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Poisson manifold

A Poisson manifold is a smooth manifold M endowed with a field \pi \in \Gamma(\wedge^2 TM) that satisfies the integrability condition [\pi, \pi]_S = 0, where [\cdot, \cdot]_S denotes the Schouten-Nijenhuis bracket, thereby defining a \{f, g\} = \pi(df, dg) on the algebra of smooth functions C^\infty(M) that obeys the Leibniz rule and . The concept of Poisson manifolds originated in the context of during the , with foundational contributions from and on the in classical dynamics, later formalized by through the . The modern geometric framework was established by André Lichnerowicz in 1977, who defined Poisson structures via bivector fields and introduced the associated Lie algebras on function spaces, enabling the study of deformations and . Subsequent developments by Alan Weinstein in the early 1980s revealed the local structure, showing that every Poisson manifold admits a splitting into and transverse components near regular points. This theory bridges and , with applications in quantization, integrable systems, and . Key properties of Poisson manifolds include the induced Hamiltonian vector fields X_f = \pi^\sharp(df), where \pi^\sharp: T^*M \to TM is the bundle map associated to \pi, forming a Lie subalgebra of vector fields isomorphic to the . The rank of \pi, defined as the dimension of the image of \pi^\sharp, is even and constant along each symplectic leaf, leading to a canonical foliation by symplectic leaves—immersed submanifolds where the restriction of \pi is nondegenerate and induces a form. Examples range from symplectic manifolds (maximal rank case) to linear Poisson structures on dual Lie algebras \mathfrak{g}^*, whose symplectic leaves are coadjoint orbits, and zero structures on arbitrary manifolds. Poisson manifolds are integrable via symplectic realizations, surjective Poisson submersions from symplectic manifolds that recover the original structure, and symplectic groupoids, which provide a categorical integration encoding the transverse geometry. Poisson cohomology, defined using the Lichnerowicz differential d_\pi on multivector fields, classifies invariants such as the modular class and obstructions to quantization. These structures generalize classical mechanics to singular settings, with ongoing research in Dirac geometry, homotopy theory, and applications to mathematical physics.

Introduction

Motivation from classical mechanics

In classical mechanics, the phase space of a system with n degrees of freedom is modeled as the cotangent bundle T^*Q of a configuration manifold Q, equipped with a canonical symplectic structure given by the closed, non-degenerate 2-form \omega = dq^i \wedge dp_i, where q^i are coordinates on Q and p_i are the corresponding momentum coordinates. This symplectic form defines Hamiltonian vector fields: for a smooth function H: T^*Q \to \mathbb{R} (the Hamiltonian), the associated vector field X_H satisfies \iota_{X_H} \omega = -dH, ensuring that the flow of X_H preserves the symplectic structure and generates the time evolution of the system via Hamilton's equations \dot{q}^i = \frac{\partial H}{\partial p_i} and \dot{p}_i = -\frac{\partial H}{\partial q^i}. The dynamics on this phase space are fundamentally encoded by the Poisson bracket \{f, g\} on smooth functions C^\infty(T^*Q), defined coordinate-wise as \{f, g\} = \frac{\partial f}{\partial q^i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q^i}. This bracket is bilinear in its arguments, skew-symmetric (\{g, f\} = -\{f, g\}), satisfies the Jacobi identity (\{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0), and obeys the Leibniz rule (\{f, gh\} = g\{f, h\} + h\{f, g\}). The time derivative of any observable f is then given by \frac{df}{dt} = \{f, H\} + \frac{\partial f}{\partial t}, linking the bracket directly to the equations of motion. Poisson manifolds generalize this framework to phase spaces where the Poisson bracket may degenerate, meaning the associated bivector field is not invertible everywhere, allowing for singular behaviors in mechanical systems such as constraints or reductions (e.g., in the limit of vanishing mass in celestial mechanics). In such cases, the phase space foliates into lower-dimensional symplectic leaves, where the dynamics restrict to Hamiltonian flows on each leaf, extending the classical setup beyond non-degenerate symplectic manifolds. To derive the Poisson bracket explicitly on cotangent bundles, start from the symplectic form \omega = dq^i \wedge dp_i, whose inverse (the Poisson tensor) yields the coordinate expression \{q^i, p_j\} = \delta^i_j, \{q^i, q^j\} = 0, and \{p_i, p_j\} = 0; extending by bilinearity and Leibniz rule to general functions gives the full bracket as above. manifolds correspond to the special non-degenerate case of manifolds.

Historical overview

The origins of Poisson geometry trace back to the early 19th century in the context of , where introduced Poisson brackets as a tool to integrate for planetary perturbations in . These brackets, developed alongside contributions from , facilitated the study of Hamiltonian systems and integrals of motion on phase spaces modeled as \mathbb{R}^{2n}. In the 1830s, rediscovered and formalized Poisson brackets, establishing their key properties including the Leibniz rule and , which linked them to symmetries in differential equations. By the late 19th and early 20th centuries, extended these ideas through his work on Lie groups and algebras, connecting linear Poisson structures to contact transformations and infinitesimal symmetries in Hamiltonian dynamics. Élie Cartan further advanced during this period by developing the theory of differential forms and exterior derivatives, providing a foundational framework for phase spaces that influenced later generalizations. The modern theory of Poisson manifolds emerged in the mid-20th century, building on rediscoveries of structures on coadjoint orbits by Alexandre Kirillov, , and Jean-Marie Souriau in the 1960s. These works highlighted Poisson structures on dual spaces, known as Lie-Poisson structures, which and explored in the 1970s to connect with . The systematic study of Poisson manifolds as geometric objects began with André Lichnerowicz in 1977, who defined Poisson structures via fields on manifolds and introduced associated Lie algebras and cohomology, emphasizing their role beyond cases. Alan Weinstein's foundational 1983 paper then established key structural theorems, including the foliation and local normal forms, solidifying the perspective and linking Poisson geometry to theory. Subsequent advancements in the 1990s and 2000s focused on integrability and global aspects, with Marius Crainic and Rui Loja Fernandes proving in 2003 that every Poisson structure integrates to a groupoid under mild conditions, resolving long-standing questions via algebroids and monodromy obstructions. Their work extended Weinstein's local results to global realizations, enabling classifications of integrable manifolds. In the 2010s, extensions like log- structures emerged, generalizing geometry to include logarithmic singularities on manifolds, as developed by Victor Guillemin, Eva Miranda, and Ana Rita Pires to model systems with degenerate leaves. Contemporary developments have applied Poisson manifolds to , particularly through Poisson sigma models coupled to topological backgrounds, which describe 2D gravity and flux compactifications in superstring theories. These applications, explored since the early , connect Poisson-Lie T-duality to non-perturbative dualities in string backgrounds. More recently, in the , the theory of Poisson manifolds of compact types has emerged, providing a broad generalization of compact structures in Poisson and Dirac .

