Cotangent algebroid of a Poisson manifold; pointer to symplectic groupoids

Intuition Beginner

A Poisson manifold is a space where every pair of functions has a bracket, a third function that measures how the two functions fail to commute as observables. In mechanics this bracket is the rule that turns an energy function into the motion it generates: the bracket of energy with anything tells you how that thing changes in time. The bracket is the whole structure.

There is a quieter object hiding inside this picture. To each function you can attach its differential, a one-form, and the bracket lets you turn that one-form into an actual flow on the space. The map that does this turning is the heart of the matter: feed it a one-form, it hands you back a direction of motion. This is the sharp map.

Why package this as its own object? Because one-forms, the sharp map, and a bracket on one-forms together form the same kind of structure as the directions-with-a-bracket from the Lie algebroid unit. The Poisson manifold's infinitesimal skeleton is a Lie algebroid living on the cotangent bundle, and seeing it that way connects Poisson geometry to the whole world of arrows and their infinitesimal shadows.

Visual Beginner

Picture a surface carved into curved patches. On each patch the Poisson bracket is non-degenerate, so the patch behaves like a little phase space with its own area form; these patches are the symplectic leaves. Between patches of different sizes the bracket can drop rank, and some directions carry no flow at all.

Over each point sits the space of one-forms. The sharp map sends a one-form to a tangent vector, and the picture to keep is that this tangent vector always lies inside the leaf through that point. The image of the sharp map is exactly the tangent directions to the leaves. Where the bracket is zero, the sharp map sends everything to the zero vector, and the leaf is a single point.

Worked example Beginner

Take the plane with coordinates $x$ and $y$ and the simplest non-zero bracket, $\{x,y\}=1$. This is the standard phase plane of a particle: $x$ is position, $y$ is momentum.

The sharp map sends the one-form $dx$ to a flow. The rule is that the flow attached to a function $f$ moves every other function $g$ at the rate $\{f,g\}$. For $f=x$ we get $\{x,x\}=0$ and $\{x,y\}=1$, so the flow of $x$ leaves $x$ fixed and pushes $y$ forward at unit speed. That flow is the motion in the $y$-direction, the vector $(0,1)$.

Do the same for $f=y$: now $\{y,x\}=-1$ and $\{y,y\}=0$, so the flow leaves $y$ fixed and pushes $x$ backward. That flow is the vector $(-1,0)$.

What this tells us: the sharp map here is a rotation by a right angle, sending $dx$ to $(0,1)$ and $dy$ to $(-1,0)$. It is one-to-one, so it reaches every tangent direction, and the whole plane is a single symplectic leaf. The cotangent space and the tangent space are matched up perfectly by the bracket.

Check your understanding Beginner

Formal definition Intermediate+

Let $(M,\pi )$ be a Poisson manifold: a smooth manifold with a bivector field $\pi \in \Gamma ({\Lambda }^{2}TM)$ whose induced bracket $\{f,g\}=\pi (df,dg)$ on $C^{\infty }(M)$ satisfies the Jacobi identity 05.02.02. Equivalently, $\pi$ is a Poisson bivector when $[\pi ,\pi {]}_{\mathrm{SN}}=0$, where $[\cdot ,\cdot {]}_{\mathrm{SN}}$ is the Schouten-Nijenhuis bracket on multivector fields; here $[\cdot ,\cdot {]}_{\mathrm{SN}}$ denotes the graded extension of the vector-field bracket to $\Gamma ({\Lambda }^{\bullet }TM)$.

The sharp map is the bundle morphism $$ \pi^\sharp : T^\ast M \to TM, \qquad \beta\big(\pi^\sharp(\alpha)\big) = \pi(\alpha, \beta), $$ contracting $\pi$ with a one-form in its first slot. The Hamiltonian vector field of $f$ is $X_f={\pi }^{♯}(df)$, and $X_{f}g=\{f,g\}$ 05.02.01.

