INTRODUCTION to POISSON GEOMETRY LECTURE NOTES, WINTER 2017 Contents 1. Poisson Manifolds 3 1.1. Basic Definitions 3 1.2. Deform

Home , Kernel (linear algebra), Poisson manifold

INTRODUCTION TO POISSON GEOMETRY LECTURE NOTES, WINTER 2017

ECKHARD MEINRENKEN

Abstract. These notes are very much under construction. In particular, the references are very incomplete. Apologies!

Contents 1. Poisson manifolds 3 1.1. Basic deﬁnitions 3 1.2. Deformation of algebras 4 1.3. Basic properties of Poisson manifolds 6 1.4. Examples of Poisson structures 7 1.5. Casimir functions 9 1.6. Tangent lifts of Poisson structures 10 2. Lie algebroids as Poisson manifolds 10 2.1. Lie algebroids 11 2.2. Linear Poisson structures on vector bundles 12 2.3. The cotangent Lie algebroid of a Poisson manifold 15 2.4. Lie algebroid comorphisms 15 2.5. Lie subalgebroids and LA-morphisms 17 3. Submanifolds of Poisson manifolds 19 3.1. Poisson submanifolds 19 3.2. Symplectic leaves 20 3.3. Coisotropic submanifolds 21 3.4. Applications to Lie algebroids 25 3.5. Poisson-Dirac submanifolds 26 3.6. Cosymplectic submanifolds 28 4. Dirac structures 29 4.1. The Courant bracket 30 4.2. Dirac structures 32 4.3. Tangent lifts of Dirac structures 33 5. Gauge transformations of Poisson and Dirac structures 34 5.1. Automorphisms of the Courant algebroid structure 34 5.2. Moser method for Poisson manifolds 37 6. Dirac morphisms 39 6.1. Morphisms of Dirac structures 39 6.2. Pull-backs of Dirac structures 41 1 2 ECKHARD MEINRENKEN

7. Normal bundles and Euler-like vector fields 42 7.1. Normal bundles 42 7.2. Tubular neighborhood embeddings 43 7.3. The Grabowski-Rotkiewicz theorem 45 8. The splitting theorem for Lie algebroids 47 8.1. Statement of the theorem 47 8.2. Normal derivative 48 8.3. Anchored vector bundles 48 8.4. Proof of the splitting theorem for Lie algebroids 49 8.5. The Stefan-Sussmann theorem 51 8.6. The Weinstein splitting theorem 52 9. The Karasev-Weinstein symplectic realization theorem 55 9.1. Symplectic realizations 55 9.2. The Crainic-M˘arcut¸formula 59 9.3. Linear realizations of linear Poisson structures 61 10. Lie groupoids 62 10.1. Basic definitions and examples 62 10.2. Left-invariant and right-invariant vector fields 64 10.3. Bisections 64 10.4. The Lie algebroid of a Lie groupoid 65 10.5. Groupoid multiplication via σL, σR 67 11. Symplectic groupoids 68 11.1. Definition, basic properties 68 11.2. Lagrangian bisections 70 11.3. The Lie algebroid of a symplectic groupoid 70 11.4. The cotangent groupoid 71 11.5. Integration of Poisson manifolds 72 11.6. Integration of Lie algebroids 74 References 75 INTRODUCTION TO POISSON GEOMETRY 3

1. Poisson manifolds 1.1. Basic definitions. Poisson structures on manifolds can be described in several equivalent ways. The quickest definition is in terms of a bracket operation on smooth functions. Definition 1.1. [27] A Poisson structure on a manifold M is a skew-symmetric bilinear map {·, ·}: C∞(M) × C∞(M) → C∞(M) with the derivation property (1) {f, gh} = {f, g}h + g{f, h} and the Jacobi identity (2) {f, {g, h}} = {{f, g}, h} + {g, {f, h}}, for all f, g, h ∈ C∞(M). The manifold M together with a Poisson structure is called a Poisson manifold. A map Φ: N → M between Poisson manifolds is a Poisson map if the pull-back map Φ∗ : C∞(M) → C∞(N) preserves brackets. Condition (2) means that {·, ·} is a Lie bracket on C∞(M), making the space of smooth functions into a Lie algebra. Condition (1) means that for all f ∈ C∞(M), the operator {f, ·} is a derivation of the algebra of smooth functions C∞(M), that is, a vector field. One calls

Xf = {f, ·} the Hamiltonian vector ﬁeld determined by the Hamiltonian f. In various physics interpreta- tions, the ﬂow of Xf describes the dynamics of a system with Hamiltonian f.

2n 1 n Example 1.2. The standard Poisson bracket on ‘phase space’ R , with coordinates q , . . . , q and p1, . . . , pn, is given by n X ∂f ∂g ∂f ∂g {f, g} = − . ∂qi ∂p ∂p ∂qi i=1 i i The Jacobi identity may be verified by direct computation, using the formula for the bracket. (Of course, one can do much better than ‘direct computation’ – see below.) The differential equations defined by the Hamiltonian vector field Xf are Hamilton’s equations

i ∂f ∂f q˙ = , p˙i = − i ∂pi ∂q from classical mechanics. Here our sign conventions (cf. Appendix ??) are such that a vector ﬁeld X ∂ X = aj(x) ∂xi j N on R corresponds to the ODE dxj = −ajx(t). dt ∞ A function g ∈ C (M) with Xf (g) = 0 is a conserved quantity, that is, t 7→ g(x(t)) is constant for any solution curve x(t) of Xf . One of Poisson’s motivation for introducing his bracket was the realization that if g and h are two conserved quantities then {g, h} is again a conserved quantity. This was explained more clearly by Jacobi, by introducing the Jacobi identity (1). 4 ECKHARD MEINRENKEN

Example 1.3. Let g be a ﬁnite-dimensional Lie algebra, with basis and corresponding structure k P k ∞ ∗ constants cij deﬁned by [i, j] = k cijk. On the C (g ), we have the bracket

X k ∂f ∂g (3) {f, g}(µ) = cijµk . ∂µi ∂µj ijk One checks that this does not depend on the choice of basis, and that {·, ·} is a Poisson bracket. (The Jacobi identity for g becomes the Jacobi identity for g∗.) It is called the Lie- Poisson structure on g∗, since it was discovered by Lie in his foundational work in the late 19th century, and is also known as the Kirillov-Poisson structure, since it plays a key role in Kirillov’s ∞ ∗ orbit method in representation theory. The Poisson structure is such that φξ ∈ C (g ) is the linear map deﬁned by a Lie algebra element ξ ∈ g, then

(4) {φξ, φζ } = φ[ξ,ζ]

The Hamiltonian vector ﬁeld Xφξ is the generating vector ﬁeld corresponding to ξ, for the ∗ P i P i coadjoint G-action on g . Writing ξ = i ξ i, we have φξ(µ) = i ξ µi, hence

X k i ∂ Xφξ = cij µk ξ . ∂µj ijk 1.2. Deformation of algebras. Classical mechanics and Lie theory are thus two of the major inspirations for Poisson geometry. A more recent motivation comes from deformation theory. Consider the problem of deforming the product on the algebra of smooth functions C∞(M), to a possibly non-commutative product. Thus, we are interested in a family of products f ·~ g depending smoothly on a parameter ~, always with the constant function 1 as a unit, and with f ·0 g the usual (pointwise) product. The commutator f ·~ g − g ·~ f vanishes to ﬁrst order in ~, let {f, g} be its linear term: d

{f, g} = (f ·~ g − g ·~ f) d~ ~=0 2 so that f ·~ g − g ·~ f = ~{f, g} + O(~ ). Then {·, ·} is a Poisson bracket. This follows since for any associative algebra A, the commutation [a, b] = ab − ba satisﬁes (5) [a, bc] = [a, b]c + b[a, c] and (6) [a, [b, c]] = [[a, b], c] + [b, [a, c]], hence the properties (1) and (2) of the bracket follow by applying these formulas for A = ∞ C (M) with product ·~, and taking the appropriate term of the Taylor expansion in ~ of both ∞ sides. Conversely, C (M) with the deformed product ·~ could then be called a quantization of the Poisson bracket on C∞(M). Unfortunately, there are few concrete examples of ‘strict’ quantizations in this sense. More is known for the so-called formal deformations of the algebra C∞(M).

Deﬁnition 1.4. Let A be an associative algebra over R.A formal deformation of A is an algebra structure on A[[~]] (formal power series in ~ with coeﬃcients in A), such that (a) The product is R[[~]]-linear in both arguments. INTRODUCTION TO POISSON GEOMETRY 5

(b) The isomorphism A[[ ]] ~ =∼ A ~A[[~]] is an isomorphism of algebras.

(Note that by (a), the subspace ~A[[~]] is a two-sided ideal in A[[~]], hence the quotient space inherits an algebra structure.) The product of A[[~]] is usually denoted ∗. We have A ⊆ A[[~]] as a subspace, but not as a subalgebra. The product ∗ is uniquely determined by what it is on A. For a, b ∈ A we have 2 a ∗ b = ab + ~F1(a, b) + ~ F2(a, b) + ··· ∞ As before, any formal deformation of A = C (M) gives a Poisson bracket {f, g} = F1(a, b) − F2(b, a). Deﬁnition 1.5. A deformation quantization of a Poisson manifold (M, {·, ·}) is given by a star ∞ product on C (M)[[~]], with the following properties: ∞ ∞ (i) (C (M)[[~]], ∗) is a deformation of the algebra structure on C (M). (ii) The terms Fi(f, g) are given by bi-diﬀerential operators in f and g. (iii) F1(f, g) − F2(g, f) = {f, g}. ∞ ∞ Conversely, we think of (C (M)[[~]], ∗) as a deformation quantization of (C (M), ·, {·, ·}). One often imposes the additional condition that 1 ∗ f = f ∗ 1 = f for all f.

∞ 2n Example 1.6. An example of a deformation quantization is the Moyal quantization of C (R ), with the standard Poisson bracket. Let µ: C∞(M) ⊗ C∞(M) → C∞(M) be the standard pointwise product. Then f ∗ g = µD(f ⊗ g) where D is the infinite-order ‘formal’ differential operator on M × M X ∂ ∂ ∂ ∂ D = exp ~ ⊗ − ⊗ . 2 ∂qi ∂p ∂p ∂qi i i i It is an exercise to check that this does indeed define an associative multiplication. Example 1.7. Let g be a finite-dimensional algebra. The universal enveloping algebra Ug is the algebra linearly generated by g, with relations XY − YX = [X,Y ] (where the right hand side is the Lie bracket). Note that if the bracket is zero, then this is the symmetric algebra. In fact, as a vector space, Ug is isomorphic to Sg, the symmetrization map 1 X Sg → Ug,X ··· X 7→ X ··· X 1 r r! s(1) s(r) s∈Sr where the right hand side uses the product in Ug. The fact that this map is an isomorphism is a version of the Poincaré-Birkhoff-Witttheorem. Using this map, we may transfer the product of U(g) to a product on S(g). In fact, we putting a parameter ~ in front of the Lie bracket, 6 ECKHARD MEINRENKEN we obtain a family of algebra structures on S(g), which we may also regard as a product on S(g)[[~]]. On low degree polynomials, this product can be calculated by hand: In particular,

X ∗ Y = XY + ~[X,Y ] 2 for X,Y ∈ g ⊆ S(g)[[~]]. The resulting Poisson structure on S(g) is just the Lie-Poisson structure, if we regard S(g) as the polynomial functions on g∗. Hence, we obtain a canonical quantization of the Lie-Poisson structure, given essentially by the universal enveloping algebra. The question of whether every Poisson structure admits a deformation quantization was settled (in the aﬃrmative) by Kontsevich, in his famous 1997 paper, “Deformation quantization of Poisson manifolds”.

1.3. Basic properties of Poisson manifolds. A skew-symmetric bilinear {·, ·} satisfying (1) is a derivation in both arguments. In particular, the value of {f, g} at any given point depnds only on the differential df, dg at that point. This defines a bi-vector field π ∈ X2(M) = Γ(∧2TM) such that π(df, dg) = {f, g} for all functions f, g. Conversely, given a bivector field π, one obtains a skew-symmetric bracket {·, ·} on functions satisfying the derivation property. Given bivector fields π1 on M1 and π2 on M2, with corresponding brackets {·, ·}1 and {·, ·}2, then a smooth map Φ: M1 → M2 is bracket preserving if and only if the bivector fields are Φ-related

π1 ∼Φ π2, that is, TmΦ((π1)m) = (π2)Φ(m) for all m ∈ M1 where TmΦ is the tangent map (extended to multi-tangent vectors). We will call π a Poisson bivector field (or Poisson structure) on M if the associated bracket {·, ·} is Poisson, that is, if it satisfies the Jacobi identity. Consider the Jacobiator Jac(·, ·, ·) defined as (7) Jac(f, g, h) = {f, {g, h}} + {g, {h, f}} + {h, {f, g}} for f, g, h ∈ C∞(M). Clearly, Jac(f, g, h) is skew-symmetric in its three arguments. Here π] : T ∗M → TM be the bundle map defined by π, i.e. π](α) = π(α, ·). Then the Hamiltonian ] vector field associated to a function f is Xf = {f, ·} = π (df). We have the following alternative formulas for the Jacobiator:

(8) Jac(f, g, h) = L[Xf ,Xg]h − LX{f,g} h = (LXf π)(dg, dh). The first equality shows that Jac is a derivation in the last argument h, hence (by skew- symmetry) in all three arguments. It follows the values of Jac(f, g, h) at any given point depend only on the exterior differentials of f, g, h at that point, and we obtain a 3-vector field 3 Υπ ∈ X (M), Υπ(df, dg, dh) = Jac(f, g, h). We hence see: INTRODUCTION TO POISSON GEOMETRY 7

Proposition 1.8. We have the equivalences,

{·, ·} is a Poisson bracket ⇔ [Xf ,Xg] = X{f,g} for all f, g

⇔ LXf π = 0 for all f ] ] ⇔ LXf ◦ π = π ◦ LXf for all f ⇔ Υπ = 0. Let us point out the following useful consequence: Remark 1.9. To check if {·, ·} satisﬁes the Jacobi identity, it is enough to check on functions ∗ whose diﬀerentials span T M everywhere. (Indeed, to verify Υπ = 0 at any given m ∈ M, we ∗ only have to check on covectors spanning TmM.)

Remark 1.10. In terms of the Schouten bracket of multi-vector fields, the 3-vector field Υπ 1 associated to a bivector field π is given by Υπ = − 2 [π, π]. Thus, π defines a Poisson structure if and only if [π, π] = 0. 1.4. Examples of Poisson structures. Example 1.11. Every constant bivector field on a vector space is a Poisson structure. Choosing 1 P ∂ ∂ a basis, this means that π = 2 Aij ∂xi ∧ ∂xj for any skew-symmetric matrix A is a Poisson structure. This follows from Remark 1.9, since we only need to check the Jacobi identity on the coordinate functions. But since the bracket of two linear functions is constant, and the bracket with a constant function is zero, all three terms in the Jacobiator are zero in that case. 2n As a special case, the bivector field on R given as n X ∂ ∂ (9) π = ∧ . ∂qi ∂p i=1 i is Poisson. Example 1.12. Similarly, if g is a Lie algebra, the bracket {·, ·} on C∞(g∗) given by (3) corresponds to the bivector field 1 X k k ∂ ∂ π = cij µ ∧ 2 ∂µi ∂µj ijk on g∗. By Remark 1.9, to verify the Jacobi identity, we only need to check on linear functions φξ, ξ ∈ g. But on linear functions, the Jacobi identity for the bracket reduces to the Jacobi identity for the Lie algebra g. Example 1.13. Any symplectic manifold (M, ω) becomes a Poisson manifold, in such a way that the Hamiltonian vector fields Xf = {f, ·} satisfy ω(Xf , ·) = −df. In local symplectic 1 n P i coordinates q , . . . , q , p1, . . . , pn, with ω = i dq ∧ dpi, the Poisson structure is given by the formula (9) above. Note that with our sign conventions, the two maps π] : T ∗M → T M, µ 7→ π(µ, ·), and ω[ : TM → T ∗M, v 7→ ω(v, ·) are related by π] = −(ω[)−1. 8 ECKHARD MEINRENKEN

Example 1.14. If dim M = 2, then any bivector ﬁeld π ∈ X2(M) is Poisson: The vanishing of Υπ follows because on a 2-dimensional manifold, every 3-vector ﬁeld is zero.

Example 1.15. If (M1, π1) and (M2, π2) are Poisson manifolds, then their direct product M1 × M2 is again a Poisson manifold, with the Poisson tensor π = π1+π2. To check that this is indeed a Poisson tensor field, using Remark 1.9 it suffices to check the Jacobi identity for functions that are pullbacks under one of the projections pri : M1 × M2 → Mi, but this is immediate. Put differently, the bracket is such that both projections pri : M1 → M2 are Poisson maps, and ∗ ∞ ∞ the two subalgebras pri C (Mi) ⊆ C (M1 × M2) Poisson commute among each other. Warning: While we usually refer to this operation as a direct product of Poisson manifolds, it is not a direct product in the categorical sense. For the latter, it would be required that whenever N is a Poisson manifold with two Poisson maps fi : N → Mi, the diagonal map N → M1 × M2 is Poisson. But this is rarely the case. Example 1.16. If A is a skew-symmetric n × n-matrix, then 1 X ∂ ∂ π = A xixj ∧ 2 ij ∂xi ∂xj ij n is a Poisson structure on R . Here are two simple ways of seeing this: (i) On the open, dense i subset where all xi 6= 0, the differentials of the functions fi(x) = log(|x |) span the cotangent space. But the Poisson bracket of two such functions is constant. (ii) Using a linear change of coordinates, we can make A block-diagonal wth 2 × 2-blocks, and possibly one 1 × 1-block 1 2 ∂ ∂ with entry 0. This reduces the question to the case n = 2; but π = x x ∂x1 ∧ ∂x2 is a Poisson structure by the preceding example. 3 Example 1.17. R has a Poisson structure π0 given as

{x, y}0 = z, {y, z}0 = x, {z, x}0 = y. The corresponding Poisson tensor ﬁeld is ∂ ∂ ∂ ∂ ∂ ∂ π = z ∧ + x ∧ + y ∧ . 0 ∂x ∂y ∂y ∂z ∂z ∂x Actually, we know this example already: It is the Poisson structure on g∗ for g = so(3) (in a 3 standard basis). Another Poisson structure on R is ∂ ∂ ∂ ∂ ∂ ∂ π = xy ∧ + yz ∧ + zx ∧ , 1 ∂x ∂y ∂y ∂z ∂z ∂x as a special case of the quadratic Poisson structures from Example 1.16. In fact, all

πt = (1 − t)π0 + tπ1 with t ∈ R are again Poisson structures. (It suﬃce to verify the Jacobi identity for f = x, g = y, h = z). This is an example of a Poisson pencil.

Exercise: Show that if π0, π1 are Poisson structures on a manifold M such that πt = (1 − t)π0 + tπ1 is a Poisson structure for some t 6= 0, 1, then it is a Poisson structure for all 2 t ∈ R. In other words, given three Poisson structures on an aﬀne line in X (M), then the entire line consists of Poisson structures. INTRODUCTION TO POISSON GEOMETRY 9

3 Example 1.18. Another Poisson structure on R : {x, y} = xy, {z, x} = xz, {y, z} = φ(x) for any smooth function φ. Indeed, {x, {y, z}} + {y, {z, x}} + {z, {x, y}} = {x, φ(x)} + {y, xz} + {z, xy} = −{x, y}z + x{y, z} + {z, x}y − {y, z}x = −(xy)z + (zx)y = 0. Example 1.19. Let M be a Poisson manifold, and Φ: M → M a Poisson automorphism. Then the the group Z acts on M × R by Poisson automorphism, generated by (m, t) 7→ (Φ(m), t + 1), and the mapping cylinder (M × R)/Z inherits a Poisson structure. 1 P i j n Example 1.20. Given a 2-form α = 2 ij αij(q) dq ∧ dq on R , we can change the Poisson 2n tensor on R to the bivector field n n X ∂ ∂ 1 X ∂ ∂ (10) π = ∧ + α (q) ∧ . ∂qi ∂p 2 ij ∂p ∂p i=1 i i,j=1 i j Is π a Poisson structure? When checking the Jacobi identity on linear functions, only the sum over cyclic permutations of {pi, {pj, pk}} is an issue. One finds ∂α {p , {p , p }} = − jk , i j k ∂qi so the sum over cyclic permutations of this expression vanishes if and only if dα = 0. This example generalizes to cotangent bundles T ∗Q (with their standard symplectic structure): 2 ∗ Given a closed 2-form α ∈ Ω (Q), we can regard α as a vertical bi-vector field πα on T Q. ∗ 2 ∗ (The constant bivector fields on Tq Q are identified with ∧ Tq Q, hence a family of such fiberwise constant vertical bivector fields is just a 2-form.) 1.5. Casimir functions. Definition 1.21. Suppose π is a Poisson structure on M. A function χ ∈ C∞(M) is a Casimir function if it Poisson commutes with all functions: {χ, f} for all f ∈ C∞(M). Note that if π is a Poisson structure, and χ is a Casimir, then χ π is again a Poisson structure. To check whether a given function χ ∈ C∞(M) is a Casimir function, it suffices to prove {f, χ} = 0 for functions f whose differentials span T ∗M everywhere.

3 Example 1.22. If M = R with the Poisson structure from example 1.17, the Casimir functions are the smooth functions of x2 + y2 + z2. Indeed, it is immediate that this Poisson commutes with x, y, z. More generally, if M = g∗, it is enough to consider diﬀerentials of linear functions

φξ with ξ ∈ g. The Hamiltonian vector ﬁelds Xφξ are the generating vector ﬁeld for the coadjoint action of G on g∗. Hence, the Casimir functions for g∗ are exactly the g-invariant functions on g∗. 10 ECKHARD MEINRENKEN

1.6. Tangent lifts of Poisson structures. Given a Poisson structure π on M, there is a canonical way of obtaining a Poisson structure πTM on the tangent bundle TM. For every smooth function f ∈ C∞(M), let ∞ fT ∈ C (TM) be its tangent lift, defined by fT (v) = v(f) for v ∈ TM. Put differently, fT is the exterior differential df ∈ Γ(T ∗M), regarded as a function on TM via the pairing. In local coordinates i i i i x on M, with corresponding tangent coordinates x , y on TM (i.e. y =x ˙ i are the ‘velocities’) we have n X ∂f f = yi. T ∂xi i=1 Theorem 1.23. Given a bi-vector field π on M, with associated bracket {·, ·}, there is a unique bi-vector field πTM on the tangent bundle such that the associated bracket satisfies

(11) {fT , gT }TM = {f, g}T , ∞ for all f, g ∈ C (M). The bivector ﬁeld πTM is Poisson if and only if π is Poisson.

Proof. From the description in local coordinates, we see that the diﬀerentials dfT span the cotangent space T ∗(TM) everywhere, except along the zero section M ⊆ TM. Hence, there is at most one bracket with the desired property. To show existence, it is enough to write the 1 P ij ∂ ∂ Poisson bivector in local coordinates: If π = 2 ij π (x) ∂xi ∧ ∂xj , X ∂ ∂ 1 X ∂πij ∂ ∂ π = πij(x) ∧ + yk ∧ TM ∂xi ∂yj 2 ∂xk ∂yi ∂yj ij ijk It is straightforward to check that this has the desired property (11). Equation (??) also implies

ΥπTM (dfT , dgT , dhT ) = (Υπ(df, dg, dh))T .

In particular, πTM is Poisson if and only if π is Poisson. ∞ Remark 1.24. Let fV ∈ C (TM) denote the vertical lift, given simply by pullback. Then

{fT , gV }TM = {f, g}V , {fV , gV }TM = 0. Example 1.25. If g is a Lie algebra, with corresponding Lie-Poisson structure on g∗, then T (g∗) inherits a Poisson structure. Under the identiﬁcation T (g∗) =∼ (T g)∗, this is the Lie-Poisson structure for the tangent Lie algebra T g = g n g. Example 1.26. If (M, ω) is a symplectic manifold, and π the associated Poisson structure, then πTM is again non-degenerate. That is, we obtain a symplectic structure ωTM on TM.

2. Lie algebroids as Poisson manifolds The Lie-Poisson structure on the dual of a finite-dimensional Lie algebra g, has the important property of being linear, in the sense that the coefficients of the Poisson tensor are linear functions, or equivalently the bracket of two linear functions is again linear. Conversely any linear Poisson structure on a finite-dimensional vector space V defines a Lie algebra structure INTRODUCTION TO POISSON GEOMETRY 11 on its dual space g := V ∗, with {·, ·} as the corresponding Lie-Poisson structure: One simply identifies g with the linear functions on V . This gives a 1-1 correspondence

n Vector spaces with o 1−1 n o (12) ←→ Lie algebras . linear Poisson structures The correspondence (12) extends to vector bundles, with Lie algebras replaced by Lie algebroids.

