PHY 745 Group Theory
11-11:50 AM MWF Olin 102
Plan for Lecture 34:
Introduction to linear Lie groups
1. Definitions and properties
2. Comparison with finite groups
3. Usefulness for physics and mathematics
Ref. J. F. Cornwell, Group Theory in Physics, Vol I
and II, Academic Press (1984)
Robert Gilmore, Lie Groups, Physics, and
Geometry, Cambridge U. Press (2008)
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Some comments from Gilmore text:
Marius Sophus Lie (1842-1899) had the vision to use group
theory to solve, analyze, simplify differential equations by
exploring relationships between symmetry and group
theory and related algebraic and geometric structures. His
worked followed that of Evariste Galois (1811-1832).
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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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Generalizations for the notion of distance – choice of
“metric”. Here we have chosen the Hilbert-Schmidt
metric:
1/ 2

2
d (T , T ')   G(T ) jk  G(T ') jk 
 j 1 k 1

m
m
Another choice of metric is based on the "regular" representation.
For an n dimensional Lie group, a regular represention is
based on n  n which satisfy the structure constant relations:
n
[ X i , X j ]   cijr X r
r 1
A "Cartan Killing" inner product is defined:
(X i , X j )CK 
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n
s r
c
 ir c js
r , s 1
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Relationships of finite groups to continuous group – notions
of “connectedness”
A maximal set of elements T of G that can be obtained
from each other by continuously varying one or more
matrix elements G(T ) jk of the faithful finite dimensional
representation G is said to form a "connected component"
of G.
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Example: O(2) -- 2  2 orthogonal matrices
 A11
A
 A21
A12 
1 0
†
with A A  


A22 
0 1
 cos x1 sin x1 
Possibility #1: A #1 ( x1 )  


sin
x
cos
x
1
1

  sin x1 cos x1 
Possibility #2: A # 2 ( x1 )  

 cos x1 sin x1 
 G( A #1 ( x1 )) are connected
 G( A # 2 ( x1 )) are connected
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Theorem: The connected component of a linear Lie group G
that contains the identity E is an invariant subgroup of G.
Example from O(2):
 cos x1 sin x1 
Possibility #1: A #1 ( x1 )  


sin
x
cos
x
1
1

1 0
A #1 ( x1  0)  

0
1


 cos x1 sin x1  cos x2 sin x2 
A #1 ( x1 ) A #1 ( x2 )  
  sin x cos x 

sin
x
cos
x
1
1 
2
2

 cos  x1  x2  sin  x1  x2  
=


sin
x

x
cos
x

x
 1 2
 1 2 

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
subgroup
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 cos x1 sin x1 
 A #1 ( x1 )  
is a subgroup G' of G

  sin x1 cos x1 
To show that G' is invariant, must show that
1
XAX  G' for all X  G
A linear Lie group is said to be “connected” if it possesses
only one connected component.
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Definition
A linear Lie group of dimension n with a finite number of
connected components is compact if the parameters
y1 , y2 , .... yn range over closed finite intervals such as
a j  y j  bj
Example from O(2):
 cos x1 sin x1 
Possibility #1: A #1 ( x1 )  


sin
x
cos
x
1
1

In this case    x1  
 compact
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With the notion of a compact linear Lie groups, we can
relate their properties, such as the great orthogonality
theorem, to those of finite groups.
Consider a complex function f of a group element T :
Left invariant:
 f T 'T    f T ''
T G
Right invariant:
 f TT '   f T '''
T G
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T ''G
T '''G
PHY 745 Spring 2017 -- Lecture 34
  f T  d lT
G
  f  T  d rT
G
17
b1
bn
b2
 f T  d T   dy  dy ....  dy
l
1
G
2
a1
a2
an
b1
b2
bn
 f T  d T =  dy  dy ....  dy
r
G
1
a1
2
a2
n
f (T ( y1. y2 ... yn )) l ( y1 , y2 ... yn )
n
f (T ( y1. y2 ... yn )) r ( y1 , y2 ... yn )
an
If  l ( y1 , y2 ... yn )   r ( y1 , y2 ... yn ), the G is called unimodular.
For G compact and unimodular and f (T ) continuous, the
integral exists and is finite.
Example:
For O (2)
 cos x1 sin x1 
Possibility #1: A #1 ( x1 )  


sin
x
cos
x
1
1

In this case    x1  
 compact
1
dT


2
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
 dx
1

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Other examples –
SU(2):
 cos y1eiy2
u= 
 iy3

sin
y
e
1

sin y1eiy3 
 iy2 
cos y1e 
 /2
1
2
2
 dT  4  dy  dy  dy
1
2
0
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2
0
3
0
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