Formal definitions

Definition via Poisson bracket

A Poisson manifold is a smooth manifold M equipped with a bilinear map \{\cdot, \cdot\}: C^\infty(M) \times C^\infty(M) \to C^\infty(M), called the Poisson bracket, that satisfies the following axioms for all f, g, h \in C^\infty(M): skew-symmetry \{f, g\} = -\{g, f\}, the Jacobi identity \{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0, and the Leibniz rule \{f, gh\} = \{f, g\} h + g \{f, h\}. The Poisson bracket induces a derivation on the algebra of smooth functions, allowing the definition of Hamiltonian vector fields. For each f \in C^\infty(M), there exists a unique vector field X_f on M such that \{f, g\} = X_f(g) for all g \in C^\infty(M). These Hamiltonian vector fields form a Lie subalgebra of the Lie algebra of all vector fields on M under the Lie bracket [\cdot, \cdot], with the Lie algebra structure given by [X_f, X_g] = X_{\{f, g\}}. In local coordinates (x^i) on an open set U \subset M, the Poisson bracket is determined by its values on the coordinate functions, \{x^i, x^j\} = \pi^{ij}(x), where the smooth functions \pi^{ij}: U \to \mathbb{R} are the structure functions satisfying \pi^{ij} = -\pi^{ji}. For general smooth functions f, g: U \to \mathbb{R}, the bracket takes the form \{f, g\}(x) = \sum_{i,j=1}^n \pi^{ij}(x) \frac{\partial f}{\partial x^i}(x) \frac{\partial g}{\partial x^j}(x). The Poisson bracket further induces bundle maps between the cotangent and tangent bundles of M. The sharp map \sharp: T^*M \to TM is defined pointwise by \sharp(\xi)(\eta) = \pi(\xi, \eta) for \xi, \eta \in T^*_x M, or equivalently \sharp(df) = X_f for f \in C^\infty(M); its adjoint, the flat map \flat: TM \to T^*M, satisfies \flat(X)(\xi) = \langle X, \sharp(\xi) \rangle. This algebraic structure on functions admits a dual geometric representation via a bivector field on M.

Definition via bivector field

A Poisson manifold (M, \pi) is defined as a smooth manifold M equipped with a bivector field \pi \in \Gamma(\wedge^2 TM), which is a section of the second exterior power of the TM, satisfying the integrability condition [\pi, \pi]_S = 0. Here, [\cdot, \cdot]_S denotes the Schouten-Nijenhuis bracket, an extension of the to fields that ensures the induced structure on functions is a . This geometric definition, introduced by Lichnerowicz, captures the Poisson structure through a contravariant skew-symmetric tensor that generalizes the form in a possibly degenerate manner. In local coordinates (x^1, \dots, x^n) on M, the bivector takes the form \pi = \frac{1}{2} \sum_{i,j=1}^n \pi^{ij}(x) \frac{\partial}{\partial x^i} \wedge \frac{\partial}{\partial x^j}, where the components \pi^{ij} = -\pi^{ji} are smooth real-valued functions on M, defining the Poisson tensor. The condition [\pi, \pi]_S = 0 manifests in these coordinates as the vanishing of the resulting trivector field, yielding the partial differential \sum_l \left( \pi^{il} \frac{\partial \pi^{jk}}{\partial x^l} + \pi^{jl} \frac{\partial \pi^{ki}}{\partial x^l} + \pi^{kl} \frac{\partial \pi^{ij}}{\partial x^l} \right) = 0 for all i,j,k, which is the coordinate expression of the enforced via the Schouten bracket. This local condition guarantees the global consistency of the Poisson structure across coordinate charts. The bivector \pi induces a bundle morphism \pi^\sharp: T^*M \to TM defined by \eta(\pi^\sharp(\alpha)) = \pi(\alpha, \eta) for all 1-forms \alpha, \eta. More directly, for a smooth function f \in C^\infty(M), the associated Hamiltonian vector field is X_f = \pi^\sharp(df), with df \in \Gamma(T^*M) the differential of f. This map \pi^\sharp, often called the Poisson tensor, encodes the entire structure by associating 1-forms to vector fields and defines the Poisson bracket as \{f, g\} = \pi(df, dg) = X_f(g). The image of \pi^\sharp determines the directions in which the manifold admits Hamiltonian flows, highlighting the role of \pi in foliating M into symplectic leaves.

Equivalence of the definitions

To establish the equivalence between the definition of a via a and via a , consider first the construction from the bracket to the . On a manifold M, suppose \{\cdot, \cdot\}: C^\infty(M) \times C^\infty(M) \to C^\infty(M) is a , satisfying skew-symmetry, the Leibniz rule, and the . In local coordinates (x^i) around a point, the components of the associated field \pi \in \Gamma(\wedge^2 TM) are defined by \pi^{ij} = \{x^i, x^j\}. This \pi is and skew-symmetric because the bracket is, and the Leibniz rule ensures that \pi acts as a on functions, yielding a genuine field. Furthermore, the for the bracket implies that the Schouten-Nijenhuis vanishes: [\pi, \pi]_S = 0. Specifically, the component form of [\pi, \pi]_S involves cyclic sums over the Jacobiator \{\{x^i, x^j\}, x^k\} + \{\{x^j, x^k\}, x^i\} + \{\{x^k, x^i\}, x^j\} = 0, confirming \pi defines a . Conversely, start with a bivector field \pi \in \Gamma(\wedge^2 TM) satisfying [\pi, \pi]_S = 0. Define a bracket on functions by \{f, g\} = \pi(df, dg), where df, dg \in \Gamma(T^*M) are the differentials. This bracket is bilinear over \mathbb{R} and skew-symmetric due to the skew-symmetry of \pi. The Leibniz rule follows directly: \{f, fg'\} = f \{g, g'\} + g' \{f, g\}, as \pi acts as a derivation in each argument via its contraction with 1-forms. The Jacobi identity holds because [\pi, \pi]_S = 0 encodes precisely the condition \{\{f, g\}, h\} + \{\{g, h\}, f\} + \{\{h, f\}, g\} = 0 for all smooth functions f, g, h, via the properties of the extended to multivectors. These constructions are inverses, establishing a one-to-one correspondence locally in coordinate charts. Applying the bracket-to-bivector to \{f, g\} = \pi(df, dg) recovers \pi, as the components match by . Similarly, the bivector-to-bracket applied to the \pi from \{x^i, x^j\} yields the original bracket on coordinate functions, hence on all functions by bilinearity and Leibniz. This local equivalence extends globally on paracompact manifolds, where smooth partitions of unity allow gluing of local expressions without ambiguity, yielding the same Poisson structure independent of choices.

Holomorphic Poisson structures

A holomorphic Poisson structure on a M is defined by a holomorphic field \pi \in \Gamma(\wedge^2 T^{1,0}M) satisfying the integrability condition [\pi, \pi]_S = 0, where [\cdot, \cdot]_S denotes the Schouten-Nijenhuis bracket in the holomorphic category. This condition ensures that the associated bundle map \pi^\sharp: T^*M \to T^{1,0}M () endows the with a Lie algebroid structure. Equivalently, a holomorphic Poisson structure corresponds to a Poisson bracket \{\cdot, \cdot\}: \mathcal{O}(M) \times \mathcal{O}(M) \to \mathcal{O}(M) on the sheaf of holomorphic functions, which is bilinear, skew-symmetric, and satisfies the Jacobi identity \{f, \{g, h\}\} + \{g, \{h, f\}\} + \{h, \{f, g\}\} = 0 for all f, g, h \in \mathcal{O}(M). The bivector \pi induces this bracket via \{f, g\} = \pi(df, dg), where df, dg are holomorphic 1-forms, and the equivalence follows from the fact that any such bracket arises from a unique holomorphic bivector satisfying the Schouten condition. Such structures relate to real Poisson manifolds through a compatible almost complex structure J on the underlying real manifold, where the real and imaginary parts of \pi, denoted \pi_R and \pi_I, satisfy \pi_R = \pi_I^\sharp \circ J^*, forming a Poisson-Nijenhuis structure. This connection allows holomorphic Poisson manifolds to be viewed as a complexification of certain real Poisson geometries, preserving the symplectic foliation in the integrable case. Examples include holomorphic symplectic manifolds, where a non-degenerate holomorphic 2-form \omega \in \Gamma(\wedge^2 T^*M) admits a holomorphic \pi as its , \pi^\sharp = -\omega^{-1}, satisfying the required conditions automatically due to d\omega = 0. The dual of such a structure on a , for instance, yields a whose real part aligns with the .

