Mathematical Physics Seminar Notes Lecture 3 Global Analysis and Lie Theory

1. Mathematical Physics Seminar Notes Lecture 3 Global Analysis and Lie Theory Wayne M. Lawton Department of Mathematics National University of Singapore 2 Science Drive 2 Singapore 117543 Email matwml@nus.edu.sg Tel (65) 6874-2749 1

2. Commuting Vector Fields Theorem. If then there exist local coordinates such that iff Proof. p 471 J. Lee Introduction to Smooth Manifolds Corollary. If is a Lie group with Lie algebra and spans an abelian subalgebra of then is an abelian subgroup of and 2

3. Lie Algebras and Lie Groups Lemma (standard homotopy result) Every connected Lie group is the quotient of a unique simply connected Lie group (obtained as its universal covering space) with a discrete central subgroup. Lie groups are locally isomorphic iff they have the same s.c. covering groups Lemma A closed subgroup of a Lie group is a L. g. Theorem (Lie). There is a 1-to-1 correspondence between Lie algebras and s. c. Lie groups. Theorem (Frobenius) There is a 1-to-1 correspondence between Lie algebras and (not necessarily closed) Lie subgroups – e.g. subgroup R of the two-dim torus 3

4. Adjoint Representation Definition Definition Theorem For 4

5. Ideals and Normal Subgroups Definition A Lie subalgebra is an ideal if is normal if Definition A Lie subgroup Theorem There is a 1-to-1 correspondence between normal connected Lie subroups of a Lie group and ideals of its Lie algebra 5

6. Killing Form Definition. Theorem Corollary 6

7. Nilpotent and Solvable Algebras nilpotent, solvable if Definition Lie algebra terminates Theorem (Lie) A subalgebra of GL(V) is solvable iff its elements are simultaneously triangulable Theorem (Engel) A Lie algebra is nilpotent iff ad(u) is nilpotent (some power = 0) for every element u 7

8. Simple and Semisimple Algebras Definition Lie algebra is simple, semisimple if it has no ideals, abelian ideals other that itself and {0} Theorem The sum of any two solvable ideals is a solvable ideal, hence every algebra has a unique maximal solvable ideal – called its radical Theorem (Levi) Every Lie algebra is the semidirect sum of its radical and a semisimple subalgebra Theorems (Cartan) A Lie algebra is solvable iff semisimple iff K is nondegenerate, so SSLA = + SI Proof D. Sattinger and O. Weaver, Lie Groups and Algebras with App. to Physics, Geom. &Mech. 8

9. Examples Heisenberg Groups Affine Groups Euclidean Motion Groups Poincare Group 9

10. Cartan’s Classification of Complex Semisimpil LA Classical Exceptional 10

11. Lagrange’s Equations in Action Lagrangian L := T – U in Action Principle of Least Action: for Lagrange Equations described as a section of T(T(M)), ie in Vect(T(M)) 11

12. Geodesics If V = 0 then L = T defines a Riemannian manifold M with metric tensor g Lagrange’s equations where the components of the Christoffel symbol describe trajectories that minimize squared magnitude of velocity, and hence minimize length and have constant speed, therefore they are geodesics 12

13. Hamilton’s Equations Hamiltonian defined by the Legendre Transformation satisfies Lagrange’s equations are equivalent to Hamilton’s and 13

14. Symplectic Structure The Liouville 1-form induces the symplectic structure on given by the nondegenerate 2-form The Hamiltonian vector field v is hence the Lie derivative satisfies 14

15. Poincare’s Recurrence Theorem If is a Hamiltonian flow then for every open set and there exists and such that Proof. Consider the (infinite) union Since the volume (induced by the symplectic form) of each set is positive and equal, they can not be disjoint, and the conclusion follows. 15

16. The Kirillov Form on Co-Adjoint Orbits Theorem. If is a Lie group with Lie algebra then the orbit of under the coadjoint antirepresentation admits a symplectic structure. Proof. Tangents u, v to M at p are represented by curves in M, hence by curves in G through 1 that define elements so the 2-form is symplectic. 16

17. Weyl-Chevalley Normal Form Theorem. If is a complex semisimple Lie algebra with Cartan subalgebra (maximal abelian with semisimple (diagonalizable) for all where are roots and then unless where is a root. 17