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By Jacques Faraut.

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It follows that the differential of the exponential map is given by (D exp) A = ∞ k=1 = ∞ k=1 = 1 (D Fk ) A k! k−1 1 k! ∞ j=0 = exp A k L k−i−1 (ad A)i i +1 A (−1)i i=0 ∞ 1 j L j! A ∞ i=0 i=0 (−1)i (ad A)i (i + 1)! (−1)i (ad A)i . (i + 1)! (c) Let us now establish the identity k−1 j=i j i = k . i +1 For k fixed put k−1 ai = j=i j . 2 Logarithm of a matrix 25 Then k−1 k−1 k−1 i=0 j=i i=0 = j=0 i=0 (z + 1) − 1 1 = z z k j k−1 j i z = i ai z i = k i=1 j i z = i k i z = i k−1 i=0 k−1 (z + 1) j j=0 k zi .

Observe that exp U = (U × {0}) ⊂ W ∩ G. 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.

In this way the properties of the group are translated in terms of the linear algebra properties of its Lie algebra. 3. Let us observe that G L(n, C) is a linear Lie group since it can be seen as a closed subgroup of G L(2n, R). In fact, to a matrix Z = X + iY in M(n, C) one associates the matrix Z˜ = X Y −Y X in M(2n, R), and the map Z → Z˜ is an algebra morphism which maps G L(n, C) onto a closed subgroup of G L(2n, R). 1 One parameter subgroups Let G be a topological group. A one parameter subgroup of G is a continuous group morphism γ : R → G, R being equipped with the additive group structure.

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