Symplectic foliation

The rank function

In a Poisson manifold (M, \pi), the rank function provides a local measure of the nondegeneracy of the Poisson structure at each point x \in M. Specifically, the of \pi at x, denoted \operatorname{rank}(\pi)_x, is defined as the of the bundle map \pi_x^\sharp: T_x^* M \to T_x M induced by the Poisson \pi, where \pi_x^\sharp(\alpha) = i_\alpha \pi_x for \alpha \in T_x^* M. This map associates to each covector its with the bivector, and its equals the dimension of the image \operatorname{im}(\pi_x^\sharp). The function is lower semicontinuous and takes only even values at every point, owing to the skew-symmetry of \pi_x^\sharp, which implies that the kernel and image have complementary dimensions in a manner preserving even dimensionality. Moreover, \operatorname{rank}(\pi)_x remains constant along the connected components of the leaves of the induced by \pi. In local coordinates (x^i) around x, the Poisson bivector \pi is represented by a skew-symmetric matrix \pi^{ij}(x), and \operatorname{rank}(\pi)_x coincides with the rank of this matrix. This matrix rank is even, and it relates to the of \pi^{ij}, which is the of the \det(\pi^{ij}) up to sign; the vanishing of the signals degeneracy, while its non-vanishing corresponds to maximal rank in the case. The image \operatorname{im}(\pi^\sharp) defines the characteristic distribution of the Poisson structure, given by \operatorname{im}(\pi^\sharp) = \operatorname{span}\{X_f \mid f \in C^\infty(M)\}, where X_f = \pi^\sharp(df) denotes the associated to f. The of this distribution at x equals \operatorname{rank}(\pi)_x, capturing the span of all Hamiltonian directions tangent to the symplectic leaves. Functions central to the Poisson algebra are the Casimir functions, which are smooth functions f \in C^\infty(M) such that \{f, g\}_\pi = 0 for all g \in C^\infty(M). Equivalently, X_f = 0, meaning df \in \ker(\pi^\sharp_x) at every x \in M. The space of Casimir functions is the center of the (C^\infty(M), \{\cdot, \cdot\}_\pi), and their level sets contain the symplectic leaves, reflecting the degeneracy encoded by the kernel of \pi^\sharp. In the dual picture, \ker(\pi^\sharp_x) has dimension \dim M - \operatorname{rank}(\pi)_x, providing a measure of the "corank" or extent of integrability by Casimirs.

Regular Poisson manifolds

A regular Poisson manifold is defined as a Poisson manifold (M, \pi) where the rank of the Poisson bivector field \pi is constant on M. This constancy of the rank function ensures that the characteristic distribution \operatorname{im}(\pi^\sharp), spanned by the Hamiltonian vector fields X_f = \pi^\sharp(df) for smooth functions f \in C^\infty(M), has constant dimension equal to the rank of \pi. The involutivity of this distribution follows from the of the , implying that [\pi, \pi]_S = 0 where [\cdot, \cdot]_S is the Schouten-Nijenhuis bracket. By the Frobenius theorem, the distribution is therefore integrable, yielding a regular of M whose leaves are the connected integral manifolds of \operatorname{im}(\pi^\sharp). These leaves, known as leaves, inherit a structure from \pi: on each leaf \mathcal{L}, the restriction \pi|_\mathcal{L} is invertible, defining a form \omega_\mathcal{L} such that \pi|_\mathcal{L} = (\omega_\mathcal{L})^{-1} in the sense that \pi^\sharp|_\mathcal{L} = -(\omega_\mathcal{L})^\flat. The dimension of each leaf equals the constant rank of \pi. Locally, near any point p \in M, the Weinstein splitting theorem provides canonical coordinates (x^1, \dots, x^k, y^1, \dots, y^k, z^1, \dots, z^{n-2k}) adapted to the , where k is half the rank of \pi and n = \dim M, such that \pi = \sum_{i=1}^k \frac{\partial}{\partial x^i} \wedge \frac{\partial}{\partial y^i}. In these coordinates, the leaves are the submanifolds defined by constant values of the transverse coordinates z^a, which are functions constant along the leaves, and the induced form on each leaf \{z = c\} is \omega = \sum_{i=1}^k \mathrm{d}x^i \wedge \mathrm{d}y^i. Globally, the symplectic foliation endows the regular Poisson manifold with the structure of a , where the fibers are the symplectic leaves (each a of dimension equal to the rank) over the base formed by the M / \mathcal{F}, parameterized by the independent Casimir functions. This bundle perspective highlights how the Poisson structure generalizes by allowing a transverse variation controlled by the Casimirs.

Singular Poisson manifolds

In singular Poisson manifolds, the rank of the Poisson bivector \pi varies across the manifold, resulting in a singular symplectic foliation where the dimensions of the leaves differ. Unlike regular cases with constant rank, the characteristic distribution \Delta = \operatorname{Im} \pi^\sharp is involutive but singular, integrating to a partition of the manifold into symplectic leaves of varying even dimensions equal to the local rank of \pi. The orbit theorem asserts that the symplectic leaves are precisely the orbits generated by the flows of all Hamiltonian vector fields, which span the characteristic distribution \Delta. Each leaf L through a point x \in M is a connected immersed with \dim L = \operatorname{rank}_\pi(x), equipped with an induced form \omega_L that makes L a . These leaves are maximal integral submanifolds of \Delta, but they are not necessarily embedded, as the leaf space may fail to be Hausdorff. Singularities arise at points where the rank of \pi drops below its generic value, leading to leaves of lower dimension or, in the case of rank zero, fixed points where \pi_x = 0 and the \Delta_x = \{0\}. At such points, the flows trivialize, resulting in zero-dimensional leaves that are isolated fixed points. functions, which Poisson-commute with all smooth functions and thus belong to the center of the , remain constant along every leaf. The center foliation, formed by the level sets of these functions, provides a coarser transverse to the foliation; in regions of minimal rank zero, these level sets contain the fixed-point singular leaves as their connected components. The Weinstein splitting describes the local normal form near a point in a singular manifold as a product of a and a transverse structure, highlighting the stratified nature of the .

Weinstein splitting theorem

The Weinstein splitting provides a local normal form for structures on a manifold near points where the is constant. Specifically, let (M, \pi) be a manifold of n, and let p \in M be a point where the of \pi is constant and equal to $2kin a neighborhood ofp. Then there exist local coordinates (x^1, \dots, x^k, y^1, \dots, y^k, z^1, \dots, z^{n-2k})aroundp$ such that the bivector takes the form \pi = \sum_{i=1}^k \frac{\partial}{\partial x^i} \wedge \frac{\partial}{\partial y^i}. In these coordinates, the bracket satisfies \{x^i, y^j\} = \delta^i_j, \{x^i, x^j\} = \{y^i, y^j\} = 0, and \{z^\alpha, x^i\} = \{z^\alpha, y^i\} = 0 for all i,j=1,\dots,k and \alpha=1,\dots,n-2k, with the brackets among the z^\alpha vanishing at p. A proof sketch proceeds by first noting that the distribution generated by Hamiltonian vector fields X_f = \pi^\sharp(\mathrm{d}f) spans the to the leaf through p, which has dimension $2k and constant rank. Since these fields commute along the [leaf](/page/Leaf) (due to the integrability of the [distribution](/page/Distribution)), the flowbox [theorem](/page/Theorem) applies to straighten a set of $2k independent Hamiltonian fields into fields \partial/\partial x^i and \partial/\partial y^i, inducing a that aligns the with the (x,y)-subspace. The remaining z^\alpha coordinates are transverse to the , and the vanishing of mixed brackets follows from the 's structure and the constancy of rank, ensuring the Poisson structure splits canonically. This local splitting implies that near such a regular point, the Poisson manifold decomposes as a product of a (on the (x,y)-directions, isomorphic to a standard symplectic \mathbb{R}^{2k}) and transverse directions parametrized by functions (the z^\alpha, which Poisson-commute with everything at p). The transverse structure inherits a Poisson of lower rank, reflecting the foliation's regularity. Extensions to singular points, where the rank varies, rely on the rank stratification of the manifold into open strata of constant rank. On each stratum of rank $2k$, the theorem applies locally as above; the strata glue together via the continuity of the Poisson bivector, yielding a piecewise splitting where transverse structures may carry induced Poisson geometries of varying rank. This stratification-based approach accommodates singularities without assuming global regularity.

