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For g ∈ G, X ∈ g, t ∈ R, g exp(t X )g −1 = exp(tg Xg −1 ). Hence g Xg −1 ∈ g. The map Ad(g) : X → Ad(g)X = g Xg −1 is an automorphism of the Lie algebra g, Ad(g)[X, Y ] = [Ad(g)X, Ad(g)Y ] (X, Y ∈ g). Furthermore Ad(g1 g2 ) = Ad(g1 ) ◦ Ad(g2 ), and this means that the map Ad : G → Aut(g) is a group morphism. 2 (i) For X ∈ g, d Ad(exp t X ) dt = ad X. t=0 (ii) Let us denote by Exp the exponential map from End(g) into G L(g). Then Exp(ad X ) = Ad(exp X ) (X ∈ g). 3 Linear Lie groups are submanifolds 41 Proof.

3 the neighbourhood V can be chosen such that exp V ∩ G = {I }. Let us show that exp U = W ∩ G. Let g ∈ W ∩ G. One can write g = exp X exp Y (X ∈ U , Y ∈ V ), and then exp Y = exp(−X )g ∈ exp V ∩ G = {I }, hence g = exp X . 5 A linear Lie group G ⊂ G L(n, R) is a submanifold of M(n, R) of dimension m = dim g. Proof. Let g ∈ G and let L(g) be the map L(g) : G L(n, R) → G L(n, R), h → gh. Let U be a neighbourhood of 0 in M(n, R) and W0 a neighbourhood of I in G L(n, R) such that the exponential map is a diffeomorphism from U onto W0 which maps U ∩ g onto W0 ∩ G.

Observe that L A RA = RA L A, ad A = L A − R A . 4 The differential of the exponential map at A is given by ∞ (D exp) A X = exp A k=0 (−1)k (ad A)k X. (k + 1)! By putting, for z ∈ C, ∞ (z) = k=0 (−1)k k 1 − e−z z = , (k + 1)! z the statement can be written I − Exp(− ad A) , ad A where Exp T denotes the exponential of an endomorphism T of the vector space M(n, R). (D exp) A = L exp A ◦ (ad A) = L exp A ◦ Proof. (a) Let us consider the maps Fk : M(n, R) → M(n, R), X → Xk, and compute the differential of Fk at A: d (A + t X )k dt (D Fk ) A X = t=0 k−1 = Ak− j−1 X A j j=0 k−1 k− j−1 = j LA R A X.

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