Home page for Math 6510 - Fall 2006, Spring 2007

Topics in Algebra: Lie Groups and Lie Algebras

Course Description
Problem Set #1 Due beginning October 12.
Problem Set #2 Due beginning January 16, 2007.
Problem Set #3 Due beginning April 25, 2007.
Lecture Notes
- August 29: Introduction, overview.
- August 31 Topological manifolds, charts, atlases, differentiable structures, examples.
- September 5 More examples, including solution set manifolds and their tangent spaces.
- September 7 Grassmanians, motivating the definition of a tangent vector.
- September 12 Smooth functions, smooth maps, definition of tangent vectors, identification of the tangent space with a dual space.
- September 14 The basis of the tangent space determined by a local chart.
- September 19 Taylor's theorem, the differential of a smooth map.
- September 21 Tangent bundle.
- September 26 Vector fields.
- September 28 Lie groups, the Lie algebra of a Lie group.
- October 3 The Lie algebra of a Lie group. An Example.
- October 5 Lie group Homomorphisms. Smooth Maps.
- October X Partitions of Unity. Inverse Function Theorem.
- November Y Submanifolds, Slices, and the Implicit Function Theorem.
- Ordinary Differential Equations General existence and uniqueness theorem for arbitrary, time-dependent systems of ordinary differential equations.
- November 16 Integral Curves and Flows of Smooth Vector Fields.
- November 21 Complete Vector Fields/Group Actions/Dynamical Systems.
- November 28 Left-invariant Vector Fields and One Parameter Subgroups of Lie Groups.
- December 14 The Exponential Map for General Linear Groups, Left-invariant Vector Fields and One Parameter Subgroups of the general Linear Group.
- January U Unitary and Orthogonal groups: SU(2) (latitudes, longitudes, conjugacy classes, maximal tori), the adjoint representation of SU(2), the Hopf fibration.
- January V Representations of SU(2).
- February 7-12 Basic representation theory.
- February 14-19 Haar measure and integration. Invariant inner products.
- February 21 Tensor products.
- February 28 Tensor products, intertwining operators, and representations; Characters; Examples -- the symmetric group on three letters, the unit circle and Fourier seres.
- March 12-14 Peter-Weyl Theorem, part 1.
- March 21 Peter-Weyl Theorem, part 2.
- April 9 The exponential map.
- April 11 The exponential map for the general linear group. Lie algebras of closed subgroups of the general linear group.
- April 18 Closed subgroups of Lie groups.

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Last modified: Tue May 1 11:00:35 CDT 2007