Examples

Trivial Poisson structures

A trivial Poisson structure on a manifold M is defined by the zero bivector field \pi = 0, which yields the vanishing bracket \{f, g\} = 0 for all functions f, g \in C^\infty(M). This structure endows every manifold with a Poisson manifold geometry where no nontrivial Hamiltonian dynamics arise, as all Hamiltonian vector fields vanish. In the trivial case, every smooth function on M qualifies as a function, since \{f, g\} = 0 holds for all g \in C^\infty(M), making the center of the Poisson algebra the entire C^\infty(M). The rank function of this structure is identically zero, implying that the associated symplectic foliation decomposes M into zero-dimensional leaves—namely, the discrete set of points comprising M itself. The Poisson algebra (C^\infty(M), \{\cdot, \cdot\}) under the trivial bracket forms an abelian Lie algebra, as the bracket satisfies the Jacobi identity trivially but induces no noncommutativity. Such structures commonly arise as the transverse Poisson geometry in local splittings of more general Poisson manifolds, particularly along submanifolds transverse to symplectic leaves of minimal rank, as described by the Weinstein splitting theorem. They also emerge in limits of degenerating families of bivector fields, where the Poisson structure flattens to zero through continuous deformation.

Symplectic Poisson structures

A symplectic Poisson structure on a smooth manifold M of even dimension $2n is defined by a field \pi whose rank equals \dim(M) at every point, ensuring that the bundle map \pi^\sharp: T^*M \to TM given by \pi^\sharp(\alpha) = i_\alpha \pi is a . This maximal rank condition implies that \pi is invertible, establishing a one-to-one correspondence between such nondegenerate structures and structures on M. The inverse of \pi, denoted \omega = \pi^{-1}, is a nondegenerate 2-form on M, and the Jacobi identity for the Poisson bracket—equivalently, [\pi, \pi]_S = 0, where [ \cdot, \cdot ]_S is the Schouten-Nijenhuis bracket—guarantees that \omega is closed, i.e., d\omega = 0. Thus, a manifold equipped with a symplectic Poisson structure is precisely a (M, \omega), where the Poisson bivector recovers the symplectic structure via \pi = \omega^{-1}. In this case, the symplectic leaves of the Poisson structure coincide with the entire manifold M, as the characteristic distribution \pi(T^*M) spans the full everywhere. By the Darboux theorem for manifolds, around any point p \in M, there exist local coordinates (x^1, \dots, x^n, y^1, \dots, y^n) such that the form takes the standard expression \omega = \sum_{i=1}^n dx^i \wedge dy^i. The dual in these coordinates is then \pi = \sum_{i=1}^n \frac{\partial}{\partial x^i} \wedge \frac{\partial}{\partial y^i}, providing a canonical local normal form for structures. This normal form underscores the equivalence, as the induced by \pi yields the standard relations \{x^i, y^j\} = \delta^i_j and \{x^i, x^j\} = \{y^i, y^j\} = 0.

Linear Poisson structures

A linear Poisson structure, also known as the Lie-Poisson structure, arises naturally on the \mathfrak{g}^* of a finite-dimensional \mathfrak{g} over \mathbb{R} or \mathbb{C}. This structure endows \mathfrak{g}^* with a Poisson manifold geometry where the Poisson field \pi is linear in the coordinates of \mathfrak{g}^*. If \{e_i\} is a basis for \mathfrak{g} with Lie bracket defined by the structure constants [e_i, e_j] = \sum_k c_{ij}^k e_k, and \{\xi^i\} is the dual basis for \mathfrak{g}^*, the bivector takes the form \pi = \frac{1}{2} \sum_{i,j,k} c_{ij}^k \xi^k \frac{\partial}{\partial \xi^i} \wedge \frac{\partial}{\partial \xi^j}. The associated Poisson bracket on smooth functions f, g \in C^\infty(\mathfrak{g}^*) is then \{f, g\}(\xi) = \left\langle \xi, \left[ \frac{\partial f}{\partial \xi}(\xi), \frac{\partial g}{\partial \xi}(\xi) \right]_\mathfrak{g} \right\rangle, where \frac{\partial f}{\partial \xi}(\xi) \in \mathfrak{g} denotes the gradient of f at \xi \in \mathfrak{g}^*, identified via the duality pairing \langle \cdot, \cdot \rangle: \mathfrak{g}^* \times \mathfrak{g} \to \mathbb{R}. For linear functions \alpha, \beta \in \mathfrak{g}, this reduces to \{\langle \xi, \alpha \rangle, \langle \xi, \beta \rangle\}(\xi) = \langle \xi, [\alpha, \beta]_\mathfrak{g} \rangle. The symplectic leaves of this Poisson structure are the coadjoint orbits of \mathfrak{g}, which are the orbits under the coadjoint action \mathrm{Ad}^*_g \xi = ( \mathrm{Ad}_{g^{-1}} )^* \xi for g \in G, where G is the simply connected Lie group integrating \mathfrak{g}. Each such orbit \mathcal{O}_\xi through \xi \in \mathfrak{g}^* inherits a canonical symplectic structure from the Kirillov-Kostant-Souriau (KKS) form, defined for tangent vectors X_{\xi}, Y_{\xi} \in T_\xi \mathcal{O}_\xi (arising from Lie algebra elements X, Y \in \mathfrak{g}) by \omega_\xi (X_\xi, Y_\xi) = -\langle \xi, [X, Y]_\mathfrak{g} \rangle. This form is G-invariant, nondegenerate on \mathcal{O}_\xi, and induces the Poisson structure restricted to the leaf. Representative examples illustrate the geometry. For an abelian Lie algebra \mathfrak{g} = \mathbb{R}^n with zero bracket, the structure constants vanish, yielding the trivial Poisson structure \pi = 0; the entire space \mathfrak{g}^* \cong \mathbb{R}^n forms a single symplectic leaf with the zero symplectic form. For the Lie algebra \mathfrak{su}(2) \cong \mathbb{R}^3 with basis elements satisfying the cross-product bracket (structure constants corresponding to the Levi-Civita symbol), the coadjoint orbits are concentric 2-spheres \mathcal{O}_\xi = \{ \eta \in \mathbb{R}^3 \mid \|\eta\| = \|\xi\| \}, each equipped with the KKS form as the standard area (symplectic) form scaled by the radius; the origin is a fixed point orbit. These linear structures extend to Poisson-Lie group structures on the corresponding Lie groups.

