By Jacques Faraut

This self-contained textual content concentrates at the viewpoint of research, assuming basically straightforward 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 ebook shape. themes lined contain the Haar degree and invariant integration, round harmonics, Fourier research and the warmth equation, Poisson kernel, the Laplace equation and harmonic capabilities. excellent for complex undergraduates and graduates in geometric research, harmonic research and illustration thought, the instruments built can also be invaluable for experts in stochastic calculation and the statisticians. With a variety of workouts and labored examples, the textual content is perfect for a graduate direction on research on Lie teams.

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**Example text**

T=0 (b) Put γ1 (t) = Exp(t ad X ), γ2 (t) = Ad(exp t X ). They are two one parameter subgroups of G L(g), and γ1 (0) = ad X, γ2 (0) = ad X. 1. 3 Linear Lie groups are submanifolds Let us recall first the definition of a submanifold in a finite dimensional real vector space. A submanifold of dimension m in R N is a subset M with the following property: for every x ∈ M there exists a neighbourhood U of 0 in R N , a neighbourhood W of x in R N and a diffeomorphism from U onto W such that (U ∩ Rm ) = W ∩ M.

Ad X ) pk (ad Y )qk (ad X )m Y. q1 ! . m! The convergence of the series is uniform for t in [0, 1]. The statement is obtained by termwise integration since 1 t q1 +···+qk dt = 0 1 . 5 1 1 1 log(exp X exp Y ) = X + Y + [X, Y ] + X, [X, Y ] + Y, [Y, X ] 2 12 12 + terms of degree ≥ 4. Proof. The terms of degree 2 and 3 are written in the following table. 5 Exercises 1. Let α be an irrational real number. (a) Show that Z + αZ is dense in R. 0 1 3 X, [X, Y ] 0 1 6 Y, [X, Y ] 48 Linear Lie groups (b) Let G be the subgroup of G L(2, C) defined by e2iπ t 0 G= 0 e2iπ αt t ∈R .

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.