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 basically hassle-free wisdom of linear algebra and easy differential calculus. the writer describes, intimately, many fascinating examples, together with formulation that have no longer formerly seemed in booklet 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 services. excellent for complicated undergraduates and graduates in geometric research, harmonic research and illustration conception, the instruments built may also be valuable for experts in stochastic calculation and the statisticians. With a variety of workouts and labored examples, the textual content is perfect for a graduate path on research on Lie teams.

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3. Let us observe that G L(n, C) is a linear Lie group since it can be seen as a closed subgroup of G L(2n, R). In fact, to a matrix Z = X + iY in M(n, C) one associates the matrix Z˜ = X Y −Y X in M(2n, R), and the map Z → Z˜ is an algebra morphism which maps G L(n, C) onto a closed subgroup of G L(2n, R). 1 One parameter subgroups Let G be a topological group. A one parameter subgroup of G is a continuous group morphism γ : R → G, R being equipped with the additive group structure. 1 Let γ : R → G L(n, R) be a one parameter subgroup of G L(n, R).

6 If two closed subgroups G 1 and G 2 of G L(n, R) have the same Lie algebra then the identity components of G 1 and G 2 are the same. 1 that the group G is discrete if and only if its Lie algebra reduces to {0}: Lie(G) = {0}. To every closed subgroup G of G L(n, R) one associates its Lie algebra g = Lie(G) ⊂ M(n, R). However, not every Lie subalgebra of M(n, R) corresponds to a closed subgroup of G L(n, R). 4 Campbell–Hausdorff formula Let G be a linear Lie group and g = Lie(G) its Lie algebra.

B) Show that the range of the exponential map exp : x z y −x x, y, z ∈ R → S L(2, R), is equal to {g ∈ S L(2, R) | tr(g) > −2} ∪ {−I }. 3. Polar decomposition of complex matrices. Show that every matrix in g ∈ G L(n, C) can be written g = k exp X with k ∈ U (n) and X ∈ Herm(n, C). 30 The exponential map Show that the decomposition is unique, and that the map (k, X ) → g = k exp X, U (n) × Herm(n, C) → G L(n, C), is a homeomorphism. 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).

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