By Jacques Faraut

This self-contained textual content concentrates at the standpoint of study, assuming simply undemanding wisdom of linear algebra and simple differential calculus. the writer describes, intimately, many fascinating examples, together with formulation that have no longer formerly seemed in publication shape. subject matters coated comprise the Haar degree and invariant integration, round harmonics, Fourier research and the warmth equation, Poisson kernel, the Laplace equation and harmonic services. excellent for complex undergraduates and graduates in geometric research, harmonic research and illustration idea, the instruments built can be worthy for experts in stochastic calculation and the statisticians. With various workouts and labored examples, the textual content is perfect for a graduate path on research on Lie teams.

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**Additional info for Analysis on Lie Groups: An Introduction**

**Example text**

1 0 and then show that there exists α > 0 such that exp(t A) ∼ αe (λ)t |t|k−1 (t → ±∞). (b) Show that tr(A2 ) ≤ 0, and, if tr(A2 ) = 0, then A = 0. 3 Exercises 33 13. Let E denote the set of real matrices X in M(n, R) whose eigenvalues λ j satisfy | Im λ j | < π. The aim of this exercise is to show that the exponential map is a diffeomorphism from E onto its image exp(E). (a) Show that the set E is open and connected, and invariant under the maps X → g Xg −1 (b) (c) (d) (e) (f ) (g ∈ G L(n, R)).

4. Polar decomposition of unitary matrices. (a) Let u ∈ M(n, C) be a complex matrix which is symmetric and unitary: u ∈ Sym(n, C) ∩ U (n). Show that there is a real symmetric matrix X ∈ Sym(n, R) such that u = exp(i X ). (b) Let the matrix u be unitary: u ∈ U (n). Show that there is a real orthogonal matrix k ∈ O(n), and a real symmetric matrix X ∈ Sym(n, R), such that u = k exp(i X ). Are the matrices k and X unique ? 5. Polar decomposition of complex orthogonal matrices. The complex orthogonal group O(n, C) is defined by O(n, C) = {g ∈ M(n, C) | g −1 = g T }.

3 Exercises 31 (b) Define W0 (t) = exp(t A), t Wk (t) = exp((t − s)A)X Wk−1 (s)ds. 0 Show that the series ∞ Wk (t) k=0 converges for every t ∈ R, and that its sum is equal to F(t). (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 , .