7.2 Poisson manifolds and their characteristics
3 min read•july 30, 2024
Poisson manifolds extend symplectic geometry, allowing for more general structures. They combine smooth manifolds with special brackets on functions, creating a framework that bridges classical mechanics and quantum physics.
These structures are defined by bivector fields and come in various types, from symplectic to . Understanding their properties, like rank and regularity, is key to grasping their role in mathematical physics and differential geometry.
Poisson Manifolds
Definition and Structure
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- consists of smooth manifold M equipped with {·,·} on space of smooth functions C∞(M)
- Poisson bracket satisfies bilinearity, antisymmetry, Leibniz rule, and Jacobi identity
- Poisson structure defined by π, section of Λ²TM satisfying [π,π] = 0 (Schouten-Nijenhuis bracket)
- Relation between Poisson bracket and bivector field expressed as {f,g} = π(df,dg) for smooth functions f and g
- Hamiltonian vector field Xf associated with function f defined by Xf = π(df,·)
- Poisson manifolds generalize symplectic manifolds allowing degenerate Poisson structures and odd-dimensional manifolds
- Local structure described by decomposing manifold into and transverse Poisson structures
Mathematical Formalism
- π# : T*M → TM induced by Poisson bivector field π
- Rank of Poisson manifold at point defined as rank of Poisson tensor at that point
- Casimir functions C Poisson-commute with all other functions {C,f} = 0 for all f ∈ C∞(M)
- Distribution of Hamiltonian vector fields integrable giving rise to symplectic leaves
- generalizes de Rham cohomology encoding information about Poisson structure
- Natural foliation by symplectic leaves where leaf through point represents orbit of all Hamiltonian vector fields
- measures extent to which Poisson structure preserves volume form
Properties of Poisson Manifolds
Rank and Regularity
- Rank of Poisson manifold may vary from point to point leading to stratification by rank
- Regular Poisson manifold has constant rank throughout manifold
- Symplectic manifolds represent regular Poisson manifolds of maximal rank equaling dimension of manifold
- has vanishing modular class
- Classification based on structure and rank distribution (symplectic, Lie-Poisson, general types)
- Singularities where rank drops crucial for understanding global behavior
- Local normal form near point of constant rank given by Weinstein splitting theorem
Tensor and Functions
- Poisson tensor π# maps T*M to tangent bundle TM
- Rank at point determines dimension of symplectic leaf passing through that point
- Casimir functions remain constant along symplectic leaves
- Hamiltonian vector fields generate symplectic leaf structure
- Poisson cohomology groups encode obstructions to deformations of Poisson structure
- Symplectic leaves form foliation of Poisson manifold with varying dimensions
- Modular class vanishes for unimodular Poisson manifolds (symplectic manifolds)
Classifying Poisson Manifolds
Structural Classification
- Symplectic manifolds represent Poisson manifolds with non-degenerate Poisson tensor
- Lie-Poisson structures arise on dual spaces of Lie algebras
- Linear Poisson structures characterized by Poisson brackets linear in coordinates
- Product Poisson manifolds inherit natural Poisson structure from factors
- of Lie groups form symplectic leaves in Lie-Poisson structures
- leads to noncommutative algebras linking classical and quantum mechanics
- characterized by complete set of Poisson-commuting functions
Analytical Classification
- provides coarse classification of Poisson manifolds
- Regular Poisson manifolds further classified by rank and dimension
- Symplectic manifolds classified by symplectic form and its cohomology class
- Lie-Poisson structures classified by underlying Lie algebra structure
- Linear Poisson structures classified by associated Lie algebra
- Singularity theory applied to classify Poisson structures near singular points
- Moser-type theorems provide local normal forms for Poisson structures
Examples of Poisson Manifolds
Geometric Examples
- Symplectic manifolds (cotangent bundles T*M of smooth manifolds M)
- Kähler manifolds with Poisson structure induced by complex structure
- Poisson structures on surfaces (area forms, magnetic fields)
- Foliated manifolds with transverse Poisson structures
- Jacobi manifolds as generalizations of Poisson manifolds
- Nambu-Poisson structures generalizing Poisson structures to higher arities
- Contact manifolds with induced Poisson structure on symplectization
Algebraic and Physical Examples
- Lie-Poisson structures on dual spaces g* of Lie algebras g
- Linear Poisson structures on vector spaces (Kirillov-Kostant-Souriau structure)
- as deformation quantizations of Poisson-Lie groups
- of classical mechanical systems ( with canonical Poisson structure)
- in symplectic reduction (Marsden-Weinstein reduction)
- Momentum maps in Hamiltonian group actions
- Poisson structures in field theory (Poisson brackets of observables)
Key Terms to Review (27)
Bivector Field: A bivector field is a mathematical construct that assigns a bivector to each point of a manifold, serving as a geometric representation of oriented areas. This concept plays a crucial role in the study of Poisson manifolds, where it helps define the Poisson bracket and provides a means to understand the underlying symplectic structure, enabling the exploration of dynamics and integrability within the manifold.
