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APPLICATIONS OF LIE THEORY
Wayne M. Lawton
Department of Mathematics
National University of Singapore
2 Science Drive 2
Singapore 117543
Email [email protected]
Tel (65) 6516-2749
http://www.math.nus.edu.sg/~matwml/courses/Undergraduate/USC/2006/USC3002
THE UTILITY OF LIE THEORY
Lie theory models nonlinear constraints and symmetry
that explains phenomena and arises in engineering
design and simulation of physical processes.
Shlomo Sternberg - Group Theory and Physics
1928 Weyl’s Gruppentheorie und Quantenmechanik
1920-30’s Chemistry and Spectroscopy
1930-40’s Nuclear and Particle Physics
1960-70’s High Energy Physics
THE GRUPPENPEST SYNDROME
Sternberg notes: the remarks on pages 60-2 in Slater, J.C.Solid-State and
Molecular Theory: A Scientific Biography. New York: Wiley, 1975.
“It was at this point that Wigner, Hund, Heitler, and Weyl
entered the picture with their “Gruppenpest”: the pest of group
theory…The authors of “Gruppenpest” wrote papers that were
incomprehensible to those like me who had not studied group
theory, in which they applied these theoretical results to the
study of the many electron problem. The practical
consequences appeared to be negligible, but everyone
felt that to be in the mainstream one had to learn about it.”
and remarks “It is, however, amazing to consider that this
autobiography was published in 1975, after the major
triumphs of group theory in elementary particle physics.”
RECOGNITION
of the utility of Lie theory in engineering design and physical
simulation may be slower than it was in physics, due to
1. the extensive mathematical background of physicists, and
Slater, John Clarke (1900-1976) American physicist who did work on
the application of quantum mechanics to the chemical bond and the
structure of substances. His name is attached to the Slater
determinant used to construct antisymmetric wavefunctions.
2. the fundamental nature of physics. But the success of Lie
theory in physics may facilitate recognition in other fields.
Calderbank and Moran have demonstrated the importance
of discrete Heisenberg-Weyl groups in coding.
Lie symmetry explains soliton/instanton phenomena that have
growing importance in material science and nanotechnology.
THREE APPLICATIONS
were chosen based on 1. my interests and 2. instructive value.
Functions with values in Lie groups
Filter Design
Dynamic Simulation
Functions (measures) on Lie groups
Localization Explanation
(for waves in random media)
CONJUGATE QUADRATURE FILTER
f : Z  C whose Fourier transform
or frequency response F : T  C defined by
sequence
F(u)  kZ f (k ) u
k
i
, u  T  {e }
satisfies the nonlinear constraint

m1
k 0
where
| F( u) |  1 , u  T
2  mZ
k
2
and
 e
2 i / m
Definition F has regularity N if | 1 - F(u) |  | 1 - u |
Regular CQF’s used for filterbanks & wavelets
N
CONJUGATE QUADRATURE FILTER
is represented by a polyphase vector
P:T  S
m
F(u)  k 1 u P(u ) k , u  T
1k
m
where
m
S  { v  C :|| v || 1 } and there exists a
m
polyphase matrix
m
~
P : T  SU (m)
whose 1st column equals
P
Theorem 1. F is a trigonometric polynomial (finite
impulse response filter) iff every component of P
~
is. Then P can be chosen to have trig. poly. entries.
CONJUGATE QUADRATURE FILTER
Lemma 1. (in Pressley and Segal’s Loop Groups)
Every continuous M : T  SU (m) can be uniformly
approximated by a trig. poly.
~
P : T  SU (m)
Proof. Trotter formula and semisimplicity of SU(m)
Theorem 2. Every continuous CQF F : T  C
can be uniformly approximated by a trig. poly. CQF
Q : T  C with the same regularity as F
Proof. [1], uses Lemma 1, Jets, and the Brower degree
GEODESIC FLOW
Euler 1765
 t u   u u  grad p, div u  0
Flow is a trajectory g : R  SDiff (D) with values
in the group of volume-preserving diffeomorphisms.
Velocity
u : D  R  R , d  2, 3
d
is the angular velocity in space
u
dg
dt
g
Moreau 1959 g is a geodesic with respect to the
right-invariant kinetic-energy Riemannian metric
 Au, v    u  v
D
-1
GEODESIC FLOW
ψ with
o
J is 90 rotation
For d = 2, div u = 0  a Stream function
u  J grad ψ
where
SDiff(D) is the Lie algebra with Poisson bracket
 u1u 2 - u 2 u1
[ψ1 , ψ 2 ]   x1 ψ1 x2 ψ 2 -  x2 ψ1 x1 ψ 2
Vorticity ω    u
   
1

Flow preserves Casimirs I p  
  [ ,  ]
p
D
i m x
For periodic flow basis Lm ( x)  e , m  Z \ {( 0,0)}
Euler equations  m ( x)  n mn n  m / n  n
nZ 2 \{( 0, 0 )}
2
GEODESIC FLOW
Quantum deformations
κ0
[Lm , Ln ] κ   1 sin(  m  n) L mn  κ L m   2 sin 2 ( m)
gives algebra derivations C* algebra (non comm. torus)
iκ
generated by A,B where A B  ε B A, ε  e
For κ  1/N ,N odd gives homomorphism onto
sl ( N , C ) with real form su (N ) yielding the
Sine-Euler Approximation by GF on
 m ( x) 

n
2
nZ N \{( 0, 0 )}
SU (N )
mn sin( 2 n  m / N ) / sin( 2 n  n / N )
 (t )  g (t ) (0) g (t ) , g (t ) g (t )   
1
1
N-Casimirs are exactly conserved ! See [2]
1
WAVE PROPAGATION
in a chain of harmonic oscillators
mn 2
u (t , n  2)
mn 1
mn
u (t , n  1)
u (t , n)
is described by the equation
 0
0 mn
M
0 0
0 0
0
0
0
0
mn 1 0
0 
mn 1
u (t , n  1)
M u(t )   Ku(t )
 1 0 0
1 2 1 0
K
0 1 2 1
0 0 1 
WAVE PROPAGATION
admits an expansion in normal modes
u (t )   a( ) cos(t    ( )) E
where
and
K E   M
2   mn
T (n)  
 1
 E (n  1)
 E (n) 
E or  E (n)   T (n)  E (n  1)
 1 are samples of a measure on

0  the Lie group SL( 2, R )
Theorem Localization of
E
as

increases.
Proof Furstenberg [3]  the Lyapunov exponent
1
lim n log max { eig (T (n)T (1)) }  0
n
REFERENCES
[1] W. Lawton, Hermite interpolation in loop groups
and conjugate quadrature filter approximation, Acta
Applicandae Mathematicae,84(3),315--349, Dec. 2004
[2] V. Arnold and B. Khesin, Topological Methods
in Hydrodynamics, Springer, New York, 1998.
[3] H. Furstenberg, Noncommuting random products,
Transactions of the AMS, 108, 377-429, 1963.
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