Mathematical Physics Seminar Notes Lecture 1 Global Analysis and Lie Theory

1. Mathematical Physics Seminar Notes Lecture 1 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. Venue: S14-03-09 (Mathematics Conference Room B) Time: 16:00-17:15 Thursday 16 September Speaker: Wayne Lawton Title: Global Analysis and Lie Theory, talk 1 of Mathematical Physics Seminar Abstract: This talk will discuss material from and supplemental to that in Chapter 2 of Loop Groups by Pressley and Segal. We start by with the following observations: sections of the tangent bundle of a smooth manifold (vector fields) induce phase flows on the manifold, the set of sections forms an infinite dimensional Lie algebra,suitable sections of the Grassmann bundle (involutive distributions) induce foliations, and use these to develop the basic relationship between Lie groups and Lie algebras using the exponential mapping and the adjoint representation. We then review the linear algebra of unitary vector spaces and develop basic Lie theory: Schur's lemma, Killing form, roots, weights, Weyl group, induced representations, complex homogeneous spaces, and the Borel-Weil theorem. 2

3. Geometrization of Analysis based on Topology/Lie Theory – invented by Henri Poincare’/Sophus Lie for Differential Equations smooth manifold, navigate with coordinate charts tangent space at a point is an equivalence class of ‘based’ smooth trajectories through the point 3

4. Tangent Bundle hence gives an isomorphism where defined by is the equivalence class of Tangent Bundle of is topologized to give homeomorphisms together with the projection 4

5. Local Coordinates smooth functions on admit local representation where is open is smooth is smooth and hence where 5

6. Vector Fields on is a smooth function that is a section, this means hence if then 6

7. Fundamental Theorem of ODE’s induces a unique local flow with and for furthermore, where applicable 7

8. Global Flows under suitable conditions, for example if is bounded and then we obtain a smooth homomorphism then we obtain a smooth homomorphism of the group of real numbers into the group of diffeomorphisms of the manifold 8

9. Integral Trajectories an integral trajectory for is such that Theorem If c is an integral trajectory then Proof From the uniqueness part of the ODE theorem Remark Ordinary Differential Equations by E. L. Ince references the work of Liouville, Cauchy, Lipschitz and others towards the modern ODE theorem 9

10. Grassmann Bundles over a Manifold Definition Let d+1>k > 0, let denote the Grassmann manifold of k-dimensional subspaces of and let be topologized in the local product manner. We will call this the Grassmann bundle over - it should not be confused with the Grassmann vector bundle over a Grassmann manifold whose fibers are subspaces – each fibers of this bundle is a Grassmann manifold ! 10

11. Distributions Definition A k-dimensional distribution on is a section of the Grassmann bundle over Definition A vector field is compatible with a distribution if 11

12. Fundamental Theorem of PDE’s Lemma is closed under commutation under which it is an infinite dimensional Lie algebra Theorem (Frobenius) A distribution is integrable (admits a foliation into integral surfaces) iff it contains the commutator of every pair of vector fields that it contains. 12