By Jacques Faraut.

ISBN-10: 0511422962

ISBN-13: 9780511422966

ISBN-10: 0521719305

ISBN-13: 9780521719308

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C) Prove the formula 1 (D exp) A X = W1 (1) = exp((1 − s)A)X exp(s A)ds. 0 (d) For X ∈ M(n, R) and g ∈ G L(n, R) one puts Ad(g)X = g Xg −1 . 2) that Exp(ad A) = Ad(exp A). Show that the above formula can be written 1 (D exp) A X = exp A Exp(−s ad A)X ds, 0 and deduce that I − Exp(− ad A) . ad A 8. Let A ∈ M(n, C) with eigenvalues λ1 , . . , λn . (a) Show that the eigenvalues of L A are the numbers λ1 , . . , λn , each of them being repeated n times. ) Show that (D exp) A = L exp A ◦ Det(L A ) = det(A)n .

Show that G = F(R) is a closed subgroup in G L(2, C), and that g = Lie(G) = RX . Show that, for g = F(2π), π g − I = 2 sin . m Show that, for m large enough, g ∈ V and g ∈ / exp(U ∩ g), hence exp(U ∩ g) ⊆ V ∩ G. 4 Lie algebras In this chapter we consider Lie algebras from an algebraic point of view. We will see how some properties of linear Lie groups can be deduced from the corresponding properties of their Lie algebras. Then we present the basic properties of nilpotent, solvable, and semi-simple Lie algebras.

The group of all automorphisms of the Lie algebra g is denoted by Aut(g). If g is finite dimensional, it is a closed subgroup of G L(g). A derivation of g is a linear endomorphism D ∈ End(g) such that D([X, Y ]) = [D X, Y ] + [X, DY ]. For X ∈ g let ad X denote the endomorphism of g defined by ad X · Y = [X, Y ]. The Jacobi identity (2) says that ad X is a derivation. The space Der(g) of the derivations of g is a Lie algebra for the bracket defined by [D1 , D2 ] = D1 D2 − D2 D1 , and the map ad : g → Der (g) is a Lie algebra morphism: ad[X, Y ] = [ad X, ad Y ].

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