

,

I!:

+

Q,

I!:,

I!:

-

Q,

I!:

+

(p+1)Q,

1!:-(q+1)Q~W.

...

,

I!:

-

(q

-

1)Qf!:

-

qQ

E

W,

(16.7)

16.2.

Actions

of

Be,

and

SU(2)(a)

~

the

Weyl

Group

229

(You

can

see

the

similarity

to

the

root

analysis in

Chapter

12).

In

particular,

there

can

be no baps in

this

sequence of weights in

W.

Now we

must

consider

the

question of multiplicities.

Consider

the

set

of ket vectors in

V

with

the

above

string

of weights, each

occurring a

certain

number

of times:

It:

+

PQ,··

.),

It:

+

(p

-

l)Q,

..

. ),

...

,

It:

+

Q,

..

. ),

It:,

...

),

It:

-

Q,

...

),

...

,

It:

-

qQ,

...

)

(16.8)

They

span

some subspace of

V

which is clearly invariant

under

action by

SU(2)(a:). So

there

must

be

present here a

certain

set of

VIR's

of SU(2)(a:)

(i.e., j - values), each occurring a

certain

number

of times.

The

string

of

h

eigenvalues involved is evidently,

m

+

p,

m

+

P -

1,

...

, m

+

1,

m,

m -

1,

...

, m -

q,

(16.9)

where m was determined in Eq.(16.5).

The

fact

that

the

chain (16.8)

terminates

at

the

indicated ends means

that

(16.10)

Evidently in

the

spectrum

of j-values present in

the

SU(2)(a:)

UIR

on

this

string

of

states,

we have

jrnax

=

m

+

P

=

-(m

-

q),

jrnax

=

(1/2)(q

+

p), m

=

(1/2)(q -

p)

(16.11)

The

VIR

with

j

=

jrnax is present as often as

the

multiplicity of

IlL

+

PQ··

.),

which

must

be

the

same

as for

IlL

-

qQ

..

.

),.

The

VIR

with

j

=

Jrnax - 1

is

present as often as

the

difference

of

the

multiplicities of

It:

+

(p

-

l)Q

.

..

),

and

IlL

+

PQ··

.),

and

so

on

. [Throughout we are considering SU(2)(a:) action on

just

the

set

of vectors (16.8)].

If

we wish, we could

arrange

to

diagonalise

E

-a:

Ea:

on these

states,

which will

then

explicitly reveal

the

spectrum

and

multiplicity

of j-values.

A characteristic feature of

UIR's

of SU(2) is

that

if

the

eigenvalue m of

h

occurs,

then

so does

-m;

in fact within SU(2) representations,

the

corresponding

vectors

are

0

rotation

about

the

y-axis:

e

irrh

Ij,

m)

'"

Ij

,

-m)

(16.12)

Applying

this

to

the

present

case of SU(2)(a:) , we see

that

if

t:

E

Wand

Q

E

9t,

then

t:

-

2QQ

.

t:/IQI2

E

W

with

the

same

multiplicity as for

t:.

This

is because

(16.13)

exp(i7rJ~a:))(set

of

states

It:

...

))

=

set

of

states

It:

-

2QQ·

t:/IQI

2

,

... )

(16.14)

230

Chapter

16.

Representations

of

Compact

Simple Lie Algebras

and

exp(

i7r

JJO'))

is evidently unitary.

We

can

summarise

these results of applying SU(2)(0')

to

!:!:.

E W:

(i)

there

exist integers p, q

2':

°

such

that

Eq.(16.7) holds;

(ii)

2!:!:.·

Q/IQI2

=

q - p

=

0,

±1, ±2,

...

;

(iii)

}1

-

2QQ . p,jIQI

2

E

W,

with

the

same

multiplicity as

!:!:..

The

reflection

operation

exp(i7rJJO')) is

an

important

and

useful one. For

each

Q

E

9't,

one such reflection is defined.

In

weight space, this is clearly

reflection

through

a plane

perpendicular

to

Q

passing

through

the

origin.

The

set

of

all such reflections is a

group

,

the

Weyl group; this is a finite

subgroup

of

G.

If}1 E

W,

any

element

of

the

Weyl

group

applied

to

}1

gives some

Ii

E

W

with

the

same

multiplicity as}1. We

say

}1,}1'

E

Ware

equivalent,

if

they

are

an

element

of

the

Weyl group. So-equivalent weights have

the

same

multiplicity,

and

Set

of

weights equivalent

to

(16.15)

16.3

Dominant

Weights,

Highest

Weight

of

a

VIR

We

are

working

with

a definite choice

and

sequence

of

the

operators

Ha

in

the

Cartan

subalgebra. We

then

say: a weight

}1

is

higher

than

a weight

}1'

if in

the

difference

}1

-

}1'

(which

may

not

be

a weight!)

the

first non-zero

component

is

positive.

So

forany

two

distinct

weights, one is definitely higher

than

the

other.

The

highest weight

in

a

set

of

equivalent weights, i.e., a

set

reflections, is called a

dominant

weight.

Given

the

UIR

1)

acting

on V,

and

the

set W

of

all weights

that

occur,

denote

by

JEW

the

highest weight.

Then

it

is a simple weight

with

no

multiplicity

or

degeneracy;

there

is

just

one ket

IJ).

We indicate

the

proof

later

on. Clearly, if J is

the

highest weight,

it

means

that

if

Q,

is

any

positive

root

,

J

+

Q

is

not

present in W:

(16.16)

Even

more economically

stated,

this

is

the

same

as

Q(a)

E

S:

En(alIJ)

=

0,

a

=

1,2,

...

,I

(16.17)

So for a general

compact

simple Lie

group

G,

the

set

of

I

"raising

operators"

En(al for

Q(a)

E

S

generalises

the

single

J+

of SU(2).

The

highest weight

state

IJ)

in

V

is simultaneously

the

maximum

magnetic

quantum

number

state

(m

=

j)

for

the

SU(2)(0') algebras for all

Q

E 9't+.

This

means:

Q

E

9't+:

2J

.

Qjlgl2

=

integer

2':

0

(16.18)

16.3.

Dominant

Weights, Highest Weight of a

UIR

We shall introduce

the

notation

Q.(a)

E

S:

2Ll·

Q.(a)

1jQ.(a)

12

=

Na

~

0,

Ll

<--------+

{N

a

}

231

(16.19)

It

turns

out

that

each

UIR

can

be

uniquely characterised by its highest weight

Ll

,

or

by giving

the

nonnegative integers N

a

.

Let

us indicate why

the

highest weight

state

ILl)

in a

UIR

is nondegenerate.

Clearly we

can

go from

ILl)

to

other

(basis) kets in

V

by action

with

En

for

Q.

E ryt_.

Now suppose

there

are two highest weight

states

ILl,

1)

and

ILl,2),

mutually

orthogonal.

With

no loss of generality:

(Ll,

21Ll,

1)

=

0

(16.20)

Then

for any

Q.

E

ryt_,

the

states

En

ILl,

1)

and

En

ILl,

2)

will also

be

orthogonal:

Q.

E

ryt-

=?

-Q.

E ryt+ :

(Ll,

21E~

En

ILl

,

1)

=

(Ll,

2IE-nEnILl,

1)

=

(Ll,

21[E-n,

EnllLl

,

1)

=

-Q.'

Ll(Ll,

21Ll,

1)

= 0

(16.21)

One

can

continue

this

argument

for chains like

E

i3

En

ILl,

1 or 2),

and

finds

Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists

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