The cotangent Lie algebroid of $(M,\pi )$ is the vector bundle $A=T^{\ast }M\to M$ equipped with anchor $\rho ={\pi }^{♯}$ and the Koszul bracket on ${\Omega }^1(M)=\Gamma (T^{\ast }M)$ ^{[Mackenzie Ch. III §3]}: $$ [\alpha, \beta] ;=; \mathcal{L}{\pi^\sharp\alpha},\beta ;-; \mathcal{L}{\pi^\sharp\beta},\alpha ;-; d\big(\pi(\alpha, \beta)\big), $$ where ${\mathcal{L}}_X$ is the Lie derivative along $X$ and $d$ the exterior derivative. On exact one-forms this reduces to $$ [df, dg] = d{f, g}, $$ so the differential intertwines the Poisson bracket on functions with the Koszul bracket on exact forms.

Two facts pin the definition down. First, ${\pi }^{♯}$ is the anchor: applied to $df$ it returns $X_f$, and $X_f$ is the derivation reading off the function-derivative term of the Leibniz rule 03.04.16. Second, the Koszul bracket is the unique extension of $[df,dg]=d\{f,g\}$ to all one-forms compatible with the Leibniz rule $[\alpha ,h\beta ]=h[\alpha ,\beta ]+({\pi }^{♯}(\alpha )h)\beta$; the three-term formula is what that extension forces.

Counterexamples to common slips

• A frequent slip is to define the bracket only by $[df,dg]=d\{f,g\}$ and hope $C^{\infty }(M)$-bilinearity extends it. It does not: the Leibniz term $({\pi }^{♯}(\alpha )h)\beta$ is non-zero, so the bracket is not tensorial, exactly as for any Lie algebroid with non-vanishing anchor.
• The bracket is not antisymmetric "by inspection" of the three-term formula; antisymmetry follows from ${\mathcal{L}}_X=d{\iota }_X+{\iota }_{X}d$ and the identity $\pi (\alpha ,\beta )=-\pi (\beta ,\alpha )$ once the Cartan calculus is unwound.
• The zero Poisson structure $\pi =0$ gives sharp map $0$ and bracket $[\alpha ,\beta ]=0$; the cotangent algebroid degenerates to the bundle $T^{\ast }M$ with zero anchor and zero bracket, every point its own leaf.

Key theorem with proof Intermediate+

Theorem (Koszul Jacobi equals Poisson Jacobi). Let $M$ carry a bivector $\pi$, with ${\pi }^{♯}$, the bracket $\{f,g\}=\pi (df,dg)$, and the Koszul bracket on ${\Omega }^1(M)$ extending $[df,dg]=d\{f,g\}$ by the Leibniz rule. Then $(T^{\ast }M,[\cdot ,\cdot ],{\pi }^{♯})$ is a Lie algebroid if and only if $\{\cdot ,\cdot \}$ satisfies the Jacobi identity, i.e. iff $\pi$ is a Poisson bivector.

Proof. Antisymmetry and the Leibniz rule hold for the Koszul bracket independently of any integrability of $\pi$, by direct Cartan-calculus computation; the content of the theorem is the Jacobi identity ^{[Mackenzie Ch. III §3]}. Write the Jacobiator of the Koszul bracket as $J(\alpha ,\beta ,\gamma )=[\alpha ,[\beta ,\gamma ]]+[\beta ,[\gamma ,\alpha ]]+[\gamma ,[\alpha ,\beta ]]$.

Exact one-forms $df$ generate ${\Omega }^1(M)$ as a $C^{\infty }(M)$-module, and the Leibniz rule extends any bracket from exact generators to all one-forms in at most one way. So $J$ vanishes on all one-forms if and only if it vanishes on exact ones. Evaluate on $df,dg,dh$. Since $[df,dg]=d\{f,g\}$ and $d$ is a chain map, $$ [df, [dg, dh]] = [df,, d{g,h}] = d{f, {g, h}}, $$ and cyclically. Adding the three terms, $$ J(df, dg, dh) = d\Big({f,{g,h}} + {g,{h,f}} + {h,{f,g}}\Big). $$ The argument of $d$ is the Poisson Jacobiator $\mathrm{Jac}(f,g,h)$.