Other constructions

Log-symplectic manifolds provide a class of singular manifolds where the bivector \Pi on a $2n-dimensional manifold M satisfies the condition that \Pi^n is transverse to the zero section of \bigwedge^{2n} T^*M, making the singular locus Z = (\Pi^n)^{-1}(0) a smooth codimension-one along which \Pi degenerates linearly. Outside Z, the structure is , and Z itself carries a corank-one structure, rendering it a . The modular class of such a structure is represented by a modular tangent to Z and transverse to its leaves. A local normal form near Z is given by \Pi = y_1 \frac{\partial}{\partial x_1} \wedge \frac{\partial}{\partial y_1} + \sum_{i=2}^n \frac{\partial}{\partial x_i} \wedge \frac{\partial}{\partial y_i}, where Z is locally \{y_1 = 0\}. On surfaces (n=1), log-symplectic structures simplify to Poisson bivectors that vanish transversally along a Z, with a representative example being the structure on S^2 given by \Pi = z \partial_\theta \wedge \partial_z in cylindrical coordinates, where the singular locus is the \{z=0\} \cong S^1. More generally, on a surface, such a bivector can take the form \pi = d \log |f| \wedge X for a non-vanishing f defining the singular and a transverse X, ensuring the linear degeneration along Z = \{f=0\}. Fibrewise linear Poisson structures arise on vector bundles, where the Poisson restricts to a linear Poisson structure on each fiber, meaning the induced bracket on fiberwise functions is graded and corresponds to a algebroid structure on the dual bundle. A prominent example occurs on the T^*Q of a manifold Q equipped with its own Poisson structure \pi_Q: the combined bivector \pi = \pi_Q^\sharp + \sum_i \frac{\partial}{\partial q^i} \wedge \frac{\partial}{\partial p_i} (where \pi_Q^\sharp denotes the appropriate horizontal lift of \pi_Q to act on momentum coordinates) yields a fiberwise linear Poisson structure, with the canonical term providing the linear form on fibers and \pi_Q^\sharp inducing the base dynamics. This construction preserves the zero section as a Poisson isomorphic to (Q, \pi_Q) and ensures that Hamiltonian vector fields for fiberwise linear functions are fiberwise linear. Almost Poisson structures generalize true Poisson bivectors by allowing a small deviation in the , specifically where the Schouten-Nijenhuis bracket [\pi, \pi]_S is a small trivector, enabling approximations of exact Poisson structures in and deformation contexts. Such structures are useful for studying and local normal forms near Poisson manifolds, as small [\pi, \pi]_S implies the existence of nearby true Poisson bivectors via methods. In and related domains, notable examples include Poisson spheres, such as the log-symplectic structure on S^2 mentioned above, which models degenerate cases in integrable systems, and quasi-Poisson manifolds, which are G-manifolds equipped with a G-invariant bivector \pi satisfying [\pi, \pi]_S = \phi, where \phi is the G-invariant trivector generated by the group's coadjoint action. Quasi-Poisson structures extend geometry to momentum maps and actions, facilitating the study of representations of compact groups on manifolds like varieties.

Cohomology and homology

Poisson cohomology

The Poisson cohomology of a Poisson manifold (M, \pi) is defined as the cohomology of the cochain complex (\wedge^\bullet TM, \delta_\pi), where \wedge^\bullet TM denotes the of smooth fields on M and the differential \delta_\pi : \wedge^k TM \to \wedge^{k+1} TM is given by \delta_\pi(\alpha) = [\pi, \alpha]_S for \alpha \in \wedge^k TM, with [\cdot, \cdot]_S the Schouten-Nijenhuis bracket. This differential satisfies \delta_\pi^2 = 0 due to the for the Schouten bracket induced by \pi, and the cohomology groups are H^k(M, \pi) = \ker \delta_\pi / \operatorname{im} \delta_\pi for each degree k \geq 0. The zeroth cohomology H^0(M, \pi) consists of the functions, which are the centers of the C^\infty(M), while higher-degree groups capture obstructions and extensions in Poisson geometry. The complex \wedge^\bullet TM carries a natural Gerstenhaber algebra , with the graded-commutative associative product given by the wedge product \wedge and the graded Lie bracket [\cdot, \cdot]_S of degree -1. The \delta_\pi is a of square zero with respect to both operations, preserving the Gerstenhaber relations and inducing a compatible on the H^\bullet(M, \pi). This algebraic extends the classical to the Poisson setting, where the \pi plays the role of a "central" element generating the via action. The second Poisson cohomology group H^2(M, \pi) classifies equivalence classes of infinitesimal deformations of the Poisson bivector \pi, where a deformation is a bivector \pi + \epsilon \beta satisfying the Maurer-Cartan equation [\pi + \epsilon \beta, \pi + \epsilon \beta]_S = O(\epsilon^2), modulo inner derivations by Hamiltonian vector fields. Equivalences between deformations are governed by H^1(M, \pi), which parametrizes infinitesimal automorphisms of \pi. This cohomology admits an interpretation via Lie-Rinehart structures, where the Poisson bivector induces a bicrossproduct combining the Lie algebroid (T^*M, [\cdot, \cdot]_\pi, \pi^\sharp) on the —with Koszul bracket [\cdot, \cdot]_\pi on 1-forms and anchor \pi^\sharp : T^*M \to TM—and the of vector fields \operatorname{Ham}(M) = \pi^\sharp(C^\infty(M)) \subset \Gamma(TM). In this view, the Poisson cohomology computes extensions and derivations in the associated Lie-Rinehart algebra (C^\infty(M), \Gamma(TM)). Lichnerowicz's theorem states that if H^1(M, \pi) = 0, then the Poisson structure \pi exhibits formal rigidity in certain analytic or settings, meaning any formal deformation of \pi is equivalent to the trivial one via a formal . This vanishing condition eliminates non-trivial automorphisms, ensuring uniqueness of the structure up to equivalence in the formal .

Modular class

The modular class of a Poisson manifold (M, \pi) is an element of the first Poisson cohomology group H^1(M; \pi). It is represented by the modular vector field X_\mu, defined relative to a nowhere-vanishing \mu \in \Omega^n(M) (where n = \dim M) by X_\mu(f) = \div_\mu(X_f) for any smooth function f \in C^\infty(M), with X_f denoting the associated to f. Here, \div_\mu(Y) = \frac{\mathcal{L}_Y \mu}{\mu} is the of the vector field Y with respect to \mu, and since X_f is tangent to the symplectic leaves of the induced by \pi, the divergence measures the infinitesimal change of \mu along the leafwise Hamiltonian flow. On each symplectic leaf, this corresponds to the leafwise \div_\omega(X_f|_{\text{leaf}}), where \omega is the induced form on the leaf. The modular vector field X_\mu is an infinitesimal , satisfying \mathcal{L}_{X_\mu} \pi = 0, and the associated class [\![\,X_\mu\,]\!] \in H^1(M; \pi) is independent of the choice of \mu. If \nu = g \mu for a nowhere-vanishing function g > 0, then X_\nu = X_\mu - X_{\log g}. The modular class vanishes there exists a Poisson-invariant on M, meaning \mathcal{L}_{X_f} \mu = 0 for all vector fields X_f (or equivalently, X_\mu = 0). In the special case of a linear Poisson structure on the dual \mathfrak{g}^* of a \mathfrak{g}, the modular class is zero precisely when \mathfrak{g} is unimodular, i.e., the trace of the vanishes on all elements. In local coordinates (x^1, \dots, x^n) where \mu = dx^1 \wedge \cdots \wedge dx^n, the components of the modular are given by X_\mu^k = \partial_i \pi^{ik}, where \pi = \frac{1}{2} \pi^{ij} \partial_i \wedge \partial_j is the bivector. This expression arises from the formula \div_\mu(X_f) = \partial_i (\pi^{ij} \partial_j f), which simplifies to (\partial_i \pi^{ij}) \partial_j f due to the antisymmetry of \pi^{ij}. More invariantly, X_\mu = \partial_\mu \pi, where \partial_\mu is the on multivector fields defined using the musical \star_\mu induced by \mu, via \partial_\mu \alpha = -(\star_\mu)^{-1} \circ d \circ \star_\mu (\alpha). Computations often involve contractions with the ; for instance, the action on functions relates to the trace-like term \pi^{ij} \partial_k \pi_{ij} in the expression for \mathcal{L}_{X_f} \pi, though the modular field isolates the cohomology class component. The modular class serves as an obstruction in Poisson geometry, particularly obstructing the existence of symplectic realizations in cases where an invariant transverse measure is required for the realizing symplectic manifold to compatibly cover the Poisson foliation. For the cotangent Lie algebroid T^*M associated to \pi, the modular class of T^*M equals twice that of (M, \pi), linking it to broader integrability conditions.

