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

162

Chapter

9.

Lie

Groups

and

Lie Algebras

(v) Given

the

Lie

algebra

£

of

dimension

r,

form its

commutator

algebra

L1

=

[£,

£],

then

L2

=

[£1,

L1]

=

commutator

algebra

of

£1, £3

=

[L2'

L2],

and

so on,

£)+1

=

[Lj, Lj]

in general.

In

the

series

of

Lie algebras

(9.70)

at

each

stage

L)+l

is

an

invariant

subalgebra

of

Lj,

and

£j

/ L)+l

is abelian.

If

for a

particular

j,

Lj+l

is a

proper

subalgebra

of

£j,

then

the

dimension of

L)+l

must

be

at

least

one less

than

that

of

£j.

On

the

other

hand,

if for some

(least)

j,

we find

£)+1

=

Lj,

then

of

course L)+l =

L)+2

=

L)+3

=

...

as well,

so

the

series stabilises

at

this

point. Therefore, in

the

series (9.70) either, (i)

the

dimension keeps decreasing

until

we

reach

Lj

for some

j

<

r,

and

thereafter

the

series is

constant

at

a nonzero "value",

£j

=

L)+l

=

L)+2

=

...

Ie

0;

or

(ii)

the

dimensions keep decreasing until, for some (least)

j

:s:

r,

£j

= 0,

and

then

of

course

£)+1

=

£)+2

=

...

=

o.

The

Lie

algebra

£

is

solvable

if for some (least)

j

:s:

r,

Lj

=

O.

(vi)

Another

sequence of

subalgebras

which we

can

form is:

we

go from

La

==

L

t.o

£1

as before,

but

write

£1

for it:

Next, we form

£2

=

[La,

L1]

=all

(real) linear

combinations

of

[x,

y]

with

x

E

La,

yELl

(9.71)

(9.72)

It

is

an

instructive

and

not

too

complicated

exercise

to

show

that

£2

is

an

invariant

sub

algebra

of

La

as well as

of

£1,

and

L1

/

L2

is abelian.

At

the

we define

£3

=

[La,

L2]

=all

(real) linear

combinations

of

[x,y]

with

x

E

Lo,y

E

£2

(9.73)

This

in

turn

is seen

to

be

an

invariant

subalgebra

of

La,

L

1

and

L2,

with

£2/

L3

being

abelian.

In

this

way

we

have

the

series:

(9.74)

L)+l

is

an

invariant

subalgebra

of

£0, £1, £2,

...

,£j

and

Lj

/£)+1

is abelian.

The

Lie

algebra

L

is

nilpotent

if

Lj

= 0 for some (least)

j

:s:

r.

(vii)

Connection

between

solvability

and

nilpotency: We have

the

two series

(9.70),(9.74)

of

Lie algebras above. While

£1

=

L1

by

definition, one

can

easily

show (by

induction

for example)

that

£j

<;;;

£j

,

j

=

2,3,

...

(9.75)

9.8. Various

Kinds

of

and

Operations

with

Lie

Algebras

163

This

means

that

if

[)

=

0 for some

j,

then

by

that

stage (or earlier) .c

j

=

0

too. So,

.c

nilpotent

=}

.c solvable,

but

not

conversely

(9.76)

While

the

above concepts

are

important

for

the

general

theory

of

Lie alge-

bras, for

most

physical applications

the

simple

and

semisimple

types

are

more

important

. We

(viii) A Lie algebra

.c is

simple (semisimple)

if

it

has

no

proper

invariant

(abelian invariant) subalgebra. We

can

immediately see

the

following relation-

ships:

.c simple

=}

.c semisimple

but

not

conversely;

.c solvable

=}

.c

1

=

proper

invariant

sub

algebra

of

.c

=}

.c

not

simple;

.c simple

=}.c

1

=

.c

=}

.c

not

solvable

(9.77)

So, simplicity

and

solvability

are

mutually

exclusive! To conclude

this

sec-

tion, let us define direct

and

semidirect sums

of

Lie algebras.

