Poisson Structures  

In General > s.a. hamiltonian; lagrangian dynamics; symplectic manifold.
$ Pre-Poisson structure: A manifold M and a Lie algebra structure on C(M) with Leibniz identity.
$ Poisson structure: A pre-Poisson structure satisfying the Jacobi identity, i.e., a pair (M, { , }), such that

f, g ∈ C(M) ,   {f, g} = Ωij (∂f/∂xi) (∂g/∂xj)

is a bilinear, skew-symmetric form satisfying the Jacobi identity.
* Relationships: A Poisson structure is canonically defined by a Lie groupoid.
* Example: A symplectic manifold, where Ωij is non-degenerate and is the inverse of the symplectic structure, which is closed by the Jacobi identity.
@ General references: Laurent-Gengoux et al 13.
@ And other structures: Balinsky & Burman JPA(94) [compatible with algebraic structure]; Landsman RVMP(97)qp/96 [transition probability]; Boucetta CRAS(01) [and pseudo-Riemannian metric], DG&A(04) [and pseudo-Riemannian Lie algebras]; Petalidou JPA(02) [and Jacobi structure]; Cattaneo et al CMP(05)m.SG/03, Dherin LMP(06)m.SG/05 [generating functions]; Cortese & García PLA(06)ht [compatibility with equations of motion]; Morchio & Strocchi LMP(08)-a0805 [Lie-Rinehart algebra of a manifold, and dynamics]; > s.a. non-commutative geometry.
@ Special cases: García-Naranjo et al LMP(15)-a1406 [on smooth four-manifolds]; Benini & Schenkel AHP(17)-a1602 [for non-linear scalar field theories, based on the Cahiers topos]; Jordan AJP(16)nov [for generators of the Galilei and Poincaré groups of spacetime transformations].
@ Quantum: Racanière JGP(06); Esposito a1502-proc; Khorasani EJTP-a1411 [derivation].
@ Related topics: Grabowski et al MPLA(93) [classification]; Hojman JPA(96) [from symmetry and conservation law]; Bering PLB(00) [boundary Poisson brackets]; Cattaneo LMP(04)m.SG/03 [integration]: Ortega & Ratiu LMP(04) [symmetry reduction]; McLachlan JPA(09) [vector fields]; Gürses et al JMP(09) [finding, for a given dynamical system]; Leclerc a1211 [symmetric, for fermion fields]; Pavelka et al PhyD(16)-a1512 [change of description and hierarchy of Poisson brackets]; > s.a. non-equilibrium thermodynamics; riemann tensor.
> Online resources: see Wikipedia page.

Jacobi Bracket / Structure on a Manifold
* Idea: A generalization of the Poisson bracket / structure, which represents a weakening the Leibniz rule.
* Generalization – Jacobi algebroid: A graded Lie bracket on the Grassmann algebra associated with a vector bundle which satisfies a property similar to that of the Jacobi brackets.
@ General references: de León et al JMP(97) [geometric quantization]; Berceanu RVMP(06) [holomorphic representation].
@ Generalizations: Pérez Bueno JPA(97)ht; Grabowski & Marmo JPA(01)m.DG, JPA(03) [algebroid].
@ Physics examples: Asorey et al a1706 [test particles].

Moyal Algebra / Brackets > s.a. algebra; deformation quantization; Dirac Bracket; Peierls Bracket; wigner functions.
* Idea: A deformation of the Poisson algebra/bracket, obtained by introducing higher-derivative terms in it.
* Freedom: The Jacobi identity fixes it almost uniquely, but it depends on a parameter κ, with { , }κ → { , }PB as κ → 0.
@ General references: Moyal PCPS(49); Fletcher PLB(90) [uniqueness]; Gozzi & Reuter MPLA(93), IJMPA(94)ht/03; Strachan JPA(95); Tzanakis & Dimakis JPA(97) [uniqueness]; Merkulov mp/00; Dias & Prata JMP(07)qp/06 [and evolution]; Hiley a1211 [and the von Neumann operator algebra].
@ For spin: Amiet & Weigert PRA(01) [spin and particle]; Heiss & Weigert PRA(01) [discrete].
@ For other theories: Fairlie MPLA(98) [in M-theory]; Finkelstein ht/99 [gauge theory].
@ Variations: Masuda & Saito MPLA(99)ht [supersymmetric]; Dimakis & Müller-Hoissen LMP(00)ht [covariant, and Seiberg-Witten maps]; Gouba et al MPLA(12)-a1106 [generalization of the Moyal and Voros products, and physical interpretation].
> Online resources: see Wikipedia pages on Moyal bracket and Moyal product.

Nambu Brackets > s.a. deformation quantization; Nambu Dynamics; Ternary Operations.
* Idea: A generalization of Poisson brackets of the form

{f1, ..., fn} = ηi1, ..., ini1 f1 ... ∂in fn ,

where η is the Nambu tensor; It is used in a modified form of classical dynamics; Its quantization is still not understood.
@ General references: Takhtajan CMP(94); Hietarinta JPA(97); Gautheron CMP(98); Pandit & Gangal JPA(98) [geometric]; Grabowski & Marmo JPA(99) [inductive definition]; Ogawa & Sagae IJTP(00) [Lagrangian formalism]; Dufour & Zhitomirskii LMP(03) [and singularities of integrable 1-forms]; Tegmen IJMPA(06)mp [with constraint functionals]; Dereli et al IJMPA(09) [3D phase space, canonical transformations].
@ Examples, systems: Yamaleev AP(00) [relativistic particle]; Guha JMP(02) [hydrodynamic models]; Salazar & Kurgansky a1011 [electromagnetic field]; Horikoshi & Kawamura a1304 [from a variant formulation of Hamiltonian dynamics].

Other Generalizations
@ Covariant: Marsden et al AP(86) [classical fields and electromagnetism]; Pol'shin IJGMP(08)-a0801 [Leibniz bracket]; D'Avignon a1510 [non-canonical bracket and physical consequences]; > s.a. symplectic structures.
@ Related topics: de Azcárraga et al JPA(96), ht/96-proc, JPA(97); Grabowski & Marmo MPLA(98) [based on 2k-forms]; Severa & Weinstein PTPS(01)m.SG [closed 3-form background]; Mokhov FAA(03)m.DG/02, TMP(02)m.DG, TMP(04)m.DG/02 [non-local, hydrodynamic]; Lavagno et al EPJC(06)qp [q-deformed]; Golmankhaneh TJP(08)-a0807 [fractional]; Khudaverdian & Voronov AIP(08)-a0808 [higher-order]; Mokhov a1001 [deformation]; Bruce JoM-a1301 [Loday-Poisson brackets].


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