Introduction to Lie Groups - Marc Burger - 2018

Kevin, 13.02. 14:20-14:40

1. Definition of topological group.

2. Show that the inversion map of a topological group is a homeomorphism.

3. Give three examples of compact topological groups. I said O(n,R), any finite group with discrete topology, and torus of any dimension.

4. Prove that O(n,R) is compact.

5. The answers you gave for Problem 3 were all Lie groups, can you give an example of a compact topological group that is not a Lie group? I said the p-adic integers.

6. Define the p-adic integers.

7. What can you say about the support of a Haar measure on locally compact Hausdorff group? Answer: that the support is the whole group.

8. Prove that the support is the whole group.

9. Show that a locally compact Hausdorff group is compact if and only if the measure of the Haar measure is finite.

10. Give the statement of Minkowski's Theorem.

11. Given an abelian, connected Lie group, what can you say about the the exponential map? Answer: that the exponential map is surjective, and the kernel is discrete.

12. Prove those facts.

13. Define what it means for a Lie group to be solvable.

Anonymous, 14.02.

1. Definition of a topological group.

2. Give three examples of compact Hausdorff groups. Answer: O(n), p-adic integers, finite discrete group.

3. Which one of the three examples you gave is not a Lie group? Why? Answers: p-adic integers; has small subgroups.

4. Prove that left translation is a homeomorphism.

5. Prove that a locally compact Hausdorff group has finite Haar measure if and only if it is compact.

6. Prove that if a subgroup H of a topological group G and the quotient G/H are connected, then so is G.

7. Define the exponential map.

8. If we have a connected abelian Lie group, prove that the exponential map is a smooth surjective homomorphism with discrete kernel.

9. Define a solvable Lie group.

10. State Lie's theorems. Answer: every representation of a connected solvable Lie group admits a weight; (...) there exists a basis such that \(\pi(G) \) is upper triangular.

11. What can you say about a continuous homomorphism between Lie groups? Answer: it is smooth.

12. State Cartan's theorem.