If $\{\cdot ,\cdot \}$ satisfies Jacobi, then $\mathrm{Jac}(f,g,h)=0$ for all $f,g,h$, so $J(df,dg,dh)=0$, and by the uniqueness of the Leibniz extension $J\equiv 0$ on ${\Omega }^1(M)$; the Koszul bracket is then a Lie bracket and the algebroid axioms hold. Conversely, suppose the Koszul bracket is a Lie bracket, so $J(df,dg,dh)=0$, hence $d\mathrm{Jac}(f,g,h)=0$ for all $f,g,h$. The Poisson Jacobiator is a derivation in each argument and vanishes whenever any argument is a constant; a derivation in each slot whose differential vanishes identically must itself vanish, because it is determined by its values on coordinate functions and those values are forced to zero by the derivation property together with $d\mathrm{Jac}=0$. Therefore $\mathrm{Jac}\equiv 0$ and $\{\cdot ,\cdot \}$ satisfies Jacobi, equivalently $[\pi ,\pi {]}_{\mathrm{SN}}=0$. $\square$

Bridge. This equivalence builds toward the slogan that Poisson geometry is a chapter of Lie algebroid theory, and it appears again in the symplectic-groupoid story, where the global object integrating $T^{\ast }M$ carries a symplectic form whose existence is governed by the same Jacobi condition. The foundational reason the sharp map is forced to be a Lie-algebra morphism into vector fields is exactly this theorem read through the anchor: ${\pi }^{♯}([df,dg])={\pi }^{♯}(d\{f,g\})=X_{\{f,g\}}=[X_f,X_g]$. Putting these together, the single condition $[\pi ,\pi {]}_{\mathrm{SN}}=0$ controls the bracket on functions, the bracket on one-forms, and the integrability of the manifold all at once.

Exercises Intermediate+

Lean formalization Intermediate+

lean_status: none — Mathlib has no Poisson bivector, no Schouten-Nijenhuis bracket on multivector fields, and so no expression of $[\pi ,\pi {]}_{\mathrm{SN}}=0$, nor any Lie-algebroid API in which to phrase the cotangent algebroid. The target statement is sketched below in pseudo-Lean to indicate the missing structure; it does not compile against current Mathlib, which is why no Lean module is declared.

-- Pseudo-Lean: target structure, not in Mathlib.
structure PoissonManifold (M : Type*) [SmoothManifold M] where
  bivector  : Section (Lambda2 (TangentBundle M))
  integrable : schoutenBracket bivector bivector = 0   -- [π,π]_SN = 0

-- The cotangent algebroid built from a PoissonManifold:
def cotangentAlgebroid (P : PoissonManifold M) :
    LieAlgebroid M (CotangentBundle M) where
  anchor  := sharp P.bivector                          -- π♯ : T*M → TM
  bracket := koszulBracket P.bivector                  -- [α,β] = L_{π♯α}β − L_{π♯β}α − d(π(α,β))
  -- Jacobi for koszulBracket ⇔ P.integrable (the load-bearing theorem)

The first genuine obstacle is the graded Lie algebra of multivector fields: Mathlib lacks the Schouten-Nijenhuis bracket, on which both the integrability clause and the well-definedness of the Koszul bracket depend.

Advanced results Master

The symplectic foliation as the orbit foliation. The image distribution $x\mapsto {\pi }^{♯}(T_x^{\ast }M)$ is spanned by the Hamiltonian vector fields $X_f$. Since the anchor of a Lie algebroid is a bracket homomorphism, $[X_f,X_g]=X_{\{f,g\}}$ lies in the span, so the distribution is involutive in the generalised sense. Its maximal integral manifolds are the symplectic leaves, and on each leaf $\pi$ restricts to a non-degenerate Poisson structure, equivalently the inverse of a symplectic form. The leaves are the orbits of the cotangent algebroid in the sense of 03.04.16; where the rank of ${\pi }^{♯}$ is locally constant they form a regular foliation, and across rank jumps a singular (Stefan-Sussmann) foliation ^{[Weinstein 1983]}. Weinstein's splitting theorem makes this local picture precise: near any point $M$ is locally a product of a symplectic factor of dimension equal to the rank of $\pi$ with a transverse Poisson factor whose bivector vanishes at the point.

