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By Bryant R.L.

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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 , .

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An Introduction to Lie Groups and Symplectic Geometry by Bryant R.L.

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