By Bryant R.L.

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7 44 Even with a minus sign, however, Proposition 1 implies that the subspace λ∗ (g) ⊂ X(M) is a (ﬁnite dimensional) Lie subalgebra of the Lie algebra of all vector ﬁelds on M. Example: Linear Fractional Transformations. Consider the M¨ obius action in1 troduced earlier of SL(2, R) on RP : a b c d ·s= as + b . cs + d A basis for the Lie algebra sl(2, R) is x= 0 1 0 0 , 1 0 0 −1 h= , y= 0 0 1 0 Thus, for example, the ﬂow Ψλy is given by Ψλy (t, s) = exp 0 t 0 0 ·s= 1 t 0 1 ·s= s = s − s2 t + · · · , ts + 1 so Yyλ = −s2 ∂/∂s.

24. For any Lie algebra g, there is a canonical bilinear pairing κ: g × g → R, called the Killing form, deﬁned by the rule: κ(x, y) = tr ad(x)ad(y) . 4 36 (i) Show that κ is symmetric and, if g is the Lie algebra of a Lie group G, then κ is Ad-invariant: κ Ad(g)x, Ad(g)y = κ(x, y) = κ(y, x). Show also that κ [z, x], y = −κ x, [z, y] . A Lie algebra g is said to be semi-simple if κ is a non-degenerate bilinear form on g. (ii) Show that, of all the 2- and 3-dimensional Lie algebras, only so(3) and sl(2, R) are semi-simple.

Moreover, we have ρg = ρh ◦ τ , thus establishing that the patches ρg overlap smoothly and hence that the patches deﬁne the structure of a smooth manifold on G · m. That the map φ: G → G· m is a smooth submersion and that the inclusion G· m → M is a smooth one-to-one immersion are now clear. It is worth remarking that the proof of Theorem 1 shows that the Lie algebra of Gm is the subspace gm . In particular, if Gm = {e}, then the map φ: G → M is a one-to-one immersion. The proof also brings out the fact that the orbit G · m can be identiﬁed with the left coset space G/Gm , which thereby inherits the structure of a smooth manifold.