By Bryant R.L.

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**Additional resources for An Introduction to Lie Groups and Symplectic Geometry**

**Example text**

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.