Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists
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182
Chapter
11. Complexification
and
Classification
of
Lie Algebras
We will prove
later
that
(Eq.(12.I5))
a,
(3,
a
+
(3
E
ryto
=}
No./3
-=F
0
(U.5I)
5.
The
hermiticity
property
(11.49) for
the
complex
quantities
Eo.
has
as a
consequence
the
relation
(11.52)
Once
we
have
arranged
gab
=
8~b,
go.,-o.
=
1,
the
relation
(U.45)
can
then
be
simplified
to
read:
1
a
E
9\0
:2l.:(aa)2
+
l.:
No.,/3N_o.,o.+/3
a=l
!3E<JiO
(n+!3E<Jiol
1
=
2l.:(aa)2
+
l.:
No.,/3N_o.-/3,o.
by Eq.(11.52)
a=l
!3E<Jio
(n+!3E<Jio)
I
=
2l.:(aa)2
+
l.:
N~,/3
by
Eq.(U.46)
a=l
=1
(11.53)
We have now assembled enough factual information
about
the
CSLA's
to
per-
mit
their
complete analysis
and
classification.
This
will
be
begun
in
the
next
Chapter.
Exercises
for
Chapter
11
1.
The
Lie algebras
of
SO(3)
and
of
the
three
dimensional homogeneous
Lorentz
group
SO(2, 1)
are
respectively
[J
l
,
J
2
]
=
ih,
[J
2
,
h]
=
iJl,
[J
3
,
J
l
]
=
iJ
2
;
[Jo,Kd
=
iK2'
[J
O
,K
2
]
=
-iKl'
[K
l
,K
2
]
=
-iJ
o
;
Show
that
they
have a
common
complex extension.
2.
For
the
two groups SO(3)
and
SO(2, 1) whose Lie algebras
are
given in
problem
(1) above, find
r,l,R
o
,Q,No./3
of
Eq.(11.47). Assume
H
=
h
or
J
o
respectively.
3.
Supply
the
proofs
of
Eqs.(11.25)-(11.28).
Chapter
12
Geometry
of
Roots
for
Compact
Simple
Lie
Algebras
We have seen
that
in
any
UIR
of
a
compact
simple Lie
algebra
£
the
hermitian
Cartan
subalgebra
generators
Ha
can
be
simultaneously
diagonalised. A
set
of
simultaneous
eigenvalues for
the
Ha
can
be
written
as
an
l-component
real
vector
/L
=
{/La}.
Such vectors
are
called
weights.
We shall explore
their
properties
in
Chapter
16.
But
it
must
be
clear
that
root
vectors
are
the
weight vectors
appearing
in
the
adjoint
representation;
and
by
one
of
the
Cartan
theorems,
the
non-zero
roots
are
nondegenerate.
Having scaled
and
arranged
our
generators
so
that
(12.1)
we
can
say
that
the
roots
of a
compact
simple Lie
algebra
of
rank
l
are
real
vectors in
an
l-dimensional
real
Euclidean
space.
The
set
9't
contains
r -
l
non-zero
distinct
root
vectors, coming
in
pairs
±Q..
As
these
vectors
must
obey
Eq.(12.1),
there
must
be
exactly
llinearly
independent
vectors in
9't
so
that,
as
we saw earlier
in
Eq.(11.40),
~(r
-
3l)
must
be
a non-negative integer.
What
arrays
or
geometrical
arrangements
of
distinct
pairs
of
vectors
±Q.
in
l-dimensional
Euclidean
space
can
arise in
the
set
of
roots
9't
of
some CSLA
of
rank
l?
It
turns
out
that
there
are
many
restrictions
on
such
arrays,
again
on
account
of
the
Jacobi
conditions. A
particularly
important
part
is played
by
a family of (in general
not
mutually
commuting)
SU(2)
subalgebras,
so let us
begin
by
recording
the
structure
of
the
SU(2) Lie
algebra
and
its
(hermitian)
irreducible
representations.
In
the
Cartan-Weyl
form,
the
elements
of
the
SU(2)
algebra
are
J+,
J_
and
184
Chapter
12.
Geometry
of
Roots
for
Compact
Simple Lie Algebras
h obeying
the
following hermiticity
and
commutation
properties:
Jl =
h,Jt
=
L,
[h,h]=±h,
and
[J+,
L]
=
2h
(12.2)
The
UIR's
are
labelled by a
quantum
number
j
which
can
take
values
0,
1/2,
1,
3/2,
....
