Mukhi S., Mukunda N. Lectures on Advanced Mathematical Methods for Physicists
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152
Chapter
9. Lie
Groups
and
Lie Algebras
Given
G
and
a(er),
in
the
coordinate system
0:
let
a(er)
have coordinates
0:(
er):
a(er)
----+
o:j(er);
o:j(O)
=
0;
j(o:(er');
o:(er))
=
o:(er'
+
er)
N ow
take
er'
to
be
an
infinitesimal
15er,
and
also
write
Then
from Eq.(9.21),
.
do:
j
t
J
=
der
(er)lo-=o,
o:j
(15er)
~
t
j
15er
o:j(er
+
15er)
=
jj(o:(15er);
o:(er))
~
jj(t15er;
o:(er))
~
o:j(er)
+
1]~(o:(er))tk15er
(9.21 )
(9.22)
(9.23)
so
the
coordinates
of
the
elements
on
an
OPS
obey
the
(generally nonlinear)
ordinary
differential equations
and
boundary
conditions
do:
j
.
k
der
(er)
=
1]~(o:(er))t
,
o:j(O)
=
0
(9.24)
We say
t
j
are
the
components
of
the
tangent
vector
to
the
OPS
at
the
identity.
Can
the
above
be
reversed? Given
G,
the
1's,
1]'S
and
fs,
suppose
we choose
an
r-component
real vector t
j
,
and
solve
the
system
(9.24)
and
get functions
o:j(er).
Will
these
be
the
coordinates on
an
OPS
and
will its
tangent
at
the
identity
be
t?
The
answer is in
the
affirmative,
and
the
proof is similar
to
that
in
the
previous section. We wish
to
prove
that,
given Eqs.(9.24)
with
solution
o:j(er),
(9.25)
Here we regard
er
as active
and
er'
as implicit in
Land
R.
Then
eqs (9.24),(9.12)
give:
dLJ
.
der
(er)
=
n~(L(er))tk,
Lj(O)
=
o:j(er');
dRj .
der
(er)
=
(H(R(er))3(0:(er))H(0:(er)))1tk,
=
1]~(R(er))tk,
Rj(O)
=
o:j(er')
(9.26)
9.3.
One-parameter
Subgroups
and
Canonical
Coordinates
153
Since
both
the
differential
equations
and
boundary
conditions
are
the
same
for
Land
R,
they
coincide
and
the
equality
(9.25) is proved.
In
summary,
a given
OPS
determines
a
tangent
vector
t
uniquely
and
vice
versa,
based
on
the
differential
equations
(9.24).
So far we
have
allowed
considerable
flexibility
in
choice of
coordinate
sys-
tems
a for
G.
The
properties
of
OPS's
motivate
the
definition
of
a special family
of
coordinate
systems,
the
so called
canonical
coordinates
(of
the
first kind).
At
first,
let
us
denote
the
solution
to
Eqs.(9.24)
by
a(O';
t),
indicating
explicitly
the
dependence
on
the
tangent
vector
t:
there
are
r
+
1
real
arguments
here.
But
a
moment's
reflection shows
that
the
dependence
can
only
be
on
the
r
products
O'tl,
O't
2
,
...
,
O't
r
,
so we will
write
the
solution
to
Eqs. (9.24) as:
aj(O';t)
==
xj(O't)
==
r
functions
of
r
real
independent
arguments;
Xj(O)
=
0,
X
j
(toO')
~
t
j
DO'
+
0(00')2.
(9.27)
In
fact
the
differential
equations
(9.24) for
an
OPS
can
be
expressed
as
a
system
of
equations
obeyed
by
the
functions
x:
k
8
X
j
.
t
8tk (t)
=
1]~(X(t))tk
X(O)
=
0
8X
j
.
8tk (t)lt=o
=
O~
(9.28)
Thus,
given
G
and
a
coordinate
system
a
around
e, we
have
arrived
at
the
system
of
functions
X(t).
Now
the
Jacobian
property
of
X
above
means
that
if
any
element
a
E
1)1
(sufficiently small)
with
coordinates
a
j
is given, we
can
find
a
unique
vector
t
j
such
that
(9.29)
In
other
words: for
any
a
E
1)1,
there
is a
unique
tangent
vector
t
such
that
the
corresponding
OPS
passes
through
a
at
parameter
value
0'
=
1.
