3. Lie Derivatives And Lie Groups
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
3.18
Introduction: How A Vector Field Maps A Manifold Into Itself
Lie Dragging A Function
Lie Dragging A Vector Field
Lie Derivatives
Lie Derivative Of A One-form
Submanifolds
Frobenius' Theorem (Vector Field Version)
Proof Of Frobenius' Theorem
An Example: The Generators Of S2
Invariance
Killing Vector Fields
Killing Vectors And Conserved Quantities In Particle Dynamics
Axial Symmetry
Abstract Lie Groups
Examples Of Lie Groups
Lie Algebras And Their Groups
Realizations And Representations
Spherical Symmetry, Spherical Harmonics And Representations
Of The Rotation Group
3.1. Introduction: How A Vector Field Maps A Manifold Into Itself
Congruence (flow):
Set of non-crossing curves that fill a part of M.
These curves are usually solutions of some vector field V.
Each point in M where V is non-singular (V0) is on 1 & only 1 curve.
dim(congruence) = dim(M) – 1
Lie derivative: Derivative along the congruence of a vector field V.
Lie dragging:
Moving each p in M by an amount Δλ along the
congruence is an auto-diffeomorphism.
These draggings form a 1-parameter Lie group.
See Choquet, pp.121 & 138.
Given
y  f  x
f : X  Y by x
one can define 2 related mappings:
f* : TP  X   T f  P  Y  by V
1. Push-forward (differential) of f :
f*V  g   V  g f 
s.t.  g : Y → R,
Given a coord basis:
V V j

x j
f*V  V j
y 
 x j  yk
k
s.t.
f * V     f* V 
x
   j dy
T f  P  Y 
TP*  X 
f


T f* P  Y 
X
f


Y
*
g
f * : T f* P  Y   TP*  X  by 
y
yj k
f    j k dx   j y j , k dx k
x
*
f*


R
Given a coord basis:
j
TP  X 
g f
f* is also written as f , D f, or d f .
2. Pull-back (reciprocal image) of f :
W  f*V
f *
f *g  g f
f*V  g  y  V  f * g 
x
f *h  h f  1
f *g  g f
f*V  g  y  V  g f  x
f * V     f* V 
x
y
f *  f*V  g    V  f * g 
See Choquet, p.144
Let σ(t,x0) be the integral curve of
i.e.,
V 
d  i  t, x 
 V i   t , x  
dt
  t ,   s, x      t  s, x 
d
dt
that passes through x0 at t = 0.
  0, x   x
with
since both satisfy the same differential eqs & same I.C.
  0,   0, x      0, x   x
Define map
 t    t,   : X  X
by
x
 t  x     t, x 
σt is called the local transformation generated by vector field V.
(c.f. Schutz’s Lie dragging. )
  t ,   s, x      t  s, x 
→
 t  s   ts
 The set of all σt is a 1-parameter transformation group.
It’s local if the range of t is not all of R.
In which case, it becomes a pseudo group since closure is violated.
3.2.
Lie Dragging A Function
Given function f : M → R
and vector field
V 
d
T M 
d
with integral curve σ(λ )
Lie-dragging f along V by Δλ gives another function
s.t.
f     P    f  P 
i.e.
f   x   f  1  x  
fΔλ is just the push-forward :
f     * f  f  1
or the pull-back :
*
1

f

f    
f


f is “Lie dragged” by V if fΔλ = f.
f  : M  R
3.3.
Given vector field
Lie Dragging A Vector Field
d
V 
T M 
d
with integral curve σ(λ )
Lie dragging a vector field W along V
= Operating on W by the push-forward of the local transformation.
  * W
     ,  
3.4. Lie Derivatives
Lie derivative = Derivative along the congruence of a vector field.
Definition: Lie derivative of a function
Given function f : M → R
and vector field
V 
d
T M 
d
 * f  x   f  x 
L V f  lim
  0
x

f   x   f  x 
 lim
  0

with integral curves σ(λ, x )
 1 * f  x   f  x 
 lim
  0

 lim
  0
f  x        f  x   
 lim
  0

df
V  f  x

d x
f    x    f  x 

LV f  V  f 
Definition: Lie derivative of a vector field W along vector field V


1
L V W  lim  t*1 W  t  x    W  x  
t0 t
x
Setting
y  t  x
→
yi
1
 lim  t*1W  W  x  
x

t0 t 
xi  t V i  x  
x j
x j 
i
 W  y    W  y  i  j  W  y  i
y
 y x j
 i
 j
W i  x 
V j  x 
k
   i  t

W  x   tV  x 
k
i
x
y


1
t*
i

j

 j
 k
W j  x 
V j  x   
i
2
 W  x   t V  x 

W
x


O
t





 j
k
i

x

x

 

 W  x   t V  x W  x   W  x V  x    O  t 2 
where the multiplication V W
VW
Also
is defined by V W  f   V W  f    C2 function f.
is not a vector field since the Leibniz rule is not obeyed.
VW  V W
V j  x  V j  x 

