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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) kZ f (k ) u k i , u T {e } satisfies the nonlinear constraint m1 k 0 where | F( u) | 1 , u T 2 mZ 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 1k 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 mn n m / n n nZ 2 \{( 0, 0 )} 2 GEODESIC FLOW Quantum deformations κ0 [Lm , Ln ] κ 1 sin( m n) L mn 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 nZ N \{( 0, 0 )} SU (N ) mn 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.