Symmetry And Group

Analysis on Lie Groups - Jacques Faraut by Jacques Faraut.

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By Jacques Faraut.

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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 . Determine the closure G¯ of G in G L(2, C). (c) Show that there does not exist any closed subgroup of G L(2, C) with Lie algebra g= it 0 0 iαt t ∈R . 2. Let G be a linear Lie group and g its Lie algebra. One assumes that G is Abelian. (a) Show that g is Abelian, that is ∀X, Y ∈ g, [X, Y ] = 0.

Proof. If A and B are two endomorphisms (exp A exp B − I )k exp A = E(k) A p1 B q 1 . . A pk B q k A m . q1 ! . m! Since (z) = ∞ k=0 (−1)k (z − 1)k z, k+1 we have Exp(ad X ) Exp(t ad Y ) Y = ∞ k=0 (−1)k Exp(ad X ) Exp(t ad Y ) − I k+1 k Exp(ad X ) Exp(t ad Y )Y. 5 Exercises 47 Observing that Exp(t ad Y )Y = Y, we obtain Exp(ad X ) Exp(t ad Y ) Y = ∞ k=0 (−1)k · k+1 t q1 +···+qk E(k) (ad X ) p1 (ad Y )q1 . . (ad X ) pk (ad Y )qk (ad X )m Y. q1 ! . m! The convergence of the series is uniform for t in [0, 1].

1 The exponential map is a homeomorphism from Sym(n, R) onto Pn . Proof. (a) Surjectivity. Let p ∈ Pn , and λ1 > 0, . . , λn > 0 be its eigenvalues. There exists k ∈ O(n) such that λ  1 .. p =k  k −1 . λn Put  log λ  1 X =k ..  k −1 . log λn Then exp X = p. (b) Injectivity. Let X and Y ∈ Sym(n, R) be such that exp X = exp Y . Let us diagonalise X and Y : λ  1 .. X = k   k −1 , . λn e λ1 .. exp X = k   k −1 , e λn 1 .. Y = h  exp Y = h   . µ   h −1 , . µn e µ1 .. k ∈ O(n), h ∈ O(n),   h −1 .

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