(ix) Direct

sum

of

Lie algebras: Let .c'

be

a

proper

invariant

sub

algebra

of a Lie

algebra.c.

If

we

can

find

another

proper

invariant

sub

algebra

.c" in .c

such

that

as vector spaces .c

=

.c'

EB.c"

in

the

sense of a

direct

sum,

then

we say

.c is

the

direct

sum

Lie algebra

of C

and

.c". Since in this

situation

the

only

element common

to

.c'

and

.c" is

the

zero element, we

must

have:

[.c'

,.c']

<:;;;

.c',

[.c',

.c"]

=

0,

[.c", .c"]

<:;;;

.c"

(9.78)

(x)

The

semidirect

sum

of Lie algebras parallels

the

semidirect

product

of groups. We say .c is

the

semi direct

sum

of

.c'

and

.c" if as vector spaces

.c

=

.c'

EB

.c",

and

as Lie algebras

[.c',

.c']

<:;;;

.c',

[L'

,.c"]

<:;;;

.c",

[.c", .c"]

<:;;;

.c",

(9.79)

This

means

that

while

both

C

and

.c"

are

sub

algebras

of

.c,

the

latter

is

an

invariant one;

and

one

can

then

see

that

the

factor

algebra

.c/ .c" is isomorphic

to

C.

From

the

second line in Eq.(9.79) we also sec

that

each element

of

C

acts

as

an

(in general)

outer

automorphism

on

.c".

Exercises

for

Chapter

9

1.

For

the

group

SO(3)

of

real

prop

er

orthogonal

rotations

in 3 dimensions,

show

that

we have various possible

parametrisations

of

group

elements:

164

Chapter

9.

Lie

Groups

and

Lie Algebras

(i)

Euler

angles:

(

co~8

-sine

n

R

E

80(3)

:R

=

R(1/J,

e,

¢)

=

sinO

cosO

0 0

(

'001'

0

~i~~

) (

cos¢

-sin¢

x

-

s~n

1/J

1

sin¢

cos¢

0

cos

1/J

0 0

o

::;

e,

¢

::;

27r,O

::;

1/J

::;

7r.

(ii) Axis angle

parameters:

R(n,

a)

=

(Rjdn,a)),

j,k

=

1,2,3,

Rjk

(n,

a)

=

Ojk

cos

a

+

njnk(1

-

cos

a)

- Ejklnl

sina,

n E

S2,

0

::;

a

::;

7r.

(iii) Homogeneous

Euler

parameters:

Rjdao,g,J

=

(a6

-

[!2)Ojk

+

2ajak -

2aOEjklal,

a

.a

22

ao

=

cos

"2'

aj

=

nj

sm

"2'

a

o

+

[!

=

1

0

)

0

1

In

each case, find

the

parameter

values for

the

identity

element. Are

they

always unique?

2.

For

the

group

80(3)

described using axis angle

parameters

R(n,

a),

show

that

by keeping

n

fixed we

obtain

a

one-parameter

subgroup.

What

is

the

range

of

a?

3. Develop analogues

of

(i), (ii), (iii)

of

problem

(1) above for

the

group

8U(2).

Find

all possible

one-parameter

subgroups

in

this

case.

Chapter

10

Linear

Representations

of

Lie

Algebras

We have reviewed in

Chapter

8 some

aspects

of

the

theory

of

group

representa-

tions.

When

applied

to

Lie groups in

particular,

they

lead

to

linear representa-

tions

of

Lie algebras. We devote

this

relatively

short

Chapter

to

this

topic.

Let

a Lie

group

G

and

its Lie

algebra

(sometimes

written

as G

and

some

times in a general

context

as

£)

be

both

given. To

get

a linear

representation

of

£,

we

must

start

with

a (real

or

complex) linear vector space

V,

of

dimension

n

say,

and

set

up

the

following association:

Elements

in

£

--+

linear

transformations

on

V,

Lie

bracket

in

£

--+

commutator

of

transformations

on

V

(10.1)

Why

is

the

abstract

Lie

bracket

in

£

"realised" as

the

commutator

of

linear

operators

in a linear

representation?

