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Lie Groupoids and Poisson Geometry LIE ALGEBROIDS, LIE GROUPOIDS AND POISSON GEOMETRY by Sebastien´ Racaniere` Abstract.— I wrote these notes for a series of lectures at Imperial College during the Summer Term 2004. The aim was to introduce symplectic groupoids, Hamiltonian actions of these groupoids and how this generalises other moment map theories. The study of these subjects in these notes is by no mean thorough. My hope is that these notes will instead consist in a nice introduction to the subject of symplectic groupoids and Poisson geometry. For a more detailed introduction, the reader might read Vaisman [8] for example. Contents 1. Lie Algebroids .......................................... 2 2. Poisson manifolds ........................................ 3 3. Lie Groupoids .......................................... 5 4. The Lie Algebroid of a Lie Groupoid ...................... 7 5. Lie’s Third Theorem ...................................... 10 6. Symplectic Lie groupoids ................................ 10 7. Hamiltonian actions of symplectic groupoids . 15 Appendix: Poisson Lie structure on a simply connected compact Lie group .............................................. 19 References ................................................ 19 I will denote , the natural pairing between a space and its dual. If Q is a manifold then TQ is its tangenth i space. Let f be a smooth map between Q and a vector space V . If X belongs to the tangent space of Q at a point p, then X acts on f and give an element X f in V . ∙ Real numbers are R, complex numbers are C, and so on . 2 SEBASTIEN´ RACANIERE` 1. Lie Algebroids A Lie algebroid is a real vector bundle with a Lie bracket on its space of sections which satisfies the Leibniz identity. More precisely, Definition 1.1.— Let Q be a smooth manifold. A (real) Lie algebroid over Q is a vector bundle A Q with an antisymmetric R-bilinear map on the space of smooth sections of A−→ Γ(A) Γ(A) Γ(A) × −→ (α, β) [α, β], 7−→ such that [[α, β], γ] + [[β, γ], α] + [[γ, α], β] = 0, for α, β, γ Γ(A) (Jacobi), ∈ and an endomorphism of vector bundles (called the anchor map) ρ : A TQ −→ which induces a homomorphism of Lie algebra between Γ(A) and X = Γ(TQ); moreover it should satisfy [α, fβ] = f[α, β] + (ρ(α) f)β (Leibniz). ∙ Remark 1.2.— It follows form the Leibniz identity that if β and β0 are two sections of A that agree on a neighbourhood of p Q, then ∈ [α, β](p) = [α, β0](p), α Γ(A). ∀ ∈ This means that [α, β](p) can be computed locally and in local coordinates, it de- pends only on α(p), β(p) and the first derivatives of α and β at p. Examples of Lie algebroids are numerous (not to say manifold). Later, we will see that Poisson manifolds can be defined in terms of Lie algebroids. Example 1.3.— Let Q be a manifold. Sections of TQ Q are vector fields. The tangent bundle TQ Q with bracket of sections the usual−→ bracket of vector fields and anchor map the−→ identity TQ TQ is a Lie algebroid called a pair algebroid. −→ Example 1.4.— A Lie algebra is a Lie algebroid over a point. Example 1.5.— Let Q be a manifold and k a real Lie algebra acting on Q; in other words we have a morphism of Lie algebras k Γ(TQ) −→ ξ vξ. 7−→ Consider A = k Q a trivial vector bundle over Q. Identify sections of A with maps Q k. Define× a bracket on sections by −→ [α, β](p) = [α(p), β(p)] + vβ p α vα p β, ( ) ∙ − ( ) ∙ LIE ALGEBROIDS, LIE GROUPOIDS AND POISSON GEOMETRY 3 and a morphism of vector bundles ρ : A TQ −→ (ξ, p) vξ(p). 7−→ These data define a Lie algebroid called an action algebroid. 2. Poisson manifolds Definition 2.1.— A Poisson structure on a manifold Q is a Lie algebroid structure on T ∗Q Q such that if [ , ] is the bracket on 1-forms and ρ : T ∗Q TQ is the anchor map,−→ then for any functions f, g on Q and p Q −→ ∈ [df, dg](p) = dp dg, ρ(df) . h i Notice that ρ is automatically antisymmetric. Indeed [df, df] = 0 for all functions f implies that d df, ρ(df) is constant for all f. Let c(f) be this value. Assume there exists f such thath c(f) = 0i, then 6 c(f 2) = 4f 2c(f), 2 c(f 2) and f is constant and equal to 4c(f) which implies that f is constant and c(f) = 0 ... Proposition 2.2.