Poisson homology

Poisson homology, also known as the canonical homology of a Poisson manifold (M, \pi), is defined as the homology of the differential complex (\Omega^\bullet(M), \partial_\pi), where \Omega^\bullet(M) denotes the space of differential forms on M and \partial_\pi is the Poisson differential given by the graded commutator \partial_\pi = [d, \iota_\pi]. Here, d is the de Rham differential, and \iota_\pi is the interior multiplication by the Poisson bivector field \pi. This satisfies \partial_\pi^2 = 0 and has -1, making (\Omega^\bullet(M), \partial_\pi) a whose homology groups are denoted H_\bullet^\pi(M). The explicit action of \partial_\pi on a k-form \alpha is \partial_\pi \alpha = \iota_\pi d\alpha - (-1)^k d \iota_\pi \alpha, which extends the de Rham differential in a way compatible with the structure. For manifolds, where \pi is invertible, this complex reduces to the de Rham complex up to isomorphism via the musical isomorphism induced by the symplectic form. In general Poisson settings, the zeroth homology H_0^\pi(M) captures invariant densities or traces associated to the Poisson structure, dual to the space of modular vector fields. When the Poisson structure is unimodular—meaning its modular class vanishes—there exists a twisted that identifies with , providing a noncommutative analogue of classical de Rham duality. This duality arises from a Serre bimodule structure on the algebra of functions and holds for both smooth and algebraic Poisson varieties. In applications to quantization, the computes the periodic cyclic homology of deformation quantizations of the , establishing an isomorphism HP_\bullet(A_\hbar) \cong H_\bullet^\pi(M) for a star product A_\hbar on the of functions, where HP_\bullet denotes periodic cyclic homology. This connection facilitates index-theoretic computations and trace formulas in . Poisson homology also classifies central extensions of Poisson algebras, where equivalence classes of such extensions correspond to elements in H_2^\pi(M), analogous to the role of Lie algebra homology in classifying central extensions of Lie algebras.

Morphisms

Poisson maps

A Poisson map between two Poisson manifolds (M, \{\cdot,\cdot\}_M) and (N, \{\cdot,\cdot\}_N) is a smooth map \phi: M \to N such that \phi^*\{f,g\}_N = \{\phi^*f, \phi^*g\}_M for all smooth functions f,g \in C^\infty(N). This condition ensures that the map preserves the algebraic structure of the Poisson bracket under pullback. Equivalently, in terms of the associated fields \pi_M on TM and \pi_N on TN, the map \phi satisfies \phi^*\pi_N = \pi_M. This formulation arises because the is given by \{f,g\}_\pi = \pi(df,dg), so the preservation condition translates to the of the bivector coinciding with the original structure on M. Another equivalent perspective is that \phi pushes forward : d\phi(X^M_h) = X^N_{\phi^*h} for any smooth function h \in C^\infty(N), where X^\pi_f = \pi^\sharp(df) denotes the associated to f via the bundle map \pi^\sharp: T^*Q \to TQ induced by \pi. Poisson maps are closed under : if \phi: M \to N and \psi: N \to P are Poisson maps between Poisson manifolds, then \psi \circ \phi: M \to P is also a Poisson map. A diffeomorphism is a bijective Poisson map whose smooth inverse is also a Poisson map, serving as an in the of Poisson manifolds. When restricted to symplectic leaves, Poisson maps induce symplectic maps between the corresponding leaves. Poisson submanifolds arise as special cases where the inclusion map is a Poisson map.

Poisson submanifolds

A Poisson submanifold of a Poisson manifold (M, \pi) is a closed embedded S \subset M such that the of the Poisson satisfies i^* \pi_M = \pi_S, where i: S \hookrightarrow M is the and \pi_S is a Poisson on S. Equivalently, this condition holds the image of \pi restricted to S is contained in the TS, i.e., \operatorname{im}(\pi|_S) \subset TS, ensuring that all Hamiltonian vector fields tangent to S remain . Such submanifolds inherit a Poisson structure directly from M, and their symplectic leaves are components of the intersection with those of M. Coisotropic submanifolds provide a broader class where the tangent space contains the image of the Poisson map restricted to the conormal directions: for S \subset M, S is coisotropic if TS \supset \operatorname{im}(\pi|_S), or equivalently, \pi^\sharp(\operatorname{ann}(TS)) \subset TS, where \pi^\sharp: T^*M \to TM is the bundle map induced by \pi and \operatorname{ann}(TS) is the conormal bundle. In this case, the vanishing ideal of functions on S forms a Poisson subalgebra, and if the characteristic distribution \pi^\sharp(\operatorname{ann}(TS)) is integrable with constant rank, the leaf space inherits a reduced Poisson structure via symplectic reduction. Poisson submanifolds are special cases of coisotropic submanifolds where the inclusion is itself a Poisson map. Dually to coisotropic submanifolds, an isotropic submanifold S \subset M satisfies TS \subset \ker(\pi|_S), meaning the Poisson bivector vanishes on the tangent directions of S, so \pi(u,v) = 0 for all u,v \in TS. This condition implies that S lies within the kernel of the induced Poisson structure, analogous to isotropic subspaces in symplectic geometry, and often results in a degenerate or trivial induced structure on S. When a S \subset M intersects the leaves of (M, \pi) cleanly—meaning S \cap L is a submanifold for each leaf L with T(S \cap L) = TS \cap TL—and is transverse to the , S inherits a reduced on the by the . This transversality ensures the of the Dirac structure associated to \pi remains , allowing for a well-defined induced without singularities.

Integration

Symplectic groupoids

A groupoid is a Lie groupoid (\Sigma \rightrightarrows M) equipped with a form \omega on \Sigma such that the graph of the partial multiplication is coisotropic and the source and target maps s, t: \Sigma \to M are relations (i.e., they pull back the structure on M to a compatible structure on \Sigma). This structure generalizes the relationship between Lie groups and Lie algebras to the setting of , where the base M inherits a bivector \pi from the infinitesimal structure of the groupoid. The integration of a Poisson manifold (M, \pi) to a exists if and only if the associated cotangent algebroid T^*M is integrable as a algebroid. This integrability condition is characterized by the monodromy groups N_x \subset \nu_x^*(L_x) (where L_x is the leaf through x \in M and \nu_x^*(L_x) is its conormal space) being uniformly discrete near each point, which corresponds to the vanishing of certain obstructions in the leafwise second group H^2(L_x, \nu_x^*(L_x)). When integrable, the source-1 \Sigma(M) \rightrightarrows M, constructed via the cotangent paths, provides a for the integration, with uniqueness up to . In a symplectic groupoid (\Sigma \rightrightarrows M), the source map s: \Sigma \to M defines a whose leaves are the source fibers s^{-1}(x), each of which is a of \Sigma. These symplectic leaves project under the target map t onto the symplectic leaves of the Poisson manifold M, thereby realizing the Poisson as the orbit space of the action. A prototypical example arises when (M, \pi) is itself symplectic, so \pi is invertible. In this case, the pair groupoid \Sigma = M \times M \rightrightarrows M, equipped with the symplectic form \omega = \omega_M \oplus (-\omega_M) (where \omega_M = \pi^{-1}) and multiplication (x, y)(y, z) = (x, z), integrates the structure canonically. Symplectic realizations correspond to special cases where the groupoid is induced by an embedding into a larger symplectic manifold.

