Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists
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172
Chapter
11.
Complexification and Classification of
Lie
Algebras
Two
such
expressions
are
the
same
only
if
the
separate
parts
are
equal:
x
+
iy
=
x'
+
iy'
{=}
x
=
x',
y
=
y'
(11.2)
For
a complex
number
a
+
ib,
with
a
and
b
real, we define
(a
+
ib)(x
+
iy)
=
ax
-
by
+
i(ay
+
bx)
(11.3)
which is
again
of
the
form (11.1). You
can
check
that
(a
+
ib)(x
+
iy)
=
0
{=}
either
a
+
ib
=
0
or
x
+
iy
=
0
or
both
(11.4 )
Finally,
then,
Lie
brackets
in
£
are
defined
by
[x
+
iy, u
+
iv]
=
[x,
u]
-
[y,
v]
+
i([x,
v]
+
[y,
u])
(11.5)
The
validity
of
the
Jacobi
identity
is obvious.
We
say
that
£
is
the
unique
complex
form of
the
real
Lie
algebra
L.
The
complex
dimension
of
£
is
the
same
as
the
real
dimension
of
L;
and
a basis
{ej}
for
L
remains
a
basis
for
£
as well.
In
this
basis,
the
structure
constants
of
£
are
the
same
as
those
of
L,
namely
C;k'
and
so
they
are
real.
Having
seen how
£
arises from
L,
we
can
now
directly
define
what
we
mean
by
a
complex
Lie algebra. A
complex
Lie
algebra
£
of
dimension r is a complex
r-dimensional
vector
space,
with
elements
z,
z', ...
E
£,
among
which a Lie
bracket
is defined:
z,z'
E
£,
.
.\,fJ E
C;
AZ
+
fJZ'
E
£;
[z,z']
E
£;
[z,
z']
=
linear,
antisymmetric;
[[z,
z'],
z"]
+
[[z',z"],z]
+
[[z",z],z']
=
0
(11.6)
This
idea
of a
complex
Lie
algebra
is
directly
defined
in
this
way,
and
not
derived
from a finite
dimensional
Lie
group
with
real
parameters.
If
{ej}
is a
basis
for
£,
in
general
we have
complex
structure
constants
C;k
obeying
antisymmetry
and
the
Jacobi
condition.
For a
general
complex Lie
algebra,
there
may
be
no
basis
in
which all
the
structure
constants
become
real!
Of
course
an
r-dimensional
complex
Lie alge-
bra
£
can
be
viewed as a real
2r-dimensional
one
with
an
additional
operation
representing
"multiplication
by
i" .
Now
let
us
mention
some
statements
which
are
intuitively
evident,
partly
repeating
what
was
said
earlier:
(i) A given
real
r-dimensional
Lie
algebra
L
leads
to
a
unique
complex
r-
dimensional
Lie
algebra
£,
its
complex form.
A basis for
L
can
be
used
as
a
basis for
£,
and
in
that
case
the
structure
constants
are
real
and
unchanged.
11.1.
Solvability, Levi's Theorem, and Cartan's Analysis of
...
173
(ii)
Many
distinct
real Lie algebras
£,
£',
...
may
lead
to
the
same
complex
one
l;
in
that
case each of
the
former is called a
real form
of
the
latter,
and
l
arises by complexification
of
any
of
£ ,
£',
....
(iii) A given complex r-dimensional Lie
algebra
l
may
have none,
or
many,
real forms.
If
it
has
none,
then
there
is
no
basis for
l
with
respect
to
which all
the
structure
constants
have real values!
While
the
use of complex Lie algebras is useful
and
indeed unavoidable for
purposes of
the
general theory,
the
process
of
"returning
to
the
real forms" is
made
quite difficult because
of
the
above facts. As
we
will see later,
though,
some
fundamental
theorems
of
Weyl help simplify
the
situation
for
the
real
compact
simple Lie algebras.
The
definitions of subalgebras, invariant subalgebras, factor algebras
etc
can
all
be
given
quite
easily for complex Lie algebras.
The
relation between
such
properties
for a real
£
and
for
its
complex form
l
is this:
£
abelian
B
l
abelian;
£
solvable
B
l
solvable;
£
simple
=I?
l
simple
(11. 7)
We have a surprise in
the
last line!
