By Jacques Faraut.

ISBN-10: 0511422962

ISBN-13: 9780511422966

ISBN-10: 0521719305

ISBN-13: 9780521719308

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B) Deduce det(exp X ) = etr X . 4. Show that in G L(n, R) there is no arbitrary small subgroup. More precisely, show that there is a neighbourhood V of I in G L(n, R) such that, if H is a subgroup contained in V , then H = {I }. 5. 1. Let U = {X ∈ M(2, C) | X < r }; the number r is chosen such that the exponential map is a diffeomorphism from U onto its image V = exp U . There exists R > 0 such that V contains the ball B(I, R) = {g ∈ G L(2, C) | g − I < R}. 5 Exercises 49 For a positive integer m put X= i 0 0 i m , and, for t ∈ R, F(t) = exp t X = eit 0 0 t ei m .

Pk , qk , m ∈ N | pi + qi > 0, i = 1, . . , k}, and E(0) = {m ∈ N}. Proof. If A and B are two endomorphisms (exp A exp B − I )k exp A = E(k) A p1 B q 1 . . A pk B q k A m . q1 ! . m! Since (z) = ∞ k=0 (−1)k (z − 1)k z, k+1 we have Exp(ad X ) Exp(t ad Y ) Y = ∞ k=0 (−1)k Exp(ad X ) Exp(t ad Y ) − I k+1 k Exp(ad X ) Exp(t ad Y )Y. 5 Exercises 47 Observing that Exp(t ad Y )Y = Y, we obtain Exp(ad X ) Exp(t ad Y ) Y = ∞ k=0 (−1)k · k+1 t q1 +···+qk E(k) (ad X ) p1 (ad Y )q1 . . (ad X ) pk (ad Y )qk (ad X )m Y.

Let N p denote the set of nilpotent matrices of order p, N p = {X ∈ M(n, C) | X p = 0}, and U p the set of unipotent matrices of order p, U p = {g ∈ G L(n, C) | (g − I ) p = 0}. Show that the exponential map is a bijection from N p onto U p , whose inverse is the logarithm map. Hint. For X ∈ N p , log(exp t X ) − t X is a polynomial in t, vanishing on a neighbourhood of 0, hence identically zero. 12. Let A ∈ M(n, C) be a complex matrix for which there exists a constant C such that ∀t ∈ R, exp(t A) ≤ C.

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