By Bryant R.L.
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Extra info for An Introduction to Lie Groups and Symplectic Geometry
Once this is done, the rest of the construction of charts with smooth overlaps follows the end of the proof of Theorem 1 almost verbatim. 6 43 Group Actions and Vector Fields. A left action λ: R × M → M (where R has its usual additive Lie group structure) is, of course, the same thing as a ﬂow. Associated to each ﬂow on M is a vector ﬁeld which generates this ﬂow. The generalization of this association to more general Lie group actions is the subject of this section. Let λ: G × M → M be a left action.
Finally, since the map v → Yv is [Yx , Yy ] = −Y[x,y] and hence must be equal to −Y[x,y] clearly linear, it follows that λ∗ is also linear. The appearance of the minus sign in the above formula is something of an annoyance and has led some authors (cf. [A]) to introduce a non-classical minus sign into either the deﬁnition of the Lie bracket of vector ﬁelds or the deﬁnition of the Lie bracket on g in order to get rid of the minus sign in this theorem. Unfortunately, as logical as this revisionism is, it has not been particularly popular.
Let λ: G × M → M be a left action. Then, for each v ∈ g, there is a ﬂow Ψλv on M deﬁned by the formula Ψλv (t, m) = etv · m. This ﬂow is associated to a vector ﬁeld on M which we shall denote by Yvλ , or simply Yv if the action λ is clear from context. This deﬁnes a mapping λ∗ : g → X(M), where λ∗ (v) = Yvλ . Proposition 1: For each left action λ: G × M → M, the mapping λ∗ is a linear antihomomorphism from g to X(M). In other words, λ∗ is linear and λ∗ ([x, y]) = −[λ∗ (x), λ∗ (y)]. Proof: For each v ∈ g, let Yv denote the right invariant vector ﬁeld on G whose value at e is v.