2.1. Lie algebroids. Definition 2.1. A Lie algebroid (E, a, [·, ·]) over M is a vector bundle E → M, together with a bundle map a: E → TM called the anchor, and with a Lie bracket [·, ·] on its space Γ(E) of sections, such that for all σ, τ ∈ Γ(E) and f ∈ C∞(M), (13) [σ, fτ] = f[σ, τ] + a(σ)(f)τ. Remarks 2.2. (a) One sometimes sees an additional condition that the map a: Γ(E) → X(M) should preserve Lie brackets. But this is actually automatic. (Exercise.) (b) It is not necessary to include the anchor map as part of the structure. An equivalent formulation is that [σ, fτ] − f[σ, τ] is multiplication of τ by some function. (In other words, adσ := [σ, ·] is a first order differential operator on Γ(E) with scalar principal symbol.) Denoting this function by X(f), one observes f 7→ X(f) is a derivation of C∞(M), hence X is a vector field depending linearly on σ. Denoting this vector field by X = a(σ), one next observes that a(fσ) = fa(σ) for al functions f, so that a actually comes from a bundle map E → TM. Some examples: • E = TM is a Lie algebroid, with anchor the identity map. • More generally, the tangent bundle to a regular foliation of M is a Lie algebroid, with anchor the inclusion. • A Lie algebroid over M = pt is the same as a finite-dimensional Lie algebra g. • A Lie algebroid with zero anchor is the same as a family of Lie algebras Em parametrized by M. Note that the Lie algebra structure can vary with m ∈ M; hence it is more general than what is known as a ‘Lie algebra bundle’. (For the latter, one requires the existence of local trivializations in which the Lie algebra structure becomes constant.) • Given a g-action on M, the trivial bundle E = M ×g has a Lie algebroid structure, with anchor given by the action map, and with the Lie bracket on sections extending the Lie bracket of g (regarded as constant sections of M × g). Concretely, if φ, ψ : M → g are g-valued functions,

[φ, ψ](m) = [φ(m), ψ(m)] + (La(φ)ψ)(m) − (La(ψ)φ)(m). • For a principal G-bundle P → M, the bundle E = T P/G is a Lie algebroid, with anchor the obvious projection to T (P/G) = TM. This is known as the Atiyah algebroid. Its sections are identified with the G-invariant vector fields on M. It fits into an exact sequence a 0 → P ×G g → T P/G → TM → 0; a splitting of this sequence is the same as a principal connection on P . 12 ECKHARD MEINRENKEN

• A closely related example: Let V → M be a vector bundle. A derivation of V is a first order differential operator D : Γ(V ) → Γ(V ), such that there exists a vector field X on M with D(fσ) = fD(σ) + X(f)σ for all sections σ ∈ Γ(V ) and functions f ∈ C∞(M). These are the sections of a certain Lie algebroid E, with anchor given on sections by a(D) = X. In fact, it is just the Atiyah algebroid of the frame bundle of V . • Let N ⊆ M be a codimension 1 submanifold. Then there is a Lie algebroid E of rank dim M, whose space of sections are the vector fields on M tangent to N. [31]. In local coordinates x1, . . . , xk, with N defined by an equation xk = 0, it is spanned by the vector fields ∂ ∂ ∂ ,..., , xk . ∂x1 ∂xk−1 ∂xk Note that it is important here that N has codimension 1; in higher codimension the space of vector fields vanishing along N would not be a free C∞(M)-module, so it cannot be the sections of a vector bundle. 2 • Let ω ∈ Ω (M) be a closed 2-form. Then E = TM ⊕ (M × R) acquires the structure of a Lie algebroid, with anchor the projection to the first summand, and with the Lie bracket on sections,

[X + f, Y + g] = [X,Y ] + LX g − LY f + ω(X,Y ). (A similar construction works for any Lie algebroid E, and closed 2-form in Γ(∧2E∗).

2.2. Linear Poisson structures on vector bundles. Given a vector bundle V → M, let κt : V → V be scalar multiplication by t ∈ R. For t 6= 0 this is a diffeomorphism. A function ∞ ∗ f ∈ C (V ) is called linear if it is homogeneous of degree 1, that is, κt f = f for all t 6= 0. A multi-vector field u ∈ Xk(V ) on the total space of V will be called (fiberwise) linear if it is homogeneous of degree 1 − k, that is, ∗ 1−k κt u = t u for t 6= 0. An equivalent condition is that u(df1,..., dfk) is linear whenever the fi are all linear. In terms of a local vector bundle coordinates, with xi the coordinates on the base and yj the coordinates on the fiber, such a fiberwise linear multi-vector field is of the form X X ∂ ∂ X X ∂ ∂ ∂ u = cj y ∧ · · · ∧ + a ∧ ∧ · · · ∧ i1···ik j i1···ik−1,r r ∂yi1 ∂yik ∂x ∂yi1 ∂yik−1 i1<···

As a special case, a bivector ﬁeld π is linear if it is homogeneous of degree −1, or equivalently LE π = −π. The following theorem gives a 1-1 correspondence

n Vector bundles with o 1−1 n o (14) ←→ Lie algebroids linear Poisson structures

∞ ∗ For any section σ ∈ Γ(E), let φσ ∈ C (E ) be the corresponding linear function on the dual bundle E∗. Theorem 2.3 (Courant [11, Theorem 2.1.4]). For any Lie algebroid E → M, the total space of the dual bundle p: E∗ → M has a unique Poisson bracket such that for all sections σ, τ ∈ Γ(E),

(15) {φσ, φτ } = φ[σ,τ]. The anchor map is described in terms of the Poisson bracket as ∗ ∗ (16) p (a(σ)f) = {φσ, p f}, for f ∈ C∞(M) and σ ∈ Γ(E), while {p∗f, p∗g} = 0 for all functions f, g. The Poisson structure on E∗ is linear; conversely, every ﬁberwise linear Poisson structure on a vector bundle V → M arises in this way from a unique Lie algebroid structure on the dual bundle V ∗.

n Proof. Let E → M be a Lie algebroid. Pick local bundle trivializations E|U = U × R over j open subsets U ⊆ M, and let 1, . . . , n be the corresponding basis of sections of E. Let x j j be local coordinates on U. The differentials of functions yi = φi and functions x yi = φx i span T ∗(E∗) everywhere, except where all yj = 0. This shows that he differentials of the ∗ linear functions φσ span the cotangent spaces to E everywhere, except along the zero section M ⊆ E∗. Hence, there can be at most one bivector field π ∈ X2(E∗) such that

(17) π(dφσ, dφτ ) = φ[σ,τ] (so that the corresponding bracket {·, ·} satisfies (15)). To show its existence, define ‘structure k ∞ P k functions’ cij ∈ C (U) by [i, j] = k cijk, and let ai = a(i) ∈ X(U). Letting yi be the n ∗ coordinates on (R ) corresponding to the basis, one finds that

1 X k ∂ ∂ X ∂ (18) π = cij yk ∧ + ∧ ai 2 ∂yi ∂yj ∂yi ijk i ∗ n ∗ is the unique bivector field on E |U = U × (R ) satisfying (17). (Evaluate the two sides on 2 ∗ dφσ, dφτ for σ = fi and τ = gj.) This proves the existence of π ∈ X (E ). The Jacobi identity for {·, ·} holds true since it is satisfied on linear functions, by the Jacobi identity for Γ(E). Conversely, suppose p: V → M is a vector bundle with a linear Poisson structure π. Let E = V ∗ be the dual bundle. We define the Lie bracket on sections and the anchor a: E → TM ∗ by (15) and (16). This is well-defined: for instance, since φσ and p f have homogeneity 1 and 0 respectively, their Poisson bracket is homogeneous of degree 1 + 0 − 1 = 0. Also, it is straightforward to check that a(σ) is a vector field, and that the map σ 7→ a(σ) is C∞(M)- linear. The Jacobi identity for the bracket [·, ·] follows from that of the Poisson bracket, while 14 ECKHARD MEINRENKEN the Leibnitz rule (13) for the anchor a follows from the derivation property of the Poisson bracket, as follows:

φ[σ,fτ] = {φσ, φfτ } ∗ = {φσ, (p f) φτ } ∗ ∗ = p (a(σ)f) φτ + (p f) φ[σ,τ].

As a simple (if unsurprising) consequence of this result, we see that if E1 → M1 and E2 → M2 are two Lie algebroids, then the exterior direct sum E1 ×E2 → M1 ×M2 is again a Lie algebroid. The corresponding Poisson manifold is the product of Poisson manifolds: ∗ ∗ ∗ (E1 × E2) = E1 × E2 . Note also that if E− is the Lie algebroid with the opposite LA-structure (that is, E− is E as a vector space, but the Lie bracket on sections given by minus the bracket on E, and with minus the anchor of E), then (E−)∗ = (E∗)− as vector bundles with linear Poisson structure, where the superscript − on the right hand side signifies the opposite Poisson structure. Example 2.4. Consider E = TM as a Lie algebroid over M. In local coordinates, the sections of TM are of the form X ∂ σ = ai , ∂qi i P i with corresponding linear function φσ(q, p) = i ai(q)p . The Lie bracket with another such P j ∂ section τ = j b (q) ∂qj is (as the usual Lie bracket of vector fields) X X ∂bk X ∂ak ∂ [σ, τ] = ai − bi ∂qi ∂qi ∂qk k i i It corresponds to k k X X i ∂b X i ∂a X ∂φσ ∂φτ ∂φσ ∂φτ φ[σ,τ] = a i − b i pk = i − i . ∂q ∂q ∂pi ∂q ∂q ∂pi k i i ik The resulting Poisson structure on T ∗M is the opposite of the standard Poisson structure. Example 2.5. Given a Lie algebra action of g on M, let E = M × g with dual bundle E∗ = ∗ ∗ M × g . The Poisson tensor on E is given by (18), with ai the generating vector fields for the action. Example 2.6. For a principal G-bundle P → M, we obtain a linear Poisson structure on (T P/G)∗. This is called by Sternberg [?] and Weinstein [?] the ‘phase space of a classical particle in a Yang-Mills field’. It may be identified with T ∗P/G, with the Poisson structure induced from the opposite of the standard Poisson structure on T ∗P . Example 2.7. For the Lie algebroid E associated to a hypersurface N ⊆ M, with local coordi- 1 k ∂ nates x , . . . , x so that N is given by the vanishing of the k-th coordinate, we have i = ∂xi for INTRODUCTION TO POISSON GEOMETRY 15

k ∂ i < k and k = x ∂xk as a basis for the sections of E. Denote by y1, . . . , yn the corresponding linear functions. The Poisson bracket reads i i k k k {x , yj} = δj for i < k, {x , yj} = x δj . 2.3. The cotangent Lie algebroid of a Poisson manifold. As a particularly important example, let (M, π) be a Poisson manifold. As we saw, the tangent bundle V = TM inherits a Poisson structure πTM such that {fT , gT }TM = {f, g}T for all f, g. The functions fT are homogeneous of degree 1, hence πTM is homogeneous of degree −1. That is, πTM is a linear Poisson structure, and hence determines a Lie algebroid structure on the dual bundle T ∗M. It ∗ is common to use the notation Tπ M for the cotangent bundle with this Lie algebroid structure. ] From {fT , gV } = {f, g}V we see that the anchor map satisﬁes a(df) = Xf = π (df). That is, ] ∗ a = π : Tπ M → TM. ∞ Since φdf = fT for f ∈ C (M), the bracket on sections is such that [df, dg] = d{f, g} for all f, g ∈ C∞(M). The extension to 1-forms is uniquely determined by the Leibnitz rule, and is given by

[α, β] = Lπ](α)β − Lπ](β)α − dπ(α, β). This Lie bracket on 1-forms of a Poisson manifold was ﬁrst discovered by Fuchssteiner [19].

2.4. Lie algebroid comorphisms. As we saw, linear Poisson structures on vector bundles V → M correspond to Lie algebroid structures on E = V ∗. One therefore expects that the category of vector bundles with linear Poisson structures should be the same as the category of Lie algebroids. This turns out to be true, but we have to specify what kind of morphisms we are using. The problem is that a vector bundle map V1 → V2 does not dualize to a vector bundle map ∗ E1 → E2 for Ei = Vi (unless the map on the base is a diﬀeomorphism). We are thus forced to allow more general kinds of vector bundle morphisms, either for V1 → V2 (if we insist that E1 → E2 is an actual vector bundle map), or for E1 → E2 (if we insist that V1 → V2 is an actual vector bundle map. Both options are interesting and important, and lead to the notions of Lie algebroid morphisms and Lie algebroid comorphisms, respectively. Deﬁnition 2.8. A vector bundle comorphism, depicted by a diagram

ΦE E1 / E2

M1 / M2 ΦM is given by a base map ΦM : M1 → M2 together with a family of linear maps (going in the ‘opposite’ direction)

ΦE :(E2)ΦM (m) → (E1)m ∗ depending smoothly on m, in the sense that the resulting map ΦM E2 → E1 is smooth. 16 ECKHARD MEINRENKEN

Given such a vector bundle comorphism, one obtains a pullback map on sections, ∗ (19) ΦE : Γ(E2) → Γ(E1) which is compatible with the pullback of functions on M. Comorphisms can be composed in the obvious way, hence one obtains a category VB∨ the category of vector bundles and vector bundle morphisms. Remark 2.9. Letting VS be the category of vector spaces, and VSop the opposite category, one ∼ has the isomorphism VS −→= VSop taking a vector space to its dual space. Taking the opposite category (‘reversing arrows’) ensures that this is a covariant functor. Similarly, taking a vector ∼ bundle to its dual is an isomorphism of categories VB −→= VB∨. In this sense, the introduction of VB∨ may appear pointless. It becomes more relevant if the vector bundles have additional structure, which is not so easy to dualize.

Deﬁnition 2.10. Let E1 → M1 and E2 → M2 be Lie algebroids. A Lie algebroid comorphism ΦE : E1 99K E2 is a vector bundle comorphism such that (i) the pullback map (19) preserves brackets, (ii) The anchor maps satisfy ∗ a1(ΦEσ) ∼ΦM a2(σ) (ΦM -related vector ﬁelds). We denote by LA∨ the category of Lie algebroids and Lie algebroid comorphisms. The second condition means that we have a commutative diagram

ΦE E1 / E2

a1 a2 TM1 / TM2 T ΦM

Note that this condition (ii) is not automatic. For instance, take M1 = M2 = M, with ΦM the identity map, let E2 = TM the tangent bundle and let E1 = 0 the zero Lie algebroid. Take X ∈ Γ(TM) be a non-zero vector ﬁeld. There is a unique comorphism ΦE : 0 99K E covering ΦM = idM ; the pull-back map on sections is the zero map, and in particular preserves brackets.

But the condition (ii) would tell us 0 ∼idM X, i.e. X = 0. Example 2.11. Let M be a manifold, and g a Lie algebra. A comorphism of Lie algebroids TM 99K g is the same as a Lie algebra action of g on M. In this spirit, a comorphism from TM to a general Lie algebroid E may be thought of as a Lie algebroid action of E on M. ∗ Remark 2.12. On the open set of all m ∈ M1 where the pullback map ΦE :(E2)Φ(m) → (E1)m ∞ is non-zero, condition (ii) is automatic. To see this let σ, τ be sections of E2, and f ∈ C (M2). Then Φ∗[σ, fτ] = [Φ∗σ, (Φ∗f)Φ∗τ]. Expanding using the Leibnitz rule, nd cancelling like terms, one arrives at the formula ∗ ∗ ∗ Φ (a2(σ)f) − a(σ1)(Φ f) Φ τ = 0. ∗ ∗ ∗ This shows that Φ (a2(σ)f) = a(σ1)(Φ f) at all those points m ∈ M1 where Φ τ|m 6= 0 for some τ ∈ Γ(E2). INTRODUCTION TO POISSON GEOMETRY 17

Now, let VBP oi be the category of vector bundles with linear Poisson structures; morphisms in this category are vector bundle maps that are also Poisson maps. (It is tempting to call these ‘Poisson vector bundles’, but unfortunately that terminology is already taken.) The following result shows that there is an isomorphism of categories

=∼ ∨ VBP oi −→ LA .

Proposition 2.13. Let E1 → M1 and E2 → M2 be two Lie algebroids. A vector bundle comor- ∗ ∗ phism ΦE : E1 99K E2 is a Lie algebroid comorphism if and only if the dual map ΦE∗ : E1 → E2 is a Poisson map.

∗ ∗ ∗ ∗ Proof. To simplify notation, we denote all the pull-back maps ΦM , ΦE, ΦE∗ by Φ . For any VB-comorphism ΦE : E1 99K E2, and any σ ∈ Γ(E2), we have that ∗ (20) φΦ∗σ = Φ φσ. ∞ Given sections σ, τ ∈ Γ(E2) and a function f ∈ C (M2), we have ∗ ∗ (21) φΦ∗[σ,τ] = Φ φ[σ,τ] = Φ {φσ, φτ },

∗ ∗ (22) φ[Φ∗σ,Φ∗τ] = {φΦ∗σ, φΦ∗τ } = {Φ φσ, Φ φτ }, and ∗ ∗ ∗ ∗ ∗ ∗ (23) p1Φ (a2(σ)f) = Φ p2(a2(σ)f) = Φ {φσ, p2f},

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ (24) p1 a1(Φ σ)(Φ f) = {φΦ∗σ, p1Φ f} = {Φ φσ, Φ p2f}

Here we have only used (20), and the description of the Lie algebroid structures of E1,E2 in ∗ ∗ terms of the Poisson structures on E1 ,E2 , see (15) and (16). ΦE being an LA-morphism is equivalent to the equality of the left hand sides of equations (21), (22) and equality of the left hand sides of equations (23), (24), while ΦE∗ being a Poisson map is equivalent to the equality of the corresponding right hand sides.

2.5. Lie subalgebroids and LA-morphisms. To deﬁne Lie algebroid morphisms FE : E1 → E2, we begin with the case of injective morphisms, i.e. subbundles. Deﬁnition 2.14. Let E → M be a Lie algebroid, and F ⊆ E a vector subbundle along N ⊆ M. Then F is called a Lie subalgebroid if it has the following properties: • If σ, τ ∈ Γ(E) restrict over N to sections of F , then so does their bracket [·, ·], • a(F ) ⊆ TN. As the name suggests, a Lie subalgebroid is itself a Lie algebroid: Proposition 2.15. if F ⊆ E is a sub-Lie algebroid along N ⊆ M, then F acquires a Lie algebroid structure, with anchor the restriction of a: E → TN, and with the unique bracket such that

[σ|N , τ|N ] = [σ, τ]|N whenever σ|N , τ|N ∈ Γ(F ). 18 ECKHARD MEINRENKEN

Proof. To show that this bracket is well-defined, we have to show that [σ, τ]|N = 0 whenever τ|N = 0. (In other words, the sections vanishing along N are an ideal in the space of sections P ∞ of E that restrict to sections of N.) Write τ = i fiτi where fi ∈ C (M) vanish on N. Then X [σ, τ] = fi|N σ, τi |N + (a(σ)fi)|N τi|N = 0 N i where we used that a(σ)fi = 0, since a(σ) is tangent to N and the fi vanish on N. Here is one typical example of how Lie subalgebroids arise: Proposition 2.16. Let E → M be a Lie algebroid, on which a compact Lie group G acts by Lie algebroid automorphisms. Then the fixed point set EG ⊆ E is a Lie subalgebroid along M G ⊆ M. Proof. Recall first that since G is compact, the fixed point set M G is a submanifold, and EG → M G is a vector subbundle. By equivariance, a(EG) ⊆ (TM)G = T (M G). Let Γ(E)G be the G-invariant sections. The restriction of such a section to M G is a section of EG, and the resulting map Γ(E)G → Γ(EG) is surjective. (Given a section of EG, we can extend extend to a section of E, and than achieve G-invariance by averaging.) But the bracket of G-invariant sections of E is again G-invariant, G and hence restricts to a section of E . Proposition 2.17. Let E → M be a Lie algebroid, and N ⊆ M a submanifold. Suppose that a−1(TN) is a smooth subbundle of E. Then a−1(TN) ⊆ E is a Lie subalgebroid along N ⊆ M. Proof. This follows from the fact that a: Γ(E) → X(M) is a Lie algebra morphism, and the Lie bracket of vector fields tangent to N is again tangent to N. Let ι: N → M be the inclusion map. We think of ι!E := a−1(TN) as the proper notion of ‘restriction’ of a Lie algebroid. Two special cases: ! (a) If a is tangent to N (i.e. a(E|N ) ⊆ TN), then ι E = E|N coincides with the vector bundle restriction. (b) If a is transverse to N, then the restriction ι!E is well-defined, with rank(ι!E) = rank(E) − dim(M) + dim(N). Note that ι!TM = TN. More generally, we can sometimes define ‘pull-backs’ of Lie algebroids E → M under smooth maps Φ: N → M. Here, we assume that Φ is transverse to a: E → TM. Then the fiber product E ×TN TM ⊆ E × TN is a well-defined subbundle along the graph of Φ, and is exactly the pre-image of T Gr(Φ). It hence acquires a Lie algebroid structure. We let ! (25) Φ E = E ×TN TM under the identification Gr(Φ) =∼ N. Remarks 2.18. (a) As a special case, Φ!(TM) = TN. (b) If Φ = ι is an embedding as a submanifold, then Φ!E coincides with the ‘restriction’. INTRODUCTION TO POISSON GEOMETRY 19

(c) Under composition of maps, (Φ ◦ Ψ)!E = Ψ!Φ!E (whenever the two sides are deﬁned). We can use Lie subalgebroids also to deﬁne morphisms of Lie algebroids.

Deﬁnition 2.19. Given Lie algebroids E1 → M1, E2 → M2, a vector bundle map

ΦE : E1 → E2 − is a Lie algebroid morphism if its graph Gr(ΦE) ⊆ E2 ×E1 is a Lie subalgebroid along Gr(ΦM ). The category of Lie algebroids and Lie algebroid morphisms will be denote LA. It will take some time and space (which we don’t have right now) to get acquainted with this deﬁnition. At this point, we just note some simple examples:

(a) For any smooth map Φ: M1 → M2, the tangent map T Φ: TM1 → TM2 is an LA- morphism. (b) For any Lie algebroid E, the anchor map a: E → TM is an LA-morphism. (c) Let E be a Lie algebroid over M, and Φ: N → M a smooth map for which the pull-back Φ!E is deﬁned. Then the natural map Φ!E → E is a Lie algebroid morphism. (d) Given g-actions on M1,M2, and an equivariant map M1 → M2, the bundle map

M1 × g → M2 × g is an LA morphism. (e) If g is a Lie algebra, then a Lie algebroid morphism TM → g is the same as a Maurer- Cartan form θ ∈ Ω1(M, g), that is, 1 dθ + [θ, θ] = 0. 2 (See e.g. [???]) Having deﬁned the category LA, it is natural to ask what corresponds to it on the dual side, in terms of the linear Poisson structures on vector bundles. The answer will have to wait until we have the notion of a Poisson morphism.

3. Submanifolds of Poisson manifolds Given a Poisson manifold (M, π), there are various important types of submanifolds. 3.1. Poisson submanifolds. A submanifold N ⊆ M is called a Poisson submanifold if the 2 2 Poisson tensor π is everywhere tangent to N, in the sense that πn ∈ ∧ TnN ⊆ ∧ TnM. Taking 2 the restrictions pointwise deﬁnes a bivector ﬁeld πN ∈ X (N), with the property that

πN ∼j π where j : N → M is the inclusion. The corresponding Poisson bracket {·, ·}N is given by ∗ ∗ ∗ {j f, j g}N = j {f, g}.

The Jacobi identity for πN follows from that for π. The Poisson submanifold condition can be expressed in various alternate ways. Proposition 3.1. The following are equivalent: (a) N is a Poisson submanifold. ] ∗ (b) π (T M|N ) ⊆ TN. 20 ECKHARD MEINRENKEN

(c) π](ann(TN)) = 0. ∞ (d) All Hamiltonian vector fields Xf , f ∈ C (M) are tangent to N. ∞ (e) The space of functions f with f|N = 0 are a Lie algebra ideal in C (M), under the Poisson bracket. Proof. It is clear that (a),(b),(c), are equivalent. The equivalence of (b) and (d) follows since ] for all m ∈ M, the range ran(πm) is spanned by the Hamiltonian vector fields Xf . Furthermore, if (d) holds, then the functions vanishing on N are an ideal since g|N = 0 implies {f, g}|N = Xf (g)|N = 0 since Xf is tangent to N. This gives (e). Conversely, if (e) holds, so that {f, g}|N = 0 whenever g|N = 0, it follows that hdg, Xf i|N = Xf (g)|N = 0 whenever g|N = 0. The differentials dg|N for g|N = 0 span ann(TN), hence this means that Xf |N ∈ Γ(TN), which gives (d). Examples 3.2. (a) If χ ∈ C∞(M) is a Casimir function, then all the smooth level sets of χ are Poisson submanifolds. Indeed, since Xf χ = {f, χ} = 0 shows that the Hamiltonian vector fields are tangent to the level sets of χ. (b) As a special case, if g is a Lie algebra with an invariant metric, defining a metric on the dual space, then the set of all µ ∈ g∗ such that ||µ|| = R (a given constant) is a Poisson submanifold. (c) For any Poisson manifold M, and any k ∈ N ∪ {0} one can consider the subset M(2k) of elements where the Poisson structure has given rank 2k. If this subset is a submanifold, then it is a Poisson submanifold. For example, if M = g∗ the components of the set of elements with given dimension of the stabilizer group Gµ are Poisson submanifolds. 3.2. Symplectic leaves. As mentioned above, the subspaces ] ran(πm) ⊆ TmM are spanned by the Hamiltonian vector fields. The subset ran(π]) ⊆ TM is usually a singular ] distribution, since the dimensions of the subspaces ran(πm) need not be constant. It doesn’t prevent us from considering leaves: Definition 3.3. A maximal connected injectively immersed submanifold S ⊆ M of a connected manifold S is called a symplectic leaf of the Poisson manifold (M, π) if ] ∗ TS = π (T M|S). ] By definition, the symplectic leaves are Poisson submanifolds. Since πS is onto TS everywhere, this Poisson structure is non-degenerate, that is, it corresponds to a symplectic 2-form [ ] −1 ωS with ωS = −(πS) . The Hamiltonian vector fields Xf are a Lie subalgebra of X(M), since

[Xf1 ,Xf2 ] = X{f1,f2}. If the distribution spanned by these vector ﬁelds has constant rank, then we can use Frobenius’ theorem to conclude that the distribution is integrable: Through every point there passes a unique symplectic leaf. However, in general Frobenius’s theorem is not applicable since the rank may jump. Nevertheless, we have the following fundamental result: Theorem 3.4. [39] Every point m of a Poisson manifold M is contained in a unique symplectic leaf S.