Casimir function: A Casimir function is a smooth function on a Poisson manifold that takes constant values on the symplectic leaves of the manifold. These functions are significant because they help to identify and characterize the structure of Poisson manifolds, which are equipped with a Poisson bracket that defines their geometric properties.
Coadjoint orbits: Coadjoint orbits are geometric objects that arise in the representation theory of Lie groups and symplectic geometry, specifically representing the action of a Lie group on the dual space of its Lie algebra. They serve as a crucial structure for understanding symplectic manifolds, especially in the context of Hamiltonian dynamics and the reduction of symplectic manifolds under group actions.
Cotangent Bundle: The cotangent bundle of a manifold is the vector bundle that consists of all the cotangent spaces at each point of the manifold, effectively capturing the linear functionals on the tangent spaces. This construction plays a crucial role in symplectic geometry as it provides a natural setting for defining symplectic structures and studying Hamiltonian dynamics.
Deformation quantization: Deformation quantization is a mathematical framework that connects classical mechanics and quantum mechanics by allowing the algebra of observables on a Poisson manifold to be 'deformed' into a non-commutative algebra. This process provides a way to construct quantum theories from classical ones, enabling the transition from the classical phase space description to the quantum mechanical framework while preserving the geometric structure of the underlying Poisson manifold.
Integrable Systems: Integrable systems are dynamical systems that can be solved exactly in terms of integrals, typically characterized by having as many conserved quantities as degrees of freedom. This means that such systems possess a high level of predictability and can be completely described using a finite set of parameters, linking them closely to energy conservation and phase space dynamics.
Jacobi Manifold: A Jacobi manifold is a smooth manifold equipped with a Jacobi bracket that generalizes the concept of Poisson brackets. It allows for the description of dynamics in systems where both the position and momentum variables are present, connecting symplectic geometry with Poisson structures and facilitating the study of integrable systems.
Kähler manifold: A Kähler manifold is a special type of complex manifold that is equipped with a symplectic form that is also compatible with the complex structure. This means it has a rich geometric structure that combines features of both complex and symplectic geometry, allowing for unique insights in various mathematical contexts such as algebraic geometry and representation theory.
Lie-Poisson: Lie-Poisson structures are mathematical constructs that arise from the interplay between Lie algebras and Poisson geometry. They provide a framework to study the dynamics of Hamiltonian systems where the phase space is a symplectic manifold endowed with a Lie algebra structure, allowing for a rich interaction between algebraic and geometric properties.
Linear Poisson Structure: A linear Poisson structure is a specific type of Poisson structure that is defined on a linear space, typically represented by a bilinear map that satisfies the Jacobi identity and is skew-symmetric. This structure helps to generalize Hamiltonian mechanics within the context of linear algebra, allowing for the study of dynamical systems through the lens of symplectic geometry. In practical applications, it provides a framework for analyzing physical systems where the state space can be described by linear equations and relationships.
Marsden-Weinstein Theorem: The Marsden-Weinstein Theorem provides a way to construct symplectic manifolds by reducing the symplectic structure of a Hamiltonian system with a symmetry, utilizing moment maps. This theorem connects the concepts of symplectic reduction and the geometry of orbits in the presence of group actions, facilitating the study of reduced spaces in symplectic geometry.
Modular Class: The modular class is an important concept in the study of Poisson manifolds, representing a cohomological invariant that captures the obstruction to integrating a given Poisson structure. It is closely linked to the symplectic structure and plays a crucial role in understanding the behavior of Hamiltonian systems on these manifolds. The modular class provides insight into the characteristics of the Poisson structure and its equivalence classes.
Momentum map: A momentum map is a mathematical tool that associates each point in a symplectic manifold with a value in a dual space of a Lie algebra, effectively capturing the action of a symmetry group on the manifold. It plays a crucial role in understanding the relationship between symmetries and conserved quantities in Hamiltonian systems, linking geometric structures with physical interpretations.
Nambu-Poisson structure: A Nambu-Poisson structure is a generalization of the Poisson structure on a manifold, defined by a multilinear map that satisfies a certain skew-symmetry and Jacobi identity. This structure allows for the description of systems with multiple independent conservation laws and extends the concept of Hamiltonian dynamics to higher dimensions. It connects deeply with the study of integrable systems and symplectic geometry.
