Download Introduction to linear Lie groups

PHY 745 Group Theory
11-11:50 AM MWF Olin 102
Plan for Lecture 33:
Introduction to linear Lie groups -continued
1. Linear Lie group and real Lie algebra
2. Fundamental theorem
3. Examples
Ref. J. F. Cornwell, Group Theory in
Physics, Vol I and II, Academic Press (1984)
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Definition of a linear Lie group
1. A linear Lie group is a group
• Each element of the group T forms a
member of the group T’’ when “multiplied” by
another member of the group T”=T·T’
• One of the elements of the group is the
identity E
• For each element of the group T, there is a
group member a group member T-1 such
that T·T-1=E.
• Associative property: T·(T’·T’’)= (T·T’)·T’’
2. Elements of group form a “topological space”
3. Elements also constitute an “analytic manifold”
Non countable number elements lying in a region
“near” its identity
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Definition: Linear Lie group of dimension n
A group G is a linear Lie group of dimension n if it satisfied
the following four conditions:
1. G must have at least one faithful finite-dimensional
representation G which defines the notion of distance.
For represent G having dimension m, the distance
between two group elements T and T ' can be defined:
1/ 2

2
d (T , T ')   G(T ) jk  G(T ') jk 
 j 1 k 1

Note that d (T , T ') has the following properties
m
m
(i) d (T , T ')  d (T ', T )
(ii) d (T , T ) = 0
(iii) d (T , T ')  0 if T  T '
(iv) For elements T , T ', and T '',
d (T , T ')  d (T , T '')  d (T ', T '')
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Definition: Linear Lie group of dimension n -- continued
2. Consider the distance between group elements T
with respect to the identity E -- d(T,E). It is possible
to define a sphere Md that contains all elements T’
such that d ( E , T ')  d .
It follows that there must exist a d  0 such that every T '
of G lying in the sphere M d can be parameterized by n
real parameters
x1 , x2 , ....xn such each T ' has a different
set of parameters and for E the parameters are
x1  0, x2  0,....xn =0
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Definition: Linear Lie group of dimension n -- continued
3. There must exist
  0 such that for every parameter set
 x1, x2 ,...xn  corresponding
to T ' in the sphere M d :
n
2
2
x


 j
j 1
4. There is a requirement that the corresponding
representation is analytic
For element T ' within M d , G(T ( x1 , x2 ,..xn )) must
be an analytic (polynomial) function of x1 , x2 ,....xn .
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Some more details
4. There is a requirement that the corresponding
representation is analytic
For element T ' within M d , G(T ( x1 , x2 ,..xn )) must
be an analytic (polynomial) function of x1 , x2 ,....xn .
Because of the mapping to the n parameters x1 , x2 ...xn to
each group element T ', G(T '( x1 , x2 ...xn ))  G( x1 , x2 ...xn ).
The analytic property of G( x1 , x2 ...xn ) also means that
derivatives
 G jk ( x1 , x2 ...xn )

 xp
must exist for all   1, 2,...
Define n m  m matrices:
a 
p
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jk

G jk ( x1 , x2 ...xn )
x p
x1  0, x2  0,.... xn  0
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Definition: Real Lie algebra
A real Lie algebra of dimension n  1 is a real vector space
of dimension n which includes a comutator [ M , N ] as
follows:
1. For all M , N in algebra, [ M , N ] is also in algebra
2. For real numbers  and  , and members M , N , O,
[ M   N , O ]   [ M , O]   [ N , O]
3. [ M , N ]  [ N , M ]
4. [ M ,[ N , O]]  [ N ,[O, M ]]  [O,[ M , N ]]  0
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Structure constants of Lie algebra
Consider the n basis matrices of the algebra a1 , a 2 ,...a n :
n
[a p , a q ]   c rpqa r for p, q =1,2 ... n
r 1
Example: G is the group SU(2) of all 2  2 unitary matrices
having determinant 1
1 0 i 
a1  