Thus, M has a decomposition into symplectic leaves. One can prove this result by obtaining the leaf through a given point m as the ‘ﬂow-out’ of m under all Hamiltonian vector ﬁelds, and INTRODUCTION TO POISSON GEOMETRY 21 this is Weinstein’s argument in [39]. We will not present this proof, since we will later obtain this result as a corollary to the Weinstein splitting theorem for Poisson structures. Example 3.5. For M = g∗ the dual of a Lie algebra g, the symplectic leaves are the orbits of coadjoint action G on g∗. Here G is any connected Lie group integrating g. Example 3.6. For a Poisson structure π on a 2-dimensional manifold M, let Z ⊆ M be its set of zeros, i.e. points m ∈ M where πm = 0. Then the 2-dimensional symplectic leaves of π are the connected components of M − Z, while the 0-dimensional leaves are the points of Z. Remark 3.7. The Poisson structure is uniquely determined by its symplectic leaves, and can sometimes be described in these terms. Suppose for instance M is a manifold with a (regular) foliation, and with a 2-form ω whose pull-back to every leaf of the foliation is closed and non- degenerate, i.e., symplectic. Then M becomes a Poisson structure with the given foliation as its symplectic foliation. The Poisson bracket of two functions on M may be computed leafwise; it is clear that the result is again a smooth function on M. (See Vaisman [?, Proposition 3.6].) Remark 3.8. Since the dimension of the symplectic leaf S through m ∈ M equals the rank ] ∗ of the bundle map πm : TmM → TmM, we see that this dimension is a lower semi-continuous function of m. That is, the nearby leaves will have dimension greater than or equal to the dimension of S. In particular, if π has maximal rank 2k, then the union of 2k-dimensional symplectic leaves is an open subset of M. 3.3. Coisotropic submanifolds. Lemma 3.9. The following are equivalent: (a) π](ann(TN)) ⊆ TN

(b) For all f such that f|N = 0, the vector ﬁeld Xf is tangent to N. (c) The space of functions f with f|N = 0 are a Lie subalgebra under the Poisson bracket. (d) The annihilator ann(TN) is a Lie subalgebroid of the cotangent Lie algebroid.

Proof. Equivalence of (a) and (b) is clear, since ann(TN) is spanned by df|N such that f|N = 0. If (b) holds, then f|N = 0, g|N = 0 implies {f, g}|N = Xf (g)|N = 0. Conversely, if (c) holds, and f|N = 0, then Xf is tangent to N since for all g with g|N = 0, Xf (g)|N = {f, g}|N = 0. ∗ If ann(TN) is a Lie subalgebroid of Tπ M, then in particular its image under the anchor is tangent to N, which is (a). Conversely, if the equivalent conditions (a),(c), hold, then ann(TN) is a Lie subalgebroid because its space of sections is generated by df with f|N = 0, and [df, dg] = d{f, g}. A submanifold N ⊆ M is called a coisotropic submanifold if it satisfies any of these equivalent conditions. Clearly, {open subsets of symplectic leaves } ⊆ { Poisson submanifolds } ⊆ { coisotropic submanifolds }. Remark 3.10. By (d), we see in particular that for any coisotropic submanifold N, the normal bundle ∗ ν(M,N) = TM|N /T N = ann(TN) inherits a linear Poisson structure πν(M,N). By the tubular neighborhood theorem, there is an diffeomorphism of open neighborhoods of N inside ν(M,N) and inside M. Hence, ν(M,N) wit 22 ECKHARD MEINRENKEN this linear Poisson structure is thought of as the linear approximation of the Poisson structure πM along N. As special cases, we obtain linear Poisson structures on the normal bundles of Poisson submanifolds, and in particular on normal bundles of symplectic leaves. Remark 3.11. There are also notions of Lagrangian submanifold and isotropic submanifold of a Poisson manifold, defined by the conditions that π](ann(TN)) = TN and π](ann(TN)) ⊇ TN. However, it seems that these notions rarely appears in practice. Example 3.12. Let E → M be a Lie algebroid, so that E∗ → M has a linear Poisson structure. ∗ For any submanifold N ⊆ M, the restriction E |N is a coisotropic submanifold. Indeed, the ∗ ∗ ∗ ∗ conormal bundle to E |N is spanned by d(p f) such that f|N = 0, but {p f, p g} for all functions on M. Example 3.13. If (M, ω) is a symplectic manifold, regarded as a Poisson manifold, then the notions of coisotropic in the Poisson sense coincides with that in the symplectic sense. Indeed, in this case π](ann(TN)) equals the ω-orthogonal space TN ω, consisting of v ∈ TM such that ω[(v) ∈ ann(TN). But TN ω ⊆ TN is the coisotropic condition in symplectic geometry. For a Poisson manifold, it follows that the intersection of coisotropic submanifolds with symplectic leaves are coisotropic.

Theorem 3.14 (Weinstein). A smooth map Φ: M1 → M2 of Poisson manifolds (M1, π1) − and (M2, π2) is a Poisson map if and only if its graph Gr(Φ) ⊆ M2 × M1 is a coisotropic − submanifold. (Here M1 is M1 with the Poisson structure −π1. ∗ ∗ Proof. The condition that π1 ∼Φ π2 means that for covectors α1 ∈ TmM1, α2 ∈ TΦ(m)M2, ∗ ] ] α1 = Φ α2 ⇒ π1(α1) ∼Φ π2(α2). ∗ ] ] But α1 = Φ α2 is equivalent to (α2, −α1) ∈ ann(T Gr Φ), while π1(α1) ∼Φ π2(α2) is equivalent ] ] to (π2(α2), π1(α1)) ∈ Gr(T Φ) = T Gr Φ. Theorem 3.14 is the Poisson counterpart to a well-known result from symplectic geometry: If M1,M2 are symplectic manifolds, then a diﬀeomorphism Φ: M1 → M2 is symplectomorphism − if and only if its graph Gr(Φ) ⊆ M2 × M1 is a Lagrangian submanifold. This leads to the idea − of viewing Lagrangian submanifolds of M2 × M1 as ‘generalized morphisms’ from M1 to M2, and idea advocated by Weinstein’s notion of a symplectic category [40]. In a similar fashion, Weinstein deﬁned:

Deﬁnition 3.15. Let M1,M2 be Poisson manifolds. A Poisson relation from M1 to M2 is a − − coisotropic submanifold N ⊆ M2 × M1 , where M1 is M1 equipped with the opposite Poisson structure. Poisson relations are regarded as generalized ‘morphisms’. We will thus write

N : M1 99K M2 for a submanifold N ⊆ M2 × M1 thought of as such a ‘morphism’. However, ‘morphism’ is in quotes since relations between manifolds cannot always be composed: Given submanifolds 0 0 N ⊆ M2 × M1 and N ⊆ M3 × M2, the composition N ◦ N need not be a submanifold. 0 Deﬁnition 3.16. We say that two relations N : M1 99K M2 and N : M2 99K M3 (given by 0 submanifolds N ⊆ M2 × M1 and N ⊆ M3 × M2) have clean composition if INTRODUCTION TO POISSON GEOMETRY 23

(i) N 0 ◦ N is a submanifold, and (ii) T (N 0 ◦ N) = TN 0 ◦ TN fiberwise. 0 By (ii), we mean that for all mi ∈ Mi with (m3, m2) ∈ N and (m2, m1) ∈ N, we have that 0 0 T(m3,m1)(N ◦ N) = T(m3,m2)N ◦ T(m2,m1)N. We stress that there are various versions of ‘clean composition’ in the literature, and the condition here is weaker (but also simpler) than the one found in [?] or [41, Definition (1.3.7)]. Our goal is to show that a clean composition of Poisson relations is again a Poisson relation. We will need some facts concerning the composition of linear relations. For any linear relation ∗ ∗ R: V1 99K V2, given by a subspace R ⊆ V2 × V1, define a relation R : V1 99K V2 of the dual spaces, by ∗ ∗ (26) R = {(α2, α1) ∈ V2 × V1 | (α2, −α1) ∈ ann(R)}.

For example, if ∆V ⊆ V × V is the diagonal (corresponding to the identity morphism), then ∆V = ∆V ∗ . The main reason for including a sign change is the following property under composition of relations. 0 Lemma 3.17. (Cf. [25, Lemma A.2]) For linear relations R: V1 99K V2 and R : V2 99K V3, 0 with composition R ◦ R: V1 99K V3, we have that 0 0 0 ∗ ∗ (R ◦ R ) = (R ) ◦ R : V1 99K V3 . Proof. It is a well-known fact in linear symplectic geometry that the composition of linear Lagrangian relations in symplectic vector spaces is again a Lagrangian relation. (No transversality assumptions are needed.) We will apply this fact, as follows. If V is a vector space, let W = T ∗V = V ⊕V ∗ with its standard symplectic structure, and let W − be the same space with the opposite symplectic structure. If S ⊆ V is any subspace, then S ⊕ ann(S) is Lagrangian in ∗ W . In our situation, let Wi = Vi ⊕ Vi . Then − 0 0 − R ⊕ R ⊆ W2 ⊕ W1 ,R ⊕ (R ) ⊆ W3 ⊕ W2 are Lagrangian relations, hence so is their composition (R0 ◦ R) ⊕ ((R0) ◦ R). This means that 0 0 (R ◦ R) = (R ) ◦ R . Put diﬀerently, the Lemma says that 0 0 (27) ann(R ◦ R) = (α3, −α1) ∃α2 :(α3, −α2) ∈ ann(R ), (α2, −α1) ∈ ann(R) . The following result was proved by Weinstein [41] under slightly stronger assumptions. 0 Proposition 3.18 (Weinstein). Let N : M1 99K M2 and N : M2 99K M3 be Poisson relations 0 0 with clean composition N ◦ N : M1 99K M3. Then N ◦ N is again a Poisson relation. Proof. We have to show that N 0 ◦ N is a coisotropic submanifold. Let 0 (α3, −α1) ∈ ann(T (N ◦ N)) 0 0 be given, with base point (m3, m1) ∈ N ◦ N. Choose m2 ∈ M2 with (m3, m2) ∈ N and (m2, m1) ∈ N. Since 0 0 T(m3,m1)(N ◦ N) = T(m3,m2)N ◦ T(m2,m1)N, 24 ECKHARD MEINRENKEN

∗ 0 Equation (27) gives the existence of α2 ∈ Tm2 M2 such that (α3, −α2) ∈ ann(TN ) and ] 0 (α2, −α1) ∈ ann(TN). Letting vi = πi (αi) we obtain (v3, v2) ∈ TN and (v2, v1) ∈ TN, 0 0 0 0 since N ,N are coisotropic. Hence (v3, v1) ∈ TN ◦ TN = T (N ◦ N), proving that N ◦ N is coisotropic.

Example 3.19. [?, Corollary (2.2.5)] Suppose Φ: M1 → M2 is a Poisson map, and N ⊆ M1 is a coisotropic submanifold. Suppose Φ(N) ⊆ M2 is a submanifold, with

(28) (TmΦ)(TmN) = TΦ(m)(Φ(N)) for all m ∈ M1. Then Φ(N) is a coisotropic submanifold. Indeed, this sat my be regarded as a composition of relations Φ(N) = Gr(Φ) ◦ N, and the assumptions given are equivalent to the clean composition assumption. Similarly, if Q ⊆ M2 is a coisotropic submanifold, such that −1 −1 −1 −1 Φ (Q) is a submanifold with Tm(Φ (Q)) = (TmΦ) (TΦ(m)Q), then Φ (Q) is a submanifold.

− If R ⊆ M2 × M1 is a Poisson relation, we can consider the transpose (or inverse) Poisson relation > − R ⊆ M1 × M2 > consisting of all (m1, m2) such that (m2, m1) ∈ R. We may then deﬁne new relations R ◦ R and R ◦ R>, provided that clean composition assumptions are satisﬁed. As a special case, suppose R = Gr(Φ) is the graph of a Poisson map Φ: M1 → M2. Then

> 0 − 0 R ◦ R = {(m, m ) ∈ M1 × M1 | Φ(m) = Φ(m )} = M1 ×M2 M1

(the ﬁber product of M1 with itself over M2). is coisotropic, provided that the composition is clean. The cleanness assumption is automatic if Φ is a submersion. In this case, one has a partial converse, which may be regarded as a criterion for reducibility of a Poisson structure.

Proposition 3.20 (Weinstein). Let Φ: M1 → M2 be a surjective submersion, where M1 is a Poisson manifold. Then the following are equivalent:

(a) The Poisson structure on M1 descends to M2. That is, M2 has a Poisson structure such that Φ is a Poisson map. − − − (b) The ﬁber product M1 ×M2 M1 ⊆ M1 × M1 is a coisotropic submanifold of M1 × M1 .

Proof. One direction was discussed above. For the converse, suppose S := M1 ×M2 M1 is coisotropic. To show that the Poisson structure descends, we have to show that functions of ∗ ∞ the form Φ f with f ∈ C (M2) form a Poisson subalgebra. For any such function f, note that ∗ ∗ ∗ ∗ ∞ − 0 ∞ F = pr1 Φ f − pr2 Φ f ∈ C (M1 × M1 ) vanishes on S. Given another function f ∈ C (M2), with corresponding function F 0, we have that {F,F 0} vanishes on S. But the vanishing of

0 ∗ ∗ ∗ 0 ∗ ∗ ∗ 0 {F,F } = pr1{Φ f, Φ f } − pr2{Φ f, Φ f } on S means precisely that {Φ∗f, Φ∗f 0} is constant along the ﬁbers of Φ. In other words, it lies ∗ ∞ in Φ (C (M2)). Remark 3.21. In [41], Weinstein also discussed the more general Marsden-Ratiu reduction procedure along similar lines. INTRODUCTION TO POISSON GEOMETRY 25

3.4. Applications to Lie algebroids. Recall that F ⊆ E is a Lie subalgebroid if and only if {σ ∈ Γ(E)| σ|N ∈ Γ(F )} is a Lie subalgebra, with {σ ∈ Γ(E)| σ|N = 0} as an ideal (the latter condition being equivalent to a(E) ⊆ TN). In the dual picture, ∗ σ|N ∈ Γ(F ) ⇔ φσ vanishes on ann(F ) ⊆ T M|N

∗ σ|N = 0 ⇔ φσ vanishes on T M|N . Proposition 3.22. Let E be a Lie algebroid, and F ⊆ E a vector subbundle along N ⊆ M. Then F is a Lie subalgebroid if and only if ann(F ) ⊆ E∗ is a coisotropic submanifold.

∗ Proof. ”⇐”. Suppose that ann(F ) ⊆ E is coisotropic. If σ|N ∈ Γ(F ) and f|N = 0, then φσ and p∗f vanish on ann(F ), hence so does their Poisson bracket ∗ ∗ {φσ, p f} = p (a(σ)(f)).

Hence a(σ)(f)|N = 0, which proves that a(σ) is tangent to N. Since σ was any section restricting to a section of N, this shows a(F ) ⊆ TN. Similarly, if σ, τ restrict to sections of F , then φσ, φτ vanish on ann(F ), hence so does

{φσ, φτ } = φ[σ,τ] which means that [σ, τ] restricts to a section of F . This shows that F is a Lie subalgebroid. ”⇒”. Suppose F is a Lie subalgebroid. Then, for all σ, τ that restrict to sections of F , and all f, g ∈ C∞(M) that restrict to zero on N, the Poisson brackets ∗ ∗ ∗ ∗ {φσ, φτ } = φ[σ,τ], {φσ, p f} = p (a(σ)(f)), {p f, p g} = 0 all restrict to 0 on ann(F ). Since these functions generate the vanishing ideal of ann(F ) inside C∞(E∗), this shows that this vanishing ideal is a Lie subalgebra; that is, ann(N) is coisotropic. Remark 3.23. Note the nice symmetry: • For a Poisson manifold (M, π), we have that N ⊆ M is a coisotropic submanifold if and only if ann(TN) ⊆ T ∗M is a Lie subalgebroid. • For a Lie algebroid E, a a vector subbundle F ⊆ E is a Lie subalgebroid if and only if ann(F ) ⊆ E∗ is a coisotropic submanifold.

∨ Deﬁnition 3.24. We denote by VBP oi the category of vector bundles with linear Poisson structures, with morphisms the vector bundle comorphisms that are also Poisson relations.

Proposition 3.25. Let E1 → M1,E2 → M2 be Lie algebroids. Then ΦE : E1 → E2 is a Lie ∗ ∗ algebroid morphism if and only if the dual comorphism ΦE∗ : E1 99K E2 is a Poisson relation. We conclude that there is an isomorphism of categories,

∨ =∼ VBP oi −→ LA.

Proof. By deﬁnition, ΦE is an LA-morphism if and only if its graph is a Lie subalgebroid. By Proposition 3.22, this is the case if and only if the dual comorphism ΦE∗ is a Poisson relation 26 ECKHARD MEINRENKEN

3.5. Poisson-Dirac submanifolds. Aside from the Poisson submanifolds, there are other classes of submanifolds of Poisson manifolds M, with naturally induced Poisson structures. For example, suppose a submanifold N ⊆ M has the property that its intersection with every symplectic leaf of M is a symplectic submanifold of that leaf. Then one can ask if the resulting decomposition of N into symplectic submanifolds defines a Poisson structure on N. This is not automatic, as the following example shows. 2 3 Example 3.26. Let M = R × R as a product of Poisson manifolds, where the first factor has ∂ ∂ the standard Poisson structure ∂q ∧ ∂p , and the second factor has the zero Poisson structure. Let N ⊆ M be the image of the embedding 3 R → M, (q, p, t) 7→ (q, p, tq, tp, t). 2 Then N contains the symplectic leaf R ×{0} ⊆ M, but intersects all other leaves transversally. The resulting decomposition of N into a single 2-dimensional submanifold together with isolated points cannot correspond to a symplectic foliation. (Cf. Remark 3.8.) See Crainic-Fernandes [?, Section 8.2] for a similar example. Definition 3.27. Let M be a Poisson manifold. A submanifold N ⊆ M is called a Poisson- ∞ ∞ Dirac submanifold if every f ∈ C (N) admits an extension fe ∈ C (M) (i.e., fe|N = f) for which X is tangent to N. fe Note that in particular, every Poisson submanifold is a Poisson-Dirac submanifold. Remark 3.28. Definition 3.27 follows Laurent-Gengoux, Pichereau and Vanhaecke, see [24, Sec- tion 5.3.2]. Cranic-Fernandes [?] use the term for any submanifold N with a Poisson structure ] ] πN such that ran(πN ) = ran(π ) ∩ TN everywhere. An equivalent condition is the following: Lemma 3.29. N ⊆ M is a Poisson-Dirac submanifold if and only if every 1-form α ∈ Ω1(N) 1 ] is the pull-back of a 1-form αe ∈ Ω (M) such that π (αe) is tangent to N. Proof. The direction ”⇒” is obvious. For the other direction, we have to show that every f ∈ C∞(M) admits an extension fe whose hamiltonian vector field is tangent to N. By using a partition of unity, we may assume that f is contained in a submanifold chart of N. Thus suppose xi, yj are local coordinates so that N is given by yj = 0. Let α = df, and choose an extension αe as in the statement of the lemma. Then αe|N has the form X j αe|N = df + cj(x)dy . j The formula X j fe(x, y) = f(x) + cj(x)y . j defines an extension of f, and since df| = α| we have that X = π](α) is tangent to N. e N e N fe e Proposition 3.30. If N is a Poisson-Dirac submanifold, then N inherits a Poisson structure via {f, g}N = {f,e ge}|N INTRODUCTION TO POISSON GEOMETRY 27 where f,e ge are extensions of f, g whose Hamiltonian vector fields are tangent to N. In terms of bivector fields,

(29) πN (α, β) = π(α,e βe)|N 1 1 ] whenever αe ∈ Ω (M) pulls back to α ∈ Ω (N) and π (αe) is tangent to N, and similarly for βe. The symplectic leaves of N with respect to πN are the components of the intersections of N with the symplectic leaves of M. Proof. To show that the bracket is well-defined, we have to show that the right hand side vanishes if g| = 0. But this follows from {f, g}| = X (g)| since X is tangent to N. e N e e N fe e N fe The Jacobi identity for {·, ·}N follows from that for {·, ·}. The formula in terms of bivector fields reduces to the one in terms of brackets if the 1-forms are all exact. To show that it is well-defined in the general case, we have to show that the right hand vanishes if the pullback of βe to N is zero, or equivalently if β|N takes values in ann(TN). But this is clear since ] π(α,e βe)|N = hβe|N , π (αe)|N i = 0 ] using that π (αe)|N takes values in TN. From the formula in terms of 1-forms, we see that ] ] πN (α) = π (αe), whenever the right hand side takes values in TN and αe pulls back to α. This ] ] shows that the range of πN is exactly the intersection of TN with the range of π . What are conditions to guarantee that a given submanifold is Poisson-Dirac? The vector ] ] −1 field π (αe) is tangent to N if and only if αe|N takes values in (π ) (TN). Hence, a necessary ∗ ∗ ] −1 condition is that the pullback map T M|N → T N restricts to a surjection (π ) (TN) → T ∗N. The kernel of this map is ann(TN), hence the necessary condition reads as ∗ ] −1 (30) T M|N = ann(TN) + (π ) (TN). Taking annihilators on both sides, this is equivalent to (31) TN ∩ π](ann(TN)) = 0.

If this condition holds, then one obtains a pointwise bivector ΠN |m for all m ∈ N, defined by the pointwise version of (29). However, the collection of these pointwise bivector fields do not define a smooth bivector field, in general. For instance, in Example 3.26 the condition (31) is satisfied, but N is not Poisson-Dirac. A sufficient condition for N to be Poisson-Dirac is the following. Proposition 3.31. The submanifold N ⊆ M is Poisson-Dirac if and only if the exact sequence ∗ ∗ (32) 0 → ann(TN) → T M|N → T N → 0, ∗ ∗ ] −1 admits a splitting j : T N → T M|N whose image is contained in (π ) (TN). That is, N is Poisson-Dirac if and only if ∗ T M|N = ann(TN) ⊕ K where K is a subbundle contained in (π])−1(TN). ∗ ∗ 1 1 Proof. Suppose such a splitting j : T N → T MN is given. Given α ∈ Ω (N), let αe ∈ Ω (M) ∗ ] be any extension of j(α) ∈ Γ(T M|N ). Then αe pulls back to α, and π (αe) is tangent to N. This shows that N is Poisson-Dirac. Conversely, suppose that N is Poisson Dirac. Given a local ∗ frame α1, . . . , αk for T N, we may choose lifts αe1,..., αek as in Lemma 3.29. These lifts span 28 ECKHARD MEINRENKEN

∗ ∗ ∗ a complement to ann(TN) in T M|N , giving the desired splitting j : T N → T M|N locally. But convex linear combinations of splittings are again splittings; and if these splittings take values in (π])−1(TN), then so does their linear combination. Hence, we may patch the local splittings with a partition of unity to obtain a global splitting with the desired property. Remark 3.32. If π](ann(TN)) has constant rank, and zero intersection with TN, then N is a Poisson-Dirac submanifold. Here is a typical example of a Poisson-Dirac submanifold. Proposition 3.33 (Damianou-Fernandes). Suppose a compact Lie group G acts on a Poisson manifold M by Poisson diffeomorphisms. Then M G is a Poisson-Dirac submanifold. Proof. We have a G-equivariant direct sum decomposition ∗ G ∗ G T M|M G = ann(TM ) ⊕ (T M) . ] ∗ G G G By equivariance of the anchor map, π ((T M) ) ⊆ (TM) = T (M ) as required. Remark 3.34. In [42], Xu introduces a special type of Poisson-Dirac submanifolds which he called Dirac submanifolds, but were later renamed as Lie-Dirac submanifolds. We will return to this later. In the case of a compact group action, Fernandes-Ortega-Ratiu [16] prove that M G is in fact a Lie-Dirac submanifold in the sense of Xu [42]. ∗ Remark 3.35. Given splitting of the exact sequence (32), with image K ⊆ T M|N such that ] π (K) ⊆ TN, the restriction of the Poisson tensor decomposes as π|N = πN + πK where 2 2 πN ∈ Γ(∧ TN) and πK ∈ Γ(∧ K). As shown in [42, Lemma 2.5], having such a decomposition already implies that πN is a Poisson tensor. Remark 3.36. Crainic-Fernandes [?] give an example showing that it is possible for a subman- ] ] ifold of a Poisson manifold M to admit a Poisson structure πN with ran(πN ) = ran(π ) ∩ TN, without admitting a splitting of (32). 3.6. Cosymplectic submanifolds. An important special case of Poisson-Dirac submanifold is the following. Definition 3.37. A submanifold N ⊆ M is called cosymplectic if ] TM|N = TN + π (ann(TN)). Remark 3.38. Compare with the definition of a coisotropic submanifold, where π](ann(TN)) ⊆ TN. Remark 3.39. If M is symplectic, then the cosymplectic submanifolds are the same as the symplectic submanifolds. Proposition 3.40. Let N be a submanifold of a Poisson manifold M. The following are equivalent: (a) N is cosymplectic ] (b) TM|N = TN ⊕ π (ann(TN)). ∗ ] −1 (c) T M|N = ann(TN) ⊕ (π ) (TN). (d) ann(TN) ∩ (π])−1(TN) = 0. ∗ (e) The restriction of π to ann(TN) ⊆ T M|N is nondegenerate. INTRODUCTION TO POISSON GEOMETRY 29

(f) N intersects every symplectic leaf of M transversally, with intersection a symplectic submanifold of that leaf. Proof. if N is cosymplectic, then the pointwise rank of π](ann(TN)) must be at least equal to the codimension of N. Hence, it is automatic that the sum in Deﬁnition 3.37 is a direct sum, which gives the equivalence with (b). Condition (c) is equivalent to (b) by dualization, and (d) is equivalent to (a) by taking annihilators on both sides. Next, condition (e) means that if α ∈ ann(TN) is π-orthogonal to all of ann(TN), then α = 0. The space of elements that are π-orthogonal to ann(TN) is ann(π](ann(TN)) = (π])−1(TN), so we see that (e) is equivalent to (d). ] Condition (b) means in particular that TM|N = TN ⊕ ran(π ), so that N intersects the ] ∗ symplectic leaves transversally. Let ωm be the symplectic form on ran(πm). If α ∈ TmM is such ] ∗ ] that v = πm(α) ∈ TmN is non-zero, then by (c) we can ﬁnd β ∈ TmM with w = πm(β) ∈ TmN ] and hβ, vi= 6 0. But this means ωm(v, w) 6= 0, thus TmN ∩ ran(πm) is symplectic. This proves (f); the converse is similar. The main example of a cosymplectic submanifold is the following: Example 3.41. Let M be a Poisson manifold. Suppose m ∈ M, and N is a submanifold passing through m with ] TmM = TmN ⊕ ran(πm). In other words, N intersects the symplectic leaf transversally and is of complementary dimension. Dualizing the condition means ∗ ] TmM = ann(TmN) ⊕ ker(πm), which shows that πm is non-degenerate on ann(TN) at the point m. But then π remains non-degenerate on an open neighborhood of m in N. This neighborhood is then a cosymplectic submanifold, with an induced Poisson structure. One refers to this Poisson structure on N near m as the ‘transverse Poisson structure’ at m.[?] Remark 3.42. Cosymplectic submanifolds are already discussed in Weinstein’s article [?], although the terminology appears later [42, 9]. They are also known as Poisson transversals [17], presumably to avoid confusion with the so-called cosymplectic structures.