The
"fly in
the
ointment" is
the
case
of
the
groups
SL(n,
C);
while
the
real Lie
algebra
SL(n,
C)
is simple,
after
complexification
it
is no longer simple.
11.2
Solvability, Levi's
Theorem,
and
Cartan's
Analysis
of
Complex
(Semi)
Simple
Lie
Algebras
It
turns
out
that
solvability is a crucial generalisation
of
abelianness,
and
it
has
a
hereditary
and
contagious
character.
For
both
real
and
complex cases one
finds solvability
of
a Lie
algebra
implies
the
same
for all its subalgebras
and
for
the
factor algebras
with
respect
to
its
invariant subalgebras.
Even
more is true:
if a (real
or
complex) Lie algebra
£
has
an
invariant
sub
algebra
.c',
and
both
.c'
and
the
factor
£/
£'
are
solvable,
then
so is
£
itself!
Levi
Splitting
Theorem
Any (real
or
complex) Lie
algebra
£
can
be
split into
the
semidirect
sum
of
a
maximal
solvable invariant
sub
algebra
S
and
a semis imp
Ie
sub
algebra
T:
L
=
S+)T,
[S,
s],
[T,
S]
<;;;
S,
[T,T]
<;;;T
(11.8)
174
Chapter
11.
Complexification
and
Classification
of
Lie
Algebras
Based
on
this
result,
can
we classify all real
or
complex Lie algebras?
No!
while
the
semis
imp
Ie
ones
can
be
classified,
this
has
so far
not
been
possible
for
the
solvable ones. However
having
seen
this
basic
theorem,
and
appreci-
ated
where
the
situation
regarding
classification now
stands,
we shall
hereafter
consider
mainly
the
semisimple case.
The
classification of complex semisimple Lie algebras
rests
heavily
on
some
fundamental
theorems
of
Cartan,
and
work
and
analysis
by
Killing,
Cartan
and
others. We give a
qualitative
account
of
these
matters,
staying
at
the
complex
level
up
to
a
certain
point,
then
switching
to
the
real
compact
simple cases. We
begin
with
Theorem
(Cartan,
1894)
A (real
or
complex) semi-simple Lie
algebra
£
is
the
direct
sum
of
mutually
commuting
simple
nonabelian
subalgebras:
£ semisimple
{o}
£
=
£'
EB
£"
EB
...
,
£',
£"
, .
..
simple, nonabelian,
[£',
£11]
=
[£',
£111]
=
[£",
£111]
=
...
=
0 (11.9)
Next, as a
result
of
the
work
of
Killing,
Cartan
and
others,
the
complex
simple
Lie
algebras have all been
found
and
classified.
There
are
four infinite
families
and
five exceptional cases. We will deal
later
with
the
compact
real
forms
of
these. For
the
present, let us see how
the
structure
of
complex simple
Lie algebras
has
been
analysed.
The
method
rests
largely
on
skillful
exploitation
of
the
Jacobi
identities for
structure
constants.
Incidentally we
can
always keep
in
mind
that
analysis
of
the
adjoint
representation
amounts
to
analysis
of
the
basic
commutation
relations
and
so of
structure
constants.
To
be
reasonably
well organized, let
us
number
the
statements
to
follow,
and
arrange
them
in
as
good
a sequence
as
we
can!
1.
Let
the
(complex r-dimensional) Lie
algebra
l
in some basis have
struc-
ture
constants
e']k
which could
be
complex (we save
the
index
I
for a
good
reason);
and
define a
"metric
tensor"
(11.10)
the
sums
on
m
and
n
being
understood.
For simplicity we avoid tildes
on
e
and
g,
though
they
may
be
complex. A
result
of
Cartan
says:
lis
semis
imp
Ie
{o}
Igjkl
=
det(gjk)
# 0
(11.11)
In
one direction,
the
proof
is very easy.
If
l
has
an
abelian
invariant
subalgebra,
we
can
choose a basis
in
l
such
that
certain
rows
and
columns
of
(gjk)
vanish identically, so
l
not
semisimple
::::}
Igl
=
0,
Igl
#
0
::::}
l
semisimple
(11.12)
11.2.
Solvability, Levi's Theorem, and Cartan's Analysis of
...
175
The
harder
thing
to
show is
that
.c
semisimple
=}
Igl
=1=
0,
Igl
=
0
=}
.c
not
semisimple (11.13)
We leave
this
as
an
exercise
to
the
interested
reader.