 O t 
 yi
 xi
L V W  VW  WV  V , W 
Properties of the Lie derivative (see Ex 3.1-3)
L V , L W   L  


V , W 
LV  W  LV  LW
  L , L  , L     L , L  , L    L , L  , L   0
Y
Z
Z
X
X
Y
 X
 Y
 Z



L V  A  B  L V A  B  A  L V B
L V W  V iW j , i  W iV j , i



j

L V W  V i ei W j   W i ei V j  e j  V iW j L ei e j
L i W  W
j
,i
j
W
W j



j
 xi
 xi
L V i  L i V   V
 xi
V j
  i j
x
( Jacobi’s identity )
( Leibniz rule: L is a derivation )
( Coordinate basis )
( General basis )
( Coordinate-free partial )
3.5.
Lie Derivative of a 1-form
see Choquet, p.148.
Definition: Lie derivative of a 1-form field ω along vector field V


1
L V   lim  t*   t  x      x  
t0 t
x
Exercise: show that

L V   L V i dx
i
V i j
L V dx 
dx
j
x
i
  L   dx
V
i
i i
V i j
V
dx  i j dx
j
x
x
j
i
c.f.
 
 i L V dx
i
V i j
 V i  dx  i j dx
x
i
 i  j
V i  j
 V
 i j  dx
i
x 
 x
The above can also be derived from
V j
L V i   i  j
x

L V  W    L V 
( c.f. Ex 3.14 )
 W    L W 
Lie derivatives of tensors can be calculated using the Leibniz rule.
V
d is an anti-derivation that changes a (mn) tensor into a (mn+1 ) tensor.
L is a derivation that leaves the rank of a tensor unchanged.
Basic formulae:
L V i  V
j
LV d  d LV
,i
L V dx i  V i , j dx j
j
( proof: operate both sides in components on f )
Example:


i
i
L V W  L V W i i  L V W  i  W L V  i
 V j W i , j  W j V i , j  i
 V j W i , j i  W iV j , i  j
3.6.
Submanifolds
A submanifold S of a manifold M is a subset of M that is also a manifold.
S is an s-D embedded submanifold of an m-D manifold M if there exists an
atlas of M s.t. in every chart, the coordinates of S are of the form
x ,
1
, x s , x s 1  0,
, xm  0

See Frankel, § 1.3d.
or, more generally,
x ,
1
, x s , y1 ,
, y ms

where the y j s are functions of the xk s.
Note: all solutions y to some set of equations involving x are of this form.
Consider a point pS.
A curve in S through p is also a curve in M.
→ A (tangent) vector in Tp(S) is also a vector in Tp(M).
Indeed, Tp(S) is a subspace of Tp(M) .
 A vector in Tp(M) is not necessarily a vector in Tp(S).
A 1-form in Tp*(M) is also a 1-form in Tp*(S) since it can map any vector in
Tp(S) to R by treating it as a vector in Tp(M).
A 1-form in Tp*(S) is not necessarily a 1-form in Tp*(M).
3.7.
Frobenius' Theorem (Vector Field Version)
On a smooth manifold M, the order of partial derivatives is irrelevant.
 i ,  j   0
since
 f, i j  0
 f C   M 
 Coordinate basis vector fields commute, i.e.,
 i ,  j  f  i   j f    j  i f   f , j i
C(M) is the ring of all C functions on M.
It is not a field because f 1 may not exist.
The set L(M) of vector fields on M is a vector space over the field K because it
is closed under linear combinations of constant coefficients.
If the coefficients of linear combinations are C functions, L(M) becomes
a module over the ring C(M).
If L(M) is closed under the Lie bracket, it becomes a Lie algebra.
A set of vector fields are linearly independent if they are lin. indep. at each point.
A set of m linearly independent, mutually commuting, vector fields can be taken
as a coordinate basis using the parametrization as coordinates.
By definition, their integral curves mesh to form a foliation (family of
submanifolds). Each (m1)-D submanifold, called a leaf of the foliation, is
specified by a fixed value of the mth coordinate.
Frobenius theorem generalize this to the case of a set of vector fields which spans
a module over C(M) that is closed under the Lie bracket.
Lemma : The set of all linear combinations (non-constant coefficients allowed)
of a set of mutually commuting vector fields is closed under the Lie bracket.
Alternative phrasing: the module over C(M) spanned by a set of mutually
commuting vector fields is closed under the Lie bracket.
Vi  , V j    0