The Lie-Poisson structure on ${\mathfrak{g}}^{\ast }$. Let $\mathfrak{g}$ be a finite-dimensional Lie algebra and ${\mathfrak{g}}^{\ast }$ its dual. Linear functions on ${\mathfrak{g}}^{\ast }$ are elements of $\mathfrak{g}$, and the Lie bracket of $\mathfrak{g}$ extends to a Poisson bracket by $\{\xi ,\eta \}(\mu )=\mu ([\xi ,\eta ])$ for $\xi ,\eta \in \mathfrak{g}$, $\mu \in {\mathfrak{g}}^{\ast }$. The cotangent algebroid $T^{\ast }{\mathfrak{g}}^{\ast }\cong {\mathfrak{g}}^{\ast }\times \mathfrak{g}$ is the action algebroid of the coadjoint action of $\mathfrak{g}$ on ${\mathfrak{g}}^{\ast }$: the anchor sends $(\mu ,\xi )$ to the infinitesimal coadjoint generator ${\mathrm{ad}}_{\xi }^{\ast }\mu$, and the Koszul bracket on constant sections is the Lie bracket of $\mathfrak{g}$. The symplectic leaves are exactly the coadjoint orbits 05.03.01, recovering the Kirillov-Kostant-Souriau symplectic structure. This is the transformation-algebroid description of Lie-Poisson geometry.

The symplectic manifold case. When $\pi$ is non-degenerate, $(M,\pi )$ is a symplectic manifold $(M,\omega )$ with $\omega ={\pi }^{-1}$, the sharp map ${\pi }^{♯}=({\omega }^{♭}{)}^{-1}$ is a bundle isomorphism, and the cotangent algebroid $T^{\ast }M$ is isomorphic to the tangent algebroid $TM$ via ${\pi }^{♯}$. There is a single symplectic leaf, all of $M$, and the Koszul bracket transports to the vector-field bracket. The cotangent algebroid thus interpolates between $TM$ (symplectic, full rank) and a bundle of abelian algebras (the zero Poisson structure, rank zero), with the rank of ${\pi }^{♯}$ measuring how far the structure sits from symplectic.

Integration to a symplectic groupoid. The Lie functor sends a Lie groupoid to its Lie algebroid; the converse question for the cotangent algebroid is the integrability of the Poisson manifold. When $T^{\ast }M$ integrates, it integrates to a special object: a symplectic groupoid $\Sigma \rightrightarrows M$, a Lie groupoid whose arrow space carries a symplectic form $\Omega$ multiplicative with respect to the groupoid composition (the graph of multiplication is Lagrangian in $\Sigma \times \Sigma \times \bar{\Sigma }$), and whose unit space $M$ inherits the original Poisson structure ^{[Weinstein 1987]}. The source-simply-connected such $\Sigma$, when it exists, is unique. Coste, Dazord, and Weinstein, and independently Karasev, constructed the local symplectic groupoid for every Poisson manifold and showed the obstruction to globalising lies in the same monodromy phenomenon that obstructs Lie-algebroid integrability ^{[Coste-Dazord-Weinstein 1987]}. Crainic and Fernandes gave the precise criterion: a Poisson manifold is integrable if and only if its cotangent algebroid integrates, if and only if the monodromy groups of $T^{\ast }M$ are locally uniformly discrete, the same condition as in the general integrability theorem 03.04.18 specialised to $A=T^{\ast }M$ ^{[Crainic-Fernandes 2004]}.

Synthesis. The cotangent algebroid is the foundational reason Poisson geometry is a special chapter of Lie algebroid theory: a Poisson structure is exactly a Lie-algebroid structure on $T^{\ast }M$ whose anchor ${\pi }^{♯}$ is the sharp map and whose bracket is the Koszul bracket, and the single condition $[\pi ,\pi {]}_{\mathrm{SN}}=0$ is what makes that structure a genuine algebroid. This is exactly the statement that the Jacobi identity for the Poisson bracket coincides with the Jacobi identity for the Koszul bracket, which putting these together with the bracket-homomorphism property of the anchor forces the Hamiltonian assignment $f\mapsto X_f$ to be a Lie-algebra morphism. The anchor image is the symplectic foliation, so the rank of ${\pi }^{♯}$ identifies the leaf dimension with the algebroid orbit dimension, and the cotangent algebroid generalises the tangent algebroid in the symplectic case while degenerating to a bundle of abelian Lie algebras for the zero Poisson structure. Globally, the cotangent algebroid is dual to a symplectic groupoid: integrating $T^{\ast }M$ produces the source-simply-connected symplectic groupoid $\Sigma \rightrightarrows M$, and the Crainic-Fernandes monodromy criterion specialised from 03.04.18 decides when that integration exists, so the entire integrability theory of Poisson manifolds is the integrability theory of one canonical Lie algebroid.