In
particular,
Eq.(l1.ll)
gives:
.c
simple
=}
Igjk
1
=1=
o.
(11.14)
Hereafter, we assume
.c
is simple, so we
can
and
will use
the
fact
that
(g)
is nonsingular.
2.
From
the
structure
constants
cjL
by lowering
the
superscript
using
the
metric, we
can
construct
a
three-subscript
symbol which is
totally
antisymmet-
ric:
(11.15)
Since
(g)
is nonsingular, we
can
always
get
back
the
true
structure
constants
from here.
The
antisymmetry
in
the
first two indices is evident.
The
Jacobi
identity
leads
to
anti
symmetry
in
the
second
and
third:
(a)
(b)
(11.16)
Here,
at
the
second
step
(a), we have used
the
Jacobi
identity
for
the
first two c's
at
the
previous step;
and
at
the
fourth
step
(b)
in
the
second
term
the
dummy
indices were changed according
to
p
--+
q
--+
n
--+
p.
Thus
we have established
(11.17)
and
the
total
antisymmetry
of
Cjkm
follows. We will use
this
later
on
at
point
no.
7.
3. Now choose
an
element
A
E
.c
and
set
up
the
"eigenvalue problem"
(ad.
A)X
==
[A,
Xl
=
pX,
X
E
.c,p
E
c.
(11.18)
One
solution
is
of
course
p
=
0, X
=
A.
In
any
case,
the
number
of
distinct
eigenvalues
or
roots
can
be
no
more
than
r-dimension
of.c.
Now
vary
A
and
choose
it
so as
to
maximise
the
number
of
distinct
roots
p.
Then
another
Theorem
of
Cartan
says
that
for such
an
A,
(a)
There
is a
set
9't
o
of
distinct
non-zero
roots
0:,
(3,
...
(b)
Each
of
these
is
nondegenerate,
i.e., for each
0:
E
9't
o
,
there
is a
unique
Ea
E.c
(unique
upto
scalar multiplication) obeying
(11.19)
176
Chapter
11. Complexification
and
Classification of Lie
Algebras
(c)
Only
the
root
p
=
0 is degenerate,
and
the
degree of degeneracy l of
this
root
is
characteristic of
.c
and
is
called
the
rank
of
.c.
(d)
The
corresponding
l
"eigenvectors" are elements
Ha
E
.c,
a
=
1,2,
...
,
l;
they
are linearly independent
and
obey
[Ha,Hb]
=0,
a,b=I,2,
...
,l
(11.20)
(e)
The
Ha
for
a
=
1,2,
...
, l
and
En
for
a
E !)to
are independent
and
give
a basis for
.c.
It
must
be clear
that
a "maximal"
A
for which all
this
happens
is
by no
means unique!
At
this
point, we see
that
!)to
consists of
r -
l distinct nonzero
(complex) numbers. We will soon refine
and
get a
better
description of
!)to.
4.
Some
immediate
consequences, on exploiting
the
Jacobi
identities, fol-
low. Since
A
is
a solution
to
(11.18) for
p
=
0,
it
is a linear combination of
the
Ha:
A=.\aH
a
,
.\
a E
C,
not
all identically zero
(11.21 )
Then,
using
the
three-term
Jacobi identity for A,
Ha
and
En,
we get
i.e.
[Ha, En]
=
aaEn,
aa
E
C (11.22)
Thus, for each (complex)
number
a E !)to,
there
is
a (complex) l-component
"root
vector"
{aa},
and
since from Eqs.(11.19),(11.21),(11.22), we get
(11.23)
each vector
{aa}
cannot
vanish identically.
In
fact we
can
say:
a
E!)to
=?
{aa}
i=
0;
a,
(3
E !)to,
a
i=
(3
=?
{aa}
i=
{(3a}.
(11.24)
The
set
!)to
consists of
r - l
distinct non-zero (complex)
numbers.
Let us
now define
!)t
to
be
the
set
of the
corresponding
r -
l
distinct
non-vanishing
(complex)
root vectors
Q,
(3,
...
with
components
aa,
(3a'....
Elements of
!)to
and
of
!)t
are
connected
by
Eq.(11.23). We
are
of course free
to
replace
the
Ha
by nonsingular linear combinations of themselves, leaving
the
En
and
the
original A unchanged.