Wa  , Wb   abi  x  Vi 

 i
Mathematical terms: if
then
 i, j  1,
,m
 W a   W ai   x  Vi 
i
Proof: See Ex 3.5.
Setting Vi   i
→ vector fields on a submanifold are closed under the Lie bracket.
Frobenius theorem:
Let L(U) be the set of all linear combinations ( with possibly non-constant
coefficients ) of a set of vector fields on a region U of a manifold M.
[ i.e, L(U) is a module of vector fields on UM over ring C(U). ]
If L(U) is closed under the Lie bracket, then the integral curves of the vector
fields mesh to form a foliation.
Proof: see §3.8.
The original version of the theorem was described in terms of differential
forms (see §4.26) and dealed with the integrability conditions of Pfaffian
systems (existence of solutions of system of partial differential equations).
Examples:
Fig.3.7
3.8
Proof of Frobenius' Theorem
Ancillary relations:
 

LV d f  d LV f
  V f 
i
,i , j

dx j

LV d f , W  LV d f W   LV d f , W  d f , LV W
d f , V , W   d f , L V W

 V i  f, j W
j

,i

 LV d f , W  d LV f , W
Strategy of proof:
Since the theorem is satisfied if the vector fields commute, we need only show
that a set of m linearly independent, mutually commuting, vector fields can be
constructed out of m lin. indep. fields that are closed under the Lie bracket.
This will be done by induction.
The case m = 1 is trivial since foliation = curve = leaf.
Assuming case m1 is valid, we shall prove that case m also holds.

Let the m lin. indep. fields be
Let
s.t.
→
V m  
V a  ; a  1,
,m

m
with
c
V a  , Vb    ab
V c 

 c 1
m 1
d
and
d  m 
X i    ji  V j 
for i = 1, … , m1
j 1
d  m , X i   0
 i  1,
,m 1
m 1
m
 X i  , X  j     V a     ikj X  k    i j V m 

 a1
k 1
a
ij
m 1
m
k
V m  , X i      V a     mi
X  k    m j V m 

 a 1
k 1
a
mi
Next, we show that
d m ,  X i  , X  j    0


 d m , Vm , X  j  




d f , V , W   LV d f , W  d LV f , W
Since
d  m  , X i   0
 

LX i  d m  d LX i  m
and



d  m  ,  X i  , X  j    LX i  d  m  , X  j   d LX i   m  , X  j 
we have
d m , Vm , X i     d m ,  X i  , Vm 




Similarly,


 LX i  d  m  ,V m   d LX i   m  ,V m 
where
 d  m 
d
0
 d i  


d  m  , Vm   d m  ,
→
d
d  m 

d  m 
d  m 
1
LX i  d  m  ,V m   LX i  1  0
0
0
m 1
 X i  , X  j      ikj X  k    i j V m 

 k 1
→
0   i j d  m  , V m 
→
0   m j d  m  , V m    m j
m 1
k
V m  , X i      mi
X  k    m j V m 

 k 1
m 1
i.e.,
 X i  , X  j     

 k 1
 i j
m 1
k
ij
V m  , X i      mk i X  k 

 k 1
X k 
 {X(i)} is a set of m1 lin. indep. vector fields closed under the Lie bracket.
Invoking the (m1) case of the theorem, {X(i)} can be transformed into a set
of m1 of lin. indep. mutually commuting vector fields {Y(i)} whose integral
curves mesh into a (m1)D foliation S.
Next, we extend {Y(i)} into the entire U by requiring it to be invariant under the
group transformation generated by V(m) ( i.e., Lie dragging along V(m) ).
Calling the resultant fields {Z(i)}, this means
with
Since
LV m Z i   V m  , Z i    0


f* V , W    f*V , f*W 
 Yi 
on S 
Z i   
 m * Yi  outside S 
we also have
 Z i  , Z  j    0


QED
An Example: The Generators of S2
3.9.
e   y ex  xe y
In general



 lz
 y
x

x
y
~
li   i j k x j

 xk
( ang mom op )
( implied sum over
repeated indices )
  i j k x j k
 li , l j    i k l  j mn  x k l , x m n 

  i k l  j mn x k  l x m  n  x m  n x k  l  x k x m  l , n 

  i k l  j mn  x k lm n  x m nk l 
  i k l  jl n x k n  i k l  j mk x ml
  i k l  j l n   i l n  j k l  x k n
   i j k n   i n k j   i j n k   i k n j  x k n
  i n k j   i k n j  x k n
 Integral curves of
 x j i  xi  j
 l ; i  x, y , z 
i
  i j k lk
form a foliation with spherical leaves.
li  r    i j k
r
x
 xk
j
 i jk
 xk 
x    0
 r 
j
 dr  li 
→ every li is tangent to the constant r surfaces.
Only 2 of them are linearly independent at each point.
Each leaf in the foliation is just S2 .
Comment: Angular momenta are generators of rotations.
Thus, they are antisymmetric tensors of rank 2.
Only in 3-D space are they equivalent to vectors.
3.10. Invariance
Principal use of Lie derivatives: Invariants of tensor fields
→ Symmetries, Lie groups …
A tensor T is invariant under a vector field V if
LV T  0
Let F = { T(1) , T(2) , … } be a set of tensor fields.
Then the set A of all vector fields under which F is invariant is a Lie algebra.
Proof:
Since
LV  W  LV  L W
and
La V  a LV
we have La V  b W Ti   aLV Ti   bL W Ti   0
 aV  bW  A
From L V , W   LV , L W 
 V ,W   A
for aR .
 V ,W  A & a , b  R
i.e., A is a vector space.
we have
L  V , W  Ti   LV , LW  Ti   0