Go

back

for a

moment

to

the

Lie

group

G:

actually

the

ensuing

operations

refer only

to

a

suitable

neighbourhood

1)1

c

G

containing

the

identity.

If

we

had

a

representation

of

G

on

V,

then

to

the

element

a

E

1)1

with

coordinates

a

we would have

the

representation

matrix

D(a)

say, obeying:

D(a)D(f3)

=

D(f(a;

(3))

D(O)

=

1

(10.2)

What

would

the

elements

on

an

OPS

look like in

the

representation?

Assume

that

the

coordinate

system

is a canonical one,

and

for small

a,

suppose

(10.3)

Thus,

given a basis for

£,

{ej}

say,

the

way

the

a's

are

enumerated

is fixed:

and

in

the

representation

on

V

we find

that

to

each

ej

there

corresponds a linear

operator

Xj

on

V:

ej

E £

--+

linear

operator

Xj

on

V

(10.4)

166

Chapter

10.

Linear

Representations

of

Lie Algebras

These

are

called

the

generators

of

the

representation

. Next, for two elements

a,

bE

91,

both

close

to

e,

the

commutator

group

element

q(a,

b)

has

coordinates

given by Eq.(9.46). Therefore we have

the

condition,

D(q(a,

b))

==

D(a)D((3)D(

-a)D(

-(3)

. k 1

r::::

1

+ c{la

(3

Xj

+ ...

(10.5)

If

we

put

in expressions like (10.3)

on the

left

hand

side

and

retain

terms

at

most

linear

in

a

and

(3,

we immediately find:

a

k

Xk(3lXl - (3lXla

k

X

k

=

c{

l

a

k

(3lX

j

,

i.e.,

[Xk'

Xd

==

XkXl

-

XlXk

=

c{lX

j

(10.6)

And

then

the

elements

on

an

OPS

ar

e of course realised as

"ordinaryexponen-

tials" :

u·

X

(u·

X)2

D(exp(u))

=

1

+

-,-

+ , + ...

1.

2.

(10.7)

One

sees how

the

up-to-now formal exponential used in relating vectors in L

to

elements in

G

via

OPS's

becomes

the

ordinary

exponential in a linear represen-

tation.

In

other

words, if we have a linear

representation

of

L

on

V,

with

the

Lie

bracket of

L

interpreted

as

the

ordinary

commutator

of

generators

acting

on

V,

then

(apart

from global aspects)

upon

exponentiation

we

get

a

representation

of

G:

in fact since every

a

E

91

does lie

on

some

OPS,

the

result

(10.7) does fix

the

way all elements in

91

C

G

are

to

be

represented.

(A

more

precise

statement about

global aspects is this. A

hermitian

repre-

sentation

of

the

Lie

algebra

L

of a Lie

group

G

yields,

upon

expon

e

ntiation

a

true

representation

of

G,

the

universal covering

group

of

G,

and

in general

not

of

G

itself).

This

situation

can

be

"turned

around

" in

the

following sense.

Many

groups

of

practical

interest

are

defined via a specific linear

representation

on

some defi-

nite

space

V

(a

defining

matrix

representation),

which is declared

to

be

globally

faithful.

Then

we

can

in

principle "look"

at

the

family

of

matrices

involved, find

independent

real

parameters

(at

least locally) for

them,

evaluate

th

e

generator

matrices

in

that

representation

,

and

from

their

commutation

properties

"read

off"

the

structure

constants

needed

to

search for

other

representations!

In

fact

this

is

the

most

practically convenient way

to

handle

the

orthogonal,

unitary

and

symplectic groups, as we will see later.

At

this

point, a

matter

of

convention

and

notation

needs

to

be

explained.

In

quantum

mechanical applications

unitary

representations of groups play

an

important

role.

