By Bryant R.L.
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Additional info for An Introduction to Lie Groups and Symplectic Geometry
This theorem “explains” the importance of the Riccati equation, and why there are so few actions of Lie groups on R. Let g ⊂ X(R) be a ﬁnite dimensional Lie algebra of vector ﬁelds on R with the property that, at every x ∈ R, there is at least one X ∈ g so that X(x) = 0.
Thus, at ﬁrst glance, it might appear that Lie’s concept of a “continuous transformation group” should correspond to what we have deﬁned as a local Lie group action. However, it turns out that Lie had in mind a much more general concept. For Lie, a set Γ of local diﬀeomorphisms in Rn formed a “continuous transformation group” if it was closed under composition and inverse and moreover, the elements of Γ were characterized as the solutions of some system of diﬀerential equations. For example, the M¨ obius group on the line could be characterized as the set Γ of (non-constant) solutions f(x) of the diﬀerential equation 2f (x)f (x) − 3 f (x) 2 = 0.
Acting on Rn by the standard aﬃne action as before. If we embed Rn into Rn+1 by the rule x , x→ 1 then the standard aﬃne action of G on Rn extends to the standard linear action of G on Rn+1 . Note that G leaves invariant the subspace xn+1 = 0, and solutions of the Lie equation corresponding to a(t) b(t) A(t) = 0 0 which lie in this subspace are simply solutions to the homogeneous equation x (t) = a(t)x(t). , the fundamental solution to X (t) = a(x)X(t) with X(0) = In . This corresponds to knowing the n particular solutions to the Lie equation on Rn+1 which have the initial conditions e1 , .
An Introduction to Lie Groups and Symplectic Geometry by Bryant R.L.