i.e., A is a Lie algebra.
 V ,W  A
Caution: A is a vector space only under combinations with constant coefficients.
Example:
l 
is linearly dependent in R3.
i
i.e.,
c  x l
i
i
0
can be satisfied with some ci (x)  0
i
→ submanifold generated by { li } is 2-D.
However,
l 
i
is linearly independent if only constant ci are allowed.
→ Lie algebra generated by { li } is 3-D.
3.11.
Killing Vector Fields
V is a Killing vector field if the metric tensor g is invariant wrt it, i.e.,

LV g  0

LV g  LV gi j dx i  dx j  V k  k gi j dx i  dx j  gi j  kV i dx k  dx j  gi j  kV j dx i  dx k
 V k  k gi j  g k j  i V k  gi k  j V k
 V k gi j , k  g k j V k , i  gi k V k , j
For V    
m
 xm
 dx
 dx
i
 dx j
i
 dx j
i
i
we have V   m
&
LV g  gi j , m dxi  dx j
 j is a Killing vector  g is independent of xj
E3, Cartesian coordinates:
E3, spherical coordinates:
Killing vectors:
g i j  i j
Killing vectors:
x ,  y , z
g  diag   r   r ,    ,      diag  1 , r 2 , r 2 sin  
  lz
Also:
lx , ly
3.12. Killing Vectors & Conserved Quantities in Particle Dynamics
Classical mechanics H = T + Φ :
Φ indep of x → px conserved
Not so for other coordinate systems
Φ indep of φ → pφ conserved
Reason: Conserved momentum must be Killing vector field of configuration space.
g is involved because the tensor form of the Newton’s equation is
ij
m x i   i    g

x j
Further discussions deferred to Chapter 5.
3.13. Axial Symmetry
Axial symmetry = invariance about an axis
Cylindrical symmetry = axial symmetry + translational symmetry along axis
L  0
Eq of motion:
L = linear operator invar under φ→ φ+ a
Caution: A solution  is not necessarily axially symmetric,
e.g., if  is the path of a particle.
  , x
Fourier analysis:
L 

 L 
m  
→
m
e
im 


j
 
m  

 m  x j  e im 

 L
m  
m
( for  a field )
m  ei m  0
L m  m  e  i m  L  m e i m  
m
Example:
1  2 
1


1
2
L  2
r
 2
sin 
 2 2
r  r  r r sin  
 r sin   2
2
Lm  m   e i m  L  m e i m  
~
2m f m  e i m  2  f m  r,   e i m  
1  2 
1


m2 
 2
r
 2
sin 
 2 2  fm
 r sin  
 r  r  r r sin  
 has axial eigenvalue m if
If  is a scalar, then
L e  i m 
Le  
i m
→  C e
= scalar axial harmonics
For axis = z, S = x-z
half-plane with x  0.
Let S be the submanifold (with boundary) φ= 0.
 e    e , e  be a 2-D basis for TP(S) at each pS.
→ e ,e 
is a basis for TP(M R3) at pS.
Lie dragging  e , e  around the axis of symmetry generates a basis  pM.
Let
j

1
2
j

j
Let σ(φ) be an integral curve of
e
The basis at a point with φ= φ0 is
e
 
0
, ej
0
e ,  0 * e j
0 * 

L e e j   e , e j   0
Since
we have
i.e.,

e
 
L e e j
, ej





   * L e e j  0
L e e
are all axially symmetric.
Note: The Cartesian components of
e
 
, ej



   *L e e  0
are functions of φ.
L e e j    L e e    0

means
 e   , e   

j
have axial eigenvalues 0.

e m    , e m  j     e i m  e   , e i m  e j   
e    e
are vector axial harmonics with axial eigenvalues m.
i.e.,
L e e m  j    L e  e i m  e j       e i m   e j    e i m  L e e j  
 i m e i m  e j    i m e m j  
Thus, the solution to
L eV  i m V
can be written as
V  a j em j    b em   
( sum over j implied )
where a1 , a2 and b are independent of φ.
The group of transformation (Lie draggings) { σφ } is
the 1-parameter Lie group SO(2).
Spherical symmetry leads to SO(3).