Symplectic realizations

A symplectic realization of a Poisson manifold (M, \pi) is an i: M \to (N, \omega) into a (N, \omega) such that the form i^*\omega on M is degenerate with kernel equal to the image of \pi^\sharp: T^*M \to TM, and the Poisson \pi is recovered as the inverse of i^*\omega on the complement of this image. This construction embeds M as a coisotropic of N, where the Poisson structure arises from the transverse to the characteristic foliation defined by \pi. Every Poisson manifold admits a (local) symplectic realization, as established independently by Karasev and Weinstein in the late through explicit constructions involving canonical relations and deformation techniques. A universal symplectic realization, which is complete and functorial, exists under conditions on the modular class of the Poisson structure; it can be constructed via the cotangent lift to T^*M equipped with a twisted symplectic form \omega_\pi = -d\theta + \pi^\flat \circ d\theta, or through leafwise completion of the foliation. The modular class, an element of the first Poisson cohomology group H^1_\pi(M), measures the failure of Hamiltonian vector fields to preserve a transverse and serves as the primary obstruction: if it does not vanish, no full (surjective) global realization exists, though realizations always do. Symplectic realizations are intimately linked to reduction procedures, where Poisson structures emerge as quotients of symplectic manifolds by group actions or foliations. In particular, the Marsden-Weinstein reduction of a coisotropic submanifold in a yields a structure on the reduced space, providing a converse construction to realizations. This framework highlights how degenerate geometries can be "resolved" into nondegenerate ones, with the modular class influencing the regularity of the reduction process.

Examples of integrations

A fundamental example of Poisson integration arises in the linear case, where the dual space \mathfrak{g}^* of a Lie algebra \mathfrak{g} is equipped with the Lie-Poisson bivector \pi, defined by \pi(\alpha, \beta) = \langle \alpha, [\beta^\flat, \alpha^\flat] \rangle for \alpha, \beta \in \mathfrak{g}^*, with \beta^\flat denoting the inverse of the anchor map. This Poisson structure integrates to the cotangent groupoid T^*G \rightrightarrows \mathfrak{g}^*, where G is the simply connected Lie group integrating \mathfrak{g}; here, T^*G carries the canonical symplectic form \omega_0 = -d\theta, with \theta the Liouville form, and the groupoid structure is induced by the cotangent lift of the group multiplication on G. For a symplectic manifold (M, \omega), the underlying Poisson structure \pi = \omega^{-1} is nondegenerate, and it integrates to the trivial (or pair) groupoid M \times M \rightrightarrows M, equipped with the symplectic form \mathrm{pr}_1^*\omega - \mathrm{pr}_2^*\omega, where \mathrm{pr}_1, \mathrm{pr}_2: M \times M \to M are the projections. This groupoid structure arises from the source and target maps \mathrm{s}(x,y) = y and \mathrm{t}(x,y) = x, with multiplication (x,y) \cdot (y',z) = (x,z) when y = y', reflecting the transitive action of M on itself. Log-symplectic manifolds, which are Poisson manifolds whose Poisson bivector is transverse to a codimension-two submanifold Z (the zero set of \pi) and induces a symplectic structure away from Z, admit symplectic groupoid integrations via specific constructions. For a proper log-symplectic structure (where Z is a smooth divisor), one method involves successive blow-ups along the preimage of Z under the source map of a local model, yielding a global symplectic groupoid whose base recovers the log-symplectic leaves. An alternative gluing construction combines local symplectic realizations over the regular part with compatible data near Z, ensuring the groupoid integrates the log cotangent algebroid T^*M(-\log Z). These approaches highlight how singularities in log-symplectic structures can be resolved while preserving integrability. Not all Poisson manifolds integrate to Lie groupoids; obstructions lie in the second cohomology group H^2 of the cotangent Lie algebroid, which vanishes if and only if the Poisson structure is integrable to a symplectic Lie groupoid. For instance, the Poisson-Heisenberg structure on \mathbb{R}^3, defined by \pi = x \partial_y \wedge \partial_z + y \partial_x \wedge \partial_z, induces a nonvanishing class in H^2 when paired with a non-prequantizable symplectic leaf, preventing global Lie groupoid integration. Similarly, Weinstein's regular Poisson structure on \mathbb{R}^3 \setminus \{0\}, given by \pi = (x^2 + y^2 + z^2) (\partial_x \wedge \partial_y + \partial_y \wedge \partial_z + \partial_z \wedge \partial_x), exhibits a nonzero obstruction due to the topology of its symplectic leaves. In such nonintegrable cases, weaker integrations exist in the form of source-1 foliations, where the source fibers of a local model foliate the base by simply connected manifolds, allowing leafwise symplectic groupoid structures despite the global failure.

Advanced topics

Deformation quantization

Deformation quantization provides an algebraic framework to "quantize" a Poisson manifold by deforming its of smooth functions into a noncommutative , preserving the Poisson structure in the classical limit. Formally, given a Poisson manifold (M, \{\cdot, \cdot\}), a star product is a bilinear map \star_\hbar: C^\infty(M) \times C^\infty(M) \to C^\infty(M)[[\hbar]] satisfying f \star_\hbar g = fg + \sum_{k \geq 1} \hbar^k B_k(f,g) for bidifferential operators B_k, with the product being associative (f \star_\hbar (g \star_\hbar h)) = ((f \star_\hbar g) \star_\hbar h) and the commutator satisfying [f, g]_\hbar = f \star_\hbar g - g \star_\hbar f = i\hbar \{f, g\} + O(\hbar^2).90225-7) This deformation, where \hbar is a formal parameter, realizes the Poisson bracket as the leading term in the quantum commutator, bridging classical and quantum mechanics.90225-7) A landmark result establishes the existence of such star products for any Poisson structure. Maxim proved that every finite-dimensional Poisson manifold admits a canonical deformation quantization, meaning the equivalence classes of star products are in one-to-one correspondence with equivalence classes of Poisson structures modulo diffeomorphisms. This construction relies on a formality theorem, which provides an L_\infty-quasi-isomorphism between the Lie algebra of polyvector fields on M (governing the Poisson structure) and the Hochschild cochains of C^\infty(M), allowing the transfer of the Gerstenhaber bracket to define the star product coefficients explicitly via graphs. On \mathbb{R}^n, the star product is unique up to gauge equivalence, where two star products \star_\hbar and \star'_\hbar are equivalent if there exists a formal series of differential operators connecting them. For general smooth manifolds, existence extends beyond \mathbb{R}^n through complementary approaches. Boris Fedosov constructed star products on manifolds using a Weyl-type and a connection, yielding a global quantization via parallel sections in a bundle of deformed algebras. Tamarkin's independent proof, based on Koszul duality and non-abelian , confirms the existence for any smooth Poisson manifold, aligning with Kontsevich's result but emphasizing homotopical algebra. The space of equivalence classes of star products is classified by the second Poisson cohomology group H^2_\mathrm{Poisson}(M), which parametrizes the possible deformations modulo transformations. These quantizations have profound applications in and physics, deforming the classical into a quantum algebra that captures semiclassical limits. For instance, on symplectic manifolds like \mathbb{R}^{2n} with the standard structure, the Weyl star product provides an explicit quantization where operators act on L^2(\mathbb{R}^n), realizing the deformation in representations and underpinning theory.