The
lh
VIR
is
of dimension
2j
+
1.
Within
this
VIR,
the
spectrum
of
J
3
is
nondegenerate
and
consists of
the
string
of eigenvalues
m
=
j,j
-
1,j
-
2,
...
,
-j
+
1,
-j:
h[j,m
>=
m[j,m
>i
(12.3)
zero occurs if
j
is
integral
and
does
not
if it
is
half
odd
integral.
The
action of
J±
is
h
[j,
m
>=
J
(j
=f
m)
(j
±
m +
1)
[j,
m
±
1
>
(12.4)
That
is, while each
[j,
m
>
is
normalised
to
unity,
the
phases
can
be chosen so
that
J±
act
in
this
way.
Let us now identify
the
different SU(2) algebras contained in a CSLA
C.
There
are in fact
~(r
-
l)
distinct ones. For each
root
vector
Q
E
91,
we have
the
subset of
commutation
relations,
and
[Ha,
En]
=
aaEn,
[Ha,
E-n]
=
-aaE_n,
[En'
E-n]
=
aaHa
We
can
see
then
that
for each
a
E
91
0
there
is
an
SU(2)(n) algebra:
(12.5)
(12.6)
Further,
the
hermiticity relations for these SU(2)(n) generators are also in
accordance
with
Eq.(12.2).
The
relation between SU(2)(n)
and
SU(2)(-n)
is
very simple:
SU(2)(n)
--+
SU(2)(-n) : h
--+
-h,
h
--+
L,
L
--+
h
(12.7)
This
is
the
same
as a
rotation
by
1r
about
the
x-axis, in
the
usual language
of
quantum
mechanics
and
angular momentum.
In
the
sequel, we choose some
a
E
91
0
,
Q
E
91
and
keep it fixed. For
this
reason
we
have
not
put
a as
an
index on
h,
J±
in Eq.(12.6). We are interested
in
the
action of SU(2)(n) in
the
adjoint
representation
of
C:
action by
J+, J_
or h on, say, some
E{3
is
the
same
as
commutation
of
J+, J_
or h
with
E(3.
Commuting
J±
with
E(3
should give as a result
E(3±n
provided
(3
±
a
E
91
0
.
We
must
also remember
that
for each
(3
E
91
0
,
E{3
is
unique
apart
from a scale,
and
185
that
the
most
general irreducible
representation
of
SU(2) is
characterised
by
a
j-value
and
has
the
form
in
Eqs.(12.3),(12.4).
Since all
this
is so, we
can
conclude: for
any
fi
E
91,
E{3
belongs
to
some
definite
UIR
of
SU(2)Ca),
carrying
definite values of j
and
m.
That
is,
with
respect
to
commutation
with
the
SU(2)Ca)
generators,
E{3
is
the
mth
component
of a
tensor
operator
of
rank
j.
(However
the
relative scales
of
the
different com-
ponents
may
not
yet
be
in accord
with
the
matrix
element values
in
Eq.(12.4)).
There
must
be
a
string
of
roots,
with
p
+
q
+
1 entries, such
that
fi
+
PQ,
fi
+
(p
-
1
)Q,
...
,
fi
+
Q,
fi,
fi
-
Q,
...
,
fi
-
(q
-
1
)Q,
fi
-
qQ
E
91,fi
+
(p
+
l)Q,fi
-
(q
+
l)Q
tf.
91
(12.8)
The
point
is
that
we
are
using
our
knowledge
of
all possible SU(2) represen-
tations,
and
the
nondegeneracy
of
each
(3
E
91,
to
maximum
advantage.
Under
commutation
with
J±,
h
the
string
of
(complex)
generators,
(12.9)
must
behave
as
the
(not
yet
properly
normalised)
components
of
an
SU(2)Ca)
tensor
operator.
In
both
Eqs.(12.8),(12.9),
it
is necessarily
true
that
P
2':
0, q
2':
0
are
integers;
and
for
the
moment
we
are
assuming
that
Q
does
not
occur
in
the
set
of
root
vectors (12.8). We
can
easily
read
off
the
m-value for
E{3,
and
the
rank
j
for
the
chain
(12.9):
[h,
E{3]
=
(Q.
fi/IQI2)E{3
=?
m
=
Q.
fi/1Q12;
.
.11
2J
+
1
=
P
+
q
+
1
=?