3.14. Abstract Lie Groups
Choquet, pp.116, 153.
A Lie group G is a group that is also a differentiable manifold s.t.
the group & differential structures are compatible.
G G  G
i.e.,
by
 x, y 
x y 1
is a differentiable map
The set { σg } is a Lie group of transformations on manifold X if the map
 :G X  X
by
 g, x 
  g, x 
is differentiable and if the set of transformations

g
: X  X ;  g  x     g, x 

is a group wrt composition, s.t.
σe is the identity transformation
with
or
 g h   gh
 g h  hg
( Left action of G on X )
→
( Right action of G on X )
 g 1   g
1
Left translation by g :
Lg : G  G
Right translation by g :
Rg : G  G
by
by
h
h
Lg  h   g h
Rg  h   h g
For X = G, the differential (push forward) of the translations are
Lg*  h  : Th G   Tg h G 
Rg*  h  : Th G   Th g G 
( Lg*(e) = Schutz’s Lg )
Left translation by g near e
A vector field V on G is left invariant if it is invariant under left translations, i.e.,
Lg*V  h   V  Lg h 
 V  gh 
 g, h  G
Note: V(h)  V|h denotes the value of V at hG.
Setting h = e gives
Lg*  V  g 
In local coordinates:
 L V h
i
g*
L  
i
For h = e :
g*
 gh 
j
 V h
h j
j   gh 

h j
  V  e
where
i
i
Left invar.

V i  gh 
V i g
he
Let σ(t , g ) be the integral curve of V that passes through g for t = 0.
The transformations σt = σ( t,  ) form an Abelian group A with group
multiplication σt  σs = σt+s .
Setting σ(t , e ) = g ( t ) turns the curve into a 1-parameter Abelian
subgroup of G isomorphic to A with group multiplication
 t  s   t s
~
Lg t  g  s   g  t  g  s   g  t  s 
g(0) = e.
The subgroup { g ( t ) =σ(t , e ) } is entirely specified by
  V  e
It is sometimes denoted by { gγ ( t ) }.
The integral curve σ(t , h ) is the left coset h { g (t ) } of { g (t ) } wrt h.
By definition:
→
 t  h     t, h   exp  t V   h 
g  t    t  e   exp  t   e 
( Exponentiation always generates a left (or
right) translation along σ(t) )
For each vector γ in Te(G), one can construct a left invariant field V & hence
a unique 1-parameter subgroup { gγ ( t ) }.
Every element on the path connected submanifold of G containing e belongs
to one of these 1-parameter subgroups.
Restriction to invariant fields guarantees manifold::group compatibility.
Since
f* V ,W    f* V , f* W 
 Lg* V ,W   h    Lg* V , Lg* W   h 
 V , W   gh 
if V, W are left invar fields
→ [ V, W ] is also a left-invariant field.
→ Left-invariant fields are closed under Lie bracket.
Lg*  aV  bW   h   a Lg* V  h   b Lg* W  h 
  aV  bW   gh 
 a, b  R
if V, W are left invar fields
→ Left (or right) invariant vector fields form a vector space over R.
 V , U  , W    U , W  , V    W , V  ,U   0
automatically
satisfied
→ Left (or right) invariant vector fields form a Lie algebra L(G).
Since each left invariant field is specified by its value at e,
L(G) is isomorphic to G , the Lie algebra of G on Te(G).
Let
 V  ; i  1,
i
,n

be a basis of G or L(G).
G or L(G) is specified by the structual constants cki j given by
 Vi  , V j    c ikj Vk 


C  c ikj 
is a (12) tensor.
Every Lie group has a unique structure tensor but not vice versa.
3.15. Examples of Lie Groups
(i) Rn
It is a manifold under the usual (n-balls) topology.
It is an Abelian Lie group under vector addition.
The 1-parameter subgroups are rays ( straight lines through 0 ).
The congruence of a left invariant field consists of lines parallel to a given ray.
A left invariant field is therefore a constant vector field: V(x) = V(0) = γ.
→ All Lie brackets of left invariant fields vanish.
The Lie algebra is Abelian.
(ii) GL(n,R) = Group of all nn invertible real matrices
Group operation = Matrix multiplication.
e = Unit matrix.
g1 = Inverse matrix.
It is a differentiable manifold (Lie group) because it is a submanifold of R
n2
Coordinates: n2 matrix elements.
Group manifold = R n \ points giving singular matrices
2
Tangent space = R n
2
1-parameter subgroup:
where generator
is isomorphic to the space of all nn real matrices.
g A  t   exp  t A  g A  0   exp  t A  e  exp  t A 
d g A t 
A
dt
R
n2
t 0
Other integral curves of the left-invariant vector field can be obtained by left
translate (multiply) { gA(t) } by some group element h  { gA(t) }.
Matrices with negative determinants don’t belong to any 1-paramter subgroup.
→ GL(n,R) is a disconnected group.
Elements path-connected to e form the component of the identity, which is just
the subgroup SL(n,R).
The Lie bracket of the left-invariant fields is defined as
 g A  t  , g B  t    g A  t   g B  t   g B  t   g A  t 
where · is the group (matrix) multiplication.
 the Lie bracket for the Lie algebra can be written as
 A , B 
e
~ AB  BA
where A is the matrix version of the vector A|e.
(iii) O(n), SO(n) = Groups of all nn orthogonal real matrices ( S means det = +1)
O(n) is disconnected.
SO(n) = component of identity of O(n).
A  GL(n,R) can be taken as a (11) tensor on Rn.
A maps a (column) vector in Rn to another vector in Rn.
Similarity transform B1AB is a basis transformation ei → B1 ei on A.
The canonical form of A  SO(n) is block diagonal with blocks of
1,  1, or
 cos 
  sin 