4. Dirac structures Dirac structures were introduced by Courant and Weinstein [?, 11] as a differential geometric framework for Dirac brackets in classical mechanics. The basic idea is to represent Poisson structures in terms of their graphs ] ∗ ∗ Gr(π) = {π (α) + α| α ∈ T M} ⊆ TM = TM ⊕ T M. The maximal isotropic subbundles E ⊆ TM arising as graphs of Poisson bivector fields are characterized by a certain integrability condition; dropping the assumption that E is the graph of a map from T ∗M to TM one arrives at the notion of a Dirac structure. Dirac geometry is extremely interesting in its own right; here we will use it mainly to prove facts about Poisson manifolds. Specifically, we will use Dirac geometry to discuss, among other things, 30 ECKHARD MEINRENKEN

(a) the Lie algebroid structure of the cotangent bundle of a Poisson manifold (b) the Weinstein splitting theorem (c) symplectic realizations and symplectic groupoids for Poisson manifolds (d) Poisson Lie groups and Drinfeld’s classiﬁcation We begin with a discussion of the Courant algebroid structure of TM.

4.1. The Courant bracket. Let M be a manifold, and ∗ (33) TM = TM ⊕ T M the direct sum of the tangent and cotangent bundles. Elements of TM will be written x = v+µ, ∗ with v ∈ TmM and µ ∈ TmM, and similarly sections will be written as σ = X + α, where X is a vector ﬁeld and α a 1-form. The projection to the summand TM will be called the anchor map

(34) a: TM → TM thus a(v + µ) = v. Let h·, ·i denote the bundle metric, i.e. non-degenerate symmetric bilinear form,

(35) hv1 + µ1, v2 + µ2i = hµ1, v2i + hµ2, v1i; ∗ 1 here v1, v2 ∈ TM and µ1, µ2 ∈ T M (all with the same base point in M). We will use this metric to identify TM with its dual; for example, the anchor map dualizes to the map ∗ ∗ ∗ ∼ a : T M → TM = TM given by the inclusion. The Courant bracket [11] (also know as the Dorfman bracket [15]) is the following bilinear operation on sections σi = Xi + αi ∈ Γ(TM),

(36) [[σ1, σ2]] = [X1,X2] + LX1 α2 − ιX2 dα1 Remark 4.1. Note that this bracket is not skew-symmetric, and indeed Courant in [11] used the skew-symmetric version [X1,X2] + LX1 α2 − LX2 α1. However, the non-skew symmetric version (36), introduced by Dorfman [15], turned out to be much easier to deal with; in particular it satisﬁes a simple Jacobi identity (see (38) below). For this reason the skew-symmetric version is rarely used nowadays.

Remark 4.2. One motivation for the bracket (36) is as follows. Using the metric on TM, one can form the bundle of Clifford algebras Cl(TM). Thus, Cl(TmM) is the algebra generated by the elements of TmM, subject to relations [x1, x2] ≡ x1x2 + x2x1 = hx1, x2i for xi ∈ TmM (using graded commutators). The Clifford bundle has a spinor module ∧T ∗M, with the Clifford action given on generators by %(x) = ι(v) + (µ) for x = v + µ; here ι(v) is contraction by v and (µ) ∗ is wedge product with µ. Hence, the algebra Γ(Cl(TM)) acts on the space Γ(∧T M) = Ω(M) of differential forms. But on the latter space, we also have the exterior differential d. The Courant bracket is given in terms of this action by

[[d,%(σ1)],%(σ2)] = %([[σ1, σ2]]). It exhibits the Courant bracket as a derived bracket. For more on this viewpoint see [?, ?, ?, 7].

1 Note that TM also has a natural ﬁberwise symplectic form, but it will not be used here. INTRODUCTION TO POISSON GEOMETRY 31

Proposition 4.3. The Courant bracket (36) has the following properties, for all sections σi, σ, τ and all f ∈ C∞(M):

(37) a(σ)hτ1, τ2i = h[[σ, τ1]], τ2i + hτ1, [[σ, τ2]]i,

(38) [[σ, [[τ1, τ2]]]] = [[[[σ, τ1]], τ2]]+ [[τ1, [[σ, τ2]]]], (39) [[σ, τ]]+ [[τ, σ]] = a∗ d hσ, τi.

Furthermore, it satisﬁes the Leibnitz rule

(40) [[σ, fτ]] = f[[σ, τ]]+ (a(σ)f) τ, and the anchor map is bracket preserving:

(41) a([[σ, τ]]) = [a(σ), a(τ)].

All of these properties are checked by direct calculation.

Generalizing these properties, one deﬁnes a Courant algebroid over M [28, ?] to be a vector bundle A → M, together with a bundle metric h·, ·i, a bundle map a: A → TM called the anchor, and a bilinear Courant bracket on Γ(A) satisfying properties (37), (38), and (39) above. One can show [37] that the properties (40) and (41) are consequences. The bundle TM is called the standard Courant algebroid over M. We will encounter more general Courant algebroids later on.

Remark 4.4. For a vector bundle V → M, denote by Aut(V ) the group of vector bundle automorphisms of V . Its elements are diffeomorphism A of the total space of V respecting the linear structure; any automorphism restricts to a diffeomorphism Φ of the base. It defines an action A: Γ(V ) → Γ(V ) on sections; here A.τ = A ◦ τ ◦ Φ−1 where on the right hand side, the section is regarded as a map τ : M → V . This has the property

A(fτ) = (Φ∗f) A(τ) for all f ∈ C∞(M) and τ ∈ Γ(V ), conversely, any such operator on Γ(V ) describes an automorphism of V . Taking derivatives, we see that the inﬁnitesimal automorphism of a vector bundle V → M may be described by operators D : Γ(V ) → Γ(V ) such that there exists a vector ﬁeld X satisfying the Leibnitz rule,

D(fτ) = fD(τ) + X(f)τ.

For a Lie algebroid, the operator given by the Lie algebroid bracket with a fixed section is such a vector bundle automorphisms; the property a([σ, τ]) = [a(σ), a(τ)] says that this automorphism preserves the anchor, and the Jacobi identity for the bracket signifies that this infinitesimal automorphism preserves the bracket. In a similar fashion, for a Courant algebroid A be a Courant algebroid (e.g., the standard Courant algebroid TM), the operator [[σ, ·]] on sections defines an infinitesimal vector bundle automorphism. The property (37) says that this infinitesimal automorphism preserves the metric, (41) says that it preserves the anchor, and (38) says that it preserves the bracket [[·, ·]] itself. 32 ECKHARD MEINRENKEN

⊥ 4.2. Dirac structures. For any subbundle E ⊆ TM, we denote by E its orthogonal with respect to the metric h·, ·i. The subbundle E is called isotropic if E ⊆ E⊥, co-isotropic if E ⊃ E⊥, and maximal isotropic, or Lagrangian if E = E⊥. The terminology is borrowed from symplectic geometry, where it is used for subspaces of a vector space with a non-degenerate skew-symmetric bilinear form. Immediate examples of Lagrangian subbundles are TM and T ∗M. Given a bivector ﬁeld π ∈ X2(M), its graph ] ∗ Gr(π) = {π (µ) + µ| µ ∈ T M} ⊆ TM, is Lagrangian; in fact, the Lagrangian subbundles E ⊆ TM with E ∩ TM = 0 are exactly the graphs of bivector ﬁelds. Similarly, given a 2-form ω its graph [ Gr(ω) = {v + ω (v)| v ∈ TM} ⊆ TM is Lagrangian; the Lagrangian subbundles E ⊆ TM with E ∩ TM = 0 are exactly the graphs of 2-forms. Note that although the Courant bracket is not skew-symmetric, it restricts to a skew- symmetric bracket on sections of Lagrangian subbundles, because the right hand side of (39) is zero on such sections.

Deﬁnition 4.5. A Dirac structure on M is a Lagrangian subbundle E ⊆ TM whose space of sections is closed under the Courant bracket.

Proposition 4.6. Any Dirac structure E ⊆ TM acquires the structure of a Lie algebroid, with the Lie bracket on sections given by the Courant bracket on Γ(E) ⊆ Γ(TM), and with the anchor obtained by restriction of the anchor a: TM → TM. Proof. By (39), the Courant bracket is skew-symmetric on sections of E, and (38) gives the Jacobi identity. The Leibnitz identity follows from that for the Courant bracket, Equation (40). The integrability of a Lagrangian subbundle E ⊆ TM is equivalent to the vanishing of the expression

(42) ΥE(σ1, σ2, σ3) = hσ1, [[σ2, σ3]]i for all σ1, σ2, σ3 ∈ Γ(E). Indeed, given σ2, σ3 ∈ Γ(E), the vanishing for all σ1 ∈ Γ(E) means ⊥ precisely that [[σ2, σ3]] takes values in E = E. Using the properties (37) and (39) of the Courant bracket, one sees that ΥE is skew-symmetric in its entries. Since ΥE is clearly tensorial in its ﬁrst entry, it follows that it is tensorial in all three entries: that is 3 ∗ ΥE ∈ Γ(∧ E ).

In particular, to calculate ΥE it suﬃces to determine its values on any collection of sections that span E everywhere. Proposition 4.7. For a 2-form ω, the graph Gr(ω) is a Dirac structure if and only if dω = 0. In this case, the projection Gr(ω) → TM along T ∗M is an isomorphism of Lie algebroids. Proof. We calculate, [ [ [[X + ω (X),Y + ω (Y )]] = [X,Y ] + LX ιY ω − ιY dιX ω

= [X,Y ] + ι[X,Y ]ω + ιY ιX dω. INTRODUCTION TO POISSON GEOMETRY 33

This takes values in Gr(ω) if and only if the last term is zero, that is, dω = 0. In fact, the ∼ calculation shows that ΥGr(ω) coincides with dω under the isomorphism Gr(π) = TM. Proposition 4.8. A bivector ﬁeld π ∈ X2(M) is Poisson if and only if its graph Gr(π) is a Dirac structure. In this case, the projection Gr(π) → T ∗M (along TM) is an isomorphism of Lie algebroids, where T ∗M has the cotangent Lie algebroid structure determined by π.

Proof. We want to show that ΥGr(π) vanishes if and only if π is a Poisson structure. It suﬃces ∞ ] to evaluate ΥGr(π) on sections of the form Xf + df for f ∈ C (M), where Xf = π (df). Thus ∞ let f1, f2, f3 ∈ C (M) and put σi = Xfi + dfi. We have [[σ , σ ]] = [X ,X ] + d L (f ), 2 3 f2 f3 Xf2 3 hence hσ1, [[σ2, σ3]]i = L (f1) + LX LX (f3) = Jac(f1, f2, f3). [Xf2 ,Xf3 ] f1 f2 The result follows. In fact, we have shown that ΥGr(π) coincides with Υπ under the isomorphism Gr(π) =∼ T ∗M. Finally, it is immediate from the formulas for the Courant bracket and the cotangent Lie algebroid that the isomorphism Gr(π) =∼ T ∗M intertwines the anchor with the map π], and takes the bracket of two sections of Gr(π) to the Lie bracket of the corresponding 1-forms,

(43) [α, β] = Lπ](α)β − ιπ](β)dα 4.3. Tangent lifts of Dirac structures. As we had explained earlier, the cotangent Lie alegbroid structure on T ∗M for a Poisson manifold (M, π) corresponds to the tangent lift to a Poisson structure on TM. What about tangent lifts of more general Dirac structures? Let p: TM → M be the bundle projection. We had defined tangent lifts and vertical lifts of functions. The tangent lift of a vector field X is characterized by XT (fT ) = X(f)T ; the vertical lift by XV (fT ) = X(f)V . In local tangent P i ∂ coordinates, if X = i a (x) ∂xi , X ∂ X ∂ai ∂ X ∂ X = ai(x) + yj ,X = ai(x) . T ∂xi ∂xj ∂yi V ∂yi i ij i We have [XT ,YT ] = [X,Y ]T , [XV ,YT ] = [X,Y ]V , [XV ,YV ] = 0. Similar formulas define the tangent and vertical lifts of multi-vector fields, e.g. for a bivector field πT (dfT , dgT ) = (π(df, dg))T , πV (dfT , dgT ) = (π(df, dg))V . With this notation, the tangent lift of a Poisson structure πTM is indeed just πT . For differential forms, we define the vertical lift αV to be simply the pull-back. The tangent lift of functions extends uniquely to a tangent lift of differential forms, in such a way that (df)T = d(fT ) and

(α ∧ β)T = αV ∧ βT + αT ∧ βV . P i For 1-forms α = i αidx , one ﬁnds, X X ∂αi α = α dyi + yj dxi. T i ∂xj i ij 34 ECKHARD MEINRENKEN

Here are some basic formulas for tangent and vertical lifts:

ι(XT )αT = (ι(X)α)T , ι(XT )αV = (ι(X)α)V = ι(XV )αT , ι(XV )αV = 0;

L(XT )αT = (L(X)α)T , L(XT )αV = (L(X)α)V = L(XV )αT , L(XV )αV = 0. For σ = X + α ∈ Γ(TM), consider σT = XT + αT ∈ Γ(T(TM)) and σV = XV + αV . From the properties of tangent and vertical lifts of 1-forms and vector ﬁelds, we obtain,

hσT , τT i = hσ, τiT , hσV , τV i = 0, hσV , τT i = hσ, τiT

[[σT , τT ]] = [[σ, τ]]T , [[σV , τV ]] = 0, [[σV , τT ]] = [[σ, τ]]V = [[σT , τV ]] and ﬁnally, a(σT ) = (a(σ))T , a(σV ) = a(σ)V . As an application, we can prove:

Theorem 4.9. For any Dirac structure E ⊆ TM there is a unique Dirac structure ET ⊆ TMT such that σT ∈ Γ(ET ) for all σ ∈ Γ(E).

Proof. For non-zero v ∈ TM, there is at least one function f such that fT (v) 6= 0. Since (fσ)T = fV σT + fT σV , we conclude that the subspace (ET )v ⊆ Tv(TM) spanned by the tangent lifts of sections of E is the same as the subspace spanned by the tangent and cotangent lifts of sections of E. Inside T(TM), we have a subbundle (TM)V , spanned by all vertical lifts of sections of TM. ∗ It is canonically isomorphic to the vector bundle pull-back of p (TM). The quotient space ∗ (TM)H := T(TM)/(TM)V is isomorphic to p (TM) as well; looking at the explicit formulas we see that it is spanned by image of horizontal lifts. The vertical lifts of sections of E span a ∼ ∗ ∼ ∗ subbundle EV = p E, while the images of tangent lifts in (TM)H span a subbundle EH = p E. It hence follows that at any v ∈ TM, the span (ET )v of the vertical and tangent lifts of sections of E has dimension at least 2 rank(E) = 2 dim M. From the properties of tangent and vertical lifts, it is immediate that this subspace is isotropic, hence its dimension is exactly 2 dim M. We conclude that ET is a subbundle, and using the Courant bracket relations of tangent and vertical lifts it is clear that ET deﬁnes a Dirac structure.

5. Gauge transformations of Poisson and Dirac structures One simple way to produce new Dirac structures from given ones is to apply a bundle automorphism of TM preserving the Courant algebroid structures. 5.1. Automorphisms of the Courant algebroid structure. Recall from Remark 4.4 that if V → M is any vector bundle, we denote by Aut(V ) the group of vector bundle automorphisms. Any such automorphisms restricts to a diffeomorphism Φ of the zero section; the kernel of the restriction map Aut(V ) → Diff(M) is denoted Gau(V ); its elements are called gauge transformations of V . We have an exact sequence, 1 → Gau(V ) → Aut(V ) → Diff(M) where the last map need not be surjective, in general. Similarly, we denote by gau(V ) the kernel of the restriction map aut(V ) → X(M), it fits into an exact sequence of Lie algebras 0 → gau(V ) → aut(V ) → X(M) → 0. INTRODUCTION TO POISSON GEOMETRY 35

Let AutCA(TM) denote the group of Courant algebroid automorphisms of TM, that is, A ∈ Aut(TM), preserves the metric, bracket and anchor. In terms of the resulting action on sections, letting Φ: M → M be the base map, this means

hAσ, Aτi = Φ∗hσ, τi, [[Aσ, Aτ]] = A[[σ, τ]], a ◦ A = T Φ ◦ a.

The group homomorphism AutCA(TM) → Diff(M) is surjective, since every diffeomorphism Φ ∈ Diff(M) defines a standard Courant algebroid automorphism

TΦ ∈ AutCA(TM) by taking the sum of the tangent and cotangent maps. This gives a split exact sequence

1 → GauCA(TM) → AutCA(TM) → Diﬀ(M), where the splitting identiﬁes the Courant automorphisms with a semi-direct product,

AutCA(TM) = GauCA(TM) o Diﬀ(M). For any 2-form ω, deﬁne an automorphism

Rω ∈ Aut(TM), x 7→ x + ιa(x)ω.

Obviously, Rω1+ω2 = Rω1 ◦ Rω2 .

Proposition 5.1. [21] The automorphism Rω preserves the metric and anchor; it preserves the Courant bracket if and only if dω = 0. The map

Ωcl(M) → GauCA(TM), ω 7→ R−ω is a group isomorphism. Proof. For σ = X + α and τ = Y + β, we have that

[[Rωσ, Rωτ]] = [X,Y ] + LX (β + ιY ω) − ιY d(α + ιX ω)

= [X,Y ] + LX β − ιY dα + LX ιY ω − ιY dιX ω

= [X,Y ] + LX β − ιY dα + ι[X,Y ]ω + ιY ιX dω

= Rω[[σ, τ]]+ ιY ιX dω

The fact that Rω preserves the metric and anchor is similar, but easier. Suppose now that A ∈ GauCA(TM) is given. In particular, a ◦ A = a. Since A preserves the metric, we have ∗ −1 −1 ∗ ∗ A = A under the identification of TM with its dual. Hence A ◦ a = a , which means that A−1, hence also A, fixed T ∗M pointwise, while on the other hand A0v − v ∈ T ∗M for all v ∈ TM. Since A preserves the metric, 0 = hv, wihAv, Awi = hAv, wi + hv, Awi for all v, w ∈ TM. Hence there is a well-defined 2-form ω such that ω(v, w) = hv, Awi, and A = R−ω. But R−ω preserves the Courant bracket if and only if ω is closed. 36 ECKHARD MEINRENKEN

Remark 5.2. The calculation applies more generally to twisted Courant brackets: Given a closed 3-form η ∈ Ω3(M), one has the η-twisted Courant bracket

[[σ, τ]]η = [X,Y ] + LX β − ιY dα + ιX ιY η. ∞ For Φ ∈ C (M) one has [[TΦ.σ, TΦ.τ]]η = TΦ.([[σ, τ]]Φ∗η). Given a 2-form ω, one has that

[[Rωσ, Rωτ]]η+dω = Rω[[σ, τ]]η. In summary, we have shown that 2 AutCA(TM) = Ωcl(M) o Diff(M), where (ω, Φ) acts as R−ω ◦ TΦ. We can similarly discuss the Lie algebra autCA(TM) of infinitesimal Courant algebroid automorphisms. Regarded as operators on sections, these are the linear maps D : Γ(TM) → Γ(TM) such that there exists a vector field X with D(fσ) = fD(σ) + X(f)σ, hDσ, τi + hσ, Dτi = Xhσ, τi, [[Dσ, τ]]+ [[σ, Dτ]] = D[[σ, τ]], a(Dτ) = [X, a(τ)]. As mentioned before, D = [[σ, ·]] has all these properties. Proposition 5.3 (Infinitesimal automorphisms of the Courant bracket). [21] (a) The Lie algebra of infinitesimal Courant automorphisms is a semi-direct product 2 autCA(TM) = Ωcl(M) o X(M), where the action of (γ, X) on a section τ = Y + β ∈ Γ(TM) is given by

(44) (γ, X).τ = [X,Y ] + LX β − ιY γ. (b) For any section σ = X + α ∈ Γ(TM), the Courant bracket [[σ, ·]] is the infinitesimal automorphism (dα, X). Proof. The proof of a) is similar to the global case. The given formula defines an injective map 2 from Ωcl(M) o X(M) to autCA(TM). To see that it is surjective, let D ∈ autCA(TM) be given, 0 with base vector field X. By subtracting LX ∈ autCA(TM), we obtain D = D − LX with corresponding vector field equal to 0. Since D0(fσ) = fD0(σ), it follows that D0 is given by an infinitesimal gauge transformation of TM, i.e. by a section of the bundle of endomorphisms of 0 ∗ TM. Since a ◦ D = 0, we see that this endomorphism takes values in T M. Dually, we obtain D0 ◦ a∗ = 0, hence D0 vanishes on T ∗M. Hence, it is given by a bundle map TM → T ∗M. Since D0 preserves metrics, hD0X,Y i + hX,D0Y i = 0. Hence, there is a well-defined 2-form γ such that γ(X,Y ) = hX,D0(Y )i, and the action of D0 is 0 D (X + α) = −ιX γ. Finally, using that D0 preserves brackets one finds that γ must be closed. Property b) is immediate from (44) and the formula for the Courant bracket. INTRODUCTION TO POISSON GEOMETRY 37

We are interested in the integration of infinitesimal Courant automorphisms, especially those generated by sections of TM. In the discussion below, we will be vague about issues of com- pleteness of vector fields; in the general case one has to work with local flows. The following result is an infinite-dimensional instance of a formula for time dependent flows on semi-direct products V o G, where G is a Lie group and V a G-representation.

Proposition 5.4. [21, 22] Let (ωt, Φt) ∈ AutCA(TM) be the family of automorphisms integrating the time-dependent inﬁnitesimal automorphisms (γt,Xt) ∈ autCA(TM). Then Φt is the ﬂow of Xt, while Z t ωt = (Φs)∗γs ds. 0

Proof. Recall (cf. Appendix ??) that the flow Φt of a time dependent vector field Xt is defined d in terms of the action on functions by dt (Φt)∗ = (Φt)∗ ◦ LXt . Similarly, the 1-parameter family of Courant automorphisms (ωt, Φt) integrating (γt,Xt) is defined in terms of the action on sections τ ∈ Γ(TM) by d (ω , Φ ).τ = (ω , Φ ).(γ ,X ).τ. dt t t t t t t Write τ = Y + β ∈ Γ(TM). Then d d (ω , Φ ).τ = (Φ ) τ − ι(Φ ) Y ω dt t t dt t ∗ t ∗ t dω = (Φ ) L τ − ι(Φ ) L Y ω − ι(Φ ) Y t . t ∗ Xt t ∗ Xt t t ∗ dt On the other hand, (ωt, Φt).(γt,Xt).τ = (ωt, Φt). LXt τ − ι(Y )γt = (Φt)∗LXt τ − ι (Φt)∗Y (Φt)∗γt − ι (Φt)∗LXt Y ωt. d Comparing, we see (Φt)∗γt = dt ωt.

This calculation applies in particular to the inﬁnitesimal automorphisms (γt,Xt) deﬁned by σt = Xt + αt ∈ Γ(TM); here γt = dαt. Note that in this case, Z t ωt = d (Φs)∗αs ds 0 is a family of exact 2-forms.

5.2. Moser method for Poisson manifolds. For any Dirac structure E ⊆ TM, and closed 2 ω 2-form ω ∈ Ωcl(M), one obtains a new Dirac structure E = Rω(E) called the gauge transformation of E by ω. We are interested in the special case that E is the graph of a Poisson bivector ﬁeld.

2 Lemma 5.5. Let π be a Poisson structure on M, and ω ∈ Ωcl(M) a closed 2-form. Then Gr(π)ω is transverse to TM if and only if the bundle map id +ω[ ◦ π] : T ∗M → T ∗M 38 ECKHARD MEINRENKEN is invertible. In this case, the Poisson structure πω defined by Gr(π)ω = Gr(πω) satisfies (45) (πω)] = π] ◦ (id +ω[ ◦ π])−1. Proof. By definition, ω ] ∗ Gr(π) = {π (µ) + µ + ιπ](µ)ω| µ ∈ T M} This is transverse to TM if and only if the projection to T ∗M is an isomorphism, that is, if and only if for all ν ∈ T ∗M there is a unique solution of [ ] ν = µ + ιπ](µ)ω ≡ (id +ω ◦ π )µ. ω ω ] ] Furthermore, in this case the resulting π is given by (π ) (ν) = π (µ), which proves (45). One calls πω the gauge transformation of π by the closed 2-form ω. Lemma 5.6. [5] The Poisson structure πω and π define the same symplectic foliation. The 2-forms on leaves are related by pull-back of ω. Proof. From (45), it is immediate that ran((πω)]) = ran(π]). For the second claim, given ] ] m ∈ M, let σ be the 2-form on Sm = ran(πm) defined by πm. If v1, v2 ∈ Sm with vi = πm(µi), we have

σ(v1, v2) = −πm(µ1, µ2) = hµ1, v2i. Similarly, let σω be the 2-form deﬁned by πω: ω σ (v1, v2) = hν1, v2i, ω ] where ν1 is such that (π ) ν1 = v1. As we saw above, [ ] ν1 = (id +ω π )µ1 = µ1 + ιv1 ω. ω Hence, σ (v1, v2) = hν1, v2i = σ(v1, v2) + ω(v1, v2). The standard Moser argument for symplectic manifolds shows that for a compact symplectic manifold, any 1-parameter family of deformations of the symplectic forms in a prescribed cohomology class is obtained by the action of a 1-parameter family of diﬀeomorphisms. The following version for Poisson manifolds can be proved from the symplectic case, arguing ‘leafwise’, or more directly using the Dirac geometric methods described above.

