Download e-book for kindle: An Introduction to Lie Groups and Symplectic Geometry by Bryant R.L.

By Bryant R.L.

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For any φ ∈ Aut(P ), there is a unique smooth map ϕ: P → G which satisfies φ(p) = p · ϕ(p). The identity ρg ◦ φ = φ ◦ ρg implies that ϕ satisfies ϕ(p · g) = g −1 ϕ(p)g for all g ∈ G. Conversely, any smooth map ϕ: P → G satisfying this identity defines an element of Aut(P ). It follows that Aut(P ) is the space of sections of the bundle C(P ) = P ×C G where C: G × G → G is the conjugation action C(a, b) = aba−1 . Moreover, it easily follows that the set of vector fields on P whose flows generate 1-parameter subgroups of Aut(P ) is identifiable with the space of sections of the vector bundle Ad(P ) = P ×Ad g.

Lie regarded these latter two examples as “infinite continuous groups”. Nowadays, we would call them “infinite dimensional pseudo-groups”. I will say more about this point of view in an appendix to Lecture 6. Since Lie did not have a group manifold to work with, he did not regard his “infinite groups” as pathological. Instead of trying to find a global description of the groups, he worked with what he called the “infinitesimal transformations” of Γ. 15 52 for each of his groups Γ, he considered the space of vector fields γ ⊂ X(Rn ) whose (local) flows were 1-parameter “subgroups” of Γ.

0 0 1 Since Sa = 0 and since S is symmetric, it follows  s11 s12  S = s12 s22 0 0 that S must be of the form  0 0. 0 Moreover, a simple calculation shows that the result of applying a change of basis of the above form is to change the matrix S into the matrix   s11 s12 0 S =  s12 s22 0  0 0 0 where s11 s12 s12 s22 = 1 1 2 A1 A2 − A12 A21 A22 −A21 −A12 A11 s12 s22 s11 s12 A22 −A12 −A21 A11 . It follows that s11 s22 − (s12 )2 = s11 s22 − (s12 )2 , so there is an “invariant” to be dealt with.

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