It

is clear from Eq.(10.7)

that

for

D(a)

to

be

unitary,

(assuming

V

is

equipped

with

the

appropriate

inner

product),

th

e

generators

Xj

must

be

antihermitian.

It

is however

usual

to

remove explicitly a factor

of

i

and

deal

167

with

hermitian generators.

Hereafter we shall uniformly adhere

to

this

quantum

mechanical convention.

In

the

context

of

unitary

(hermitian) representations of

G(£),

we

shall instead of

the

previous Eqs.(10.3),(1O.6),(1O.7)

adopt

the

follow-

ing practice:

Unitary

r

ep

resentation of G

<->

Hermitian

represe

ntation

of

£

ej

--->

Xj

=

xJ;

[ej,

ek]

=

C;ke1

--->

[Xj,

Xk]

=

iC;kX1

(a)

(b)

(c)

(d)

(e)

(10.8)

Alternatively we could say

that

the

abstract

Lie bracket in

£

goes into

-i

times

the

commutator.

A real orthogonal re

presentation

of G (on a real

V

with

appropriate

inner

product)

leads

to

generator

matrices

Xj

which are

hermitian

and

antisymmetric,

hence purely imaginary. To emphasize

that

a common

quantum

mechanical

convention

is

being used, we express

the

situation

thus:

Unitary

group

repr

ese

ntation

<->

he

rmitian

generators,

XJ

"

=

xt.

J'

Real orthogonal group representations

<->

hermitian

antisymmetric generators,

X

t

-X

XT

-

X·

j -

j,

j - -

j,

Real repres

en

tation

<->

pure

imaginary generators,

X;

= -

X

j

.

(10.9)

In

fact, even for non

unitary

non-real orthogonal representations,

the

same

conventions (10.8) apply.

In

this

most general case,

Xj

are

neither

hermitian

nor

antisymm

et

ric imaginary.

For Lie algebra representations, all

the

notions of invariant subspaces, irre-

ducibility, reducibility, decomposability, direct

sum

etc

can

be

directly carried

over from

the

discussions of

Chapter

8.

Other

than

these,

the

passages

to

the

contragredient, adjoint,

and

complex conjugate group representations

are

re-

flected

at

the

generator

level thus:

D

--->

(DT)-l

:

Xj

--->

-XJ;

D

--->

(Dt)-l:

Xj

--->

xJ;

D

--->

D* :

Xj

--->

-x;

(10.10)

In

these rules,

the

"quantum

mechanical

i"

has been properly respected.

We finally

take

up

in this

Chapter

a

particular

real

representation

of a Lie

algebra

£

and

"its Lie

group

G" (locally), which

is

intrinsic

to

the

structure

of

168

Chapter

10.

Linear

Representations

of

Lie Algebras

.c

and

is

called

the

adjoint

representation.

The

beauty

of

this

representation

is

that

.c

itself serves as

the

representation

space

V,

on

which

both

.c

and

G are

made

to

act! For each

x

E

£,

define a linear

transformation

ad

·X

which

acts

on

£

in

this

way:

(ad.

x)

: y

--->

y'

=

[x,

y],

i.e., (ad.

x)y

=

[x,

y]

(10.11)

Here

the

abstract

Lie bracket in

£

has been used,

and

the

linearity prop-

erties of

this

bracket ensure

that

ad. x

is

a linear transformation. Do

we

have

here a representation of

.c?

Yes, by Jacobi!

(ad. x)(ad.

y)z -

(ad. y)(ad.

x)z

=

[x,

[y,

zll-

[y,

[x,

zll

=

[[x,

y],

z]

=

(ad.

[x,

y])z,

[ad.

x,

ad.

y]

=

ad.

[x,

y]

(10.12)

(The

brackets on

the

two sides of

this

last line

are

formally different!).