2 Theorem 5.7. [1, 2] Suppose πt ∈ X (M) is a 1-parameter family of Poisson structures related by gauge transformations, ωt πt = (π0) , 2 where ωt ∈ Ω (M) is a family of closed 2-forms with ω0 = 0. Suppose that dω t = −da , dt t 1 ] with a smooth family of 1-forms at ∈ Ω (M), defining a time dependent vector field Xt = πt (at). Let Φt be the flow of Xt. Then (Φt)∗πt = π0. INTRODUCTION TO POISSON GEOMETRY 39

Proof. Let bt = at − ι(Xt)ωt, so that X + t + b + t = R−ωt (Xt + at). Since Xt + at is a section of Gr(πt) = Rωt Gr(π0) it follows that Xt + bt is a section of Gr(π0). Hence, Courant bracket with Xt + bt preserves Γ(Gr(π0)). Equivalently, the family of inﬁnitesimal automorphisms (dbt,Xt) ∈ autCA(TM) preserves Gr(π0), hence so does its ﬂow (ut, Φt) ∈ AutCA(TM). By Proposition 5.4, the 2-forms ut are given in terms of their derivative by d u = (Φ ) db dt t t ∗ t = (Φt)∗(dat − L(Xt)ωt) dω = −(Φ ) ( t + L(X )ω ) t ∗ dt t t d = − ((Φ ) ω ). dt t ∗ t

Thus, ut = −(Φt)∗ωt. It follows that

Gr(π0) = R−ut ◦ TΦt(Gr(π0))

= TΦt ◦ Rωt (Gr(π0)) = TΦt(Gr(πt)) = Gr (Φt)∗πt which shows that π0 = (Φt)∗πt.

6. Dirac morphisms We still have to express ‘Poisson maps’ in terms of Dirac geometry.

6.1. Morphisms of Dirac structures. Let ϕ ∈ C∞(N,M) be a smooth map. Recall that ∗ ∗ ∗ the vector bundle morphism T ϕ: TN → TM dualizes to a comorphism T ϕ: T N 99K T M, ∗ ∗ ∗ given fiberwise by maps in the opposite direction, Tϕ(n)M → Tn N. The comorphism T ϕ, or rather its graph Gr(T ∗ϕ) ⊆ T ∗M × T ∗N defines a relation from T ∗N to T ∗M. Similarly, we define Tϕ: TN 99K TM as a relation from TN to TM; its graph is the sum of the graphs of the tangent and cotangent maps. We write y ∼ϕ x ⇔ (x, y) ∈ Gr(Tϕ). Given x = v + µ ∈ TmM, y = w + ν ∈ TnN this means m = ϕ(n) and v = (Tnϕ) w, ν = ∗ (Tnϕ) µ. For sections σ, τ ∈ Γ(TM) we write

τ ∼ϕ σ ⇔ (σ, τ) restricts to a section of Gr(Tϕ). 2 ∗ For σ = X + α, τ = Y + β, this means Y ∼ϕ X (related vector ﬁelds ) and β = ϕ α.

2 Recall that vector ﬁelds Y ∈ X(N) and X ∈ X(M) are ϕ-related (written Y ∼ϕ X) if (Tnϕ)(Yn) = Xϕ(n) for all n ∈ N. If Y1 ∼ϕ X1 and Y2 ∼ϕ X2 then [Y1,Y2] ∼ϕ [X1,X2]. 40 ECKHARD MEINRENKEN

Lemma 6.1. The relation Tϕ preserves Courant brackets, in the sense that ∗ τ1 ∼ϕ σ1, τ2 ∼ϕ σ2 ⇒ hτ1, τ2i = ϕ hσ1, σ2i,

τ1 ∼ϕ σ1, τ2 ∼ϕ σ2 ⇒ [[τ1, τ2]] ∼ϕ [[σ1, σ2]]

τ ∼ϕ σ ⇒ a(τ) ∼ϕ a(σ) Proof. Straightforward computation, using that if two pairs of vector fields are related, then their Lie brackets are related. ∞ Definition 6.2. Let F ⊆ TN and E ⊆ TM be Dirac structures. Then ϕ ∈ C (N,M) defines a (forward) Dirac morphism Tϕ:(TN,F ) 99K (TM,E) if it has the following property: For every n ∈ N and x ∈ Eϕ(n), there exists a unique y ∈ Fn such that y ∼ϕ x. Remark 6.3. We will use the term weak Dirac morphism for a similar definition where we omit ∗ ∗ the uniqueness condition. For instance, Tϕ:(TN,T N) 99K (TM,T M) is a Dirac morphism, but Tϕ:(TN,TN) 99K (TM,TM) is only a weak Dirac morphism.

Proposition 6.4. Let (N, πN ) and (M, πM ) be Poisson manifolds. Then ϕ: N → M is a Poisson map if and only if Tϕ:(TN, Gr(πN )) 99K (TM, Gr(πM )) is a Dirac morphism. Proof. ϕ is a Poisson map if and only if

∗ ∗ πN (ϕ µ1, ϕ µ2) = πM (µ1, µ2) ∗ for all µ1, µ2 ∈ T M. Equivalently, this means

] ∗ ] T ϕ(πN (ϕ µ)) = πM (µ) for all µ ∈ T ∗M, i.e. ] ∗ ∗ ] πN (ϕ µ)) + ϕ µ ∼ϕ πM (µ) + µ ∗ for all µ ∈ T M. This precisely means that for every x ∈ Gr(πM )ϕ(n) there exists y ∈ Gr(πN )n, necessarily unique, with y ∼ x.

Let Tϕ:(TN,F ) 99K (TM,E) be a Dirac morphism. Using the uniqueness assumption in the deﬁnition, one obtains a linear maps Eϕ(n) → Fn, taking x ∈ Eϕ(n) to the unique y ∈ Fn such that y ∼ϕ x. It is not hard to see that this depends smoothly on n, and hence gives a comorphism of vector bundles.

Proposition 6.5. Any Dirac morphism Tϕ:(TN,F ) 99K (TM,E) deﬁnes a Lie algebroid comorphism F 99K E. Proof. We have to check that (i) the pull-back map ϕ∗ : Γ(E) → Γ(F ) preserves brackets, and (ii) the anchor satisﬁes T ϕ(a(y)) = a(x) for y ∼ϕ x. But both properties are immediate from the previous Lemma. INTRODUCTION TO POISSON GEOMETRY 41

6.2. Pull-backs of Dirac structures. In general, there is no natural way of pulling back a Poisson structure under a smooth map ϕ: N → M. However, such pull-back operations are deﬁned for Dirac structures, under transversality assumptions.

Proposition 6.6. Suppose E ⊆ TM is a Dirac structure, and ϕ: N → M is transverse to the anchor of E. Then ! ϕ E = {y ∈ TN| ∃x ∈ E : y ∼ϕ x} is again a Dirac structure. ! Put differently, ϕ E is the pre-image of E under the relation Tϕ. Proof. Consider first the case that ϕ is the embedding of a submanifold, ϕ: N,→ M. The ! transversality condition ensures that ϕ E is a subbundle of TN of the right dimension; since Tϕ preserves metrics it is isotropic, hence Lagrangian. It also follows from the transversality ! that for any y ∈ ϕ E, the element x ∈ E such that y ∼ϕ x is unique; this defines an inclusion ! ϕ E,→ E|N with image a−1(TN) ∩ E. Hence, every section τ ∈ Γ(ϕ!E) admits an extension to a section σ ∈ Γ(E); thus τ ∼ϕ σ. Conversely, given σ ∈ Γ(E) such that a(σ) is tangent to N we have ! τ ∼ϕ σ for a (unique) section τ. Suppose τ1, τ2 are sections of ϕ E, and choose σi ∈ Γ(E) such ! that τi ∼ϕ σi. Then [[τ1, τ2]] ∼ϕ [[σ1, σ2]], so that [[τ1, τ2]] is a section of ϕ E. This proves the proposition for the case of an embedding. In the general case, given ϕ consider the embedding of N as the graph of ϕ, j : N → M × N, n 7→ (ϕ(n), n). It is easy to see that ϕ!E = j!(E × TN) ! as subsets of TN. Since j (E × TN) is a Dirac structure by the above, we are done. In particular, if π is a Poisson structure on M, and ϕ: N → M is transverse to the map ] ! π , we can define the pull-back ϕ Gr(π) ⊆ TN as a Dirac structure. In general, this is not a Poisson structure. We have the following necessary and sufficient condition. Proposition 6.7. Suppose (M, π) is a Poisson manifold, and ϕ: N → M is transverse to π]. ! Then ϕ Gr(π) ⊆ TN defines a Poisson structure πN if and only if ϕ is an immersion as a cosymplectic submanifold. That is, ] (46) TM|N = TN ⊕ π (ann(TN)). ! Proof. ϕ Gr(π) defines a Poisson structure if and only if it is transverse to TN ⊆ TN. But ! ϕ Gr(π) ∩ TN contains in particular elements y ∈ TN with y ∼ϕ 0. Writing y = w + ν with a tangent vector w and covector ν, this means that ν = 0 and w ∈ ker(T ϕ). Hence, it is necessary that ker(T ϕ) = 0. Let us therefore assume that ϕ is an immersion. For w ∈ TN ⊆ TN, we have ] ] ∗ w ∼ϕ π (µ) + µ ∈ Gr(π) ⇔ (T i)(w) = π (µ), µ ∈ ker(ϕ ) = ann(TN). ! ] Hence, the condition i E ∩ TN = 0 is equivalent to π (ann(TN)) ∩ TN = 0. 42 ECKHARD MEINRENKEN

7. Normal bundles and Euler-like vector fields In this section we will develop some diﬀerential geometric machinery, in preparation for our approach to the Weinstein splitting theorem. 7.1. Normal bundles. Consider the category of manifold pairs: An object (M,N) in this category is a manifold M together with a submanifold N ⊆ M, and a morphism Φ: (M1,N1) → (M2,N2) is a smooth map Φ: M1 → M2 taking N1 to N2. The normal bundle functor ν assigns to (M,N) the vector bundle ν(M,N) = TM|N /T N, over N, and to a morphism Φ: (M1,N1) → (M2,N2) the vector bundle morphism

ν(Φ): ν(M1,N1) → ν(M2,N2). Under composition of morphisms, ν(Φ0 ◦ Φ) = ν(Φ0) ◦ ν(Φ). Example 7.1 (Tangent functor). Given a pair (M,N), the tangent functor gives a new pair (TM,TN) with a morphism p:(TM,TN) → (M,N) deﬁned by the bundle projection. Applying the normal functor, we obtain an example of a double vector bundle ν(TM,TN) / ν(M,N)

TN / N On the other hand, by applying the tangent functor to ν(M,N) → N we obtain a similar double vector bundle, T ν(M,N) / ν(M,N)

TN / N There is a canonical identiﬁcation T ν(M,N) =∼ ν(TM,TN) identifying these two double vector bundles, in such a way that for morphisms Φ of pairs, ν(T Φ) = T ν(Φ). See [?] for details.

Example 7.2. Suppose X is a vector ﬁeld on M such that X|N is tangent to N. Viewed as a section of the tangent bundle p: TM → M, it deﬁnes a morphism X :(M,N) → (TM,TN), inducing ν(X): ν(M,N) → ν(TM,TN) = T ν(M,N).

From p ◦ X = idM we get

T ν(p) ◦ ν(X) = ν(T p) ◦ ν(X) = ν(idM ) = idν(M,N) . That is, ν(X) is a vector field on ν(M,N). It is called the linear approximation to X along N. In local bundle trivializations, the linear approximation is the first order Taylor approximation in the normal directions. Lemma 7.3. For a vector bundle V → M, there is a canonical identification ν(V,M) =∼ V . Proof. The restriction of TV → V to M ⊆ V splits into the tangent bundle of the fiber and the tangent space to the base: TV |M = V ⊕ TM. Hence ν(V,M) = TV |M /T M = V . INTRODUCTION TO POISSON GEOMETRY 43

7.2. Tubular neighborhood embeddings. Given a pair (M,N), Lemma 7.3, applied to ν(M,N) → N, gives an identification ν(ν(M,N),N) = ν(M,N). Definition 7.4. A tubular neighborhood embedding is a map of pairs ϕ:(ν(M,N),N) → (M,N) such that ϕ: ν(M,N) → M is an embedding as an open subset, and the map ν(ϕ) is the identity. Definition 7.5. Let X be a vector field on M that is tangent to N. By a linearization of the vector field X along N, we mean a tubular neighborhood embedding ϕ taking ν(X) to X on a possibly smaller neighborhood of N. The problem of C∞-linearizability of vector fields is quite subtle; the main result (for N = pt) is the Sternberg linearization theorem [34] which proves existence of linearizations under non- resonance conditions.

2 Example 7.6. The vector ﬁeld on R given as ∂ ∂ x + (2y + x2) ∂x ∂y does not satisfy Sternberg’s non-resonance conditions, and turns out to be not linearizable.3. On the other hand, ∂ ∂ 2x + (y + x2) ∂x ∂y is linearizable.

n Example 7.7. A vector field on R of the form X ∂ x + higher order terms i ∂x i i satisfies the non-resonance condition, and hence is always linearizable. We will make the following definition. Definition 7.8. [8] A vector field X ∈ X(M) is called Euler-like along N if it is complete, with X|N = 0, and with linear approximation ν(X) the Euler vector field E on ν(M,N).

Remark 7.9. (a) In a submanifold chart, with coordinates x1, . . . , xn on N and y1 . . . , yk in the transverse direction, an Euler-like vector ﬁeld has the form X ∂ X ∂ X ∂ X = y + g (x, y) + h (x, y) , i ∂y i ∂y j ∂x i i i i j j

where gi vanish to second order for y → 0, while hj vanish to ﬁrst order.

3See http://mathoverﬂow.net/questions/76971/nice-metrics-for-a-morse-gradient-ﬁeld-counterexample- request 44 ECKHARD MEINRENKEN

(b) Another coordinate-free characterization of Euler-like vector fields is as follows [?] Let I ⊆ C∞(M) be the ideal of functions vanishing along N. Its powers Ik are the functions vanishing to order k along N. Then a complete vector field X is Euler-like along N if and only if for all f ∈ I, we have that LE f equals f modulo functions vanishing on N to second order (or higher). That is, 2 LX − id: I → I . More generally, this property implies that k k+1 LX − k id: I → I for all k = 0, 1, 2,.... An Euler-like vector field determines a tubular neighborhood embedding: Theorem 7.10. If X ∈ X(M) is Euler-like along N, then X determines a unique tubular neighborhood embedding ϕ: ν(M,N) → M such that

E ∼ϕ X. Proof. The main point is to show that X is linearizable along N. Start out by picking any tubular neighborhood embedding to assume M = ν(M,N). Since ν(X) = E, it follows that the difference Z = X −E vanishes to second order along N. Let κt denote scalar multiplication by t on ν(M,N), and consider the family of vector fields, defined for t 6= 0, 1 Z = κ∗Z t t t Since Z vanishes to second order along N, this is well-defined even at t = 0. Let φt be its (local) flow. 4 On a sufficiently small open neighborhood of N in ν(M,N), it is defined for all |t| ≤ 1. The flow Ψs of the Euler vector field E is

Ψs = κexp(−s) by substitution t = exp(s) this shows that d κ∗ = t−1κ∗ ◦ L . dt t t E Consequently, d d (tZ ) = (κ∗Z) = L Z = [E,Z ]. dt t dt t E t t Therefore, d (φ ) (E + tZ ) = (φ ) (L (E + tZ ) + [E,Z ]) = 0, dt t ∗ t t ∗ Zt t t which shows that (φt)∗(E +tZt) does not depend on t. Comparing the values at t = 0 and t = 1 −1 we obtain (φ1)∗X = E, so that (φ1) giving the desired linearization on a neighborhood of N. In summary, this shows that there exists a map from a neighborhood of the zero section of ν(M,N) to a neighborhood of N in M, intertwining the two vector fields E and X, and hence also their flows. Since X is complete, we may use the flows to extend globally to a tubular neighborhood embedding of the full normal bundle. This proves existence.

4 d Thus dt (φt)∗ = (φt)∗ ◦ LZt as operators on tensor ﬁelds (e.g., functions, vector ﬁelds, and so on). INTRODUCTION TO POISSON GEOMETRY 45

For uniqueness, suppose that a tubular neighborhood embedding ψ satisfying E ∼ψ X is given. Let Ψs be the flow of E and Φs the flow of X. We have that κt = Ψ− log(t) for t > 0; accordingly we define λt = Φ− log(t). Since νN is invariant under κt for all t > 0, its image U = ψ(νN ) is invariant under λt for all t > 0. Furthermore, since limt→0 κt is the retraction p from νN onto N ∈ νN , we have

(47) U = {m ∈ M| lim λt(m) exists and lies in N ⊆ M}. t→0 We want to give a formula for the inverse map ψ−1 : U → ν(M,N). For all v ∈ ν(M,N), with base point x ∈ N, the curve κt(v) in ν(M,N) has tangent vector at t = 0 equal to v itself (using the identiﬁcation T ν(M,N)|N = TN ⊕ ν(M,N)). Hence d v = κt(v) mod TxN. dt t=0

Using λt ◦ ψ = ψ ◦ κt, and writing ψ(v) = m, this shows,

−1 d (48) ψ (m) = λt(m) mod TxN ∈ TxM/TxN dt t=0 Formulas (47) and (48) give a description of the tubular neighborhood embedding directly in terms of the ﬂow of X, which proves the uniqueness part.

Example 7.11. Let V → M be a vector bundle. Its Euler vector ﬁeld EV determines a tubular neighborhood embedding ν(V,M) → V. This is just the ‘canonical identiﬁcation’ from Lemma 7.3.

Proposition 7.12. Let Φ:(M1,N1) → (M2,N2) be smooth map of pairs. Suppose Xi are Euler-like vector ﬁelds for these pairs, and that X1 ∼Φ X2. Then the following diagram, where the vertical maps are the tubular neighborhood embeddings deﬁned by the Xi, commutes:

M Φ / M O 1 O 2

ϕ1 ϕ2

ν(M1,N1) / ν(M2,N2) ν(Φ)

Proof. Let Ui be the images of the tubular neighborhood embeddings ϕi. Since Φ intertwines the Euler-like vector ﬁelds, it follows that Φ(U1) ⊆ U2. We need to show that −1 −1 ν(Φ) ◦ ϕ1 = ϕ2 ◦ Φ, −1 but this is immediate from the explicit formula for ϕi . 7.3. The Grabowski-Rotkiewicz theorem. This result has the following remarkable consequence for vector bundles, due to Grabowski-Rotkiewicz [20, Corollary 2.1].

Theorem 7.13 (Grabowski-Rotkiewicz). Let V1 → M1 and V2 → M2 be vector bundles, and Ψ: V1 → V2 a smooth map. Then Ψ is a vector bundle morphism if and only if Ψ intertwines the Euler vector ﬁelds. 46 ECKHARD MEINRENKEN

Proof. Proposition 7.12 gives a commutative diagram

V Ψ / V O1 O2

ϕ1 ϕ2

ν(V1,M1) / ν(V2,M2) ν(Ψ) Here the vertical maps, given as the tubular neighborhood embeddings for the Euler(-like) vector ﬁelds, are just the standard identiﬁcations of the normal bundle of the zero section inside a vector bundle, with the vector bundle itself. In particular, they are vector bundle isomorphisms. Since the lower horizontal map is a vector bundle map, the upper horizontal map is one also. This result shows that a smooth map of vector bundles is a vector bundle morphism if and only if it intertwines the scalar multiplications – the fact that it intertwines additions is automatic. We may thus characterize vector bundles as manifold pairs (V,M) together with a smooth map action κt : V → V of the multiplicative group R>0 such that • for all v ∈ V , limt→0 κt(v) ∈ M, • the action preserves M, i.e. κt :(V,M) → (V,M), • the resulting action ν(κt): ν(V,M) → ν(V,M) is the standard scalar multiplication by t > 0.

Indeed, letting E be the vector field on V with flow s 7→ κexp(−s), the second condition shows that E is tangent to M, the last condition shows that E is Euler-like (in particular, it vanishes along M), and the first condition guarantees that the resulting tubular neighborhood embedding ν(V,M) → V is surjective. The manifold V inherits the vector bundle structure via this identification with ν(V,M). Grabowski-Rotkiewicz [20] also have the following attractive characterization of vector subbundles. Proposition 7.14. Let V → M be a vector bundle. A subset L ⊆ V is a vector subbundle if and only if it is invariant under scalar multiplication κt, for all t ≥ 0 (including t = 0) Proof. It is a general result (see e.g. [?]) that if Q is a manifold and Φ: Q → Q is a smooth projection (i.e., Φ◦Φ = Φ), then the image Φ(Q) ⊆ Q is a submanifold. In our case, κ0 : V → V is such a projection, and so is its restriction to L. It follows that κ0(L) = κ0(V ) ∩ L = M ∩ L is a smooth submanifold of L. The Euler vector field of V restricts to L, and is Euler-like along M ∩ L. Hence, L acquires the structure of a vector bundle over M ∩ L. Since the inclusion L → V intertwines Euler vector fields, it is a vector bundle morphism. Remark 7.15. One of the main applications of the Grabowski-Rotkiewicz theorem is a simple characterization of double vector bundles. A double vector bundle is a commutative square D / A

B / M where all maps are vector bundle maps, with suitable compatibility conditions between the horizontal and vertical vector bundle structures. In the original deﬁnition, this was given INTRODUCTION TO POISSON GEOMETRY 47 by a long list of conditions for vertical and horizontal addition and multplication. Accord- ing to Grabowski-Rotkiewicz, the compatibility conditions are equivalent to stating that the horizontal and vertical Euler vector ﬁelds (equivalently the horizontal and vertical scalar multiplication) commute! A typical example is the tangent bundle of a vector bundle, TV / V

TM / M

8. The splitting theorem for Lie algebroids 8.1. Statement of the theorem. Our goal in this section is to prove the following result. Theorem 8.1. Let (E, a, [·, ·]) be a Lie algebroid over M, and N ⊆ M a submanifold transverse to the anchor. Then there exists a tubular neighborhood embedding ϕ: ν(M,N) → U ⊆ M with an isomorphism of Lie algebroids,

! ! p i E / E|U ⊆ E

/ ν(M,N) ϕ U ⊆ M Here p: ν(M,N) → N and i: N → M are the projection and inclusion. See (25) for the deﬁnition of a pull-back of a Lie algebroid under a smooth map transverse to the anchor. In this case, we have that i!E = a−1(TN), and ! ! ! p i E = i E ×TN T ν(M,N). Remark 8.2. If the normal bundle is trivial, ν(M,N) = N × S where S is a vector space, then p!i!E is simply the direct product i!E × TS. Hence, we obtain an isomorphism ! ∼ i E × TS = E|U which justiﬁes the name ‘splitting theorem’. Note that this isomorphism also shows ! a(i E) × TS = a(E)|U .

Hence, the leaves (if any) of the singular distribution a(E)|U are of the product form L × S, where L is a leaf of the singular distribution i!E. Remark 8.3. We can use this to show that there exists an integral submanifold (leaf) through every given m ∈ M. Indeed, take N ⊆ M to be any submanifold passing through m, with

TmM = a(Em) ⊕ TmN. Taking N smaller if necessary, we can assume that N is transverse to a everywhere, and that ! ! ν(M,N) = N × S as above. Then a(i E)m = 0, so that i E has the single point {m} as an integral submanifold. We conclude that {m} × S is an integral submanifold of E|U . 48 ECKHARD MEINRENKEN

8.2. Normal derivative. The key idea in the proof of the splitting theorem 8.1 is to choose a section ∈ Γ(E) such that the vector field X = a() is Euler-like. The tubular neighborhood embedding ϕ will be defined by X, and the bundle map lifting ϕ will be determined by the choice of . To prove the existence of , we need yet another characterization of Euler-like vector fields. Let V → M be a vector bundle. If a section σ ∈ Γ(V ) vanishes along N ⊆ M, then by applying the normal functor to σ :(M,N) → (V,M), and recalling ν(V,M) = V , we obtain a map N d σ : ν(M,N) → V |N called the normal derivative of σ along N. Using partitions of unity, it is easy to see that any bundle map ν(M,N) → V |N arises in this way, as the normal derivative of a section. Remark 8.4. The normal derivative of σ can be characterized in several other ways. for example, ∗ ∗ note that for τ ∈ Γ(V ), the restriction dhσ, τi|N ∈ Γ(T M|N ) vanishes on vector tangent to N, hence it is a section of ann(TN), and is tensorial in τ. Hence

dhσ, ·i|N ∈ Γ(ann(TN) ⊗ V |N ), and this is the normal derivative. If σ1, . . . , σr is a local frame of sections of V , so that P σ = i fiσi with fi|N = 0, then N X d σ = dfi ⊗ σi|N . i Example 8.5. If X is a vector field vanishing along N, then X is Euler-like if and only if its normal derivative N d X : ν(M,N) → TM|N ∞ defines a splitting of the quotient map TM|N → ν(M,N). To see this, let f ∈ C (M) with f|N = 0. Then N hd X, df|N i = dhX, dfi|N = dX(f)|N This coincides with df|N if and only if X(f) = f modulo functions vanishing to second order. 8.3. Anchored vector bundles. For the following considerations, the Lie bracket on sections of E does not play a role, hence we will work in the more general context of anchored vector bundles. An anchored vector bundle is a vector bundle E → M together with a bundle map a: E → TM, called the anchor. Morphism of anchored vector bundles are defined in the obvious way. We denote by AutAV (E) the bundle automorphisms Φb (with base map Φ) compatible with the anchor, i.e. a ◦ Φb = T Φ ◦ a; its Lie algebra is denoted autAV (E) and consists of infinitesimal vector bundle automorphisms D : Γ(E) → Γ(E), with corresponding vector feld X, such that a(Dσ) = [X, a(σ)]. In this section, we prefer to regard the elements of aut(E) has vector fields Xb on the total space of E, homogeneous of degree 0 and with base vector field X. The compatibility with the anchor a: E → TM is then expressed as the property

(49) Xe ∼a XT , where XT ∈ X(TM) is the tangent lift of X. Given a submanifold N ⊆ M that is transverse to a, we can deﬁne a ‘pull-back’ i!E = a−1(TN); it is an anchored subbundle of E. INTRODUCTION TO POISSON GEOMETRY 49

Proposition 8.6. There exists a section ∈ Γ(E) such that |N = 0, and with a() Euler-like.