We
express
this
relationship
in
the
(at
this
point
somewhat
formal)
exponential notation:
a
with
coordinates
a
j
:
a
=
exp(t),
(9.30)
Thus
in
this
notation
the
elements
a(O')
on
an
OPS
with
tangent
vector
tare
exp(O't),
and
the
OPS
property
means
exp(O't)
exp(O"t)
=
exp((O'
+
O")t).
All
a
E
1)1
are
obtained
by
suitably
varying
t
in
exp(t).
In
particular,
e
=
exp(O),
(exp(t))-l
=
exp(
-t)
(9.31)
(9.32)
154
Chapter
9.
Lie
Groups
and
Lie Algebras
Can
we now exploit
these
results
to
pass from
the
initial
arbitrarily
chosen
coordinates
oj
to
a new
system
of
coordinates
in
which
not
only
are
t1
the
components
of
the
tangent
vector
to
the
OPS
passing
through
a given
a
E
l)1
at
(J
=
1,
but
also serve as
the
coordinates of
a?
Yes, because once
the
functions
X(t)
are
in
hand
, we
can
in principle
turn
the
equations
(9.33)
"inside out" (the
Jacobian
property
of
X
in
Eq. (9.28) ensures this),
and
express
t
in
terms
of
0:
t
j
=
(j
(0)
==
new coordinates
a
j
(9.34)
The
distinguishing characteristics of these new coordinates
a
for
l)1
c
G
are
that
on
an
OPS
a((J):
a(
(J)
~
old coordinates
oj
((J)
=
X
j
((Jt)
~
new coordinates
a
j
((J)
=
(Jt
j
(9
.35)
Such
coordinates
are
called canonical coordinates.
If
we were
to
write
/,
fj,
~
for
the
functions associated
with
them,
the
characterisation
(9.35) means
that
now
the
solution
to
Eqs.(9.24) in
the
new coordinates is very simple:
da
j
. .
d(J
((J)
=
fjk(a((J))tk, aJ(
O)
=
0
=}
a
j
(
(J)
=
(Jt
j
=}
fjk(a)a
k
=
a
j
,
i.e.,
H(a)a
=
a.
(9.36)
We
thus
have two direct ways
of
defining a canonical
coordinate
system: ei-
ther
(i)
the
functions
fj
must
have
the
functional
prop
e
rty
appearing
in Eq.(9.36)
or
(ii)
the
elements
on
an
OPS
have
coordinates
(Jt.
The
passage from a general
coordinate
system
0
to
the
canonical
system
a
uniquely associated with
it
has
the
feature
that
upto
linear
terms
they
agree,
and
(possibly) diverge only thereafter:
a
j
'::'
oj
+
0(0
2
),
oj
'::'
a
j
+
0(a
2
)
(9.37)
(In
particular,
if
0
were already canonical,
then
a
=
o!) We
can
say
that
while
there
are
"infinitely many" general
coordinate
systems for
l)1
c
G
re
lated
to
one
another
by functional transformations,
there
is a many-to-one passage from
general
to
canonical
coordinate
systems;
these
are
"far fewer"
but
still infinite
in
number
,
and
related
to
one
another
by linear transformations.
Of
course
the
set
of
canonical
coordinate
systems
appears
already as a subset
of
the
set
of
all
general
coordinate
systems;
and
all these relationships hinge
on
the
concept
and
properties
of
OPS's.
9.4.
Integrability
Conditions
and
Structure
Constants
9.4
Integrability
Conditions
and
Structure
Constants
155
Now we consider
the
conditions
on
the
ry's
and
Cs
which ensure
that
Eqs.(9.12)
for
1's
can
be
solved. Regarding
the
(3's
as fixed
param
eters,
these
integrability
conditions are:
o
0
p_
0 0
p
oa
m
oak - oak
oam
'
i.e.
o:m
(17f
(f)E~(a))
=
o~k
(171
(f)Ez",(a))
(9.38)
We
can
develop
both
sides using Eqs.(9.12) again,
and
then
rearrange
terms
so
that
all expressions evaluated
at
f
are
on
one side, those evaluated
at
a
on
the
other.
Partial
derivatives
are
indicated
by indices
after
a comma,
and
the
result
is:
Ez",
(f)
[17j
(f)17',;n
(f)
-
17'k
(f)17'!:n
(f)]
=
ryj(a)17'k(a)[Ez""n(a) -
E;',m(a)]
(9.39)
Since
the
two sides
are
evaluated
at
different
points
in
1)1
(remember
the
(3's!),
each side
must
be
a constant indexed
by
j,
k and
l.