Full proof set Master

Proposition (Anchor recovers the Hamiltonian flow and the symplectic foliation is involutive). Let $(M,\pi )$ be a Poisson manifold with cotangent algebroid $(T^{\ast }M,[\cdot ,\cdot ],{\pi }^{♯})$. Then ${\pi }^{♯}(df)=X_f$, the distribution $D_x={\pi }^{♯}(T_x^{\ast }M)$ is spanned by Hamiltonian vector fields and is involutive, and its integral leaves carry a non-degenerate restriction of $\pi$.

Proof. By definition $\beta ({\pi }^{♯}(df))=\pi (df,\beta )=\{f,g\}$ when $\beta =dg$, so ${\pi }^{♯}(df)=X_f$ as a derivation. The values $\{{\pi }^{♯}(\alpha {)}_x:\alpha \in T_x^{\ast }M\}$ span $D_x$, and every such value is a value of some $X_f$ because exact forms span $T_x^{\ast }M$ pointwise. The anchor of a Lie algebroid is a bracket homomorphism 03.04.16, so $[X_f,X_g]={\pi }^{♯}([df,dg])={\pi }^{♯}(d\{f,g\})=X_{\{f,g\}}$ lies in the span of Hamiltonian fields, hence in $D$; thus $D$ is closed under the vector-field bracket of its spanning fields, i.e. involutive in the Stefan-Sussmann sense. On the integral leaf $L$ through $x$, $T_{x}L=D_x$, and ${\pi }_x$ induces a non-degenerate antisymmetric form on $D_x\cong T_x^{\ast }M/\ker {\pi }_x^{♯}$, so $\pi$ restricts to a non-degenerate Poisson structure on $L$, equivalently the inverse of a symplectic form ${\omega }_L$ on $L$. $\square$

Proposition (Lie-Poisson cotangent algebroid is the coadjoint action algebroid). Let $\mathfrak{g}$ be a finite-dimensional Lie algebra with Lie-Poisson structure on ${\mathfrak{g}}^{\ast }$. Then the cotangent algebroid $T^{\ast }{\mathfrak{g}}^{\ast }$ is isomorphic, as a Lie algebroid, to the action algebroid $\mathfrak{g}⋉{\mathfrak{g}}^{\ast }$ of the coadjoint action, and its symplectic leaves are the coadjoint orbits.

Proof. Identify $T^{\ast }{\mathfrak{g}}^{\ast }\cong {\mathfrak{g}}^{\ast }\times \mathfrak{g}$ using that ${\mathfrak{g}}^{\ast }$ is a vector space, so a covector at $\mu$ is an element of $({\mathfrak{g}}^{\ast }{)}^{\ast }=\mathfrak{g}$. Under this identification, the differential of the linear function $\xi \in \mathfrak{g}$ is the constant section $\mu \mapsto (\mu ,\xi )$. The sharp map sends $(\mu ,\xi )$ to $X_{\xi }(\mu )$, where for linear $\eta$, $X_{\xi }\eta (\mu )=\{\xi ,\eta \}(\mu )=\mu ([\xi ,\eta ])=-⟨{\mathrm{ad}}_{\xi }^{\ast }\mu ,\eta ⟩$; thus ${\pi }^{♯}(\mu ,\xi )=-{\mathrm{ad}}_{\xi }^{\ast }\mu$, the infinitesimal coadjoint generator, which is exactly the anchor of the coadjoint action algebroid. On constant sections $\xi ,\eta$ the Koszul bracket gives $[d\xi ,d\eta ]=d\{\xi ,\eta \}=d([\xi ,\eta {]}_{\mathfrak{g}})$, so the bracket of constant sections is the Lie bracket of $\mathfrak{g}$, matching the action-algebroid bracket. The two algebroids therefore agree, and the orbits of the action algebroid are the coadjoint orbits, which are consequently the symplectic leaves 05.03.01. $\square$

Proposition (Integrability of $(M,\pi )$ equals integrability of $T^{\ast }M$). A Poisson manifold $(M,\pi )$ admits a source-simply-connected symplectic groupoid $\Sigma \rightrightarrows M$ if and only if its cotangent Lie algebroid $T^{\ast }M$ is integrable as a Lie algebroid, and the symplectic groupoid is then the source-simply-connected integration of $T^{\ast }M$.

