By Bryant R.L.
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Additional resources for An Introduction to Lie Groups and Symplectic Geometry
Xn of g∗ , we can write the expression for the Lie bracket as an element β ∈ g ⊗ Λ2 (g∗ ), in the form β = 12 cijk xi ⊗ xj ∧ xk . 21 1 6 m m cij cm xm ⊗ xi ∧ xj ∧ xk . k + cjk ci + cki cj 32 Exercise Set 2: Lie Groups 1. Show that for any real vector space of dimension n, the Lie group GL(V ) is isomorphic to GL(n, R). ) 2. Let G be a Lie group and let H be an abstract subgroup. Show that if there is an open neighborhood U of e in G so that H ∩ U is a smooth embedded submanifold of G, then H is a Lie subgroup of G.
It follows that the equation h (t) = Rh(t) B(t) is a Lie equation for h. In other words in order to ﬁnd the fundamental solution of a Lie equation for G when the particular solution with initial condition g(0) = m ∈ M is known, it suﬃces to solve a Lie equation in Gm ! This observation is known as Lie’s method of reduction. It shows how knowledge of a particular solution to a Lie equation simpliﬁes the search for the general solution. ) Of course, Lie’s method can be generalized. If one knows k particular solutions with initial values m1 , .
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. 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).
An Introduction to Lie Groups and Symplectic Geometry by Bryant R.L.