Let us
write
-C;k for
these
constants:
then
the
int
egrability conditions for Eqs.(9.12)
are
that
there
must
be
these
constants
arid
Ez",(a)
[17j(a)ry',;n
(a) -
17'k
(a)17;:'n
(a)]
=
-C;k:
17j(a)17'k(a)[Ez""n(a)
-
E;',m(a)]
=
-C;k
(a)
(b) (9.40)
These two conditions are identical in content,
on
account
of
(17)
-1
being
(E).
It
is convenient
to
express
them
in
the
forms
17j(a)17',;n(a)
-
17'k(a)17'J~(a)
=
-c;k17l(a)
(a)
Ll~n(a)
==
~z""n(a)
-
E;',m(a)
+
c;kE!n(a)E~(a)
=
0 (b)
(9.41 )
An
"acceptable"
set
of functions
17(a)
must
obey
these
partial
differen-
tial
equations
for some numerical
constants
C;k;
if
they
do,
then
we
can
solve
Eqs.(9.12)
to
get
p
(a;
(3),
and
these
will
determine
the
group
structure
(at
least
locally) .
The
C;k
are
called
"struct
ure
constants" for
the
group
G.
What
are
accept-
able
sets
of
structure
constants?
They
cannot
be
chosen freely.
If
the
group
G
were given,
the
corresponding
C;k
can
be
calculated,
and
will
be
found (as we
shall see)
to
obey
certain
algebraic conditions.
In
section 9.6, we will see
that
all sets of
constants
obeying these algebraic conditions
are
acceptable.
We outline
in
sequence
the
properti
es
of
the
r3
constants
C;k'
which by
definition
are
real numbers, assuming
that
G is given:
156
Chapter
9.
Lie
Groups
and
Lie Algebras
(i)
Antisymmetry
in
the
subscripts: obvious from Eqs.(9.40):
(9.42)
(ii)
Taking
a
=
0 in Eq.(9.41a), we have
the
explicit expression
(9.43)
Therefore if we
expand
f(f3;
a)
in a Taylor series as
(9.44)
then,
(9.45)
(iii)
Relation
to
commutators
of
group
elements:
take
two elements
a,
b
E
l)1
both
close
to
e,
and
calculate
the
coordinates
of
the
commutator
q(a,
b)
(Eq.(7.16)):
retaining
leading
terms
only, we find:
c=q(a,b)
:
. . k I
,J
=
q,la
f3
+ ...
(9.46)
This
shows
that
if
G
is abelian, all
the
structure
constants
vanish;
the
converse
is also
true,
as we will find later.
(iv)
Jacobi condition:
we exploit
the
integrability condition (9.41a)
on
Ti
by multiplying
it
by two
tangent
vectors
tju
k
and
writing
terms
in
a suggestive
form:
(9.47)
From
experience
in
handling linear first
order
differential
operators
in
quantum
mechanics, we see
that
we
can
express this as a
commutation relation:
[
j
n(
) B
k
m(
)
B]
I
m(
)
B
t
Tij
a
Ban' u
Tik
a
Bam
=
-v
Til
a
Bam'
That
is, if for each
tangent
vector
t
we define
the
linear
operator
then
the
integrability conditions
on
Ti
are,
[Ti[t]
, Ti[u]l
==
Ti[t]Ti[u] - Ti[u]Ti[t]
=
Ti[v]
(9.48)
(9.49)
(9.50)
9.5. Definition
of
a (real) Lie Algebra: Lie
Algebra
of
a given Lie
Group
157
where
v
is
another tangent
vector formed
out
of
t
and
u
using
the
structure
constants:
v
=
[t,
u]
=
Lie bracket
of
t
with
u,
v
l
=
c;k
tjuk
(9.51 )
We have
introduced
here
the
notion
of
the
Lie bracket,
an
algebraic
operation
determining
a
tangent
vector
[t,
u]
once
t
and
u
are
given,
at
this
moment
to
be
distinguished from
the
commutator
bracket
occurring
in
Eq.(9.50).
But,
since
commutators
among
operators
do
obey
the
Jacobi
identity, Eqs.(9.50),(9.51)
lead
to
the
result
that
Lie brackets
must
also
obey
them! For
any
three
vectors
t,
u
and
w,
we have
L
[[l7[tl'
l7[ul],
l7[wl]
=
0
*
cyclic
L
[l7[t,ul,
l7[wl]
=
0
*
cyclic
L
l7[[t,u],wl
=
0
*
cyclic
[[t,
u],
w]
+
[[u,
w],
t]
+
[[w,
t],
u]
=
0
(9.52)
Here we have used
the
fact
that
l7[t]
depends linearly
on
t.
