By Hossein Abbaspour and Martin Moskowitz
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As we shall see, when properly understood, these examples are typical. 3. Prove that for an n × n matrix X and for m a positive X m ) . integer, one has Exp X = limm→∞ (I + m Finally, notice also that the derivative of Exp at zero is the identity, This is because the power series expression for Exp(X) shows that X2 Exp(X) = Exp(0) + (X − 0)(I) + (X − 0)( X 2! + 3! + . ). By the linear X2 approximation theorem if X 2! + 3! +. . tends to 0 as X → 0, we conclude 2 2 ||X|| ||X|| X d0 Exp = I. But || X 2!
Then f can be extended to all of G. Local homomorphism means that f is defined on an open neighborhood of the identity element and f (ab) = f (a)f (b) on the neighborhood. Proof. Let U be the open neighborhood where f is defined. We may assume that U is symmetric. If not, we can replace U by U ∩ U −1 . Let D ⊂ G × G be the subset consisting of (p, q) such that qp−1 ∈ U . Evidently D contains the diagonal and is connected. To every p ∈ G, we associate the map φpq : x → f (qp−1 )x on H. One checks directly that D, Mp = H and φpq satisfy the conditions of the previous theorem.
Therefore f (G) is open in H which proves (2). 50 Chapter 1 Lie Groups Proof of (3). e f (expG (tX)) ≡ 1. That is, expH tf ′ (X) ≡ 1. Differentiating gives f ′ (X) = 0. Conversely if f ′ (X) = 0, then f (expG (tX)) ≡ 1. Therefor the Lie algebra of Ker f is Ker f ′ . Finally, (4) follows from the inverse function theorem. 13 and is left to the reader. 3. Let G be a connected Lie group and e and f be two smooth homomorphisms G → H, with e′ = f ′ . Then e ≡ f . As the next result, we show that for simply connected Lie groups, there is a one-to-one correspondence between Lie homomorphisms and Lie algebra homomorphisms.
Basic Lie Theory by Hossein Abbaspour and Martin Moskowitz