Proof. The transversality condition means precisely that the map E|N → ν(M,N), given by the anchor map to TM|N followed by the quotient map, is surjective. Its kernel is the subbundle i!E. We obtain a commutative diagram

! 0 / i E / E|N / ν(M,N) / 0

a a = 0 / TN / TM|N / ν(M,N) / 0 N Choose a section ∈ Γ(E) with |N = 0, whose normal derivative d : ν(M,N) → EN splits the map E|N → ν(M,N). Its image under the anchor is a vector field X = a(), with X|N = 0, N such that d X splits the map TM|N → ν(M,N). That is, a() is Euler-like. The definition of i!E generalizes to any smooth map Φ: N → M that is transverse to E. ! Indeed, transversality implies that the fiber product Φ E = E ×TM TN is a vector bundle over N, with anchor given by projection to TN. It comes with a morphism of anchored vector bundles Φ!E → E. Lemma 8.7. Suppose (E, a) is an anchored vector bundle over M, and N ⊆ M is a submanifold transverse to a. Then there is a canonical isomorphism (of double vector bundles) ν(E, i!E) =∼ p!i!E, where i: N → M is the inclusion and p: ν(M,N) → N is the projection. Proof. By applying the normal functor to (E, i!E) → (TM,TN), we obtain a double vector bundle ν(E, i!E) / ν(TM,TN)

i!E / TN A dimension count shows that this is a ﬁber product diagram. But the ﬁber product of i!E ! ! and ν(TM,TN) = T ν(M,N) over TN s just the pullback p i E. 8.4. Proof of the splitting theorem for Lie algebroids. For any Lie algebroid, the anchor map on sections has a canonical lift

autLA(E) 9

ea / Γ(E) a X(M) here ea(σ) = [σ, ·], viewed as an infinitesimal Lie algebroid automorphism. In particular, our Euler-like vector field X = a() gets lifted to Xe = ea(). We will think of Xe as a linear vector field on E (that is, Xe is homogeneous of degree 0). Lemma 8.8. The vector field Xe is Euler-like for (E, i!E). 50 ECKHARD MEINRENKEN

Proof. We have to show that ν(Xe) is the Euler vector ﬁeld for ν(E, i!E) → i!E. Since Xe preserves the anchor, we have that Xe ∼a XT , the tangent lift of X. Hence, under the map a:(E, i!E) → (TM,TN),

ν(Xe) ∼ν(a) ν(XT ) = ν(X)T = ET where ET ∈ X(T ν(M,N)) is the tangent lift of E ∈ X(ν(M,N). But the tangent lift of an Euler vector field on a vector bundle V → M is just the Euler vector field of TV → TM. We conclude that ν(Xe) is ν(a)-related to the Euler vector field of ν(TM,TN) → TN. But by the pullback diagram 8.7, the bundle ν(E, i!E) → i!E is just the pull-back of the bundle ! ν(TM,TN) → TN under a: i E → TN. We conclude that ν(Xe) is an Euler vector field. We are now in position to prove the splitting theorem for Lie algebroids, Theorem 8.1.

Proof of the splitting theorem. Let Φs, Φes be the ﬂow of X, Xe, respectively. Put

λt = Φ− log(t), λet = Φe− log(t).

Let φ: ν(M,N) → U be the tubular neighborhood embedding defined by X. Since λet covers the flow of λt, it is defined over E|U even for t = 0. Consider the following diagram, defined for all 0 ≤ t ≤ 1, ! ! ! E|U / λ (E|U ) / φ λ (E|U ) =∼ t =∼ t

U / U / ν(M,N) id φ−1 Here the ﬁrst upper horizontal arrow is given by the Lie algebroid morphism ! E|U → λt(E|U ) ⊆ TU × E|U , v 7→ (a(v), λet(v)).

This map is an isomorphism for all t: If t > 0, this is clear since λet is an isomorphism then. If t = 0 we note that it is an isomorphism along N ⊆ U, hence also on some neighborhood of N, and using e.g. λ0 = λ0 ◦ λt we conclude that it is an isomorphism over all of U. ! ! We hence obtain a family of Lie algebroid isomorphisms φ λt(E|U ) → E|U , all with the base map φ. For t = 0, we have λ0 ◦ φ = φ ◦ κ0 = i ◦ p, so we obtain the desired Lie algebroid isomorphism ! ! φe: p i (E|U ) → E|U , with base map φ. Remark 8.9. The map φe can itself be regarded as a tubular neighborhood embedding. Indeed, ! ∼ ! ! ! under the isomorphism ν(E, i E) = p i E = Tν(M,N) ×TN (i E), the inverse of the tubular neighborhood embedding is given by ! ∼ ! −1 E|U → ν(E, i E) = T ν(M,N) ×TN (i E), v 7→ T φ (a(v)), λf0(v) ! (where we regard λf0 as a map to i E ⊆ E|U ). Indeed, this is the unique map of anchored −1 ! vector bundles, with base map φ , such that the i E-component is λf0(v). But this is just the description of φe−1. INTRODUCTION TO POISSON GEOMETRY 51

8.5. The Stefan-Sussmann theorem. The idea of proof of the splitting theorem for Lie algebroids also works for anchored vector bundles, provided that they satisfy the following condition. Definition 8.10. [8] An anchored vector bundle (E, a) is called involutive if Γ(a(E)) ⊆ X(M) is closed under Lie brackets. For example, Lie algebroids are involutive, due to the property [a(σ), a(τ)] = a([σ, τ]) for sections σ, τ ∈ Γ(E). Courant algebroids are involutive as well. Remark 8.11. Stefan [33] and Sussmann [36] defined a ‘singular distribution’ on a manifold M to be a subset D ⊆ TM spanned locally by a finite collection of vector fields. They developed necessary and sufficient conditions of integrability for such singular distributions; in terms of the submodule D ⊆ X(M) of vector fields taking values in D. However, their results contain some errors that were corrected by Balan [4]. Androulidakis-Skandalis [3] take a ‘singular distribution’ to be a locally finitely generated submodule C ⊆ X(M), and call it ‘integrable’ if C is involutive. Our definition of involutive anchored vector bundles is very similar to this viewpoint. Theorem 8.12. [8] Let (E, a) be an involutive anchored vector bundle over M, and N ⊆ M a submanifold transverse to the anchor. Then there exists a tubular neighborhood embedding φ: ν(M,N) → U ⊆ M, which is the base map for an isomorphism of anchored vector bundles,

! ! p i E → E|U . The discussion for Lie algebroids in Remark 8.3 applies to the more general setting, and shows that every point of m ∈ M is contained in a leaf S of a(E). The proof of this theorem is parallel to that for Lie algebroids, once the following result is established: Proposition 8.13. [8] An anchored vector bundle (E, a) is involutive if and only if the map a: Γ(E) → X(M) lifts to a map ea: Γ(E) → autAV (E). In this case, one can arrange that the lift satisﬁes

(50) ea(fσ)τ = fea(σ)τ − (a(τ)f)σ for all σ, τ ∈ Γ(E).

Proof. Given such a lift ea, the submodule a(Γ(E)) is involutive because [a(σ), a(τ)] = a ea(σ)τ (by deﬁnition of autAV (E)). In the other direction, one constructs ea with the help of a connection. (See [8].) Having chosen such a lift, and having chosen a section ∈ Γ(E) such that X = a() is Euler-like, one proves as in the case of Lie algebroids that Xe = a() is Euler-like. The same approach as for Lie algebroids, using the ﬂow of Xe, gives an isomorphism of anchored vector ! ! bundles p i E → E|U . 52 ECKHARD MEINRENKEN

8.6. The Weinstein splitting theorem. We begin with a statement of the theorem. Theorem 8.14 (Weinstein splitting theorem [39]). Let (M, π) be a Poisson manifold, and 1 k 1 r m ∈ M. There exists a system of local coordinates q , . . . , q , p1, . . . , pk, y , . . . , y centered at m in which π takes on the following form: k r X ∂ ∂ 1 X ∂ ∂ π = ∧ + cij(y) , ∂qi ∂p 2 ∂yi ∂yj i=1 i i,j=1 where cij = −cji are smooth functions with cij(0) = 0. ∼ 2k Thus, π splits into a sum π = πS + πN on S × N, where S = R with the standard non- r degenerate Poisson structure πS, and N = R with a Poisson structure πN having a critical point at y = 0. The remarkable fact is that one can eliminate any ‘cross-terms’. Of course, the transverse Poisson structure πN can still be quite complicated. A direct consequence of the splitting theorem is the existence of a symplectic foliation: Corollary 8.15 (Symplectic leaves). Let M be a Poisson manifold, and m ∈ M. Then there exists a unique maximal (injectively immersed) integral submanifold S ⊆ M of the singular distribution ran(π]) ⊆ TM. Proof. In the model, it is immediate that the submanifold S ⊆ M given by yi = 0 is a symplectic leaf. (The passage from local integrability to global integrability merely involves patching of the local solutions, and is the same as in the standard proofs of Frobenius’ theorem.) To give a coordinate-free formulation of Weinstein’s theorem, let ] ∗ S := π (TmM). 2 2 2 By definition, πm ∈ ∧ S ⊆ ∧ TmM, defining a constant bivector field πS ∈ Γ(∧ T S). It is non- degenerate, corresponding to a symplectic form on S. Let N ⊆ M be a submanifold through m, with the property that TmM = TmN ⊕ S. As we saw in Example 3.41, taking N smaller if necessary, it is a cosymplectic submanifold and hence inherits a Poisson structure πN (see Proposition 6.7). This is referred to as the transverse Poisson structure. The coordinate-free formulation of the splitting theorem is as follows: Theorem 8.16 (Weinstein splitting theorem, II). The Poisson manifold (M, π) is Poisson diffeomorphic near m ∈ M to the product of Poisson manifolds, N × S ] where S is the symplectic vector space ran(πm), and N is a transverse submanifold as above, equipped with the transverse Poisson structure. More precisely, there exists a Poisson diffeomorphism between open neighborhoods of m in M and of (m, 0) in N × S, taking m to (m, 0), and with differential at m equal to the given decomposition TmM → TmN ⊕ S. Weinstein’s theorem has been generalized by Frejlich-M˘arcut¸[17] to a normal form theorem around arbitrary cosymplectic submanifolds N ⊆ M. Their result is best phrased using some Dirac geometry. Recall again that any cosymplectic manifold inherits a Poisson structure πN such that ! Gr(πN ) = i Gr(πM ) INTRODUCTION TO POISSON GEOMETRY 53

(as subbundles of TN). See Proposition 6.7. On the other hand, the vector bundle V = π](ann(TN)) → N has a fiberwise symplectic structure ωV, defined by the restriction of π to ann(TN). Since it is a complement to TN in TM|N , we will identify V =∼ ν(M,N); in particular, the projection map will be denoted p: V → N. As for any symplectic vector bundle, it is possible to find a closed 2-form ω on the total space of V, such that ω(v, ·) = 0 for v ∈ TN, and such that ω pulls back to the given symplectic form on the fibers. (A particularly nice way of obtaining such a 2-form is the ‘minimal coupling’ construction of Sternberg [35] and Weinstein [38].) Remark 8.17 (Minimal coupling). The following construction is due to Sternberg [35] and Weinstein [38]. Let P → B be a principal G-bundle, and Q is a manifold with an invariant closed r-form ωQ. Then the associated bundle P ×G Q has a fiberwise form induced by ωQ. One may wonder if it is possibly to extend ωQ to a global closed r-form on P ×G Q which pulls back to the given forms on the fibers. The minimal coupling gives such a construction if r = 2, and the 2-form ωQ admits a ∗ moment map ΦQ : Q → g , that is, ιξQ ωQ = −hdΦ, ξi. (Actually, the construction generalizes to arbitrary r, provided that ωQ has an equivariant extension in the sense of de Rham theory.) 1 ∗ Choose a principal connection θ ∈ Ω (P, g); thus θ has the equivariance property Agθ = Adg θ for g ∈ G (where Ag denotes the action of g on P ), and ι(ξP )θ = ξ (where ξP are the generating vector fields for the action). Then the 2-form 2 ωQ − dhθ, Φi ∈ Ω (P × Q) is G-invariant for the diagonal action. In fact it is G-basic, by the calculation

ι(ξQ)hdθ, Φi = −dι(ξQ)hθ, Φi = −dhΦ, ξi = ι(ξQ)ωQ (where we used Cartan’s identity and the invariance of hθ, Φi ∈ Ω1(P × Q)). Hence it descends to a closed 2-form 2 ωeQ ∈ Ω (P ×G Q) pulls back to ωQ on the fibers. As a special case, one can apply this construction to symplectic vector bundles V → M. Any such vector bundle is an associated bundle 2k V = P ×Sp(2k) R 2k where P is the associated symplectic frame bundle, and R has the standard symplectic structure. That is, the fiberwise symplectic form extends to a global closed 2-form on the total space of V. ! Consider the pull-back Dirac structure p Gr(πN ) ⊆ TV. It is not the graph of a Poisson structure (see Proposition 6.7), indeed ! p Gr(πN ) ∩ T V = ker(T p). However, once we take a gauge transformation by ω ∈ Ω2(V), the resulting ! (51) Rω p Gr(πN ) ⊆ TV 54 ECKHARD MEINRENKEN is transverse to T V near N ⊆ V, hence it defines a Poisson structure πV on a neighborhood of N. In fact, this Poisson structure agrees with π along N, in terms of the identification ∼ TM|N = TN ⊕ V = V|N . The Poisson structures for different choices of ω are related by the Moser method (Theorem 5.7). Theorem 8.18 (Frejlich-M˘arcut¸[17]). Let N ⊆ M be a cosymplectic submanifold of a Poisson manifold, with normal bundle V. Define a Poisson structure πV on a neighborhood of N in V, as explained above. Then there exists a tubular neighborhood embedding V ,→ M which is a Poisson map on a possibly smaller neighborhood of N ⊆ V. If the bundle V admits a trivialization V = N × S, then the 2-form ω can simply be taken as ! pull-back of ωS under projection to the second factor. Furthermore, p Gr(πN ) = Gr(πN ) × T S in this case, with ! Rω(p Gr(πN )) = Gr(πN ) × Gr(ωS). In particular, we recover the Weinstein splitting theorem. For the proof of the Frejlich-Marcut theorem, we will use: Lemma 8.19. Let N ⊆ M be cosymplectic. Then there exists a 1-form α ∈ Ω1(M), with ] α|N = 0, such that the vector field X = π (α) is Euler-like. Proof. This is a special case of Proposition ??, applied to the cotangent Lie algebroid E = ∗ ∗ Tπ M, with the section ∈ Γ(Tπ M) interpreted as a 1-form. Indeed, the condition that N is ] ∗ cosymplectic means in particular that π : T M → TM is transverse to N. Proof of the Frejlich-M˘arcut¸theorem 8.18, after [8]. Choose a 1-form α as in the Lemma. The Euler-like vector field X = π](α) gives a tubular neighborhood embedding ψ : ν(M,N) → M, with E ∼ψ X. Using this embedding, we may assume M = ν(M,N) is a vector bundle, with X = E the Euler vector field. Let Φs be the flow, and κt = Φ− log(t) as before. Consider the infinitesimal automorphism ∼ (dα, X) ∈ Ωcl(M) o X(M) = autCA(TM) defined by σ = X + α ∈ Γ(Gr(π)). By Proposition 5.4, the corresponding 1-parameter group of automorphisms is ∼ (−ωt, Φt) ∈ Ωcl(M) o Diff(M) = AutCA(TM), where Z t Z t Z 1 ∗ 1 ∗ ωt = −d (Φs)∗α ds = −d (Φ−s) α ds = d κvα dv. 0 0 exp(t) v

Since σ is a section of Gr(π), the action Rωt ◦ TΦt of this 1-parameter group preserves Gr(π). That is, ! Rωt (Φ−t) Gr(π) = Gr(π) INTRODUCTION TO POISSON GEOMETRY 55 for all t ≥ 0. Consider the limit t → −∞ in this equality. Since α vanishes along N, the family 1 ∗ of forms v κvα extends smoothly to v = 0. Hence ω := ω−∞ is well-deﬁned: Z 1 1 ∗ ω = d κvα dv. 0 v On the other hand, Φ∞ = κ0 = i ◦ p where p: ν(M,N) → N is the projection, and i: N → M is the inclusion. Thus ! ! Rω(p i Gr(π)) = Gr(π). ! ! Since i Gr(π) = Gr(πN ), the left hand side is Rω(p Gr(πN )), which coincides with the model Poisson structure πV near N. Remark 8.20. A similar argument can be used to prove more general normal form theorems for Dirac structures whose anchor is transverse to a submanifold N ⊆ M.

9. The Karasev-Weinstein symplectic realization theorem 9.1. Symplectic realizations. Our starting point is the following deﬁnition, due to Wein- stein.

Definition 9.1. [39] A symplectic realization of a Poisson manifold (M, πM ) is a symplectic manifold (P, ωP ), with associated Poisson structure πP , together with a surjective submersion ϕ: P → M such that ϕ:(P, πP ) → (M, πM ) is a Poisson map . Remark 9.2. In Weinstein’s original definition, it is not required that ϕ is a surjective submersion. For instance, the inclusion of a symplectic leaf would be a symplectic realization in the more general sense. The definition above is what Weinstein calls a full symplectic realization. We will drop ‘full’ to simplify the terminology. 2 ∂ ∂ Examples 9.3. (a) (See [14].) Let M = R with the Poisson structure π = x ∂x ∧ ∂y . ∗ 2 A symplectic realization is given by P = T R with the standard symplectic form P2 i ω = i=1 dq ∧ dpi, and 1 2 1 2 1 ϕ(q , q , p1, p2) = (q , q + p1q ). To check that this is indeed a realization, we calculate the Poisson brackets: 1 2 1 1 1 1 {q , q + p1q }P = {q , p1}P q = q ,

corresponding to {x, y}M = x. (b) Let G be a Lie group, with Lie algebra g. The space M = g∗, with the Lie-Poisson structure, has a symplectic realization ϕ: T ∗G → g∗, where T ∗G has the standard symplectic structure, and the map ϕ is given by left ∂ ∂ trivialization. (The ﬁrst example (a) may be seen as a special case, using that x ∂x ∧ ∂y is a linear Poisson structure, corresponding to a 2-dimensional Lie algebra.) 56 ECKHARD MEINRENKEN

(c) Let (P, ωP ) be a symplectic manifold, with a proper, free action preserving the symplectic form ωP . Then the Poisson structure πP descends to a Poisson structure πM on the quotient space M = P/G. (Indeed, smooth functions on M are identiﬁed with G-invariant smooth functions on P , and these are a Poisson subalgebra of C∞(P ).) The manifold P is then a symplectic realization of M. (Example (b) is a special case, with G acting on T ∗G by the cotangent lift of the left-multiplication.) (d) Let M be a manifold with the zero Poisson structure. Then the cotangent bundle, with its standard symplectic structure, and with ϕ the cotangent projection τ : T ∗M → M, is a symplectic realization. (e) Let M be a manifold with a symplectic structure. Then P = M, with ϕ the identity map, is a symplectic realization. (f) Every symplectic manifold P can be regarded as a symplectic realization of M = pt with the zero Poisson structure. Does every Poisson manifold admit a symplectic realization? Before addressing this question, let us ﬁrst consider the opposite problem: when does a symplectic structure descend under a surjective submersion.

Proposition 9.4. Let (P, πP ) be a Poisson manifold, and ϕ: P → M a surjective submersion with connected ﬁbers.

(a) Then πP descend to a bivector ﬁeld πM on M if and only if ann(ker T ϕ) ⊆ T ∗P is a subalgebroid of the cotangent Lie algebroid. (b) (Libermann’s theorem [26].) If πP is the Poisson structure for a symplectic form ωP , then πP descends if and only if the ωP -orthogonal distribution to ker(T ϕ) is involutive in the sense of Frobenius.

Proof. (a) “⇒”. Suppose πP descends to πM . Then πM necessarily is a Poisson structure, and ϕ is a Poisson map. Then ∗ ∗ ∗ ∗ ∗ [d ϕ f, d ϕ g] = d {ϕ f, ϕ g}P = d ϕ {f, g}M , for all f, g ∈ C∞(M). Since the bundle ann(ker T ϕ) is spanned by all d ϕ∗f with f ∈ C∞(M), this shows that ann(ker T ϕ) is a Lie subalgebroid. ∞ ∗ ∗ “⇐”. If ann(ker T ϕ) is a Lie subalgebroid, it follows that for all f, g ∈ C (M), d{ϕ f, ϕ g}P vanishes on ker(T ϕ). Since the fibers of ϕ are connected, this means that the function ∗ ∗ {ϕ f, ϕ g}P is fiberwise constant, and hence is the pull-back of a function on M. Taking this function to be the definition of {f, g}M , it follows that there is a unique bilinear form {·, ·}M on M such that ∗ ∗ ∗ {ϕ f, ϕ g}P = ϕ {f, g}M ∞ for all f, g ∈ C (M). Since {·, ·}P is a Poisson structure, it follows that {·, ·}M is a Poisson structure, and the identity above shows that ϕ is a Poisson map. ] ∗ (b) The map πP : T P → TP is a Lie algebroid isomorphism, taking ann(ker(T ϕ)) to the ωP - orthogonal bundle of ker(T ϕ). The latter being a Lie subalgebroid is equivalent to Frobenius integrability. INTRODUCTION TO POISSON GEOMETRY 57

Libermann’s theorem shows that if ϕ: P → M is a symplectic realization, then the foliation given by the ϕ-ﬁbers is symplectically orthogonal to another foliation.

Example 9.5. Let (P, ωP ) be a symplectic manifold with a free, proper G-action. As we saw, M = P/G inherits a Poisson structure, and the quotient map is a symplectic realization. In this case, the transverse distribution is given by the ω-orthogonal spaces to the G-orbit directions:

{v ∈ TpP | ∀ξ ∈ g: ωP (ξP (p), v) = 0} If the action admits an equivariant moment map Φ: P → g∗, then this foliation is given exactly by the level sets of Φ. Indeed, for v tangent to a level set, and any ξ ∈ g,

ωP (ξP (p), v) = −ι(v) dhΦ, ξi = 0. Note that the moment map Φ is again a Poisson map (possibly up to reversing the sign of the Poisson structure – this depends on sign conventions). The assumption that G acts freely means that it is a submersion. That is, Φ (viewed as a map to Φ(M) ⊆ g∗) provides a symplectic realization of g∗. (The assumption on existence of a moment map is not very restrictive; at least locally, on a neighborhood of a G-orbit, a moment map always exists.) This is a good time to state the Karasev-Weinstein theorem. Theorem 9.6 (Karasev [23], Weinstein [39]). Let (M, π) be a Poisson manifold. Then there exists a symplectic manifold (P, ω), with an inclusion i: M,→ P as a Lagrangian submanifold, and with two surjective submersions t, s: P → M such that t ◦ i = s ◦ i = idM , and • t is a Poisson map, • s is an anti-Poisson map, • The t-fibers and s-fibers are ω-orthogonal. In fact, much more is true: There exists a structure of a local symplectic groupoid on P , having s, t as the source and target maps, and i as the inclusion of units. We postpone the discussion of the multiplicative structure to Section ?? below. Let us first illustrate the theorem for some of the Examples 9.3. 2 ∂ ∂ ∗ 2 Examples 9.7. (a) For M = R with π = x ∂x ∧ ∂y , and P = T (R ) with the standard symplectic form, 1 2 1 2 1 1 2 1 2 t(q , q , p1, p2) = (q , q + p1q ), s(q , q , p1, p2) = (q exp(p1), q ). (Note in particular that s is anti-Poisson, and that the component functions of t Poisson commute with the component functions of s.) Here i is the inclusion as the zero section. (b) For M = g∗, we may take P = T ∗G, with t the left trivialization, s the right trivialization, and i the inclusion as units. (c) For M with the zero Poisson structure, we take P = T ∗M, with t = s = τ the cotangent projection and i the inclusion as units. (d) For a symplectic manifold M, the choice of P = M with ϕ = id is a symplectic realization, but it does not have the properties described in the Karasev-Weinstein theorem 9.6. Instead we may take P = M × M −, where the minus sign signifies the opposite symplectic structure. Here t is projection to the first factor, s is projection to the second factor, and i is the diagonal inclusion. 58 ECKHARD MEINRENKEN

The three conditions that t be Poisson, s anti-Poisson, and the t- and s-ﬁbers being symplectically orthogonal can be combined into a single condition that the map (t, s): P → M × M − be Poisson. Here M − indicates M with the opposite Poisson structure. We also have the following Dirac-geometric characterization of the condition. Lemma 9.8 (Frejlich-M˘arcut¸[18]). Let (P, ω) be a symplectic manifold, i: M → P a La- grangian submanifold, and t, s: P → M two surjective submersions such that t◦i = s◦i = idM . Let πM be a Poisson structure on M. Then (t, s):(P, πP ) → (M, πM ) × (M, −πM ) is Poisson if and only if ! ! Rω(t Gr(π)) = s Gr(π), as Dirac structures in TP .

