Statements (51)
| Predicate | Object |
|---|---|
| gptkbp:instanceOf |
gptkb:mathematical_concept
differential geometric structure |
| gptkbp:definedIn |
smooth manifold equipped with a Poisson bracket
|
| gptkbp:field |
differential geometry
mathematical physics |
| gptkbp:generalizes |
symplectic manifold
|
| gptkbp:has_local_model |
cotangent bundle of a manifold
|
| gptkbp:hasApplication |
gptkb:classical_mechanics
gptkb:Lie_bialgebras gptkb:Lie_groupoids integrable systems noncommutative geometry quantization deformation theory foliation theory |
| gptkbp:hasProperty |
local structure described by Weinstein splitting theorem
bracket defines a Lie algebra structure on smooth functions Poisson tensor is a bivector field Poisson tensor satisfies Schouten–Nijenhuis bracket condition bracket is a derivation in each argument bracket is antisymmetric bracket is bilinear bracket satisfies Jacobi identity can be degenerate can be described by Casimir functions can be described by Dirac structures can be described by Lie algebroids can be described by Poisson cohomology can be integrated to symplectic groupoids can be quantized to noncommutative algebras can be regular or singular every symplectic manifold is a Poisson manifold foliated by symplectic leaves global structure can be complicated may have singular symplectic leaves not every Poisson manifold is symplectic Casimir functions are central in the Poisson algebra Poisson bracket is a Lie bracket on smooth functions |
| gptkbp:hasSpecialCase |
symplectic manifold
|
| https://www.w3.org/2000/01/rdf-schema#label |
Poisson manifold
|
| gptkbp:introduced |
gptkb:André_Lichnerowicz
1977 |
| gptkbp:relatedTo |
gptkb:Hamiltonian_mechanics
gptkb:Lie_algebroid Poisson algebra |
| gptkbp:structure |
Poisson bracket
|
| gptkbp:studiedBy |
gptkb:geometry
gptkb:topology mathematical physics |
| gptkbp:bfsParent |
gptkb:deformation_quantization
|
| gptkbp:bfsLayer |
6
|