Reading: Syllabus. Text (Kirillov): Chapter 2 through Section 2.5.
Problems from Chapter 2 (Section 2.8):: 2.1, 2.2, 2.3, 2.5 (but use GL(n,R) instead of O(n,R)).
Problem A: Is the Closed Subgroup Theorem (a.k.a., Cartan's Theorem) 2.9 true for complex Lie groups?
Problem B: Let G be a Lie group. Define the structure of a 0-dimensional, i.e., discrete Lie group on the set π0(G) of connected components by multiplying representatives of connected components. Prove that this group is isomorphic to the quotient group G/G0 by the connected component of the identity.
Problem C: Explain why the kernel of the universal cover, see Theorem 2.7, is isomorphic to the fundamental group π1(G) of G as a group.
Last modified: (2015-09-19 00:44:41 CDT)