J
=
2"
(q
+
p),
m
=
2"
(q
-
p)
(12.10)
Incidentally,
the
general
structure
of
SU(2)
representations
tells us
that
there
can
be
no
gaps
either
in
the
string
of
root
vectors (12.8)
or
generators
(12.9).
In
('ase
the
vector
Q
occurs in
the
string
(12.8), we
cannot
say
that
all
the
vectors
in
the
string
belong
to
91.
But
you
can
easily convince yourself
that
the
corresponding
generator
would have
to
be
Q .
HI
IQI
2
,
namely
h.
From
here
we
can
only
move
"up
or
down" one
step,
to
Ea
or
E_
a
,
so
that
j
=
1 in
this
case.
What
we have therefore done is
to
use SU(2)
representation
theory
to
anal-
yse
the
geometry
of
the
root
vectors as we move
up
and
down
in
l-dimensional
Euclidean
space
by
amounts
±Q, ±2Q, ±3Q,
....
We find
that
the
triplet
E±a,Q·
H
forms
an
(unnormalised)
j
=
1
operator
under
SU(2)Ca).
Every
(3
E
91
other
than
±Q
belongs
to
some
chain
(12.8)
of
non-zero roots;
this
does
not
conflict
with
m
=
0 being a possible value
of
m if
j
is
an
integer, since
it
only
means
that
Q
and
fi
are
perpendicular.
Some
information
on
the
N
a{3
can
be
obtained,
independent
of
the
relative
normalisations
of
the
E{3's.
With
j
and
m identified
as
in Eq.(12.1O), we
do
have
(12.11)
186
Chapter
12.
Geometry
of
Roots
for
Compact
Simple Lie Algebras
since
[Eo:,
Ej3l
=
No:j3
E
o:+j3,
[E_o:,
Eo:+j3l
=
N_o:,o:+j3Ej3,
and
N
-o:,o:+j3
=
N
o:,j3
We
can
get
the
magnitude
of
No:j3:
N;j3
=
1~12
p(q
+
1)
So we
can
definitely
say
a,
(3,
a
+
(3
E
91
0
=}
P
::::
1
=}
No:j3
=I
0
This
property
of
No:(3
was
quoted
earlier
in
Eq.(1l.51).
(12.12)
(12.13)
(12.14)
The
fact
that
m
appearing
in
Eq.(12.10) is
quantised
is
the
origin
of
the
se-
vere geometrical restrictions
on
possible systems
of
root
vectors. Before deriving
those restrictions, let us record
the
following
additional
properties
of roots:
Q
E
91
=}
±2Q,
±3Q,
...
~
91,
Ii
=
Q
in Eq.(12.1O)
=}
P
=
O,q
=
2,j
=
m
=
1;
Ii
=
-Q
in Eq.(12.1O)
=}
P
=
2,
q
=
O,j
=
-m
=
1 (12.15)
Now all
that
we did
to
E(3
by
commutation
with
the
SU(2)(O:)
generators
can
be
repeated
with
the
roles of
a
and
(3
interchanged! So for
any
two
root
vectors
Q,
(3
E
91
distinct
or
the
same,
SU(2)(O:)
analysis tells us
there
must
exist
integ;s
p, q
::::
0;
while SU(2)((3) analysis tells us
there
must
exist integers
p',
q'
::::
0;
and
then
2
1
n
Q .
iii
IQI
="2
(q
-
p)
=
2
=
positive
or
negative
half
integer
or
zero;
. 1 '
Q.
lillQI2
=:=
"2(q'
- p') =
~
=
positive
or
negative
half
integer
or
zero.
If
0
is
the
angle between
Q
and
Ii,
these
restrictions mean:
llil
n
IQI
n'
IQI
cosO
=
2'
llil
coso
=
2'
nn'
cos
2
o
=
4
We
can
draw
several conclusions
on
the
possible values
of
nand
n':
n
=
0
{=}
n'
=
0
{=}
0
=
90
0
;
n
>
0
{=}
n'
>
0;
n
<
0
{=}
n'
<
0;
n,
n'
=
0,
±1,
±2,
±3,
±4;
0:<:;
nn':<:;
4.
(12.16)
(12.17)
(12.18)
187
In
those
cases
where
both
nand
n'
are
non-zero,
I.~I/IQI
=
yin'
/n
(12.19)
We see t
hat
the
possible values of angles
e
between
roots,
and
ratios
of
lengths
of
roots,
are
severely limited.
Let
us see
what
are
the
allowed possibilities.