Linearization problem

The problem for a \pi on a smooth manifold M concerns the existence of local coordinates (x^1, \dots, x^n) around a point p \in M such that \pi takes the form of a constant field, i.e., \pi = \sum_{i<j} c^{ij} \frac{\partial}{\partial x^i} \wedge \frac{\partial}{\partial x^j} with constant coefficients c^{ij}. This is equivalent to finding a mapping the given to the linear induced by a \mathfrak{g} (the isotropy algebra at p) on the \mathfrak{g}^*. The problem, first systematically studied by Weinstein, arises naturally in understanding the local geometry of manifolds and extends the Darboux theorem from to settings. Near points where the rank of \pi is constant, the Weinstein splitting theorem provides a local model U \times V, where U is (corresponding to the symplectic leaf through p) and V is a transverse Poisson manifold; this decomposition is a prerequisite for addressing . Around symplectic leaves, the structure is linearizable via an adaptation of the Moser-Weinstein deformation method, which constructs an of Poisson structures deforming \pi to a model where the leaf is straightened to a linear symplectic subspace, while preserving transversality. For nondegenerate () cases, analytic follows from Moser's path method applied to the induced symplectic forms on leaves, ensuring in suitable topologies. In the category, reduces to solving an in H^2(\mathfrak{g}, \mathfrak{g}^*), which vanishes for semisimple algebras by the second Whitehead lemma, allowing a formal solution via homological perturbation. Conn established analytic linearizability around zeros (zero-dimensional leaves) when the isotropy algebra is semisimple, using a Moser-type deformation that converges due to analyticity. For smooth structures, holds for compact semisimple isotropy via the Nash-Moser , but smooth global linearization can be obstructed by nontrivial cohomology classes even when formal solutions exist. In singular cases, where the rank of \pi varies, full linearization may fail, but partial linearization is achievable along the strata of the foliation: coordinates can be chosen to linearize the transversely to each while restricting to the induced symplectic form on the leaf. This stratified approach generalizes Conn's to higher-dimensional singular leaves that are compact and admit exact symplectic forms.

Poisson-Lie groups

A –Lie group is a G equipped with a bivector field \pi on G such that the multiplication map m: G \times G \to G is a map, meaning m_* (\pi \oplus \pi) = \pi \circ Tm, where Tm is the tangent map of m. This compatibility ensures that the structure interacts naturally with the group law, generalizing the linear – structure on the dual of a . The concept was introduced by Drinfeld in the context of Hamiltonian structures on s and their relation to the classical . Infinitesimally, at the e \in G, the \mathfrak{g} = T_e G becomes a bialgebra (\mathfrak{g}, [\cdot, \cdot], \delta), where the cobracket \delta: \mathfrak{g} \to \wedge^2 \mathfrak{g} is obtained as the derivative of \pi along left-invariant vector fields. This equivalence holds because the multiplicativity of \pi implies that \delta satisfies the co-Jacobi identity, making (\mathfrak{g}^*, [\cdot, \cdot]_\delta) a dual to \mathfrak{g}. Such structures are equivalently described via Manin triples. A Manin triple consists of a \mathfrak{d} equipped with an ad-invariant, nondegenerate \langle \cdot, \cdot \rangle, together with two Lie subalgebras \mathfrak{g}, \mathfrak{b} \subset \mathfrak{d} that are isotropic (\langle \mathfrak{g}, \mathfrak{g} \rangle = 0 = \langle \mathfrak{b}, \mathfrak{b} \rangle) and complementary as vector spaces (\mathfrak{d} = \mathfrak{g} \oplus \mathfrak{b}). For a Poisson–Lie group G with Lie bialgebra (\mathfrak{g}, \delta), the associated Manin triple is (\mathfrak{d}, \mathfrak{g}, \mathfrak{g}^*), where \mathfrak{d} = \mathfrak{g} \oplus \mathfrak{g}^* is the Drinfeld double, with the Lie bracket on \mathfrak{d} extending those on \mathfrak{g} and \mathfrak{g}^* via the pairing \langle X + \xi, Y + \eta \rangle = \eta(X) + \xi(Y), and the bracket satisfying \langle [X, Y]_{\mathfrak{d}}, \eta \rangle + \langle \xi, [X, Y]_{\mathfrak{d}} \rangle = 0 for mixed terms. The dual Lie bialgebra (\mathfrak{g}^*, [\cdot, \cdot]_\delta) arises from the cobracket \delta, and integrating the Manin triple yields the Poisson–Lie structure on G. In the coboundary case, where \delta(X) = [\![ r, X ]\!] for a classical r-matrix r \in \wedge^2 \mathfrak{g}, the Poisson bivector on G takes the form \pi(g) = L_{g*} r - R_{g*} r, with L_g, R_g denoting left and right translations by g \in G. The induced Poisson bracket on smooth functions C^\infty(G) is known as the Sklyanin bracket, defined by \{f, h\}(g) = \left\langle \frac{\partial f}{\partial g}, \left[ \frac{\partial h}{\partial g}, \pi(g) \right] \right\rangle, or equivalently via the r-matrix as \{f, h\}(g) = r(\mathrm{d} f^L, \mathrm{d} h^R) - r(\mathrm{d} h^L, \mathrm{d} f^R), where \mathrm{d}^L, \mathrm{d}^R are left- and right-Poisson differentials. This bracket satisfies the compatibility condition with multiplication, ensuring \{f_1 f_2, h\}(g) = \{f_1, h\}(g) f_2(g_1) + f_1(g_1) \{f_2, h\}(g_2) for g = g_1 g_2. The leaves of a –Lie group G are the orbits under the dressing action of the dual –Lie group G^* on G. Specifically, for a \in G^*, the dressing transformation is given by the \xi_a(g) = -L_{g*} \Pi_+ (\mathrm{Ad}_{g^{-1}}^* a), where \Pi_+: T^*G \to TG is the associated to the , and these orbits coincide with the connected components of the sets G \cdot a G (double cosets) or projections of left cosets under the map \Pi_+: G \times G^* \to G. Each leaf inherits a structure from the form restricted to the spaces spanned by the of the functions. A key example is the dual of a Poisson–Lie group: if (G, \pi) is Poisson–Lie, then the dual group G^* carries a compatible Poisson structure \pi^* such that the pairing map G \times G^* \to \mathbb{R} is a Poisson map, and the Poisson bracket on G^* preserves multiplicativity of functions. That is, if f, h: G^* \to \mathbb{R} are multiplicative (i.e., f(gh) = f(g) f(h)), then \{f, h\} is also multiplicative. For instance, the dual of the trivial Poisson structure on a compact semisimple Lie group G (where \pi = 0) is the linear Lie–Poisson structure on \mathfrak{g}^*, with symplectic leaves as coadjoint orbits. Another example is G = \mathrm{SL}(2, \mathbb{R}) with the factorizable r-matrix r = \frac{1}{8} (H \wedge H + 4 X \wedge Y), whose dual G^* \cong \mathrm{SB}(2, \mathbb{C}) (the Poincaré group) has symplectic leaves as dressing orbits corresponding to hyperboloids in Minkowski space.