sin  
cos  
(see Ex.3.14)
where the number of 1 blocks must be even.
→ Every A  SO(n) represents a rotation in some 2-D plane.
SO(n) = group of rotations.
The other component of O(n) represents inversions that change
the handedness of basis.
Convention: Algebras of subroups of GL(n,K) are denoted by lower case letters.
A etB
Let
then
1
A A  e
T
t B
e
t BT
 BT   B
The Lie algebra o(n) of group O(n) consists of all anti-symmetric nn matrices.
The anti-symmetric condition gives n diagonal & n(n1)/2 off-diagonal relations.
→ Dimensions of both o(n) & so(n) are
n2  n 
1
1
n  n  1  n  n  1
2
2
The basis of so(3) is
0 0 0 
L1   0 0 1
0 1 0 


L 
or
i
jk
  i j k
 0 1 0 
L3   1 0 0 
 0 0 0


 0 0 1
L2   0 0 0 
 1 0 0 


so that
L L 
i
j mn
  i mk  j k n   i n m j   i j mn
 Li , L j    i n m j   j n mi   i j k  k n m   i j k  Lk 
mn
mn
L 
2
i
mn
 i n mi   i i  mn


  1   i  n   mn
 Li , L j    i j k Lk
(iv) SU(n) = Group of all nn unitary matrices with unit determinants
SU(n) is a subgroup of GL(n,C).
SU(n) is the component of identity of U(n).
Let
U  etA
then

U 1  U   e t A  e t A
 A   A
 u(n) consists of all nn anti-Hermitian matrices.
The anti-Hermitian condition imposes n(n1)/2 off-diagonal complex,
& n imaginary diagonal, conditions.
→ (Real) dimension of u(n) is
2n2  n  n  1  n  n 2
The (real) dimension of group U(n) is n2.
 su(n) consists of all nn anti-Hermitian traceless matrices.
The (real) dimension of SU(n) is n21.
3.16. Lie Algebras & Their Groups
Every Lie group G has its Lie Algebra G.
Every element g of G is on one of the integral curves of a left invariant field
specified by a vector in Te(G).
If g is not a member of a 1-par subgroup, the integral curve that passes through
it can be obtained by a left or right translation of some subgroup.
Hence, not all Lie groups can be obtained solely from their Lie algebras.
Definition: Lie Algebra
A Lie algebra is a vector space endowed a bilinear Lie bracket s.t.
 A , B     B , A 
  A , B  , C     B , C  , A   C , A , B   0

 
 

Example:
 A , B   A  B
 R3
M is disconnected if  2 disjoint open sets A1 & A2 s.t. A1  A2 = M.
M is connected   & M are the only sets that are both open & closed.
M is locally connected if every neighborhood of every point contains a
connected neighborhood.
( M, π ) is a covering space of N if (see Choquet, p.19)
 M is connected & locally connected.
 π : M → N is onto & continuous.
 π1 : N → M is a multi-valued homeomorphism.
Which means  pN,  neighborhood N(p) s.t. the restriction of
π to each connected component Cα of π1 (N(p)) is a homeomorphism.
Schutz’s notation: N is covered by M
M is simply connected if every closed curve can be shrunk continuously to a point.
Example: R is simply connected.
Example: S1 is covered by R
 : R  S1 by x
  x    cos x, sin x 
C   2 , 2   1  
S1 is multiply-connected.
Sn is simply connected for n  2.
Theorem:
1. Every Lie algebra is the Lie algebra of 1 & only 1 simply-connected Lie group.
2. Every Lie group with the same Lie algebra is covered by the simply-connected one.
SU(2) is simply connected
Proof:
→
Let
b 
 a
H  set of all matrices of the form 


b
*
a
*


det A  a  b
2
 A H
2
SU(2) = H with
where
H\{O} is a Lie subgroup of GL(n,C)
a  b 1
2
2
 0 0
O