Phase Spaces: Phase spaces are mathematical constructs that represent all possible states of a physical system, including positions and momenta of particles. They provide a framework for analyzing dynamical systems in physics and mathematics, allowing for the visualization and study of system behaviors over time. In symplectic geometry, phase spaces play a crucial role in understanding Hamiltonian mechanics and its applications.
Poisson bracket: The Poisson bracket is a binary operation defined on the algebra of smooth functions over a symplectic manifold, capturing the structure of Hamiltonian mechanics. It quantifies the rate of change of one observable with respect to another, linking dynamics with the underlying symplectic geometry and establishing essential relationships among various physical quantities.
Poisson Cohomology: Poisson cohomology is a mathematical tool used to study the properties and structures of Poisson manifolds by analyzing the cohomological aspects of their Poisson brackets. This concept helps in understanding the deformation theory of Poisson structures, as well as providing insights into the relationship between symplectic geometry and algebraic structures. By investigating the cohomology groups associated with a Poisson manifold, one can derive important invariants and classify Poisson structures effectively.
Poisson manifold: A Poisson manifold is a smooth manifold equipped with a Poisson bracket, which is a bilinear operation that satisfies certain properties, turning the manifold into a geometric structure that encapsulates both symplectic and algebraic properties. It allows for a general framework to study Hamiltonian dynamics in a broader context than just symplectic geometry. Understanding Poisson manifolds enables connections between classical mechanics and modern geometry.
Poisson Tensor: A Poisson tensor is a bilinear, skew-symmetric map that defines a Poisson bracket on a smooth manifold, enabling the formulation of Hamiltonian dynamics. It plays a critical role in the study of Poisson manifolds by providing the structure needed to define the geometric properties of these spaces, including their symplectic nature and the relationship between smooth functions and Hamiltonian flows.
Product Poisson Manifold: A product Poisson manifold is a type of Poisson manifold formed by taking the Cartesian product of two or more Poisson manifolds, inheriting a Poisson structure from each component. This construction allows for the combination of different Poisson structures, creating a richer geometric framework that captures interactions between the manifolds. The resulting manifold retains key characteristics from each individual Poisson structure, facilitating the study of complex systems where multiple dynamics are present.
Quantum Groups: Quantum groups are algebraic structures that generalize the concept of groups in the framework of quantum mechanics, allowing for the study of symmetries in quantum systems. These groups emerge from the study of Hopf algebras and provide a way to incorporate quantum phenomena into the framework of geometry and representation theory, particularly in relation to Poisson manifolds.
Rank Stratification: Rank stratification refers to the classification of a Poisson manifold based on the rank of its Poisson structure, which indicates how many independent functions can be derived from it. This classification is significant because it provides insight into the manifold's geometric and dynamical properties, allowing for a better understanding of its underlying symplectic and algebraic characteristics.
Reduced Spaces: Reduced spaces refer to the quotient spaces obtained by taking a symplectic manifold and factoring out the action of a Lie group that preserves the symplectic structure. These spaces play a crucial role in the study of Poisson manifolds, where they help in understanding the reduction of symplectic forms and the corresponding dynamics, simplifying the analysis of systems with symmetries.
Symplectic leaves: Symplectic leaves are the connected components of the symplectic foliation in a Poisson manifold. They can be thought of as the 'slices' or 'layers' of the manifold where the symplectic structure is well-defined and behaves nicely. Understanding symplectic leaves is crucial for exploring the relationship between symplectic geometry and Poisson structures, as they reveal how these structures can vary across the manifold.
Symplectic Manifold: A symplectic manifold is a smooth, even-dimensional differentiable manifold equipped with a closed, non-degenerate differential 2-form called the symplectic form. This structure allows for a rich interplay between geometry and physics, especially in the formulation of Hamiltonian mechanics and the study of dynamical systems.
Unimodular poisson manifold: A unimodular Poisson manifold is a special type of Poisson manifold where the Poisson bivector field has a divergence-free property with respect to a volume form. This means that the manifold supports a natural volume that remains invariant under the flow generated by any Hamiltonian vector field. Unimodularity is significant because it allows for the study of symplectic structures and their relationships to dynamical systems while ensuring conservation laws are preserved.
Weinstein Splitting Theorem: The Weinstein Splitting Theorem states that any symplectic manifold can be decomposed into a product of a symplectic submanifold and a Lagrangian submanifold, under certain conditions. This theorem provides a foundational understanding of the structure of symplectic manifolds, allowing for the analysis of their properties through simpler components. The significance of this theorem extends to various applications in Poisson geometry and Hamiltonian dynamics, where understanding the underlying symplectic structure is crucial.