To
start,
in
principle
the
pair
(n,
n')
can
have 17 possible values:
(n,
n')
=
(0,0);
±(1,
1);
±(1,
2);
±(1,
3);
±(1,
4);
±
(2,1);
±(2,
2);
±(3,
1);
±(4,
1)
(12.20)
But
som
e
of
these
are
actually
ruled
out!
The
inadmissible
ones
are
(n,
n')
=
±(1
,4)
and
±(4
, 1).
For
instance
,
because
of
Eq.(12.15),
n
=
1,
n'
=
4
'*
e
=
0
'*
q
=
2Q
tJ-
9't
(12.21 )
and
similarly
in
the
other
three
cases.
That
leaves 13 possibilities.
Of
these,
(n,n')
=
±(2,2)
'*
e =
0
or
180°, q
=
±Q
(12.22)
Leaving
these
aside as well, we
are
left
with
11
significant possibilities,
which we
present
in
tabular
form:
Geometrical
conditions
on
roots
in
a
CSLA
n
n'
e
Iql/IQI
0
0
90°
unspecified
1
1
60°
1
-1
-1
120°
1
1
2
45°
Vi
2
1
45°
1/Vi
-1
-2
135°
Vi
-2
-1
135°
1/Vi
1 3
30°
V3
3
1
30°
1/V3
-1
-3
150°
V3
-3
-1
150°
1/V3
To find
and
classify all possible
CSLA
means
to
find all possible
sets
of
real
root
vectors
in
Euclidean
l-dimensional
space, for various
ranks
l,
obeying
all
these
geometrical
restrictions.
This
is
made
tractable
by
the
concepts
of
positive
and
simple
roots
,
to
which we
devote
the
next
Chapter.
This page intentionally left blankThis page intentionally left blank
Chapter
13
Positive
Roots,
Simple
Roots,
Dynkin
Diagrams
13.1
Positive
Roots
Let us make a choice of
the
Cartan
subalgebra generators
H
a
,
a
=
1,2,
...
,
l
in a definite sequence (of course maintaining
gab
=
6
a
b),
and
hereafter keep
the
sequence unchanged. We shall say
that
a
root
Q
E
9't
is
positive
if in its
set
of
components
Q
=
{al'
a2,
...
, az},
the
first non-vanishing
entry
is
positive.
Otherwise, we shall say
Q
is
negative.
The
set of all positive
roots
will
be
denoted
by
9't+,
while
the
set of negative ones will
be
denoted
9't_.
Remembering
that
every
Q
E
9't
is
nonzero,
and
that
all
roots
come in pairs
±Q,
it
is clear
that
every
Q
E
9't
is
definitely either positive
or
negative,
and
each of
9't+
and
9't_
has
precisely
~(r
-l)
vectors:
9't
=
9't+
U
9't_,
9't
=
set of
r -
l
distinct
root
vectors,
1
9't+
(9't_)
=
set of
"2
(r -
l)
distinct positive (negative)
root
vectors:
(13.1)
13.2
Simple
Roots
and
their
Properties
Now we define simple roots: these
are
a subset
of
the
set
9't+
of positive roots,
so
the
simple
roots
form a set
S
C
9't+.
A
root
Q
E
9't+
is called a
simple root,
Q
E
S,
if
and
only if
Q
cannot
be
expressed as a non-negative integral linear
combination
of
other
positive roots. We have
the
inclusion relations
9't
=
all
roots
::)
9't+
=
all positive
roots
::)
S
=
all simple roots.
(13.2)
A finer subdivision of
9't+
will
appear
later
on.
190
Chapter
13.
Positive Roots, Simple Roots, Dynkin Diagrams
The
properties
of
S
are
really remarkable
and
beautiful. We describe
and
prove
them
one by one.
(i)
The
angles between simple
roots
are
further
restricted, beyond
the
re-
strictions listed in
the
previous
Chapter.
Let
Q,
f3
be
two
distinct
simple roots.
Then
neither
Q -
f3
nor
f3
- Q
can
be
a root. Otherwise, one of
them
would
be
in
9\+
and
the
other
in
5L.
If
Q -
Ii
E
9\+,
we
are
able
to
express
(13.3)
so
Q
could
not
have
been
simple. Similarly,
f3
-
Q
E
9\+
would have led
to
the
conclusion
that
Ii
is
not
simple. We
can
state
the
result as
Q,
Q
E
S, Q
i=
Q
=?
