Introduction to Lie Groups
Spring 2011

Course Information

Introduction to Lie groups. Topics include Lie groups, Lie algebras, Lie groups and symmetric spaces, root space decomposition, Cartan subalgebras, root systems, Weyl groups, Peter-Weyl theorem, KAK and Iwasawa decompositions, Cartan decompositions, advanced strcture theory and its applications to dynamics on symmetric spaces.


Time: M W 5:00 p.m. - 6:15 p.m.
Classroom: 500-L309

Instructor: Seonhee Lim  email
Office: building 27, room 227
Office Hours: M W 2:50 p.m. - 3:50 p.m. and by appointments.
There is no textbook for this course.

References:

·  Lie groups, Lie algebras and representations, an elementary introduction, by B. Hall

·  Lie groups beyond an introduction, by A. Knapp

·  Linear algebraic groups, by J. Hymphreys

·  Differential Geometry, Lie groups and symmetric spaces, by S. Helgason

·  Lie groups and algebras with applications to physics, geometry and mechanics by D. Sattinger and O. Weaver.

Assignments

  1. Show that Sping(3) is isomorphic to SU(2).
  2. Show that the Mobius group is not simply connected, and find its universal cover.
  3. Show that the Mobius transformations preserves the Poincare metric on the upper half plane.
  4. Check that the set of derivations of a given associate algebra is a Lie algebra.  
  1. When does the exponential map of Lie groups coincide with the exponential map of the Riemannian metric? (Here choose an inner product at the origin and define a left invariant metric. Is it always bi-invariant? Prove or give a counter-example. Reference : exercises A.5,6 B.1 of Lie groups and Lie algebra section of Helgason.)
  2. What is the relation between Ricci curvature and the killing form? (Reference: Semi-Riemannian geometry and general relativity, by Shlomo Sternberg)
  3. Explain why the diagonalizable part and the nilpotent part of the Jordan decomposition of any complex matrix are polynomials of the given matrix. (Reference : Linear Algebra, by Hoffmann & Kunze, p. 222)

 

 

Exams

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