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

3. Lie Algebras and Lie Groups 3 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 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 Lemma A closed subgroup of a Lie group is a L. g.

4. Adjoint Representation 4 Definition Theorem For

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

6. Killing Form 6 Definition. Theorem Corollary

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

8. Simple and Semisimple Algebras 8 Definition Lie algebra is simple, semisimple if it has no ideals, abelian ideals other that itself and {0} Theorems (Cartan) A Lie algebrais solvable iff Proof D. Sattinger and O. Weaver, Lie Groups and Algebras with App. to Physics, Geom. &Mech. 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 semisimple iff K is nondegenerate, so SSLA = + SI

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

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

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 describe trajectories that minimize squared magnitude of velocity, and hence minimize length and have constant speed, therefore they are geodesics 12 where the components of the Christoffel symbol

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

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

15. Poincare’s Recurrence Theorem If is a Hamiltonian flow then for every open set and there exists 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. Ifis a Lie group with Lie algebra of then the orbit under the coadjoint antirepresentation Proof. Tangents u, v to M at p are represented by curves in M, hence by curves in G through 1 that define elementsso the 2-form is symplectic. admits a symplectic structure. 16

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