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Algebraic topology. Errata (web draft, Nov. 2004) by Hatcher A.

By Hatcher A.

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47) equals − β, [π (α), T (X)] + T ([π (α), X]) . 49) for v˜ constant; in this case, the equation is just the invariance of π . 48) equals (α) (ρ ∨ dJ (X)). 24 H. Bursztyn and M. 50) J )(X) . On the other hand, for all vector fields Y on M and g-valued 1-forms ν on G, we have iX LY (J ∗ ν) = LdJ (X) (ν(dJ (Y ))) + (dν)(dJ (Y ), dJ (X)). 47) and prove the claim. 41) evaluated at a vector field X is b, dρ ∨ ((σ ∨ )∗ a, V ) − a, dρ ∨ ((σ ∨ )∗ b, V ) − a, LV (D)b . 52) where we have used again that dJ π = −((σ )∨ )∗ (ρM )∗ .

On the other hand, if a Lie algebroid is integrable, then there exists a canonical source-simply-connected integration G(A); see [14]. If M is a point, then a Lie groupoid over M is a Lie group, and the associated Lie algebroid is its Lie algebra. 1 (transformation Lie groupoids). Let G be a Lie group acting from the left on a manifold M. The associated transformation Lie groupoid , denoted by G M, is a Lie groupoid over M with underlying manifold G × M, source map s(g, x) = x, target map t(g, x) = g · x, and multiplication (g, x) · (g , x ) = (gg , x ).

In Poisson geometry, Poisson maps are always associated with Lie algebroid actions: If (Q, πQ ) and (P , πP ) are Poisson manifolds, then any Poisson map J : Q → P induces a Lie algebroid action of T ∗ P on Q by 1 (P ) −→ X (Q), α → πQ (J ∗ α). 16) When the target P is the dual of a Lie algebra, we recover a familiar example. 8 (infinitesimal Hamiltonian actions). Consider g∗ equipped with its Lie– Poisson structure. 5) is that of a transformation Lie algebroid g g∗ with respect to the coadjoint action.

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