Directly
in
terms
of
the
structure
constants,
the
Jacobi
condition is
(9.53)
The
complete
set
of
conditions
on
structure
constants
which makes
them
ac-
ceptable
are: reality, antisymmetry, Eq.(9.42);
and
the
Jacobi
identities (9.53).
There are no more conditions.
All
these
properties
of
structure
constants,
re-
flected also
in
the
way
the
tangent
vector
v
is formed
out
of
t
and
u,
motivate
us
to
define
the
concept
of
the
Lie Algebra.
9.5
Definition
of
a (real) Lie
Algebra:
Lie
Algebra
of
a
given
Lie
Group
A (real) Lie
algebra
is a real linear vector space,
.c
say,
with
elements
u, v,
t,
w,
...
among
which a
Lie bracket
is defined.
This
is a rule which associates
with
any
two given elements
of
.c
a
third
element,
subject
to
certain
conditions:
(i)
u,v
E.c
*
[u,v]
E.c;
(ii)
[u,v]=-[v,u];
(iii)
(iv)
[u,
v]
linear in
both
u
and
v;
u,
v,
w
E
.c
*
[[u,
v],
w]
+
[[v,
w],
u]
+
[[w,
u],
v]
=
0
(9.54)
158
Chapter
9. Lie
Groups
and
Lie
Algebras
If
a basis
{ej},
j
=
1,2,
...
, r is chosen for
L,
the
dimension of
L
being
r,
then
because
of
the
linearity
property
(9.54) (iii),
u
=
ujej,v
=
vjej
=?
. k ]
[u,
v]
=
uJv
[ej, ek
. k l
=
uJv
Cjk
el
,
[ej,
ek]
=
C;ke1,
i.e.,
[u,
v]l
=
C;kujvk
(9.55)
For
these
structure
constants
C;k
of
L,
properties
(9.54) (ii),(iv))
then
im-
mediately
give
back
the
properties
(9.42),(9.53)
we
had
found for
the
structure
constants
of a Lie group!
We
thus
see
that
the
definition
of
a Lie
algebra
is designed
to
capture
in intrinsic form all
those
properties
that
are
known
to
be
possessed
by
the
structure
constants
of
any
Lie group, in
particular
the
process
by
which
in
Eqs.(9.50),(9.51),
any
two
tangent
vectors
determine
a
third.
We
can
now say:
given a Lie
group
0
of dimension
r,
the
tangent
vectors
u, v,
W,
t,
...
at
e
to
all possible
OPS's
in
0
taken
together
form
an
r-dimensional
Lie
algebra
de-
termined
by
O.
We call
it
the
Lie algebra
oj
O.
It
is sometimes
denoted
by
0,
sometimes
by
g.
Changing
the
coordinate
system
0:
over
1)1
c
G for
the
calculation
of
the
~tructure
constants
C;k
results
in a linear
transformation,
or
rather
a change of basis, in
O.
It
is
the
group
structure
for elements in
0
infinitesimally close
to
e
that
determines
the
bracket
structure
in
G.
Therefore
if we have two (or more)
Lie
groups
G,
0',
...
which
are
locally isomorphic
to
one
another
in
suitable
neighbourhoods
of
their
respective
identity
elements,
they
will all
lead
to,
and
so
they
will all share,
the
same
Lie
algebra
L.
One
can
thus
at
best
expect
that
a given Lie
algebra
can
determine
only
the
local
structure
of
any
of
the
Lie
groups
that
could have led
to
it. We discuss
this
reconstruction
process in
the
next
section.
9.6
Local
Reconstruction
of
Lie
Group
from
Lie
Algebra
Suppose a Lie
algebra
L,
that
is,
an
acceptable
set
of
structure
constants
C;k'
is
given.
Can
we
then
say
that
we
can
compute
the
functions
7)~
(o:)?
If
the
answer
were
in
the
affirmative,
then
the
analysis
in
Section 9.2 tells us
that
we
can
go
on
to
compute
j(o:;
(3)
too,
so from
the
structure
constants
we would have
built
up
a Lie
group
G
at
least
locally.