Q -
Q,
Q -
Q
ef.
9\
(13.4)
Now let
us
interpret
this
in
the
light
of
the
actions of
the
SU(2)(a)
and
SU(2)C
8
)
algebras
on
9\.
Using
the
notations
of
the
previous
Chapter,
SU(2)(a) applied
to
E(3
:
q
=
O,p
2':
0;
SU(2)((3) applied
to
En
:
q'
=
O,p'
2':
0;
Q.f3
p
Q.f3
i
IQI2
=
-2
:::;
0,
Ilil2
=
-"2
:::;
0
(13.5)
Referring
to
Eqs.(12.16)-(12.19),
we
have
n
=
-p
and
n'
=
-p'.
So
the
angle
e
between
Q
and
Ii
must
obey
(13.6)
(We
must
rule
out
e
=
180
0
,
since
that
means
f3
=
-Q,
which conflicts
with
both
being positive roots!)
The
possible
angles-and
length
ratios
for simple
roots
form
the
following subset
of
the
table
in
Chapter
12:
Geometrical
conditions
on
simple
roots
in
a
CSLA
n
n'
e
IQI/IQI
0
0
90
0
unspecified
-1
-1
120
0
1
-1
-2
135
0
V2
-2
-1
135
0
1/V2
-1
-3
150
0
V3
-3
-1
150
0
1/V3
(ii)
The
set
of simple
roots
is linearly independent. For, suppose
there
is a
nontrivial
relation
LXaQ=O
~ES
(13.7)
among
the
simple roots. Since each
Q
here is in
9\+,
there
must
be
some
Xa
>
0
and
other
Xa
<
o.
(Naturally
terms
with
Xa
=
0
can
be
ignored).
Write
x~
13.2. Simple
Roots
and
their
Properties
191
for
the
former,
-y~
for
the
latter,
and
splitting
the
two sets
of
terms,
write
Eq.(13.7) as
I
==
L
x~Q:
=
L
y~Q:
;;t
0
(13.8)
gES gES
Here
the
simple
roots
occurring in
the
two sums
are
definitely distinct.
One
then
has
o
<
1112
=
L
x~y~Q:'
~
:::;
0
g,(!..ES
(13.9)
because each
x~,
y~
is
strictly
positive,
and
from
point
(i) above
Q:'
~
is non-
positive. Since
situation
(13.9) is
an
impossibility,
the
result is proved.
It
is
interesting
to
note
that
the
Euclidean
geometry
for
the
space of
roots
was used
in
the
argument,
since
that
is
what
ensures
I
;;t
0
=}
III
>
O.
(iii)
Each
positive
root
Q:
E
91+
can
be
written
uniquely as a linear combi-
nation
of
simple
roots
with
non-negative integer coefficients. For, let
Q:
E
91+.
If
Q:
E
S,
we
are
done.
If
Q:
t/:.
s,
it
can
be
written
as a positive integral combi-
nation
of
oth
er positive
roots
(definition
of
S!).
Do so.
If
each
term
occurring
involves a simple root, we
are
done.
If
not,
the
terms
involving nonsimple
roots
can
again
be
written
as a positive integral combination
of
positive roots.
This
process
must
end
with
a positive integral combination
of
simple
roots
because
at
each stage all coefficients
are
positive integers, all vectors
are
positive, so
there
cannot
be
an
indefinite
or
unending build-up
of
terms.
The
uniqueness
of
such
an
expression follows from
property
(ii) above of linear independence
of
simple
roots. We
can
summarise:
Q:
E
91+
=}
Q:
=
unique non-negative integer linear combination
of
simple
roots
,
Q:
E
91_
=}
Q:
=
unique non-positive integer linear combination
of
simple
roots
(iv)
If
we
now combine
properties
(ii)
and
(iii) above
with
the
fact,
noted
at
Eq.(12.
1)
,
that
there
are
certainly enough
independent
vectors in
91
to
span
I-dimensional Euclidean space, we arrive
at
the
nice result
that
there
are
exactly
I
simple
roots
in
S!
Hereafter we
write
and
the
contents
of
point
(iii) above are:
I
Q:
E
91+
=}
Q:
=
L
naQ:(a),
na
2:
0,
integer, unique for
Q:i
a=l
I
Q:
E
91_
=}
Q:
=
L
naQ:(a),
na
:::;
0, integer, unique for
Q:.
a=l
(13.10)
(13.11)