If
we do
this
we will
regard
the
vectors
Q,
(3,
...
E !)t
as
not
having changed,
but
as being resolved in a new basis. -
5.
The
information so far collected
on
the
structure
constants
cjk
is this:
each index
j,
k,
m,
...
=
1,2,
...
,
r
goes
partly
over
a,
b,
...
=
1,2,
...
, l,
and
partly
over
the
r
-l
distinct non-zero numbers a E !)to (or if we wish,
Q
E !)t).
This
is
the
result of using
Ha
and
En
as a basis for
.c.
Then,
(a)
Property
(3d) above
=?
c:J,
=
0
(11.25)
11.2. Solvability, Levi's
Theorem,
and
Cartan's
Analysis
of
..
.
(b)
Property
(4) above
=?
c;;:'a
=
aaom,a
(c)
If
a,;3, a
+;3
E
91
0
,
use
of
the
Jacobi
Identity
on
A , E
a
,
E(3
gives
[Ea,
E(3]
=
N
a
,(3E
a
+(3,
N
a
,(3
=
-N(3,a,
c";(3
=
N a,
(30m,a+(3
(d)
If
a,;3
E
91
0
,
a
+;3
i-
0, a
+;3
~
910,
then
necessarily
[Ea,E(3]
=
0,
c";(3
=
0
177
(11.26)
(11.27)
(11.28)
It
may
be
mentioned
that
(c)
and
(d) above
are
necessary consequences of
the
Jacobi
identities, in advance of knowing
the
contents
of
the
set
910.
6. Now we will show
that
a
E
910
implies
-a
E
91
0
as well.
In
turn,
and
this
is nontrivial,
it
will
mean
that
Q
E
91
implies
-Q
E
91
too. Consider
the
subset
of
(r -
l)
rows
of
the
matrix
(gjk) in which
j
=
a
E
91
0
.
Indicating
all
summations
explicitly, we have, using results
gathered
above:
gak
=
L~mck;
j,m
=
L
(t~aC~j
+
L
~(3C~j)
J
a=l
(3E~o
=
L
(-
t
aaOj,aC~j
+
L
Na,(30j,a+(3C~j
+
(c!.,-aCkn)
j
a=l
~
E
~O
(
"+
~E
~o)
= -
L
aac~a
L
Na(3C~,a+(3
+
L(~,-aCkn
(11.29)
a=l
~ E
~O
j
(Q+~E~o)
In
the
preceding two lines
we
have
put
inside parentheses a
term
which is
present only if
-a
E
91
0
,
not
otherwise.
If
indeed
-a
E
91
0
,
then
the
Jacobi
identity
for
A,
Ea and
E_
a
gives
[Ea,
E-a]
=
aa
H
a
,
aa
E
C
if
j
=
a,
=0
if
j
=
;3
E
91
0
(11.30)
Now going
back
to
Eq.(11.29), we
can
take
either
k
=
b
=
1,2,
...
,
l
or
k
= ,
E
910.
In
each case we
can
list
contributions
from each
of
the
three
terms,
assuming in
the
latter
case
that,
i-
-a:
k
=
b,
gab
=
0
+
0
+
(0),
k
=
,i-
-a,
ga,,(
=
0
+
0
+
(0).
(11.31)
178
Chapter
11. Complexification
and
Classification
of
Lie
Algebras
Here, we have used Eqs.(11.25),(11.26),(l1.30) when necessary. We
can
conclude
that
a
E
910,
-a
¢:.
910
=}
gak
=
0 for all
k
=}
Igjkl
=
0,
(11.32)
which
contradicts
Cartan's
theorem
(11.14)! Therefore
910
necessarily
consists
of equal
and
opposite non-zero
pairs
of
numbers
±a,
±j3,
... ;
and
in
the
ath
row
of
(gjk)
the
only non-zero element occurs for
k
=
-a:
ga,-a
= -
L
aac':.a,a
+
L
Na{3N-a
,a+{3
+
L
c~,_ac=~,a
(11.33)
a=l
i3E'R
O
a=l
(o+
i3
E'R
O
)
The
evaluation of
the
last
term
here involves some subtlety.
It
is
necessary
to
prove
that
the
I-component vector
(-a)a
in
91,
associated
with
-a
E
910
is
indeed
-aa.
To see
this
we
start
with
Eq.(11.30)
and
apply
the
Jacobi identity
for
H
a
,
Ea
and
E_
a
.