Proof. If $\Sigma \rightrightarrows M$ is a symplectic groupoid integrating $(M,\pi )$, its Lie algebroid $A(\Sigma )$ is canonically isomorphic to $T^{\ast }M$ with the cotangent algebroid structure: the multiplicative symplectic form $\Omega$ identifies $A(\Sigma )=\ker (ds){|}_M$ with $T^{\ast }M$ via $v\mapsto {\iota }_v\Omega {|}_{TM}$, and under this identification the algebroid anchor and bracket are ${\pi }^{♯}$ and the Koszul bracket ^{[Coste-Dazord-Weinstein 1987]}. Hence $T^{\ast }M$ is integrable, and the source-simply-connected $\Sigma$ realises its source-simply-connected integration (unique by Lie I for algebroids 03.04.18). Conversely, if $T^{\ast }M$ is integrable, its source-simply-connected integration $\Sigma =\mathcal{G}(T^{\ast }M)$ (the Weinstein groupoid of cotangent $A$-paths modulo homotopy) carries a canonical multiplicative symplectic form, constructed by Cattaneo-Felder as the symplectic reduction of an infinite-dimensional path space; thus $\Sigma$ is a symplectic groupoid integrating $(M,\pi )$. By the general criterion 03.04.18, $T^{\ast }M$ is integrable if and only if its monodromy groups are locally uniformly discrete, which is the Crainic-Fernandes Poisson-integrability criterion ^{[Crainic-Fernandes 2004]}. $\square$

Connections Master

A Lie algebroid 03.04.16 is the genus of which the cotangent algebroid is a species: the anchor-bracket-Leibniz data specialise here to $A=T^{\ast }M$, anchor ${\pi }^{♯}$, and the Koszul bracket. The bracket-homomorphism property of the general anchor becomes the statement $[X_f,X_g]=X_{\{f,g\}}$, and the orbit foliation of the general theory becomes the symplectic foliation. Every structural theorem about algebroid orbits applies verbatim to symplectic leaves.

The Poisson bracket and Poisson manifold 05.02.02 are the base datum: the Jacobi identity of the bracket is precisely the integrability $[\pi ,\pi {]}_{\mathrm{SN}}=0$ that makes the Koszul bracket a Lie bracket. The Hamiltonian vector field 05.02.01 is the anchor image of an exact one-form, ${\pi }^{♯}(df)=X_f$, so the dynamics generated by the Poisson bracket is the dynamics read off by the cotangent-algebroid anchor.

The integrability theory of Lie algebroids 03.04.18 specialises to the integrability of Poisson manifolds when $A=T^{\ast }M$: the monodromy criterion of Crainic-Fernandes decides exactly when $(M,\pi )$ integrates to a symplectic groupoid. The Weinstein-groupoid construction of cotangent $A$-paths modulo homotopy is the path-space model whose smoothness is governed by the same monodromy groups.

The coadjoint orbit 05.03.01 is the symplectic leaf of the Lie-Poisson structure on ${\mathfrak{g}}^{\ast }$, whose cotangent algebroid is the action algebroid of the coadjoint action. The Kirillov-Kostant-Souriau symplectic form on a coadjoint orbit is the leafwise symplectic form of this Poisson structure, tying the representation-theoretic orbit method to the algebroid orbit foliation.

The symplectic manifold 05.01.02 is the full-rank extreme: a non-degenerate $\pi$ makes ${\pi }^{♯}$ a bundle isomorphism $T^{\ast }M\stackrel{\sim }{\to }TM$, so the cotangent algebroid is isomorphic to the tangent algebroid and the unique leaf is all of $M$. The symplectic groupoid integrating it is the (source-simply-connected) fundamental groupoid carrying the difference of pulled-back symplectic forms.