But
actually
we
kno'w
in
advance
that
the
7)~
(0:)
cannot
be
found if only
C;k
are
given! Suppose
you
already
had
a Lie
group
0
in
hand;
clearly
there
is infinite freedom
to
change
the
coordinate
system
on
1)1
C
0 leaving
the
coordinates
around
e
unchanged
to
first
order
only. Such changes
of
coordinates
9.6. Local
Reconstruction
of
Lie
Group
from Lie
Algebra
159
over
!)1
would definitely change
the
7]t(a)
but
not
the
e;k'
So unless
we
restrict
the
coordinate
system
in some way, we
cannot
hope
to
get
7]~
(a)
starting
from
I
ejk'
The
key is
to
use canonical
coordinates
of
the
first kind.
It
is possible, given
an
acceptable
set
of
structure
constants
e;k'
to
solve uniquely
the
combined
system
of
partial
differential equations
and
algebraic equations,
L1~n(a)
==
~~.n(a)
-
~~,m(a)
+
e;k~t,,(a)~~(a)
=
0,
~k(a)ak
=
a
j
,
~k(O)
=
ot
(9.56)
While
the
details
can
be
found in various references, here we simply describe
the
results. We define a real r x r
matrix
C(a)
by
(9.57)
the
superscript
(subscript) being
the
row (column) index.
Then
the
matrix
3(a)
is given as
an
infinite power series, in fact
an
entire function, in
C(
a):
.
(e
Z
-
1)
3(a)
==
(~Ua))
=
-z-
z=C(a)
00
=
1
+
2:)C(a))n
/(n
+
1)!
(9.58)
n=l
Because of
the
antisymmetry
property
of
the
structure
constants,
the
condition
reflecting
the
use of a canonical
coordinate
system
is satisfied.
It
is in
the
course of developing
this
solution (9.58)
that
one sees
that
no more conditions
need
to
be
imposed
on
structure
constants
beyond
antisymmetry
and
the
Jacobi
identities.
The
matrix
H(a)
inverse
to
3(a)
is again a power series which (in
general)
has
a finite
radius
of
convergence:
H(a)
==
(7]t(a))
=
(z/(e
Z
-
1))z=C(a)
=
(-~
+
~coth~)
2 2 2
z=C(a)
(9.59)
In
all
these
expressions,
the
basic ingredient is
the
matrix
C(a).
Its
intrinsic
property
is
that
acting
on
a vector
{3,
it
produces
the
Lie bracket
of
a
with
{3:
C(
a){3
=
[a,
(3J
(9.60)
Now
that
an
acceptable
set
of
functions
7]~
(a)
has
been
constructed
in
canonical coordinates,
it
is a
routine
matter
to
solve
the
partial
differential
equations (9.12),
and obtain
the
group
composition functions
f(a;
(3),
also in
canonical coordinates.
The
result is
the
so-called
Baker-Campbell-Hausdorff
160
Chapter
9.
Lie
Groups and
Lie
Algebras
series.
Remembering
that
we
are
using canonical coordinates, so
that
a
j
,
(3j,
•..
are
both
coordinates
for
group
elements
and
components
of
tangent
vectors, one
finds
1 1
f(a;
(3)
=
a
+
(3
+
"2
[a,
(3]
+
12
[a
-
(3,
[a,
(3]]
+ ...
(9.61 )
The
intrinsic,
coordinate
free form for
this
composition law for
G
reads:
u,v,
...
E.c
-7
exp(u),exp(v),
...
E
G:
1 1
exp(u)
exp(v)
=
exp(u
+
v
+
"2[u,
v]
+
12
[u
- v,
[u,
v]]
+ ... )
(9.62)
In
summary, we saw in sections 9.4
and
9.5
that
a given Lie
group
G
with
coordinates
a
over a neighbourhood
1)1
allows calculation
of
its
structure
constants,
and
thus
the
setting
up
of
its
Lie
algebra
G. Conversely
in
this section
we
have
indicated
how, given
the
structure
constants
and
working
in
a canonical
coordinate
system, we
can
build
up
~(a),
T/(a),
f(a;
(3)
for small enough
a,
(3,
and
thus
locally
reconstruct
the
group.
We find
in
both
forms (9.61 ),(9.62)
of
the
group
composition law
that
after
the
two leading
terms,
all higher
order
terms
involve (repeated) Lie brackets.
This
immediately establishes
the
converse
to
a
statement
we
made
in
section
9.4
(at
Eq.(9.46)): if
the
structure
constants
vanish,
the
group
is Abelian.