0=
[Ha,
[Ea, E-a]]
=
[Ea,
[Ha,
E-all-
[E_
a
,
[Ha,
Eall
=
(-a)a[Ea,
E-al
+
aa[Ea,
E-al
=
(aa
+
(-a)a)a
b
Hb
(11.34)
We will show (in point seven
to
follow)
that
{aa}
cannot
vanish identically, so
we
can
conclude:
c;;
,
'=-a
=
(-a)a
=
-aa
no
sum
on
a
(11.35)
Using
this
in
Eq.(l1.33),
we see
that
ga,-a
=
2
L
aaaa
-
L
N
a
{3N
a
+{3,
- a
i=
0
(11.36)
a
i3E'Ro
(o+13E'Ro)
and
the
nonsingular
r
x
r
matrix
(gjk)
has a block diagonal form:
gab
0
()
0
o
( )
o
±
ath
2 x 2 block in lower right
hand
block
= (
o
ga,-a
(11.37)
11.2. Solvability, Levi's Theorem, and Cartan's Analysis of
...
179
It
must
then
be
the
case
that
det(gab)
of.
0,
r - l
= even integer
(11.38)
We
can
get
an
expression for
the
matrix
elements
gab:
=
L L
c?"aCbj
j
oE!){o
(11.39)
Since
(gab)
is nonsingular,
this
means
that
there
are
enough
independent
vectors
among
the
(r
-
l)
vectors
{aa}
E
!J\
to
span
l-dimensional space;
further,
since
these
vectors come
in
pairs
±Q,
it
must
be
that
~(r-l»l
2 - ,
i.e.,
1 .
2(r
-
3l)
=
mteger
~
0
(11.40)
7.
Let
us
relate
the
l-component
quantities
aa
occurring
in Eq.(11.30)
to
the
earlier
aa;
the
former
are
connected
with
[Ea,
E-a],
the
latter
with
[Ha, Ea].
We exploit
the
total
antisymmetry
of
Cjkm
for
this
purpose:
Ca
,
-a
,a
==
-Co:,a,-a
i.e.,
C!x,-agja
=
-~,agj,-a
i.e.,
gabab
=
ga,-aaa
(11.41)
Therefore,
as
already
mentioned
in
point
six above,
(11.42)
8.
Keeping
the
Ha
unchanged,
what
happens
if we exploit
the
freedom
to
change
the
scale
of
each
Ea
independently?
Under
the
change,
(11.43)
we clearly get:
I
aa
=
aa,
I
gab
=
gab,
la
a
a =
nan-aa
I
ga,-a
=
nan_aga
,
-a,
N~j3
=
nanj3Naj3/na+j3
(11.44)
180
Chapter
11. Complexification
and
Classification
of
Lie Algebras
If
we therefore choose
the
numerical factors
na
to
achieve
ga,-a
=
1,
then
we get:
b
aa
=
gaba ,
aa
=
gab
ab
,
(gab)
=
(gab)-l,
2
L
aaaa
-
L
N
a
{3N
a
+{3,-a
=
1
a
(11.45)
9.
Some useful
information
on
the
No:{3
be
obtained
from
the
Jacobi
identities.
Suppos
e
a,
(3,
a
+
(3
E
910,
a
nd
they
are
all distinct;
then
the
Jacobi
identity
for
Eo:,
E{3,
E_O:-{3
and
the
fact
that
the
vectors
Q,
(i,
are
independent
gives:
(11.46)
10. We
can
now
put
together
all
the
information
we
have
gathered
on
the
structure
of
the
Lie
Brackets
in
a complex simple Lie
algebra
l.
Th
e
algebra
is
spanned
by
H
a
,
a
=
1,2,
...
,
I
and
Eo:,
a
E
910,
and
the
brackets
among
them
ar
e:
[Ha,H
b
]
=
0,
[Ha,
Eo:]
=
aaEa,
[Eo:,
E-
o:]
=
aa H
a
,
[Eo:,
E{3]
=
No:
,{3
Eo:+{3,
a,
(3,
a +
(3
E
910
(11.47)
These
equations
are
called
th
e
Cartan-
Weyl
form for
any
complex simple Lie
algebra.
One
can
proceed in
this
way
to
get
more
information
on
the
roots
Q,
(3,
. . .
