Analysis on Lie Groups: An Introduction by Jacques Faraut

By Jacques Faraut

This self-contained textual content concentrates at the point of view of study, assuming purely simple wisdom of linear algebra and simple differential calculus. the writer describes, intimately, many attention-grabbing examples, together with formulation that have now not formerly seemed in e-book shape. issues lined comprise the Haar degree and invariant integration, round harmonics, Fourier research and the warmth equation, Poisson kernel, the Laplace equation and harmonic capabilities. ideal for complicated undergraduates and graduates in geometric research, harmonic research and illustration conception, the instruments built may also be worthwhile for experts in stochastic calculation and the statisticians. With quite a few workouts and labored examples, the textual content is perfect for a graduate path on research on Lie teams.

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The differential of p˜ at A is defined by d p˜ (A + t X ) t=0 . dt One assumes here that the matrix A = is diagonal. Show that (D p˜ ) A (X ) = (D p˜ ) (X ) = M p, (X ). Hint. Consider first the case of p(λ) = λm . Recall that d (A + t X )m dt m−1 t=0 = Am−k−1 X Ak . k=0 (iii) Show that, if A = k k , where k is an orthogonal matrix and diagonal, then T (D p˜ ) A (X ) = k M p, (k T X k)k T . (iv) Show that, if f ∈ C 1 (R), then the map f˜ is differentiable, and that, if is diagonal, (D f˜ ) (X ) = M f, (X ).

If G ⊂ G L(n, R) is a linear Lie group, then g = Lie(G) is a subalgebra of M(n, R), it is the Lie algebra of G. 2 Lie algebra of a linear Lie group 39 Examples. Lie G L(n, R) = M(n, R), Lie S L(n, R) = X ∈ M(n, R) | tr X = 0 , Lie S O(n) = X ∈ M(n, R) | X T = −X , Lie Sp(n, R) = A C B −A T A ∈ M(n, R), B, C ∈ Sym(n, R) , Lie U (n) = X ∈ M(n, C) | X ∗ = −X . Consider G = S L(2, R) and let g = sl(2, R) be its Lie algebra. The following matrices constitute a basis of g: H= 1 0 0 −1 , E= 0 0 1 0 , 0 1 F= 0 0 , and [H, E] = 2E, [H, F] = −2F, [E, F] = H.

It follows that X and Y are diagonalisable with respect to the same basis: one can take h = k, and then eλi = eµi , hence λi = µi . 22 The exponential map (c) Continuity. The exponential map is continuous. For α > 0 let E be the closed ball E = {X ∈ Sym(n, R) | X ≤ α}. 3). The exponential maps continuously and injectively the compact set E onto the compact set F, and hence is a homeomorphism from E onto F. It follows that it is a homeomorphism from Sym(n, R) onto Pn . 2 Every matrix g ∈ G L(n, R) can be written g = k exp X , with k ∈ O(n), X ∈ Sym(n, R), and the map (k, X ) → k exp X, O(n) × Sym(n, R) → G L(n, R), is a homeomorphism The exponential map is real analytic, hence C ∞ .

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