By Robert Hermann
Hermann R. Geometric computing technology (Math Sci Press, 1991)(ISBN 0915692414)
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9. Let (G, µ) be a Lie group. Using the canonical identification T(a,b)(G×G) = Ta G⊕Tb G, prove the formula µ (a, b)(v, w) = Rb (a)(v) + La (b)(w) for all v ∈ Ta G and w ∈ Tb G. 10. Complete the proof of Proposition 3 by explicitly exhibiting the map c as a composition of known smooth maps. ) 11. Show that, for any v ∈ g, the left-invariant vector field Xv is indeed smooth. Also prove the first statement in Proposition 4. (Hint: Use Ψ to write the mapping Xv : G → T G as a composition of smooth maps.
This (partly) explains why the Riccati equation holds such an important place in the theory of ODE. In some sense, it is the first Lie equation which cannot be solved by quadratures. ) In any case, the sequence of subalgebras {gk } eventually stabilizes at a subalgebra gN whose Lie algebra satisfies [gN , gN ] = gN . A Lie algebra g for which [g, g] = g is called “perfect”. Our analysis of Lie equations shows that, by Lie’s reduction method, we can, by quadrature alone, reduce the problem of solving Lie equations to the problem of solving Lie equations associated to Lie groups with perfect algebras.
Show that, for the homomorphism det: GL(n, R) → R• , we have det (In )(x) = tr(x), where tr denotes the trace function. Conclude, using Theorem 1 that, for any matrix a, det(ea ) = etr(a) . 14. Prove that, for any g ∈ G and any x ∈ g, we have the identity g exp(x) g −1 = exp Ad(g)(x) . ) Use this to show that tr exp(x) ≥ −2 for all x ∈ sl(2, R). Conclude that exp: sl(2, R) → SL(2, R) is not surjective. (Hint: show that every x ∈ sl(2, R) is of the form gyg −1 for some g ∈ SL(2, R) and some y which is one of the matrices 0 ±1 0 0 , λ 0 0 −λ , or 0 λ −λ 0 , (λ > 0).