By Bryant R.L.

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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 ad: g → End(g) actually has its image in der(g), and that this image is an ideal in der(g). What is the interpretation of this fact in terms of “inner” and “outer” automorphisms of G? ) Show that if the Killing form of g is non-degenerate, then [g, g] = g. (Hint: Suppose that [g, g] lies in a proper subspace of g. Then there exists an element y ∈ g so that κ [x, z], y = 0 for all x, z ∈ g. ) Show that if the Killing form of g is non-degenerate, then der(g) = ad(g). This shows that all of the automorphisms of a simple Lie algebra are “inner”.