Proof. We will write s∗ = T s, and similar, to simplify notation. ! ! “⇐”. Observe ker(t∗) ⊆ t Gr(π), ker(s∗) ⊆ s Gr(π). For v ∈ ker(t∗) and ω ∈ ker(s∗), we have ω(v, w) = hv + ιvω, wi = hRω(v), wi = 0, ! since both w and Rω(v) are in the Lagrangian subbundle s Gr(π) ⊆ TP . This shows that the subbundles ker(t∗), ker(s∗) ⊆ TP are ω-orthogonal; for dimension reasons ker(t∗) is exactly ∗ the ω-orthogonal bundle to ker(s∗). We next show that s is anti-Poisson. Given µ ∈ Ts(p)M, deﬁne v ∈ TpP by ∗ ιvω = s µ. That is, ] ∗ v = −πP (s µ) ∈ TpP. ∗ Since ιvω = s µ pairs to zero with all vectors of ker(s∗), it follows that v is in the ω-orthogonal space to ker(s∗). Hence v ∈ ker(t∗). But ] ∗ ∗ ! ! −πP (s µ) + s µ = v + ιvω = Rω(v) ∈ Rω(t Gr(π)) = s Gr(π) ∗ ∗ is s-related to some element of Gr(π)s(p). Since the T P -component is s µ, that element must be π](µ) + µ ∈ Gr(π). This shows that ] ∗ ] s∗(−πP (s µ)) = π (µ), hence s is anti-Poisson. A similar argument shows that t is Poisson. “⇒” For the converse, suppose that (t, s): P → M × M is a Poisson map with respect to ∗ ∗ (πM , −πM ). Equivalently, for all v, w ∈ TpP, µ ∈ Tt(p)M, ν ∈ Ts(p)M ∗ ] (52) ι(v)ω = −t µ ⇒ t∗v = π (µ), s∗v = 0, ∗ ] ι(w)ω = −s ν ⇒ t∗w = 0, s∗w = −π (ν)(53) Consider the direct sum decompositions ! ! t Gr(π) = ker(t∗) ⊕ (t Gr(π) ∩ Gr(πP ))(54) ! ! (55) s Gr(π) = ker(s∗) ⊕ (s Gr(π) ∩ Gr(πP )). INTRODUCTION TO POISSON GEOMETRY 59

Elements in the second summand of (54) are of the form v + t∗µ, with v uniquely determined ∗ ∗ by ιvω = −t µ. Elements in the second summand of (55) are of the form w + s ν with ι(w)ω = −s∗ν. ∗ ! ∗ ∗ Let v + t µ ∈ (t Gr(π) ∩ Gr(πP )). The property ιvω = −t µ shows Rω(v + t µ) = v, by (52) this lies in ker(s∗). Hence ! Rω(t Gr(π) ∩ Gr(πP )) = ker(s∗). Similarly, R−ω is an isomorphism from the second summand of (55) to the first summand of (54); equivalently, ! Rω(ker t∗) = (s Gr(π) ∩ Gr(πP ))). ! ! This shows Rω(t Gr(π)) = s Gr(π); in fact, Rω interchanges the two summands in the decompositions (54) and (55). 9.2. The Crainic-M˘arcut¸formula. Our proof of Theorem 9.6 will use an explicit construction of the realization due to Crainic-M˘arcut¸[13], with later simplifications due to Frejlich- M˘arcut¸[18]. As the total space P for the symplectic realization, we will take a suitable open neighborhood of M inside the cotangent bundle τ : T ∗M → M. Definition 9.9 (Crainic-M˘arcut¸[13]). Let (M, π) be a Poisson manifold. A vector field X ∈ X(T ∗M) is called a Poisson spray if it homogeneous of degree 1 in fiber directions, and for all µ ∈ T ∗M, ] (Tµτ)(Xµ) = π (µ). ∗ The homogeneity requirement means κt X = tX, where κt is fiberwise multiplication by t 6= 0. In local coordinates, for a given Poisson structure 1 X ∂ ∂ (56) π = πij(q) ∧ , 2 ∂qi ∂qj ij a Poisson spray is of the form X ∂ 1 X ∂ (57) X = πij(q) p + Γij(q) p p i ∂qj 2 k i j ∂pk ij ijk ij ji where pi are the cotangent coordinates, and Γk = Γk are functions. Lemma 9.10. Every Poisson manifold (M, π) admits a Poisson spray. Proof. In local coordinates, Poisson sprays can be defined by the formula above (e.g., with ij Γk = 0). To obtain a global Poisson spray, one patches these local definitions together, using a partition of unity on M. ∗ Let X be a Poisson spray, and Φt its local flow. Since X vanishes along M ⊆ T M, there exists an open neighborhood of M on which the flow is defined for all |t| ≤ 1. On such a neighborhood, put Z 1 ω = (Φs)∗ωcan ds, 0 where ωcan is the standard symplectic form of the cotangent bundle. 60 ECKHARD MEINRENKEN

Lemma 9.11. The 2-form ω is symplectic along M. Proof. For m ∈ M ⊆ T ∗M, consider the decomposition ∗ ∗ Tm(T M) = TmM ⊕ TmM. Since the vector field X is homogeneous of degree 1, it vanishes along M. In particular, ∗ its flow Φt fixes M ⊆ P , hence T Φt is a linear transformation of Tm(T M). Consequently, (TmΦt)(v) = v for all m ∈ M and v ∈ TmM. Again by homogeneity, the linear approximation along M ⊆ T ∗M vanishes: ν(X) = 0 as a vector field on ν(T ∗M,M) =∼ T ∗M. (This is not to be confused with the linear approximation of X at {m}, which may be non-zero.) Consequently, ν(Φt) = idT ∗M , which shows that

(TmΦt)(w) = w mod TmM ∗ for all w ∈ TmM. Hence ((Φs)∗ωcan)(v, ·) = ωcan(v, ·) for all v ∈ TmM, and therefore

ω(v, ·) = ωcan(v, ·). ∗ Since TM is a Lagrangian subbundle (in the symplectic sense!) of T (T M)|M with respect to ωcan, this implies that the 2-form ω is symplectic along M. Theorem 9.12 (Crainic-M˘arcut¸[13]). Let P ⊆ T ∗M be an open neighborhood of the zero section, with the property that Φt(m) is deﬁned for all m ∈ P and |t| ≤ 1, and such that ω is symplectic on P . Let i: M,→ P be the inclusion as the zero section, and put

s = τ, t = τ ◦ Φ−1. Then the symplectic manifold (P, ω) together with the maps t, s, i has the properties from the Karasev-Weinstein theorem 9.6. Proof. [18]. Let α ∈ Ω1(T ∗M) be the canonical (Liouville) 1-form. That is, for all µ ∈ T ∗M, ∗ αµ = (Tµτ) µ. P i P i Recall that ωcan = −dα. In local cotangent coordinates, α = i pidq and ωcan = i dq ∧ dpi. Given a Poisson spray X, observe that ∗ X + α ∈ Γ T(T M) ! ∗ is a section of τ Gr(π) ⊆ T(T M). Indeed, the definition of a spray (and of the canonical 1-form α) means precisely that for all µ ∈ T ∗M, ] Xµ + αµ ∼τ π (µ) + µ ∈ Gr(π). ∗ The infinitesimal automorphism (dα, X) ∈ aut T(T M) defined by the section X +α preserves τ ! Gr(π). By Proposition 5.3, the (local) 1-parameter group of automorphisms exponentiating ∗ (dα, X) ∈ aut T(T M) is given by (−ωt, Φt), where Φt is the (local) flow of X, and Z t Z t ωt = −d (Φs)∗α = (Φs)∗ωcan. 0 0 We conclude ! ! Rωt ◦ TΦt(τ Gr(π)) = τ Gr(π). INTRODUCTION TO POISSON GEOMETRY 61

Putting t = 1 in this identity, and use the deﬁnition of t, s, ω, together with the fact that ! ∗ ! ! T Φ1(E) = (Φ−1) E for any Dirac structure E ⊆ T(T M), we obtain Rω ◦ t Gr(π) = s Gr(π). By Lemma 9.8, this is equivalent to the conditions from the Karasev-Weinstein theorem. 9.3. Linear realizations of linear Poisson structures. Let V → M be a vector bundle with a linear Poisson structure. The tangent bundle TV and the cotangent bundle T ∗V are double vector bundles TV / TM T ∗V / V ∗

V / M V / M h v We denote by κt , κt the horizontal and vertical scalar multiplications. By definition, a Poisson ∗ v ∗ spray X ∈ X(T V ) satisfies (κt ) X = tX; we will call X a linear Poisson spray if it is linear h ∗ in the horizontal direction, (κt ) X = X. Lemma 9.13. For every linear Poisson structure on V → M there exists a linear Poisson spray X ∈ X(T ∗V ). Proof. It suffices to construct a linear Poisson spray for any local trivialization of V ; a global linear Poisson spray is then obtained by using a partition of unity on M. Using local vector r bundle coordinates, with q the coordinates on the base and yi those on the fiber, we obtain (see (18)) 1 X k ∂ ∂ X r ∂ ∂ π = cij(q) yk ∧ + ai (q) ∧ r . 2 ∂yi ∂yj ∂yi ∂q ijk ir r k with ai , cij smooth functions. A corresponding spray is given by

X k i ∂ X r i ∂ X r ∂ X = cij(q) ykη + ai (q)η ∧ r − ai (q)pr ; ∂yj ∂q ∂yi ijk ir ir i r here pr, η are the cotangent variables corresponding to q , yi. The horizontal and vertical scalar multiplications are, in coordinates, h v κt (q, y, p, η) = (q, ty, tp, η), κt (q, y, p, η) = (q, y, tp, tη). We hence see that the spray given above is indeed homogeneous of degree 0 in horizontal direction.

Let Φs be the (local) ﬂow of the linear Poisson spray X. The homogeneity properties of X give that h h v v Φs ◦ κt = κt ◦ Φs, Φs ◦ κt = κt ◦ Φst. ∗ P r P k The canonical 1-form on T V is given by α = prdq + η dyk; its homogeneity properties are h ∗ v ∗ (κt ) α = tα, (κt ) α = tα. Hence, ωcan = −dα has similar scaling properties, and for the Crainic-Marcut form ω = R 1 0 (Φs)∗ωcan ds, one obtains h ∗ v ∗ (κt ) ω = tω, (κt ) ω = ωt 62 ECKHARD MEINRENKEN

∗ (where ωt is given by the integral from 0 to t). Passing to the Poisson structure on T V (deﬁned h ∗ on a κt -invariant neighborhood of the zero section V ⊆ T V ) corresponding to ω, this means that h ∗ 1 (κ ) π ∗ = π ∗ . t T V t T V That is, it is a linear Poisson structure on T ∗V , relative to the vector bundle structure T ∗V → V ∗. The source and target map s, t: T ∗V → V are compatible with the linear structure as well: h h h s ◦ κt = κt ◦ s, t ◦ κt = κt ◦ t; here κt is the scalar multlication for V .

10. Lie groupoids For a more detailed discussion of Lie groupoids, we refer to the books [29] or [32].

10.1. Basic definitions and examples. The definition of a Lie groupoid G ⇒ M involves a manifold G of arrows, a submanifold i: M,→ G of units (or objects), and two surjective submersions s, t: G → M called source and target such that t ◦ i = s ◦ i = idM . One thinks of g as an arrow from its source s(g) to its target t(g), with M embedded as trivial arrows. Using that t, s are submersions, one finds that the set of k-arrows, for k = 1, 2,... (k) k G = {(g1, . . . , gk) ∈ G | s(gi) = t(gi+1)} is a smooth submanifold of Gk. For k = 0 one puts G(0) = M. The definition of a Lie groupoid also involves a smooth multiplication map, defined on composable arrows (i.e., 2-arrows) (2) MultG : G → G, (g1, g2) 7→ g1 ◦ g2, such that s(g1 ◦ g2) = s(g2), t(g1 ◦ g2) = t(g1). It is thought of as a concatenation of arrows. Note that when picturing this composition rule, it is best to draw arrows from the right to the left.

Definition 10.1. The above data define a Lie groupoid G ⇒ M if the following axioms are satisfied:

1. Associativity: (g1 ◦ g2) ◦ g3 = g1 ◦ (g2 ◦ g3) whenever s(g1) = t(g2), s(g2) = t(g3). 2. Units: t(g) ◦ g = g = g ◦ s(g) for all g ∈ G. 3. Inverses: Every g ∈ G is invertible: There exists h ∈ G such that s(h) = t(g), t(h) = s(g), and such that g ◦ h, h ◦ g are units. Remark 10.2. The inverse of an element is necessarily unique. Denoting this element by g−1, we have that g ◦ g−1 = t(g), g−1 ◦ g = s(g). Remark 10.3. Note that the entire groupoid structure is encoded in the graph of the multiplication map, 3 (2) Gr(MultG) = {(g1 ◦ g2, g1, g2) ∈ G | (g1, g2) ∈ G }

Thus, the set of all (g, g1, g2) such that g = g1 ◦ g2 determines all the structure maps. For instance, the units are recovered as those elements m such that m = m ◦ m, while the source s(g) of an element g ∈ G is the unique unit m such that g ◦ s(g) is deﬁned. INTRODUCTION TO POISSON GEOMETRY 63

There are obvious notions of Lie subgroupoid of a Lie groupoid, and of a morphism of Lie 0 0 0 0 groupoids ΦG : G → G from G ⇒ M to G ⇒ M (with base map ΦM : M → M ). The graph of a Lie groupoid morphism is a subgroupoid of 0 0 G × G ⇒ M × M. Example 10.4. A Lie group G is the same as a Lie groupoid with a unique unit, G ⇒ pt. At the opposite extreme, every set M can be regarded as a trivial groupoid M ⇒ M where all elements are units. Example 10.5. A Lie groupoid for which the source and target map coincide is the same as a bundle of Lie groups: A surjective submersion with a fiberwise group structure such that the fiberwise multiplication depends smoothly on the base point. The fibers need not be isomorphic as Lie groups, or even diffeomorphic.

Example 10.6. For any manifold M, one has the pair groupoid Pair(M) = M × M ⇒ M, with 0 0 0 0 0 (m , m) = (m1, m1) ◦ (m2, m2) ⇔ m1 = m, m1 = m2, m2 = m. Here units are given by the diagonal embedding M,→ M × M, and the source and target of 0 0 (m , m) are m and m , respectively. The target and anchor of a Lie groupoid G ⇒ M combine into a Lie groupoid morphism (58) (t, s): G → Pair(M). This groupoid morphism is sometimes called the (groupoid) anchor; it is related to the anchor of Lie algebroids as we will see below. Example 10.7. Another natural Lie group associated to any manifold M is the fundamental groupoid Π(M), consisting of homotopy classes of paths [γ] relative to fixed end points. The source and target maps are s([γ]) = γ(0), t([γ]) = γ(1), and the groupoid multiplication is concatenation of paths. The natural map Π(M) → Pair(M) is a local diffeomorphism; if M is simply connected it is a global diffeomorphism. 5 Example 10.8. Given an smooth action of a Lie group G on M, one has the action groupoid G ⇒ M. It may be defined as the subgroupoid of G × Pair(M) ⇒ M, consisting of all (g, m0, m) ∈ G × (M × M) such that m0 = g.m. Using the projection (g, m0, m) 7→ (g, m) to identify G =∼ G × M, the product reads as

(g, m) = (g1, m1) ◦ (g2, m2) ⇔ g = g1g2, m = m2, m1 = g2.m2. Generalizing group actions, one can also consider groupoid actions of G ⇒ M on sets X with a map Φ: X → M. Such an action is given by an action map

AX : G s ×Φ X → X, (g, p) 7→ g ◦ p with Φ(g ◦ p) = t(g), such that (g1 ◦ g2) ◦ p = g1 ◦ (g2 ◦ p) whenever the compositions are deﬁned, and such that units act trivially: Φ(p) ◦ p = p. Similar to the groupoids themselves, the action is determined by its graph Gr(AX ) ⊆ X × G × X; for example, Φ(p) ∈ M is the

5 If M is connected, and Mf is a simply connected covering space, with covering map Mf → M, one has a Lie groupoid homomorphism Pair(Mf) = Π(Mf) → Π(M). By homotopy lifting, this map is surjective. Let Λ be the discrete group of deck transformations of Mf, i.e., diﬀeomorphisms covering the identity map on M. Then M = M/f Γ, and Π(M) = Π(Mf)/Γ = (Mf × Mf)/Γ, a quotient by the diagonal action. 64 ECKHARD MEINRENKEN unique unit such that p = Φ(p)◦p. Given such a groupoid action, one can again form an action groupoid, as a subgroupoid of G × Pair(X) ⇒ M × X. Sometimes we use subscripts for the action operation, to avoid confusion with the groupoid multiplication. Every Lie groupoid G ⇒ M acts on itself by left multiplication g ◦L a = lg(a) = g ◦ a (here −1 Φ = t), and by right multiplication g ◦R a = rg−1 (a) = a ◦ g (here Φ = s), and it also acts on its units, by g ◦M m = t(g) (here Φ = idM ).

10.2. Left-invariant and right-invariant vector fields. Given a Lie groupoid G ⇒ M, the left multiplication by a groupoid element g ∈ G with source s(g) = m and target t(g) = m0 is a diffeomorphism −1 −1 0 lg : t (m) → t (m ), a 7→ g ◦ a A vector field X ∈ X(G) is called left-invariant if it is tangent to the t-fibers and for all g ∈ G, 0 the restrictions of X to the corresponding t-fibers of m = s(g), m = t(g) are related by lg. The left-invariant vector fields form a Lie subalgebra X(G)L of the Lie algebra of vector fields on G, and they are a module over s∗C∞(M). Similarly a vector field is called right-invariant if it is tangent to all s-fibers, and for all g as above the restriction to the s-fibers of m0, m R are related by rg−1 . The right-invariant vector fields form a Lie subalgebra X(G) , and are a module over t∗C∞(M). The flow of a left-invariant vector field is by right translations; in particular, left-invariant vector fields commute with right-invariant ones. Example 10.9. For the pair groupoid Pair(M) = M × M, a left -invariant vector field is of the form (0,X) where X is a vector field on M, while right-invariant vector fields are of the form 0 0 (X, 0). The flow of a left-invariant vector field is (m , m) 7→ (m , Φt(m)) where Φt is the flow of −1 X. This is right multiplication by the element (m, Φt(m)) = (Φt(m), m) . Similarly, a right 0 0 invariant vector field has flow (m , m) 7→ (Φt(m ), m). 10.3. Bisections. A bisection of a Lie groupoid is a submanifold S ⊆ G such that both t, s restrict to diffeomorphisms S → M. For example, M itself is a bisection. The name indicates that S can be regarded as a section of both s and t. We will denote by Γ(G) the set of all bisections. It has a group structure, with the multiplication given by 0 0 S ◦ S = MultG(S s ×t S ). The identity for this multiplication is the unit bisection M, and the inverse is given by S−1 = −1 InvG(S) = {g | g ∈ S}. This group comes with a homomorphism

Γ(G) → Diff(M),S 7→ ΦS where for all g ∈ S ⊆ G, ΦS(s(g)) = t(g). Examples 10.10. (a) For a Lie group G ⇒ pt, regarded as a Lie groupoid, a bisection is simply an element of G, and Γ(G) = G as a group. (b) For a vector bundle V → M, regarded as groupoid V ⇒ M, a bisection is the same as a section. More generally, this is true for any bundle of Lie groups. (c) For the ‘trivial’ groupoid M ⇒ M the only bisection is M itself. The resulting group Γ(M) consists of only the identity element. (d) For the pair groupoid Pair(M) ⇒ M, a bisection is the same as the graph of a diffeomorphism of M. This identifies Γ(G) =∼ Diff(M). INTRODUCTION TO POISSON GEOMETRY 65

(e) Let P → M be a principal G-bundle, and G ⇒ M the associated (Atiyah) groupoid. A bisection of G is the same as a principal bundle automorphism ΦP : P → P . That is, Γ(G) = Aut(P ). (f) Given a G-action on M, a bisection is a smooth map f : M → G for which the map m 7→ f(m).m is a diffeomorphism. The group of bisections has natural actions on G, by left multiplication: L AS (g) = h ◦ g, with the unique element h ∈ S such that s(h) = t(g). Similarly, there is an action by right multiplication, R 0 −1 AS (g) = g ◦ (h ) , with the unique element h0 ∈ S such that s(h0) = s(g). The left and right multiplication actions combine into an adjoint action, 0 −1 AdS(g) = h ◦ g ◦ (h ) by groupoid automorphisms of G. The restriction of the adjoint action to M ⊆ G is the diffeomorphism ΦS. In general, there may not exist a global bisection passing through a given point g ∈ G. example? However, one can always find a local bisection S ⊆ M, that is, t, s restrict to local diffeomorphisms to open subsets t(S) = V, s(S) = U of M. The left-action of a local bisection is then defined as a diffeomorphism L −1 −1 AS : t (U) → t (V ). Lemma 10.11. A vector field is left-invariant if and only if for every local bisection S,

X|t−1(U) ∼ L X|s−1(U). AS Proof. We have to show that if a vector field X satisfies the above property for all local bisections, then X is in particular tangent to the t-fibers. Consider Xm for m ∈ M ⊆ G. Let L S be a local bisection passing through m; thus m is a fixed point for the action AS . If X is L 0 00 0 invariant under AS , then Xm must be fixed. Decompose Xm = Xm +Xm, where Xm is tangent L −1 to the t-fiber, while Xm00 is tangent to M. The action AS fixes the fiber t (m) pointwise, 0 00 hence it takes Xm to itself. Hence Xm must be fixed. On the other hand, the action of S 00 maps U ⊆ M to a submanifold of S. We can choose S such that the image of Xm is no longer 00 tangent to TmM. Thus, we must have Xm = 0. This shows that X is tangent to t-fibers along M. Since the left-action by local bisections preserves tangent spaces to t-fibers, X is tangent to t-fibers everywhere. 10.4. The Lie algebroid of a Lie groupoid. Let G be a Lie groupoid, and let A = ν(G,M) be the normal bundle of M in G. In the case of a pair groupoid, we identify ν(Pair(M),M) =∼ TM by the map taking the class of (v2, v1) ∈ TM (Pair(M)) to v1 − v2. The groupoid anchor (58) is a map of pairs aG = (t, s):(G,M) → (Pair(M),M), 66 ECKHARD MEINRENKEN hence it induces a vector bundle morphism

a = ν(aG): A → TM. called the anchor map. Equivalently, a is the map induced by

T s − T t: T G|M → TM. For σ ∈ Γ(A), there are unique vector fields σL ∈ X(G)L, σR ∈ X(G)R mapping to σ under restriction to M followed by the quotient map T G|M → A. Lemma 10.12. We have that L R L R a(σ) ∼i σ − σ , σ ∼s a(σ), σ ∼t −a(σ). Proof. Since σL is tangent to the t-fibers, we have that L L a(σ) = (T s − T t)(σ |M ) = (T s)(σ |M ), Similarly, a(σ) = −T t(σR). Finally, note that σL − σR is tangent to M, hence restricts to a vector field on M. Since L R T s((σ − σ )|M ) = a(σ), we see that the restriction is a(σ). We will use the identification of Γ(A) with left-invariant vector fields on G to define a Lie bracket on Γ(A). 6 Lemma 10.13. With these data, (A, a, [·, ·]) is a Lie algebroid. Proof. Let f ∈ C∞(M) and σ, τ ∈ Γ(A). Using (fτ)L = (s∗f) τ L and σL(s∗f) = s∗a(σ)f L (since σ ∼s a(σ)) we compute, [σ, fτ]L = [σL, s∗f τ L] = s∗f [σL, τ L] + s∗(a(σ)f)τ L = f[σ, τ] + a(σ)f τL, proving [σ, fτ] = f[σ, τ] + a(σ)f τ. One calls A the Lie algebroid of the Lie groupoid G; conversely, one refers to G as an integration of A. Lemma 10.14. The left and right invariant vector fields satisfy the bracket relations, for σ, τ ∈ Γ(A): (59) [σL, τ L] = [σ, τ]L, [σL, τ R] = 0, [σR, τ R] = −[σ, τ]R.

