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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 ....
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- Spring '09