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1 Integrating Lie algebroids and inﬁnitesimal actions Lie groupoids are the global counterparts of Lie algebroids. In order to ﬁx our notation, we recall that a Lie groupoid over a manifold M consists of a manifold G together with 30 H. Bursztyn and M. , [8]. To simplify our notation, we will often identify an element x ∈ M with its image ε(x) ∈ G. 1) with anchor ρ = dt : ker(ds)|M → T M and bracket induced from the Lie bracket on X (G) via the identiﬁcation of sections (ker(ds)|M ) with right-invariant vector ﬁelds on G tangent to the s-ﬁbers.

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 ).

This concludes the proof of the claim. 27. 27, it follows that s is a −J ∗ φ G -IM form for A. To conclude that L = Im(r, s) is a Dirac structure, we must still prove that L has rank n = dim(M). 20). 32. The sequence j (r,s) 0 −→ T ∗ G −→ A −→ L −→ 0 is exact, where j (a) = (−J ∗ a, σ ∨ (a)), a ∈ T ∗ G. Proof. 31). We deﬁne the maps U : A −→ T ∗ G, i : L −→ A, 1 ∗ U (α, v) = − (ρ ∨ )∗ ρM (α) + σ (v), 4 1 i(X, α) = α, ρ ∨ dJ (X) . 57) 26 H. Bursztyn and M. 58) and these identities imply that the sequence is exact.