By Jacques Faraut
This self-contained textual content concentrates at the point of view of study, assuming purely straightforward wisdom of linear algebra and easy differential calculus. the writer describes, intimately, many attention-grabbing examples, together with formulation that have no longer formerly seemed in publication shape. subject matters lined contain the Haar degree and invariant integration, round harmonics, Fourier research and the warmth equation, Poisson kernel, the Laplace equation and harmonic services. ideal for complex undergraduates and graduates in geometric research, harmonic research and illustration concept, the instruments built can be precious for experts in stochastic calculation and the statisticians. With a number of routines and labored examples, the textual content is perfect for a graduate direction on research on Lie teams.
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The critical goal of this ebook is to offer an advent to harmonic research and the speculation of unitary representations of Lie teams. the second one version has been mentioned so far with a few textual alterations in all the 5 chapters, a brand new appendix on Fatou's theorem has been further in reference to the bounds of discrete sequence, and the bibliography has been tripled in size.
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Extra info for Analysis on Lie Groups: An Introduction
A 24 The exponential map We will establish below (see (c)) the identity: k−1 j i j=i k . i +1 = Then k−1 (D Fk ) A = k L k−i−1 (ad A)i . i +1 A (−1)i i=0 (b) By (a) (D Fk ) A ≤ k A k−1 , ( (D Fk ) A denotes the norm of the endomorphism (D Fk ) A of the normed vector space M(n, R)) and the series of the differentials ∞ k=1 1 (D Fk ) A k! converges uniformly on every ball of M(n, R). 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!
The group of automorphisms of the Lie algebra g is denoted by Aut(g). Let G be a linear Lie group, and g = Lie(G) its Lie algebra. By the definition of the Lie algebra of G, the exponential map maps g into G: exp : g → G. For g ∈ G, X ∈ g, t ∈ R, g exp(t X )g −1 = exp(tg Xg −1 ). Hence g Xg −1 ∈ g. The map Ad(g) : X → Ad(g)X = g Xg −1 is an automorphism of the Lie algebra g, Ad(g)[X, Y ] = [Ad(g)X, Ad(g)Y ] (X, Y ∈ g). Furthermore Ad(g1 g2 ) = Ad(g1 ) ◦ Ad(g2 ), and this means that the map Ad : G → Aut(g) is a group morphism.
Let us prove that the function F satisfies the differential equation F (t) = Exp(ad F(t) Y. One can write exp F(t) = exp X exp tY. Taking the derivative at t: (D exp) F(t) F (t) = (exp X exp tY )Y. 4, we obtain ad F(t) F (t) = Y. Since ad F(t) < log 2 this can be written F (t) = Exp(ad F(t)) Y. 46 Linear Lie groups We can also write F (t) = Ad(exp F(t) Y = Ad(exp X ) Ad(exp tY ) Y = Exp(ad X ) Exp(ad tY ) Y. Furthermore F(0) = log(exp X ) = X , and 1 F(1) = F(0) + F (t)dt, 0 hence 1 log(exp X exp Y ) = X + Exp(ad X ) Exp(t ad Y ) Y dt.