Answer Happy

C 6),te 1) [1 pt each, maximum 8] List as many examples of subgroups of Lie groups as you can. Give a very brief explanation of one sentence or a few words for each example. You will get more points for more examples, but will get marked down 1 point if you write an example which is not a subgroup. 2) [8 pts) In the set of 2 by 2 matrices with complex entries, determine whether matrices of the form te R is an ideal. Write one sentence to explain how you are determining the answer, and then do a computation. 3) [6 pts] If G is an Abelian matrix Lie group, show that the set {g?,9 E G} is a matrix Lie subgroup. Think carefully about definitions. 4) [5 pts] Show that every 2 by 2 matrix X with trace 0 satisfies X2 = -(det X)/ 0 b 5) [10 pts) Let g be the Lie algebra of 3 by 3 matrices of the form ( 0 0 с 0 0 0/ is nilpotent. You do NOT have to check the definition of Lie algebra. Just show it's nilpotent 6) [6 pts] Given a representation II of a Lie group G acting on a vector space V, consider the dual representation II* acting on V*. If W is an invariant subspace of V, show how to get an invariant subspace of V* Bonus: Are these the only invariant subspaces of V*? (I don't know. This is a challenge question). a