This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Differential Geometry 2 Test 3 Define what is meant by a Lie group homomorphism : G H and con struct its derivative # : g h . (a) Show that exp G = exp H # and explain briefly why is uniquely determined by its derivative # when G is connected. (b) Show that if : g h is a Lie algebra homomorphism then there need not exist : G H such that # = . Suggestion : (b) Let G = H = S 1 . Let : Lie( S 1 ) Lie( S 1 ) be given by = 1 2 Id; assume = # and take the square. Solutions : When G and H are Lie groups, a Lie group homomorphism is a smooth, group homomorphism from G to H . Being a group homomor phism, maps e = e G to e = e H ; being smooth, has a tangent map * : T e G T e H . Now the derivative # : g h is defined to be the com posite g T e G * T e H h in which the first map is evaluation at e G and the last is the inverse of evaluation at e H ....
View
Full
Document
 Spring '09
 Robinson

Click to edit the document details