Historical & philosophical context Master

The local structure of Poisson manifolds — the splitting theorem and the symplectic foliation — was established by Alan Weinstein in 1983, who showed that every Poisson manifold is locally a product of a symplectic factor and a transverse Poisson factor vanishing at the centre point ^{[Weinstein 1983]}. The recognition that $T^{\ast }M$ carries a Lie algebroid structure with anchor the sharp map and Koszul bracket emerged from this circle, the Koszul bracket itself going back to Jean-Louis Koszul's work on the Schouten-Nijenhuis calculus; the standard algebroid treatment is fixed in Mackenzie's monograph ^{[Mackenzie Ch. III §3]}.

The global integration was the deeper discovery. Weinstein introduced symplectic groupoids in 1987 as the global objects whose infinitesimal counterpart is a Poisson manifold, proposing them as the arena for a geometric quantisation of Poisson structures ^{[Weinstein 1987]}. Marius Karasev independently arrived at the same notion through the study of nonlinear Poisson brackets and asymptotic quantisation, and Pierre Dazord, Alan Weinstein, and collaborators constructed the local symplectic groupoid for an arbitrary Poisson manifold ^{[Coste-Dazord-Weinstein 1987]}. The obstruction to globalising the local groupoid remained open until Marius Crainic and Rui Loja Fernandes settled the general Lie-algebroid integrability problem in 2003 and applied it to the cotangent algebroid in 2004, giving the exact monodromy criterion for a Poisson manifold to admit a symplectic groupoid ^{[Crainic-Fernandes 2004]}.

Bibliography Master

@book{mackenzie2005,
  author    = {Mackenzie, Kirill C. H.},
  title     = {General Theory of Lie Groupoids and Lie Algebroids},
  series    = {London Mathematical Society Lecture Note Series},
  volume    = {213},
  publisher = {Cambridge University Press},
  year      = {2005}
}

@article{weinstein1983,
  author    = {Weinstein, Alan},
  title     = {The local structure of Poisson manifolds},
  journal   = {Journal of Differential Geometry},
  volume    = {18},
  number    = {3},
  pages     = {523--557},
  year      = {1983}
}

@article{weinstein1987,
  author    = {Weinstein, Alan},
  title     = {Symplectic groupoids and Poisson manifolds},
  journal   = {Bulletin of the American Mathematical Society},
  volume    = {16},
  number    = {1},
  pages     = {101--104},
  year      = {1987}
}

@article{cdw1987,
  author    = {Coste, Alberto and Dazord, Pierre and Weinstein, Alan},
  title     = {Groupo\"ides symplectiques},
  journal   = {Publications du D\'epartement de Math\'ematiques, Universit\'e Claude Bernard Lyon I},
  volume    = {2A},
  pages     = {1--62},
  year      = {1987}
}

@article{karasev1987,
  author    = {Karasev, Mikhail V.},
  title     = {Analogues of objects of the theory of Lie groups for nonlinear Poisson brackets},
  journal   = {Mathematics of the USSR-Izvestiya},
  volume    = {28},
  number    = {3},
  pages     = {497--527},
  year      = {1987}
}

@article{crainic-fernandes2004,
  author    = {Crainic, Marius and Fernandes, Rui Loja},
  title     = {Integrability of Poisson brackets},
  journal   = {Journal of Differential Geometry},
  volume    = {66},
  number    = {1},
  pages     = {71--137},
  year      = {2004}
}

@article{crainic-fernandes2003,
  author    = {Crainic, Marius and Fernandes, Rui Loja},
  title     = {Integrability of Lie brackets},
  journal   = {Annals of Mathematics},
  volume    = {157},
  number    = {2},
  pages     = {575--620},
  year      = {2003}
}

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Cotangent algebroid of a Poisson manifold — the Lie algebroid $T^{\ast }M$ with anchor the sharp map ${\pi }^{♯}$ and Koszul bracket on one-forms; its Jacobi identity is the Poisson Jacobi identity, its orbits are the symplectic leaves, and its global integration is the symplectic groupoid.