6Recall that for Lie groups, one uses left-invariant vector fields to ensure that for matrix Lie groups, the Lie bracket is the commutator. Let θL ∈ Ω1(G, g) be the left-Maurer-Cartan form on G. For a matrix Lie group, L −1 L 1 L L this is θ = g dg, which satisfies the Maurer-Cartan equation dθ + 2 [θ , θ ] = 0, with the bracket denoting the matrix commutator. For an abstract Lie group, we may take this equation as the definition of the Lie bracket on g. The left-invariant vector fields are defined by ι(ξL)θL = ξ, for ξ ∈ g. We obtain, using the Cartan calculus, 1 [ξ, ζ] = ι(ζL)ι(ξL) [θL, θL] = −ι(ζL)ι(ξL)dθL = ι([ξL, ζL])θL. 2 where we used that., e.g. L(ξL)ι(ζL)θL = L(ξL)ζ = 0. That is, our sign conventions give [ξ, ζ]L = [ξL, ζL]. INTRODUCTION TO POISSON GEOMETRY 67

Proof. The bracket relations for the left-invariant vector fields hold by definition; those for the L R right-invariant vector fields follow since σ ∼ InvG − σ . The fact that left-and right-invariant vector fields commute follows because their flows commute. Remark 10.15. One can also understand the construction of the Lie algebroid in terms of bisections. Given a 1-parameter family of embedded submanifolds St ∈ Γ(G) with S0 = M, the t-derivative at t = 0 is a section of the normal bundle ν(G,M) = A. In particular, this applies to 1-parameter families of bisections. Conversely, the flow of a left-invariant vector field σL defined by σ ∈ Γ(A) is a (locally defined) 1-parameter group of bisections7 Hence, the Lie algebra of Γ(G) is given by Γ(A), identified with left-invariant vector fields. Not every Lie algebroid A admits an integration to a Lie groupoid; the precise obstructions to integrability were determined by Crainic-Fernandes [12]. If an integration exists, then (as one can show) there exists a unique integration with connected and simply connected s-fibers; every other integration is then obtained by taking a quotient by a finite group. On the other hand, every Lie algebroid admits an integration to a local Lie groupoid. We will not give the precise definition of a local Lie groupoid, for which we refer to [10]. Roughly, a local Lie groupoid is again given by a manifold G with a smooth submanifold M and two surjective submersions t, s which are projections from G to M. However, the groupoid multiplication is not necessarily defined on all of G(2), but only on some neighborhood of M, and the associativity and unit axioms only hold for elements in a sufficiently small neighborhood of M. The first explicit proof of integrability to a local Lie groupoid was given by Crainic-Fernandes, although one might say it is implicit in the integration theorem for Poisson manifolds (by viewing a Lie algebroid in terms of the linear Poisson structure on the dual bundle). 10.5. Groupoid multiplication via σL, σR. For i = 0, 1, 2 and σ ∈ Γ(A), the vector fields on G × G × G, 0 R R 1 L R 2 L L (60) Xσ = (−σ , −σ , 0),Xσ = (0, σ , −σ ),Xσ = (σ , 0, σ ). are all tangent to Gr(MultG). For instance, the invariance of the graph under the (local) flow 1 −1 of Xσ follows from the fact that g1 ◦ g2 = (g1 ◦ h ) ◦ (h ◦ g2) whenever s(h) = t(g2). The vector fields satisfy bracket relations i j i [Xσ,Xτ ] = X[σ,τ] δi,j for σ, τ ∈ Γ(A) and i, j = 0, 1, 2. If G is t-connected, then the graph is generated from M ⊆ Λ (embedded as m 7→ (m, m, m)) by the flow of these vector fields. In fact, it is already obtained i using the flows of the Xσ’s for any two of the indices i ∈ {0, 1, 2}. For reference in Section 9, let us note the following partial converse. Proposition 10.16. Let (A, a, [·, ·]) be a Lie algebroid over M. Suppose i: M → P is an embedding, with normal bundle ν(P,M) =∼ A, and suppose σL, σR ∈ X(P ) are vector fields L R on P , mapping to σ ∈ Γ(A) under the quotient map TP |M → A, with a(σ) ∼i σ − σ , and satisfying the bracket relations (59). Then a neighborhood of M in P inherits a structure of a local Lie groupoid integrating A, in such a way that σL, σR are the left, right invariant vector fields.

7In contrast to Lie groups, the left-invariant vector fields on a groupoid need not be complete. For example, for a pair groupoid Pair(M) the left-invariant vector fields correspond to arbitrary vector fields on M. 68 ECKHARD MEINRENKEN

L Sketch of proof. Since σ |M maps to σ under T G|M → ν(G,M), the restrictions of the left- invariant vector fields to M span a complement to TM in TP |M . In a particular, on a neighborhood of M they determine a distribution of rank equal to that of A. By the bracket relations, this distribution is Frobenius integrable. A similar argument applies to the vector fields −σR. Taking P smaller if necessary, we can assume that these foliations define surjective submersions t, s: P → M, with σL, σR tangent to the respective fibers, and with t ◦ i = s ◦ i = id. Define i vector fields Xσ in P × P × P as above, and embed M,→ P × P × P by m 7→ (m, m, m). 0 2 Along M, hence also on some neighborhood of M inside P × P × P , the vector fields Xσ,Xτ span a distribution of rank equal to twice the rank of A. The bracket relations guarantee that this distribution is integrable, hence they define a foliation. Since the intersection of this distribution with the tangent bundle of M ⊆ P × P × P is trivial, we conclude that the flow-out of M under these vector fields defines a (germ of a) submanifold Λ ⊆ P × P × P , of dimension 2 rank(A) + dim M = 2 dim(A) − dim M This is our candidate for the graph of the multiplication map. By construction, Λ ⊆ P × P × P contains M, and is invariant under the local flow of all 0 2 1 vector fields Xσ,Xσ. In fact, it is also invariant under the local flow of Xτ for τ ∈ Γ(A), since these vector fields are tangent to Λ along M, and hence everywhere since the commute with 0 2 all Xσ,Xσ. 0 2 Under projection P × P × P → P × P, (p, p1, p2) 7→ (p1, p2), the vector fields Xσ,Xσ are related to (−σR, 0) and (0, σL), respectively. Hence, this projection restricts to a diffeomorphism (2) from Λ onto a neighborhood of the diagonal embedding of M in P = P s ×t P ⊆ P × P . Taking the inverse map, followed by projection to the first P -factor, defines a multiplication (2) (2) map MultP : P → P ; strictly speaking it is defined only on some neighborhood of M in P . By construction, Λ = Gr(MultP ). Letting idP : P → P be the identity relation (given by the diagonal in P ×P , the associativity of the groupoid multiplication means that

Λ ◦ (Λ × idP ) = Λ ◦ (idP ×Λ) as relations P × P × P 99K P , where the circle means composition of relations. In fact, we can see that both sides are given by Λ[2] ⊆ P × (P × P × P ), [3] the submanifold generated from the diagonal M∆ (consisting of elements (m, m, m, m)) by the action of vector ﬁelds of the form R R L R L R (−σ , −σ , 0, 0), (0, σ , −σ , 0), (0, 0, σ , −σ ).

11. Symplectic groupoids 11.1. Deﬁnition, basic properties. Deﬁnition 11.1 (Weinstein [40]). Let (M, π) be a Poisson manifold. A symplectic groupoid is a Lie groupoid G ⇒ M, equipped with a symplectic structure ω such that that the graph of the groupoid multiplication Gr(MultG) is a Lagrangian submanifold of G × G × G. Here the line indicates the opposite symplectic structure; thus G × G × G has the symplectic ∗ ∗ ∗ structure pr1 ω −pr2 ω −pr3 ω, with pri the projection to the i-th factor. ‘Lagrangian’ is meant INTRODUCTION TO POISSON GEOMETRY 69

3 in the sense of symplectic geometry: Gr(MultG) has the middle dimension 2 dim G, and the pullback of the symplectic form vanishes. The condition that Gr(MultG) is a Lagrangian submanifold may be rephrased as a multi- plicativity condition for the symplectic form. Note that T G is a groupoid over TM, with the groupoid multiplication MultT G = T MultG. Lemma 11.2. Let ω be a symplectic 2-form on a Lie groupoid G ⇒ M. Then (G, ω) is a (2) symplectic groupoid if and only if for all (v1, v2), (w1, w2) ∈ T G , with the same base points (2) (g1, g2) ∈ G , we have that

(61) ω(v1 ◦ v2, w1 ◦ w2) = ω(v1, w1) + ω(v2, w2). (2) Proof. For (v1, v2) ∈ T G , we have (v1 ◦ v2, v1, v2) ∈ T Gr(MultG). Hence, if (G, ω) is a symplectic groupoid, then the multiplicative property (61) follows from 0 = ωG×G×G (v1 ◦ v2, v1, v2), (w1 ◦ w2, w1, w2) = ω(v1 ◦ v2, w1 ◦ w2) − ω(v1, w1) − ω(v2, w2).

Conversely, if the multiplicative property holds then this calculation shows that Gr(MultG) is isotropic in G × G × G. Hence (2) 2 dim G − dim M = dim G = dim Gr(MultG) ≤ 3/2 dim G, 1 hence dim M ≥ 2 dim G. On the other hand, by applying the identity to v ◦ v = v, w ◦ w = w ∈ 1 TM, we also see that M is isotropic in G, hence dim M ≤ 2 dim G. We have thus proved that the inequalities above are all equalities, hence Gr(MultG) is Lagrangian. (At the same time, we have proved that M is Lagrangian in G.) Theorem 11.3 (Coste-Dazord-Weinstein [10]). For any symplectic groupoid (G, ω) the space M of objects acquires a Poisson structure, in such a way that t, s, i satisfy the Karasev-Weinstein conditions from Theorem 9.6. That is, the map (t, s): G → M × M − is Poisson, where M − stands for M with the opposite Poisson structure −π. The inversion map of a symplectic groupoid is anti-symplectic.

One calls (G, ωG) a symplectic groupoid integrating the Poisson manifold (M, π). Proof. We have already shown, in the proof of Lemma 11.2, that M ⊆ G is Lagrangian. If (2) (v1, v2), (w1, w2) ∈ G are tangent to the graph of the inversion map, then v1 ◦ v2, w1 ◦ w2 are both tangent to M, hence

0 = ω(v1 ◦ v2, w1 ◦ w2) = ω(v1, v2) + ω(w1, w2).

This shows that the inversion is anti-symplectic. If v ∈ ker(t∗) and w ∈ ker(s∗), with the same 0 0 base point g ∈ G, then v = 0t(g) ◦ v, while w = w ◦ 0g, for suitable w ∈ ker(s∗)t(g) deﬁned by this equation. (Namely, w0 is the image w under right transation by g−1.) It follows that ω(v, w) = ω(0, w0) + ω(v, 0) = 0. 1 Since ker s∗, ker t∗ both have rank 2 dim G, this shows that they are ω-orthogonals of each other. By Libermann’s theorem, we conclude that M admits a Poisson structure such that t is Poisson. Since inversion on G is anti-symplectic, and interchanges the source and target maps, it follows that s is anti-Poisson relative to the same Poisson structure. 70 ECKHARD MEINRENKEN

The basic examples of symplectic groupoids are as follows: Example 11.4. If (M, ω) is a symplectic manifold (regarded as a Poisson manifold), then the pair groupoid Pair(M) = M × M ⇒ M is a symplectic groupoid integrating M. Here M indicates M with the opposite symplectic structure −ω. Example 11.5. [10] Suppose G is a Lie group, with Lie algebra g. Then the cotangent bundle T ∗G, with its standard symplectic structure, has the structure of a symplectic groupoid ∗ ∗ T G ⇒ g ∗ ∗ where for µi ∈ Tgi G, i = 1, 2 and µ ∈ Tg G, ∗ µ = µ1 ◦ µ2 ⇔ g = g1g2, (µ1, µ2) = (Tg1,g2 MultG) µ.

Here MultG : G × G → G is the group multiplication map of the group G. The space of units ∗ ∗ is g embedded as the fiber Te G, while source and target map are given by left trivialization, respectively right trivialization. The symplectic groupoid T ∗G integrates g∗ equipped with the Lie-Poisson structure. Indeed, the map t is the symplectic moment map for the cotangent lift of the left G-action on itself, and moment maps are always Poisson maps. 11.2. Lagrangian bisections. A bisection S ⊆ G which is a Lagrangian submanifold is called a Lagrangian bisection. The units M are such a Lagrangian bisection; more generally, if S is obtained from M by the flow of a Hamiltonian vector field on G, then S is a Lagrangian bisection as long as the flow is sufficiently small (so that S is a section of both t and s). The (2) Lagrangian bisections form a subgroup ΓLag(G) of Γ(G): If (v1, v2), (w1, w2) ∈ T G , with 0 0 v1, w1 ∈ TS and v2, w2 ∈ TS , then v1 ◦ v2, w1 ◦ w2 ∈ T (S ◦ S ) satisfy

ω(v1 ◦ v2, w1 ◦ w2) = ω(v1, w1) + ω(v2, w2) = 0, as required. A similar argument shows that the action of ΓLag(G) on G (by left multiplication, right multiplication, or adjoint action) is by symplectic transformations. It is often useful to have a Lagrangian bisection passing through a given element g ∈ G. This is not always possibly, but one can always ﬁnds a local Lagrangian bisection: Lemma 11.6. For every g ∈ G, there exists a Lagrangian submanifold S ⊆ G such that t, s restrict to diﬀeomorphisms from S onto open subsets t(S), s(S) ⊆ M.

Proof. By elementary symplectic geometry, one can find a Lagrangian subspace of TgG that is ∗ transverse to ker(s )g. It is then automatic that this Lagrangian subspace is also transverse to ∗ ker(t∗)g, since ker(s )g and ker(t∗)g are ω-orthogonal. Take S to be a Lagrangian submanifold through g, with this Lagrangian subspace as its tangent space at g. Replacing S with a sufficiently small neighborhood of g inside S, it follows that t, s restrict to diffeomorphisms.

11.3. The Lie algebroid of a symplectic groupoid. Suppose G ⇒ M is a symplectic ] groupoid integrating the Poisson manifold (M, π). Since M is Lagrangian in G, the map −πG takes ann(TM) onto TM; it hence induces an isomorphism ∗ ∗ (62) T M = T G|M / ann(TM) → ν(M,G) = T G|M /T M. Give T ∗M the Lie algebroid structure as a cotangent Lie algebroid of M. INTRODUCTION TO POISSON GEOMETRY 71

Proposition 11.7. The map (62) is an isomorphism of Lie algebroids. Under this isomorphism, the left-invariant and right-invariant vector fields defined by α ∈ Ω1(M) are given by R ] ∗ L ] ∗ (63) α = −πG(t α), α = −πG(s α). The satisfy the bracket relations, (64) [αL, βL] = [α, β]L, [αR, βR] = −[α, β]R, [αL, βR] = 0 where the bracket of 1-forms is defined using the Lie algebroid bracket of T ∗M. Proof. The two identities (63) hold along M ⊆ G, by definition of the isomorphism. To show ] ∗ that they hold true over all of G, it suffices to show that the vector field πG(s α) is left- ] ∗ invariant, while πG(t α) is right-invariant. For this, it suffices to verify the invariance under L local Lagrangian bisections S ⊆ G. The left-translation AS by a Lagrangian bisections is a ] symplectic transformation, hence pull-back under such a transformation commutes with πG. ∗ L L ] ∗ But the 1-form s α is invariant under pull-back by AS , since s◦AS = s. It follows that πG(s α) ] ∗ is left-invariant, and similarly that πG(t α) is right-invariant. The bracket relations follow since t is Poisson, s is anti-Poisson. They also show in particular that the identification T ∗M → A is a Lie algebroid isomorphism. 11.4. The cotangent groupoid. Let G ⇒ M be a Lie groupoid. Then the tangent bundle T G ⇒ TM is a VB-groupoid. T G // TM

G / M The cotangent bundle T ∗G has a VB-groupoid structure as well. The groupoid multiplication may be described as follows: µ = µ1 ◦ µ2 if and only if the base points satisfy g = g1 ◦ g2, and for all tangent vectors v, v1, v2 at those base points, with v = v1 ◦ v2, we have that

hµ, vi = hµ1, v1i + hµ2, v2i. Equivalently, the graph of the groupoid multiplication ∗ ∗ ∗ Gr(MultT ∗G) ⊆ T G × T G × T G is the annihilator of that of T G, followed by sign changes in the second and third factor. In the notation of (26), Gr(MultT ∗G) = (Gr(MultT G)) . It is thus Lagrangian, and hence T ∗G is a symplectic groupoid. 8 The base of this groupoid is the annihilator of TM inside T G, i.e. the dual of A = ν(G,M). We obtain a VB-groupoid

T ∗G // A∗

G // M

8This generalizes to the fact that the dual of any LA-groupoid is a Poisson groupoid. 72 ECKHARD MEINRENKEN

Note that Poisson structure on T ∗G is linear with respect to the vector bundle structure over G. Hence, the resulting Poisson structure on A∗ is linear as well.

∗ ∗ Theorem 11.8. The symplectic groupoid T G ⇒ A is an integration of the Poisson manifold A∗ with its natural Lie-Poisson structure.

∞ ∗ The proof will require the following Lemma. Given σ ∈ Γ(A), let φσ ∈ C (A ) be the ∗ ∞ ∗ corresponding linear function, and s φσ ∈ C (T G) the left-invariant function obtained by pull-back. By (63), the corresponding Hamiltonian vector field is the left-invariant vector field ∗ ∗ 1 ∗ on the symplectic groupoid T G ⇒ A defined by the 1-form dφσ ∈ Ω (A ): ] ∗ L −πT ∗G(ds φσ) = (dφσ) . L ∗ Proposition 11.9. The left-invariant vector field (dφσ) ∈ X(T G) is the cotangent lift of the left-invariant vector field σL ∈ X(G). In particular, L L (65) σ ∼j (dφσ) , where j is the inclusion.

∗ ∗ ∗ ∗ ∗ Proof. The canonical decomposition TA |M = A ⊕TM dualizes to T A |M = A⊕T M. Since ∗ φσ is linear with respect to the scalar multiplication on A , the 1-form dφσ vanishes on tangent ∗ ∗ ∗ vectors to M ⊆ A . Hence (dφσ)|M takes values in the first summand, A ⊆ T A |M . Using local bundle coordinates for A, it is clear that this restriction is just σ. This proves (??), and L L hence shows that the linear vector field (dφσ) is a lift of σ . Since it is a Hamiltonian vector ∗ ∞ ∗ field with respect to a linear function f = s φσ ∈ C (T G), it is the cotangent lift. ∗ Proof of Theorem 11.8. Let {·, ·}1 be the Poisson structure on A induced by the symplectic ∗ ∞ ∗ groupoid T G, and {·, ·}2 the Lie Poisson-structure. Let σ, τ ∈ Γ(A), and φσ, φτ ∈ C (A ) the corresponding functions. Since s is anti-Poisson with respect to {·, ·}1 we have that L L ] ∗ ] ∗ [(dφσ) , (dφτ ) ] = [πT ∗G(ds φσ), πT ∗G(ds φτ )] ] ∗ = −πT ∗G(ds {φσ, φτ }1) On the other hand, L ] ∗ (dφ[σ,τ]) = −πT ∗G(ds φ[σ,τ]) ] ∗ = −πT ∗G(ds {φσ, φτ }2) L L L L L L Since [σ , τ ] = [σ, τ] , their cotangent lifts satisfy [(dφσ) , (dφτ ) ] = (dφ[σ,τ]) . This shows ∗ ∗ that the Hamiltonian vector fields corresponding to s {φσ, φτ }1, s {φσ, φτ }2 are equal. Hence the difference of these functions is locally constant, and in fact is zero since they are linear ∗ functions on T G → G. This shows {φσ, φτ }1 = {φσ, φτ }2. 11.5. Integration of Poisson manifolds. Not every Poisson manifold admits an integration to a symplectic groupoid. But one always has a local symplectic groupoid integrating the Poisson structure. Local Lie groupoids are a generalization of Lie groupoids, where the groupoid multiplication is only defined on some open neighborhood of the diagonally embedded M ⊆ G(2). (For details, see [10, Definition III.1.2].) INTRODUCTION TO POISSON GEOMETRY 73

Theorem 11.10 (Coste-Dazord-Weinstein [10], Karasev [23]). Every Poisson manifold (M, π) admits a local symplectic groupoid G ⇒ M integrating it. In fact, it is shown in [10] that any symplectic realization τ : P → M for which τ is a retraction onto a Lagrangian submanifold i: M,→ P , can be given the structure of a local symplectic groupoid over M. The following discussion is similar to their approach. It shows that given the data from the Karasev-Weinstein theorem, one automatically gets the local groupoid structure.

Proposition 11.11. Let (M, πM ) be a Poisson manifold, and (P, ωP ) a symplectic manifold with an inclusion i: M → P as a Lagrangian submanifold, and with two surjective submersions t, s: P → M, satisfying the properties of the Karasev-Weinstein theorem 9.6. Replacing P with a smaller neighborhood of M if necessary, it has a unique structure of a local symplectic groupoid, with s and t as the source and target map, and i the inclusion of units. The idea is to generating the graph of the groupoid multiplication by flows of vector fields, as in Proposition 11.7. For all α ∈ Ω1(M), define vector fields αL, αR ∈ X(P ) by L ] ∗ R ] ∗ (66) α = −πP (t, s) (0, α) , α = −πP (t, s) (α, 0) . L ] ∗ R ] ∗ − Equivalently, α = −πP (s α), α = −πP (t α). Since (t, s): P → M × M is a Poisson map, we have that L ] L R ] R α ∼s πM (α), α ∼t 0, α ∼t −πM (α) α ∼s 0 and the bracket relations [αL, βL] = [α, β]L, [αR, βR] = −[α, β]R, [αL, βR] = 0. Furthermore, L L ∗ R R ∗ L R (67) ω(α , β ) = −s πM (α, β), ω(α , β ) = t πM (α, β), ω(α , β ) = 0. By Proposition 11.7, the bracket relations among the vector fields αL, αR give P the structure of a local groupoid, where Λ = Gr(MultP ) is obtained as the flow-out of M ⊆ P × P × P under (αL, 0, αL), (−αR, −αR, 0). To conclude the argument that it is a local symplectic groupoid, it remains to check: Proposition 11.12. The submanifold Λ is a Lagrangian submanifold (in the sense of symplectic geometry) of P × P × P . Proof. Recall that Λ is generated from the isotropic submanifold M ⊆ P × P × P by the flow-outs of vector fields of the form (−αR, −αR, 0), (αL, 0, αL). Indeed, it is already generated by vector fields of this form such that α is exact. Since these are Hamiltonian vector fields, they preserve the symplectic form. It therefore suffices to check that the tangent space to Λ is Lagrangian along M. The latter is spanned by the restrictions of the vector fields of the form above, together with TM ⊆ T (P × P × P ). Using the relations (67), one checks that this subbundle is isotropic, and since it has rank 3 dim M it is Lagrangian. 74 ECKHARD MEINRENKEN

11.6. Integration of Lie algebroids. Suppose A → M is a Lie algebroid, and V = A∗ the dual bundle with associated linear Poisson structure. Consider the double vector bundle T ∗V / V

V ∗ / M ∗ The choice of a linear Poisson spray deﬁnes a symplectic realization T V ⊇ P ⇒ V , where P is a vector subbundle of T ∗V → V ∗ over Q = P ∩ V ∗. The source and target map are both vector bundle morphisms, so that we have a local VB-groupoid

T ∗V ⊇ P // V

V ∗ ⊇ Q / M The symplectic form on the vector bundle P → Q is linear; by the following Lemma, this means that P =∼ T ∗Q. Lemma 11.13 (Vector bundles with linear symplectic structures). (a) Let W → Q be a vector bundle, equipped with a symplectic structure ω that is homogeneous of degree 1. Then there is a unique vector bundle isomorphism W =∼ T ∗Q taking ω to the canonical symplectic form, and restricting to the identity map on Q. ∗ ∗ − (b) If L ⊆ T Q2 × T Q1 is a Lagrangian relation, which is also a vector subbundle along S ⊆ Q2 × Q1, then ∗ ∗ − L = ann(TS) := {(ξ2, −ξ1)| (ξ2, ξ1) ∈ ann(TS)} ⊆ T Q2 × T Q1 .

Proof. Consider the restriction ω|Q to TW |Q = TQ ⊕ W , where the first summand corresponds to tangent vectors to base and the second to tangent vectors to the fibers The R>0-action defined by scalar multiplication is trivial on the first summand, and multiplication by t on the second summand. Since ω is homogeneous of degree 1, it follows from this scaling behaviour that the symplectic form restricts to zero on each of the subbundles T Q, W ⊆ TW |Q. Hence both are Lagrangian subbundles, and the symplectic form identifies W =∼ T ∗Q as vector bundles. Under this isomorphism, ω becomes a linear symplectic struc- ∗ ture on T Q, which agrees with the canonical symplectic structure ωcan along Q. Since both are closed 2-forms that are homogeneous of degree 1, it follows that ω = ωcan. This proves ∗ (a). For (b), let L = {(ξ2, −ξ1)| (ξ2, ξ1) ∈ L}, as a subbundle of W = T (Q2 × Q1). Using TL |S = TS ⊕ L ⊆ TW |Q = TQ ⊕ W , we see that L = ann(TS). The application to symplectic groupoids gives:

Proposition 11.14. Let H ⇒ V be a symplectic groupoid, which is also a VB-groupoid over G ⇒ M, and such that the symplectic form ω on H is homogeneous of degree 1. Then the identiﬁcation H = T ∗G from Lemma 11.13 preserves the groupoid structure. In particular, the Lie algebroid A = V ∗ coincides with the Lie algebroid of G.

Proof. The condition that H is a symplectic groupoid means that Gr(MultH) ⊆ H × H × H is a Lagrangian submanifold, and since H is a VB-groupoid this is a vector subbundle along the INTRODUCTION TO POISSON GEOMETRY 75 submanifold Gr(MultG) ⊆ G × G × G. By the Lemma, it follows that under the identiﬁcation ∗ H = T G, we have Gr(MultH) = ann(T Gr(MultG)) . But this is exactly the standard groupoid ∗ structure for T G. (For details, see [25, Appendix C].) A similar result holds for local symplectic VB-groupoids P ⇒ V , with the same proof. We hence conclude that the local symplectic groupoid P ⇒ V constructed above is just the cotangent groupoid P = T ∗Q. Let us now explain how to construct an integration of a Lie algebroid A → M, using the integration results for Poisson manifolds. Let V = A∗ → M be the dual bundle, and consider the double vector bundle T ∗V / V

V ∗ / M The bundle V → M has a linear Poisson structure. Choose a Poisson spray X ∈ X(T ∗V ) which is linear relative to the vertical vector bundle structure. (This is always possible: e.g., choose X locally in bundle charts for T ∗V → V ∗, and then patch the local definitions using a partition of unity.) The symplectic realization (P, ω) defined by the Crainic-Marcut theorem [13] is then linear, i.e., P ⊆ T ∗V is a vector subbundle for the vertical vector bundle structure, with base an open neighborhood Q ⊆ V ∗ of M, and the symplectic form ω is homogeneous of degree 1. As explained in [30], one obtains the local groupoid structure on P by describing the left and right invariant vector fields. The base of the local symplectic VB-groupoid P ⇒ V is ∗ a local Lie groupoid Q ⇒ M, and by Proposition 11.14 it defines an integration of A = V . Remark 11.15. Alternatively, given any local symplectic groupoid P ⇒ V integrating the Poisson manifold V (e.g., as in [10]), one can always lift the R>0-action on V to a (local) action on P , such that the symplectic form is homogeneous of degree 1. See [6].

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