In

a basis

{ej}

for

£,

the

generator

matrices are

the

structure

constants

(times a factor of

i):

(ad.

ej

)ek

=

[ej,

ek]

=

C;kel

=}

(X}adj))~

=

i

x coefficient of

el

in (ad.

ej)ek

. I

=

~Cjk

(10.13)

Then

the

Jacobi identity (9.53) for

the

structure

constants

amounts

to

the

gen-

erator

commutation

relations:

[x

(ad

j

)

X(adj)]

_ .

I

X(adj)

j

'k

-

~Cjk

I

(10.14)

In

the

expressions in Eq.(10.13), while one of

the

subscripts

on

the

structure

constant

is

used

up

to

enumerate

the

X's,

the

remaining superscript (subscript)

acts

as a

matrix

row (column) index.

Exponentiation

of

the

adjoint representation of

£

leads of course

to

a real

local

representation

of G on

£

itself. This, in

its

global aspect,

can

be

called

the

adjoint representation of

the

simply connected universal covering group

0,

which

.c

determines unambiguously. Alternatively, one also often defines as

the

adjoint

group

of

.c

that

Lie group which possesses

£

as its Lie algebra,

and

for

which by definition

the

set of all exponentials of

the

generators

iu

j

X}ad

j

),

and

products

thereof, give a faithful representation.

The

adjoint

representation

will play

an

important

role in all

our

consider-

ations.

169

Exercises

for

Chapter

10

1.

Show

that

among

functions

f

(q,

p),

g(

q, p),

..

.

on a classical mechanical

real 2n-dimensional

phase

space,

the

Poisson bracket

{J

g}

-

~

(Of

~

_

of

Og)

, -

~

oq·

op op

oq

J=l

J J J J

is a realisation of

the

Lie bracket concept. Similarly show

that

among

three-dimensional real vectors

Q,

...

,

the

vector

product

operation

is also a realisation of

this

concept.

2.

From

th

e

quantum

theory

of

angular

momentum

we know

that

the

Lie

algebra

commutation

relations for generators of SO(3) are

Ejkl

=

completely antisymmetric,

E123

=

1.

Using Eq.(10.13), verify

that

the

adjoint

representation

of

80(3)

is

its real 3 dimensional defining rep-

resentation.

This page intentionally left blankThis page intentionally left blank

Chapter

11

Complexification

and

Classification

of

Lie

Algebras

So far we have

dealt

with

real

(finite dimensional) Lie algebras, as

they

arise

while

analysing

Lie

groups

whose elements

can

be

parametrised

with

an

essen-

tial

and

finite

number

of

real

independent

coordinates.

The

task

of

trying

to

systematically

classify all such Lie algebras is a very difficult

and

complex one,

and

it

would definitely

be

out

of place

to

attempt

to

describe

here

in

all

detail

the

results

known

in

this

area.

On

the

other

hand,

it

seems worthwhile

at

least

introducing

some

of

the

key concepts

that

are

used

in

this

branch

of

mathemat-

ics,

and

conveying

the

flavour of

the

subject.

We

attempt

to

do

no

more

than

this

in

the

present

Chapter.

To go

further,

the

reader

is advised

to

study

one

of

the

works

in

the

list

of

references given

at

the

end.

We describe

the

process

of

complexification

of

a real Lie algebra,

and

give

the

definition

of

a complex Lie algebra. We also

indicate

how

the

properties

of

solvability, semisimplicity

and

simplicity

are

used in

the

classification pro-

gramme.

Our

final

aim

is

to

arrive

at

and

deal

with

the

series

of

Compact

Simple Lie Algebras (CSLA),

and

their

associated

Lie groups.

11.1

Complexification

of

a

Real

Lie

Algebra

Let

I:

be

a

real

r-dimensional

Lie algebra.

By

a

straightforward

procedure

we

can

complexify

it

and

arrive

at

a complex Lie

algebra

C.

To begin, we complexify

I:

as

a

vector

space.

Thus

the

complex

r-dimensional

vector space

C

consists

of

formal expressions

of

the

form,

z

=

x

+

iy

E

C,

x,y

E

I:

(11.1)

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

Подождите немного. Документ загружается.