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By Shafarevich I.R.

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In particular the induced maps E ∗ (g)θ=0 → H ∗ (g, R) → H ∗ (G, R) are isomorphisms. Sketch of proof. Let dg be a normalized Haar measure on G, that is, a normalized invariant measure. Since G is compact, we can use this measure to average any differential form with respect to the left action of G over itself. Namely we define a morphism ρ : E ∗ (G, R) → E ∗ (G, R)L by g ∗ ω dg. 1. 3]). To prove that j is a homotopy equivalence we can consider the action T of G × G on G given by Ta,b (x) = axb−1 and also use an averaging argument.

13. Let f : h → g be a morphism of Lie algebras. Then the induced morphism f ∗ : E ∗ (g) → E ∗ (h), restricts to a morphism of Hopf algebras f ∗ : E ∗ (g)θ=0 → E ∗ (h)θ=0 . In particular f ∗ P ∗ (g) = P ∗ (h). Let G be a compact Lie group, and let g be its Lie algebra. The cohomology H ∗ (G, R) has a structure of Hopf algebra induced by the multiplicative structure of G. 2 there is a natural isomorphism H ∗ (G, R) → H ∗ (g, R). 14. Let G be a compact Lie group, and let g be its Lie algebra. Then the natural isomorphism H ∗ (G, R) → H ∗ (g, R) is a Hopf algebra isomorphism.

P ) = Tr(φ1 ··· φp ). Observe that the homogeneous polynomials 1 Cp (φ) = Cp (φ, . . , φ) = Tr(∧p φ) p! are the coefficients of the characteristic polynomial of the endomorphism φ. 26. The trace coefficients of the complex vector space V are the elements Trp ∈ I 2p gl(V ) defined by Tr0 = dim V and Tr(φσ(1) ◦ · · · ◦ φσ(p) ). Trp (φ1 , . . 3]) Tr1 = C1 , n−1 Trn = (−1)n+1 nCn + (−1)j+1 Cj Trn−j , for n > 1. 27. Let Un be the unitary group. Then the elements ∗ ⊗ C as an algebra. Moreover, the class Cp , p = 0, .

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