2  i 0
1  0 1
a2  

2  1 0 
1i 0 
a3  

2  0 i 
Structure constants for this case:
[a1 , a 2 ]  a3
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[a 2 , a3 ]  a1
[a3 , a1 ]  a 2
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Fundamental theorem –
For every linear Lie group there exisits a
corresponding real Lie algebra of the same
dimension. For example if the linear Lie group has
dimension n and has mxm matrices a1, a2,….an then
these matrices form a basis for the real Lie algebra.
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One-parameter subgroup of a linear Lie group
A "one-parameter subgroup" of a linear Lie group
fullfills the requirements of a Lie group as a subgroup
whose elements T (t ) depend on a real parameter t with
  t   where
T ( s )T (t )  T ( s  t )
Note that T ( s )T (t )  T (t )T ( s ) so that the subgroup
is Abelian. It follows that T (t  0)  E
Theorem: Every one-parameter subgroup of a linear Lie
group of m  m matrices is formed by exponentiation of
m  m matrices.
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A(t )  e
ta
dA
a=
dt t  0
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Theorem: Every one-parameter subgroup of a linear Lie
group of m  m matrices is formed by exponentiation of
m  m matrices.
A (t )  e
at
dA
a=
dt t  0

dA (t )
 A(t  s )  A(t ) 
 A( s )  A(0)  
 lim 
  lim  A (t ) 

dt
s
s
s 0 
 s 0 


=A (t )a
A (t )  eat
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Correspondence between each linear Lie group G and
a real Lie algebra L Simplify the consideration to G
consisting of mxm matrices T=A and G(T)=A.
As part of the definition of the linear Lie group, there are
n parameters x1 , x2 ,...xn such that all A(x1 , x2 ,...xn ) are
analytic functions of the parameters, and the n m  m
matrices
a 
p
jk
A

x p
x1  x2  .. xn  0
form the basis of an n-dimensional real vector space.
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Definition: Analytic curve in G
Suppose x1 (t ), x2 (t )....xn (t ) are analytic functions of
0  t  t0 within which
n
2
2
x
(
t
)


, then the set
 j
j 1
of m  m matrices A(t)=A(x1 (t ), x2 (t )....xn (t )) are
said to form an analytic curve in G.
Definition: Tangent vector of an analytic curve in G
The tangent vector of an analytic curve A (t ) in G is defined
dA (t )
to be the m  m matrix a =
.
dt t  0
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Theorem
The tangent vector of any analytic curve in G is a
member of the real vector space having the matrices
a1 , a 2 ,.....a n as its basis. Conversely, every member
of the real vector space is the tangent vector of some
analytic curve in G.
Theorem
If a and b are the tangent vectors of the analytic curves
A (t ) and B(t ) in G , then c=[a,b] is the tangent vector
of the analytic curve C(t ) in G , where

C(t )  A( t )B( t ) A( t )
Note that
A( t )  1+a t +O(t )


  B( t ) 

1
1

C(t )  1+a t ... 1+b t ... 1  a t ... 1  b t ...
 1+abt  bat....
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

dC(t )
 [a,b]
dt t  0
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Fundamental theorem:
For every linear Lie group G there
exists a corresponding real Lie algebra L of the same dimension.
More precisely, if G has dimenison n then the m  m matrices
a1 , a 2 ,....a n form a basis for L.
Converse to fundamental theorem: Every real Lie algebra is
isomorphic to the real Lie algebra of some linear Lie group.
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Example:
SO(3)
In this case, it is convenient to consider rotations about the
3 orthogonal directions:
R ( ,  ,  )  R( , xˆ ) R (  , yˆ ) R( , zˆ )
0
1
=  0 cos 
 0 sin 

 cos 
 sin  
 0
cos  
  sin 
0
0
R
  0
      0 
0
0
R
  0
      0 
 1
0
R
  1
      0 
0
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0
0
1
0
0
0
1
0
0
0 sin    cos 
1
0   sin 
0 cos    0
0
1  a1
0 
1
0   a 2
0 
0
0   a3
0 
PHY 745 Spring 2017 -- Lecture 33
 sin 
cos 
0

0 
1 
Note that:
[a1 , a 2 ]  a3
[a 2 , a3 ]  a1
[a3 , a1 ]  a 2
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