0
0


H is a 4-D real vector space with basis
 1 0
0  I  

0 1
Thus
→
0 1
1  

 1 0
 0 i 
2  

i 0 
3
b 
 a
 0  3



A
  i i 

b
*
a
*

 i0
 1  i2


SU  2    A






1


i0


1
3  
0
1  i2 
0  3 
0
1
αi R
3
2
i
 SU(2) is simply connected
= S3 in 4-D α-space
i.e., every element of SU(2) can be
generated by exponentiation from e
The generators Jk of a unitary matrix must be anti-hermitian.
The Pauli matrix σk are hermitian → Jk  i σk
 i ,  j   2i
The Pauli matrices satisfy
i
Jk  k
2
Setting
→
 J i , J j  
3

k 1
i jk
k
 i, j  1, 2,3
3
i j k J k
Note: J2 in eq(3.65),
Schutz, is wrong.
k 1
( The structure constants should be real since the vector space & algebra are real )
3
A vector in Te(SU(2)) can be written as
   ti J i
→ left invar. fields
i 1
g  t   exp  t 
Elements of a 1-parameter subgroup of SU(2) are given by
E.g.
g J k  t   exp  t J k  



n 0


n 0
n

tn
n
 J k    1  i t   k n
n!
n  0 n!  2 
n
2 n 1
   n  t 2 n


t

  I i
  k
 2n  1!  2 
  2n  !  2 




 I cos
 k2  I
t
t
 i  k sin
2
2
L 
The generators for SO(3) are (see §3.15):
L 
2
i
3
i

mn
mn
  i j k


  1   i  n   mn   I with the i, i  th element set to 0


 1   i  k   mk  i k n
Li 2 n 1    
n 1
g Lk  t   exp  tLk   I  
n1

I
n1
  i mn
→
Li 2 n    
Li

Li 3   Li
n 1
n 1
    n 1 2 n 1

  t 2n L 2
t Lk 

k
2
n

1
!
2
n
!





0
1
g L1  t    0 cos t
 0 sin t

n = 1,2,…
Li 2
tn
n
L
 k
n!
 I  Lk sin t  Lk 2 cos t  Lk 2
E.g.
jk
 Li , L j    i j k Lk
so that
L 
i
0 
 sin t 
cos t 



Setting
 : SU  2  SO  3
by
gJk t 
  gJ t   gL t 
k
k
→ (SU(2) , π) is a covering space of SO(3).
The domain of
gJk t 
is t  [ 0, 4 π). That of
g Lk  t 
is t  [ 0, 2 π).
Restricting π to either t  [ 0, 2 π) or t  [ 2 π, 4 π)
makes it a homeomorphism.
→ (SU(2) , π) is a double covering space of SO(3).
i.e., SO(3) is doubly covered by SU(2).
The projection of S3 onto the 3-D space of coordinates
( α1 , α2 , α3 ) is the closed ball of radius 1.
( c.f. projection
of S2 onto R2. )
Only the outermost sphere with = 1 can be taken to represent actual points on S3.
All other spheres with r < 1 are projections.
2
2
All points on a sphere have the same coordinate  0  1  r
Every point in the ball can be taken as e, different choices merely reflect
different perspectives for the projection.
Every great circle C on the r = 1 sphere is also a great circle of S3.
By choosing e to lie on the r = 1 sphere, each C on it represents a 1parameter subgroup of SU(2).
Going once around C changes t by 4π.
For SO(3), each pair of diametrically opposite points on C are identical.
(The pair is also diametrically opposite on S3. )
This creates a 2nd kind of closed path that cannot be shrunk continuously
to a point : any curve joining a diametrically opposite pair of points.
→ SO(3) is doubly connected.
SU(2) & SO(3) share the same Lie algebra → they are identical near e.
SU(2) is the only simply-connected Lie group associated with the Lie algebra.
A rotation in n-D space ( an element of SO(n) ) can be represented by a vector
in the Lie algebra, which is also a vector space of dimension n(n1)/2.
Thus, the association of an n-D rotation with an n-D angular momentum
vector is possible only if n = n(n1)/2, i.e. n = 3.
Objects whose rotations are elements of SU(2) are called spinors.
A vector in the Lie algebra ( internal space of the particle ) is called a spin.
The association of a SU(2) element with a 3-D vector allows the association
of spin with an (angular momentum) vector in R3.
The Lie algebra of the Lie group Rn is an n-D Abelian Lie algebra A.
Since Rn is simply-connected, all other Lie groups G with Lie algebra A
must be covered by Rn.
Also Rn is Abelian & identical to G near e. → G must be Abelian.
 All Lie groups of an Abelian algebra are Abelian.
3.17. Realizations & Representations
Abstract group : Group defined by manifold & group structure.
Group in physics : symmetry transformations (operations).
Realization of a group : A map T : G → L(M), where L(M) is the space
of operators on M, that preserves the group structure, i.e.,
T  gh   T  g  T  h 
 g, h  G
T  e   I  identity transform on M
T  g 1   T 1  g 
The realization is a representation if L(M) is the space of
linear operators on a vector space M.
The realization / representation is faithful if T is 1-1.
Example (i):
The abstract group SO(3) is defined by its Lie algebra & topology.
Consider the group G of all 33 orthogonal matrices with unit determinant.
Given any column matrix x corresponding to a point xR3 and a matrix R G,
Rx represents a rotation of x.
It’s straightforward to show that there is a 1-1 correspondence of G & SO(3) that
preserves the group operations.
Since R3 is a vector space, G(R3) is a faithful representation of SO(3).
The above still holds if G is applied to the 2-sphere S2.
However, since S2 is a manifold, G(S2) is a faithful realization of SO(3).
Historically, G(R3) is studied first & then “abstracted” to SO(3).
Most abstract groups are established in a similar manner.
Example (ii):
Every group has at least 2 faithful realizations :
The left- & right- translations of itself.
(h → gh) is called the progressive realizations &
(h → hg1 ) the retrograde realizations.
Example (iii): Adjoint Representations
Given group G, the map
Ig : G → G by h  g h g1 where gG
is called the adjoint realization of G on G.
Ig is an inner automorphism of G.
The group adjoint realization need not be faithful.
E.g., Ig = id  g if G is Abelian.
The differential (push-forward) Ig* of Ig at e is
called the adjoint map Adg.
Since
I g  e  geg 1  e
I g*  Ad g : Te  TI g  e  Te
 gG
Thus, Ig maps a 1-par subgroup fX (t) to another 1-par subgroup
i.e.,
e
t Ad g X
 f Ad g X  t   I g f X  t   I g e t X
 g e t X g 1
See Aldrovandi, §8.4.
Taking t → 0 gives (see Ex 3.23)
g  s   e sY
→
Ad g  s  X  e
f Ad g X  t 
sLY
X
The map Ad: G → L(G,G) is called the adjoint representation of G on G.
L(G,G) = space of all linear automorphisms of G.
The adjoint rep is used in the definition of connection for fibre bundles.
3.18.
Spherical Symmetry, Spherical Harmonics &
Representations of the Rotation Group
A manifold M with metric g is spherically symmetric if the Lie algebra of its
Killing vector fields has subalgebra isomorphic to so(3).
Reminder: A Killing vector field generates (isometric) group of transformations.
I.e., g is invariant under translation along the integral curves of the field.
The above definition avoids the possible embarassment that the “center
of spherical symmetry” does not exist because it lies outside M.
It was shown in §3.9 that the Killing vector fields on S2 are
with commutators
l
1
, l2 , l3