9.7
Comments
on
the
G
---+
G
Relationship
It
was
already
mentioned in section 9.5
that
the
Lie
algebra
G associated
with
a Lie
group
G is
determined
by
the
structure
of G "in
the
small" , i.e.,
near
the
identity
element. Therefore two (or more) locally isomorphic groups G,
G',
...
,
which all "look
the
same"
near
their
identities,
but
possibly differ globally, will
share,
or
possess,
the
same
Lie algebra: G
=
G'
= ....
The
global
properties
of
G do
not
show
up
in G
at
all.
Examples
familiar from nonrelativistic
and
relativistic
quantum
mechanics
are:
SU(2)
and
SO(3); SL(2,C)
and
SO(3,1).
If
then
some Lie
algebra.c
is given,
in
the
reconstruction
process, which
of
the
many
possible globally defined Lie
groups
can
be
uniquely singled
out,
as
the
result
of
the
reconstruction?
Of
all
the
Lie groups G,
G',
...
possessing
the
same
Lie
algebra
.c, only one is simply
connected,
and
it
is called
the
universal covering
group
of
all
the
others. For
instance,
SU(2)
(SL(2,C))
is
the
universal covering
group
of SO(3) (SO(3,1)).
Denoting
this
topologically distinguished
group
by
C,
we
can
say
that
.c leads
unambiguously
to
C.
In
other
words,
structure
near
e plus simple connectivity
fixes
C
completely.
All
of
the
general
group
theoretical
notions,
when
specialised
to
Lie groups,
in
turn
lead
to
corresponding notions for Lie algebras. While we look
at
many
of
them
in
the
next
section, we dispose
of
the
simplest one now.
This
is
the
question:
When
are
two Lie algebras
.c,.c'
to
be
regarded
as
"the
same"?
How
do we define
isomorphism
of
Lie algebras?
This
requires
the
existence
of
a one-
to-one,
onto
(hence invertible) linear
map
'P
:
.c
-7
.c' (so as real vector spaces,
9.8. Various
Kinds
of
and
Operations
with
Lie Algebras
161
.e
and
.e'
must
have
the
same
dimension)
compatible
with
the
two Lie
bracket
operations:
We
can
then
say:
x,y
E.e:
cp(x),cp(y)
E
.e';
cp([x,
Y]d
=
[cp(x),
cp(y)]e
G
and
G'
locally
isomorphic
{:}
G
and
G'
isomorphic
(9.63)
(9.64)
In
a similar way, homomorphisms
and
automorphisms
for Lie algebras
can
easily
be
defined - we leave
these
as exercises for
the
reader.
9.8
Various
Kinds
of
and
Operations
with
Lie
Algebras
We now briefly go over
the
notions
of
subalgebras, invariant, solvable,
etc
as
applied
to
Lie algebras.
(i)
A
Lie
algebra
.e
is
abelian
or
commutative
if all Lie brackets vanish:
[x,
y]
=
0
for all
x,
y
E
.e
(9.65)
(ii)
A
subset
.e'
c
.e
is a
subalgebm
if,
firstly,
it
is a linear subspace
of
.e
viewed as a real linear vector space,
and
secondly,
it
is a Lie
algebra
in its own
right:
x,y
E
.e',a,b
E
R
=}
ax
+
by
E
.e',
[x,y]
E.e'
(9.66)
It
is a
proper
subalgebra, if
it
is neither
.e
nor
the
zero vector alone.
(iii)
A
sub
algebra
.e'
c
.e
is
an
invariant
or
a
normal
subalgebra
(also called
an
ideal),
if
x
E
.e',y
E.e
=}
[x,y]
E.e'
(9.67)
This
can
also
be
suggestively expressed as
[.e,
.e']
c
.e'
(9.68)
In
such a case, if we
project
.e
as a vector space
with
respect
to
.c',
and
go
to
the
quotient
.ej.e'
which is a vector space,
it
is also a Lie algebra.
It
is
called
the
factor algebra. (Verify this!).
(iv) Given a Lie
algebra
.e,
its
commutator
subalgebra
.e
1
consists
of
all
(real) linear combinations of Lie brackets
[x,
y]
for all choices
of
x,
y
in
.e.
It
is
also called
the
derived
algebra
of
.e,
and
can
be
suggestively
written
as
(9.69)
You
can
easily verify
that
.e
1
is
an
invariant
sub
algebra
of
.e,
and
also
that
.ej.e
1
is
an
abelian
subalgebra.