Poisson Structures  

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

f, g Cinfty(M) ,   {f, g} = ij (f/xi) (g/xj)

is a bilinear, skew-symmetric form satisfying the Jacobi identity.
* Relationships: One is canonically defined by a Lie groupoid.
* Example: A symplectic manifold, where ij is non-degenerate, and is the inverse of the symplectic structure, which is closed by the Jacobi identity.
@ 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 m.SG/05 [generating functions]; Cortese & García PLA(06)ht [compatibility with equations of motion]; Morchio & Strocchi a0805 [Lie-Rinehart algebra of a manifold, and dynamics]; > s.a. non-commutative geometry.
@ 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]; Racanière JGP(06) [quantization]; McLachlan JPA(09) [vector fields].

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.
@ Representations: Berceanu RVMP(06) [holomorphic].
@ Generalizations: Pérez Bueno JPA(97)ht; Grabowski & Marmo JPA(01)m.DG, JPA(03) [algebroid].
@ Quantization: de León et al JMP(97) [geometric].

Moyal Brackets > s.a. algebra; deformation quantization; Dirac Bracket; Peierls Bracket; wigner functions.
* Idea: A deformation of the Poisson bracket, obtained by introducing higher-derivative terms in it.
* Freedom: The Jacobi identity fixes it almost uniquely, but it depends on a parameter , with { , }kappa → { , }PB as → 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].
@ 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].

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

{f1, ..., fn} = i_1, ..., i_n i_1 f1 ... i_n fn ,

where is the Nambu tensor; Used in a modified form of classical dynamics; The 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 def]; 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 equation of motion]; Guha JMP(02) [hydrodynamic].

Other Generalizations
@ Covariant: Marsden et al AP(86) [classical fields and electromagnetism]; Pol'shin IJGMP(08)-a0801 [Leibniz bracket]; > s.a. symplectic structures.
@ Related topics: de Azcárraga et al JPA(96), ht/96, JPA(97); Grabowski & Marmo MPLA(98) [based on 2k-forms]; Severa & Weinstein PTPS(01)m.SG [closed 3-form background]; Mokhov m.DG/02, m.DG/02, m.DG/02 [non-local, hydrodynamic]; Lavagno et al EPJC(06)qp [q-deformed]; Golmankhaneh a0807 [fractional]; Khudaverdian & Voronov a0808-in [higher-order].

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