 li , l j      i j k lk
k
Reminder: lj are lin. indep. under linear combinations of constant coefficients.
On S2 of radius a :
g  diag  g , g   diag  a 2 , a 2 sin 2  
Switching to the basis
 V   l 
i
i
gives commutators
Vi  ,V j      i j k V k 

 k
i.e., the Lie algebra of Killing fields on S2 the same as so(3).
Conversely,  a foliation in a spherically symmetric M with spherical leaves.
Let L2(S2) be the Hilbert space of all square-integrable functions on S2.
The norm of f  L2(S2) is defined as
f 

S2
f  ,   sin  d d
2
2

1/ 2
A realization of SO(3) on S2 is given by the map g  σg ,
where σg is the group of transformation
σg (x) = σ( g, x ), x  S2 , g  SO(3)
s.t. σg h = σg σh .
This induces the pull back of σg s.t.
  f   x   f   x  
*
g
g
 f  L2  S 2 
In physics, one usually defines the function transformed by R by
For the transformation Rg corresponding to σg , this means
 Rf  Rx   f  x 
Rg   g* 1
Note that Rg : L2(S2) → L2(S2) is an automorphism on linear space L2(S2).
Hence the map R by g  R(g) = Rg is a representation of SO(3) on L2(S2).
A representation is irreducible if it contains no invariant subspaces.
One task of theoretical group theory is to find all irreducible representations of a group.
By definition, SO(3) has at least one faithful 3-D representation in terms of
orthogonal matrices.
Since L2(S2) is an infinite dimensional space, the representation of SO(3) on it
must be reducible.
Indeed, it can be decomposed into a countable set of finite dimensional
irreducible representations (IR) each specified by an integer L = 0, 1, 2, ….
The Lth IR has dimension 2L+1 and basis { YLM , M = L, L+1, …, L}.
RgYLM    RgL 
By definition:
N
where
R 
L
g MN

S
2
MN
YLN
 gSO(3)
*
YLM
 ,   Rg YLN  ,   d 
and Rg is some differential operator representing g.
Historically, YLM is chosen to be the pherical
harmonics, which are eigenfunctions of lz and L2

  L lk

2
k
(See Ex.3.24-5 )
Since SU(2) doubly covers SO(3), the representation of SU(2) is double-valued.
Thus, a faithful representation of SU(2) consists of 2 disjoint parts, each of which
is a faithful representation of SO(3).
IRs of SU(2) are labeled by half-integers J = 0, ½, 1, 3/2, ….
Every element is on some 1-par subgroup parametrized by t = [0, 4π) which can
be splitted into 2 disjoint parts, [0, 2π) and [2π, 4π).
For J = L = integers, the representation on both parts are identical.
For J = odd half integers, it behaves differently.