— Let Q be a Poisson manifold. There exists a Lie bracket on its algebra of smooth functions that is a R-bilinear antisymmetric map C∞(Q) C∞(Q) C∞(Q) × −→ (f, g) f, g 7−→ { } which satisfies for all functions f, g and h fg, h = f g, h + g f, h (Leibniz), { } { } { } and f, g , h + g, h , f + h, f , g = 0 (Jacobi). {{ } } {{ } } {{ } } The converse is also true, that is given a bracket on the algebra of smooth functions that satisfies the Leibniz and Jacobi identities, then Q is a Poisson manifold. This result is left as an exercise for those interested (alternatively, you can read [4]). Proof. — Define the bracket in the following way: let f and g be functions on Q and set f, g (p) = dpg, ρ(dpf) { } h i = ρ(dpf) g. ∙ 4 SEBASTIEN´ RACANIERE` This bracket is clearly R-bilinear. It is antisymmetric because f, g (p) = dpg, ρ(dpf) { } h i = ρ(dpg), dpf h− i = g, f (p). −{ } It satisfies the Leibniz identity because fg, h (p) = dph, ρ(dp(fg)) { } h i = dph, ρ(f(p)dpg + g(p)dpf) h i = f(p) g, h (p) + g(p) f, h (p). { } { } Finally, it satisfies the Jacobi identity because f, g , h = dh, ρ([df, dg]) {{ } } h i and g, h , f + h, f , g = df, ρ(d(ρ(dg) h)) + dg, ρ(d(ρ(df) h)) {{ } } {{ } } h ∙ i h − ∙ i = ρ(df) (ρ(dg) h) + ρ(dg) (ρ(df) h) − ∙ ∙ ∙ ∙ = dh, [ρ(df), ρ(dg)] , −h i so that f, g , h + g, h , f + h, f , g = dh, ρ([df, dg]) [ρ(df), ρ(dg)] {{ } } {{ } } {{ } } h − = 0, because ρ is a homomorphism of Lie algebra. Definition 2.3.— If Q and P are Poisson manifolds, a smooth map ϕ : P Q is −→ called Poisson if for all functions f, g in C∞(Q) ϕ∗ f, g = ϕ∗f, ϕ∗g . { } { } It is called anti-Poisson if ϕ∗ f, g = ϕ∗f, ϕ∗g . { } −{ } Particular examples of Poisson manifolds are the symplectic ones. Definition 2.4.— A symplectic manifold (Q, ω) is a manifold Q with a non-degenerate closed 2-form, that is dω = 0 and the map TQ T ∗Q −→ ξ ω(ξ, ) 7−→ ∙ is an isomorphism. Proposition 2.5.— A symplectic manifold (Q, ω) is naturally a Poisson manifold. Proof. — Use the isomorphism between TQ and T ∗Q to define a Lie algebroid structure on T ∗Q. LIE ALGEBROIDS, LIE GROUPOIDS AND POISSON GEOMETRY 5 3. Lie Groupoids In the language of categories, a groupoid is a small category in which all mor- phisms are invertible. A Lie groupoid is then a groupoid with a nice smooth structure. Definition 3.1.— A Lie groupoid is given by – two smooth manifolds G (the morphims or arrows) and Q (the objects or points), – two smooth maps s : G Q the source map and t : G Q the target map, – a smooth embedding ι :−→Q G (the identities or constant−→ arrows), −→ 1 – a smooth involution I : G G, also denoted x x− , – a multiplication −→ 7−→ m : G2 G −→ (x, y) x y, 7−→ ∙ where G2 = Gs t G = (x, y) G G s(x) = t(y) , × { ∈ × | } such that the source map and target map are surjective submersions (hence G2 is a smooth manifold because t and s are submersions), the multiplication is smooth and 1. s(x y) = s(y), t(x y) = t(x), 2. x (∙y z) = (x y) ∙z, 3. ι is∙ a section∙ of∙ both∙ s and t, 4. ι(t(x)) x = x = x ι(s(x)), 1 ∙ ∙1 5. s(x− ) = t(x), t(x− ) = s(x), 6. x x 1 = ι(t(x)), x 1 x = ι(s(x)), ∙ − − ∙ whenever (x, y) and (y, z) are in G2. I will identify Q with its image in G using ι; so that if p Q then also p G. Lie groupoids are almost as numerous as Lie algebroids.∈ Here are few examples.∈ Example 3.2.— Let Q be a manifold. Let G = Q Q and define × s(p, q) = q t(p, q) = p (p, q) (q, r) = (p, r) ∙ I(p, q) = (q, p) ι(p) = (p, p). These data define a Lie groupoid called a pair groupoid. Example 3.3.— A Lie group is a Lie groupoid with Q a singleton. 6 SEBASTIEN´ RACANIERE` Example 3.4.— Let Q be a manifold and K a Lie group acting on the left on this manifold. Let G = K Q and define × 1 s(k, p) = k− p ∙ t(k, p) = p 1 (k, p) (h, k− p) = (kh, p) ∙ ∙ 1 1 I(k, p) = (k− , k− p) ∙ ι(p) = (e, p) where e is the unit of K. These data form a Lie groupoid called an action groupoid. Example 3.5.— A particular and important case of the last example is when Q is k∗ the dual of a Lie algebra k and K, a Lie group integrating k, acts on k∗ by the co- adjoint action. The total space of the action groupoid is K k .
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