E
91,
their
geometrical
properties,
the
structure
constants
Na;{3
etc.
For
instance,
if
Q,
(3
E
91,
one
can
ask
under
what
conditions
(3
±
Q,
(3
±
2Q,
...
belong
to
91
as
well. Such
an
analysis is
presented
for
example
by
-Racah.
At
this
point
,
however, we will switch
attention
to
the
real
simple
compact
Lie algebras.
11.3
The
Real
Compact
Simple
Lie
Algebras
To descend from
Cartan
's classification
of
complex simple Lie algebras
to
the
real
compact
simple ones, we need
to
depend
on
some
fundamental
theorems:
I:
Every
complex simple Lie
algebra
£
has
bases
in
which all
structure
constants
become real.
Thus
£
definitely
can
be
obtained
by
a process
of
com-
plexification
of
(several)
real
Lie
algebras
£,
£',
....
II. (Weyl):
Every
complex simple Lie
algebra
£
has
a
unique
compact
simple
real
form
£.
Compactness
here
means
that
the
Lie
groups
G
associated
with
£
are
compact.
11.3.
The
Real
Compact
Simple Lie
Algebras
181
III
(Weyl):
If
G is a (real)
compact
simple Lie group,
and
G' is locally
isomorphic
to
G,
then
G' is also
compact.
Thus
compactness
of a real simple
Lie
group
can
be
read
off from
its
Lie algebra, which
further
illuminates
Theorem
II
above.
We will now
state
various
properties
of (real)
compact
simple Lie algebras,
which
taken
together
with
the
results
already
obtained
in
the
complex case, will
permit
a complete classification
of
all possible CSLA's.
In
the
following,
£
will
denote
some
real
compact
simple Lie
algebra
of dimension
r
and
rank
l.
1.
There
exist bases for
£
(real,
of
course) in which
the
(real)
structure
constants
c'Ik
become
totally antisymmetric
and
can
be
written
as
Cjkm.
This
is because
the
real
metric
tensor
gjk
is positive definite
and
we
can
choose
the
basis so
that
it
becomes
the
unit
tensor:
gjk
=
6jk
c'Ik
=
Cjkm
(11.48)
Therefore
the
adjoint
representation
generators
Xj
are
i
times
real
antisymmet-
ric matrices, so
at
the
level
of
the
group
the
adjoint
representation
matrices
are
real
orthogonal.
Incidentally
the
adjoint
representation
is irreducible because
£
is simple.
2.
All
matrix
representations
of
£
are
by
hermitian
matrices;
in
particular,
any
irreducible
representation
is,
with
no
loss of generality,
by
finite dimensional
hermitian
matrices.
These
statements
of
hermiticity
refer of course
to
the
real
elements
of
12.
The
Cartan-Weyl
basis however
has
the
following behaviour:
H
t.=
H
Et
- E
a
a,
a -
-a
(11.49)
Thus,
while
the
Ha
are
indeed
real
elements
of
12,
the
Ea
are
complex
combinations
of
real
elements,
and
in
that
sense,
strictly
speaking,
they
are
not
elements
of
£
at
all.
They
are
analogous
to
the
raising
and
lowering
operators
in
angular
momentum
theory. Nevertheless we will use
them
in
our
analysis.
3.
Since
the
Ha
are
hermitian
and
mutually
commute,
in
any
UIR
(to
save
on
words we use
the
same
description
as
for a group)
they
can
all
be
simul-
taneously
diagonalised.
They
can
be
taken
as
part
of a
complete
commuting
set.
The
Ha
are
a
maximal
commuting
or
abelian
subalgebra
of
12;
it
is called
a
Cartan subalgebra.
4.
The
quantities
aa, aa
and
Na{3
are.all
real. Now from
the
expression
(11.39) for
gab,
we
can
see
that
by
subjecting
the
Ha
to
a
real
orthogonal
rotation
in
l
dimensions
and
then
rescaling
them,
we
can
arrange
for
gab
to
become 6
a
b.
We
can
also reduce each ga,-a
to
unity.
When
these
have
been
done,
the
Cartan-Weyl
form of
the
Lie
brackets
is
[Ha,Hb]
=
0,
[Ha,
Ea]
=
aaEc"
[Ea,
E-a]
=
aaHa,
[Ea,
E{3]
=
Na{3Ea+{3
(11.50)