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

By Bryant R.L.

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In terms of the corresponding map ϕ: P → G, we have A · φ = ϕ∗ (ωG ) + Ad ϕ−1 (A). It follows by direct computation that FA·φ = φ∗ (FA ) = Ad ϕ−1 (FA ). We say that A is flat if FA = 0. It is an elementary ode result that A is flat if and only if, for every m ∈ M, there exists an open neighborhood U of m and a smooth map τ : π −1 (U) → G which satisfies τ (p · g) = τ (p)g and τ ∗ (ωG ) = A|U . In other words A is flat if and only if the bundle-with-connection (P, A) is locally diffeomorphic to the trivial bundle-with-connection (M × G, ωG ).

Show that SL(2, R) is not a matrix group! In fact, show that any homomorphism φ: SL(2, R) → GL(n, R) factors through the projections SL(2, R) → SL(2, R). (Hint: Recall, from earlier exercises, that the inclusion map SL(2, R) → SL(2, C) induces the zero map on π1 since SL(2, C) is simply connected. Now, any homomorphism φ: SL(2, R) → GL(n, R) induces a Lie algebra homomorphism φ (e): sl(2, R) → gl(n, R) and this may clearly be complexified to yield a Lie algebra homomorphism φ (e)C : sl(2, C) → gl(n, C).

18. Show that a connected Lie group G is abelian if and only if its Lie algebra satisfies [x, y] = 0 for all x, y ∈ g. Conclude that a connected abelian Lie group of dimension n is isomorphic to Rn /Zd where Zd is some discrete subgroup of rank d ≤ n. (Hint: To show “G abelian” implies “g abelian”, look at how [, ] was defined. ) ˜ → G be 19. ) Let G be a connected Lie group and let π: G ˜ can be regarded as the space the universal covering space of G. ) Show ˜×G ˜→G ˜ for which the homotopy class of that there is a unique Lie group structure µ ˜: G ˜ is the identity and so that π is a homomorphism.

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