International Journal of Geometric Methods in Modern Physics, ISSN 0219-8878, 09/2013, Volume 10, Issue 8

We review origins and main properties of the most important bracket operations appearing canonically in differential geometry and mathematical physics in the...

Nijenhuis tensor | Poisson bracket | Lie bracket | Courant bracket | Lie algebroid | HAMILTONIAN-FORMALISM | CALCULUS | NIJENHUIS | PHYSICS, MATHEMATICAL | MECHANICS | NAMBU-POISSON | COHOMOLOGY | ISOMORPHISMS | LIE ALGEBROIDS | DIRAC STRUCTURES | DYNAMICS | Mathematics - Differential Geometry

Nijenhuis tensor | Poisson bracket | Lie bracket | Courant bracket | Lie algebroid | HAMILTONIAN-FORMALISM | CALCULUS | NIJENHUIS | PHYSICS, MATHEMATICAL | MECHANICS | NAMBU-POISSON | COHOMOLOGY | ISOMORPHISMS | LIE ALGEBROIDS | DIRAC STRUCTURES | DYNAMICS | Mathematics - Differential Geometry

Journal Article

1993, Translations of mathematical monographs, ISBN 0821845969, Volume 119., xi, 366

Book

Journal of Geometry and Physics, ISSN 0393-0440, 10/2018, Volume 132, pp. 358 - 362

We give a method to construct Poisson brackets on Banach manifolds , for which the value of at some point may depend on higher order derivatives of the smooth...

Banach manifolds | Operational vector fields | Queer Poisson brackets | Higher order derivations | MATHEMATICS | PHYSICS, MATHEMATICAL

Banach manifolds | Operational vector fields | Queer Poisson brackets | Higher order derivations | MATHEMATICS | PHYSICS, MATHEMATICAL

Journal Article

Journal of Algebra, ISSN 0021-8693, 12/2017, Volume 492, pp. 212 - 233

We propose a non-skew-symmetric generalization of the original definition of double Poisson Bracket by M. Van den Bergh. It allows one to explicitly construct...

Noncommutative geometry | Integrable systems | Poisson brackets | Representation algebras | MATHEMATICS | ALGEBRAS | Mathematics - Quantum Algebra

Noncommutative geometry | Integrable systems | Poisson brackets | Representation algebras | MATHEMATICS | ALGEBRAS | Mathematics - Quantum Algebra

Journal Article

Letters in Mathematical Physics, ISSN 0377-9017, 07/2004, Volume 69, Issue 1, pp. 61 - 87

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We...

Odd Poisson supermanifold | Courant algebroid | Vinogradov bracket | derived bracket | Courant bracket | Mathematical and Computational Physics | Poisson structure with background | Physics | Geometry | Loday–Leibniz algebra | Lie algebroid | Group Theory and Generalizations | Statistical Physics | FIELDS | Loday-Leibniz algebra | LIE BIALGEBROIDS | PHYSICS, MATHEMATICAL | odd Poisson supermanifold | FORMS | Derived bracket | ALGEBRAS | COHOMOLOGY | DIRAC STRUCTURES | POISSON MANIFOLDS | OPERATORS | GEOMETRY

Odd Poisson supermanifold | Courant algebroid | Vinogradov bracket | derived bracket | Courant bracket | Mathematical and Computational Physics | Poisson structure with background | Physics | Geometry | Loday–Leibniz algebra | Lie algebroid | Group Theory and Generalizations | Statistical Physics | FIELDS | Loday-Leibniz algebra | LIE BIALGEBROIDS | PHYSICS, MATHEMATICAL | odd Poisson supermanifold | FORMS | Derived bracket | ALGEBRAS | COHOMOLOGY | DIRAC STRUCTURES | POISSON MANIFOLDS | OPERATORS | GEOMETRY

Journal Article

Annals of Mathematics, ISSN 0003-486X, 3/2003, Volume 157, Issue 2, pp. 575 - 620

In this paper we present the solution to a longstanding problem of differential geometry: Lie's third theorem for Lie algebroids. We show that the...

Mathematical manifolds | Homomorphisms | Tangents | Algebra | Mathematical theorems | Vector fields | Lie groups | Mathematical vectors | Mathematical transitivity | Lie groupoid | Lie algebroid | MATHEMATICS | TRANSVERSALLY COMPLETE FOLIATIONS | ALGEBROIDS | INTEGRATION | GROUPOIDS | HOLONOMY | POISSON MANIFOLDS

Mathematical manifolds | Homomorphisms | Tangents | Algebra | Mathematical theorems | Vector fields | Lie groups | Mathematical vectors | Mathematical transitivity | Lie groupoid | Lie algebroid | MATHEMATICS | TRANSVERSALLY COMPLETE FOLIATIONS | ALGEBROIDS | INTEGRATION | GROUPOIDS | HOLONOMY | POISSON MANIFOLDS

Journal Article

Journal of Geometry and Physics, ISSN 0393-0440, 04/2017, Volume 114, pp. 404 - 419

We compute the Poisson cohomology of a scalar Poisson bracket of Dubrovin–Novikov type with independent variables. We find that the second and third cohomology...

Deformations of Poisson structures | Poisson cohomology | Poisson brackets of hydrodynamic type | Mathematical Physics | Physics and Astronomy(all) | Geometry and Topology | MATHEMATICS, APPLIED | HAMILTONIAN OPERATORS | HYDRODYNAMIC TYPE | PHYSICS, MATHEMATICAL | Mathematics

Deformations of Poisson structures | Poisson cohomology | Poisson brackets of hydrodynamic type | Mathematical Physics | Physics and Astronomy(all) | Geometry and Topology | MATHEMATICS, APPLIED | HAMILTONIAN OPERATORS | HYDRODYNAMIC TYPE | PHYSICS, MATHEMATICAL | Mathematics

Journal Article

Physica D: Nonlinear Phenomena, ISSN 0167-2789, 11/2016, Volume 335, pp. 54 - 69

Reversible evolution of macroscopic and mesoscopic systems can be conveniently constructed from two ingredients: an energy functional and a Poisson bracket....

Non-equilibrium thermodynamics | Hierarchy | Poisson bracket | Projection | GENERIC | Lie–Poisson equation | COMPLEX FLUIDS | MATHEMATICS, APPLIED | PHYSICS, MULTIDISCIPLINARY | EQUATIONS | RHEOLOGY | Lie-Poisson equation | PRINCIPLE | FORMULATION | PHYSICS, MATHEMATICAL | KINETIC-THEORY | GENERAL FORMALISM | DYNAMICS | DIFFUSION | Thermodynamics

Non-equilibrium thermodynamics | Hierarchy | Poisson bracket | Projection | GENERIC | Lie–Poisson equation | COMPLEX FLUIDS | MATHEMATICS, APPLIED | PHYSICS, MULTIDISCIPLINARY | EQUATIONS | RHEOLOGY | Lie-Poisson equation | PRINCIPLE | FORMULATION | PHYSICS, MATHEMATICAL | KINETIC-THEORY | GENERAL FORMALISM | DYNAMICS | DIFFUSION | Thermodynamics

Journal Article

International Journal of Geometric Methods in Modern Physics, ISSN 0219-8878, 11/2018, Volume 15, Issue 11

We propose an extension of n-ary Nambu-Poisson bracket to superspace R-n vertical bar m and construct by means of superdeterminant a family of Nambu-Poisson...

Nambu-Poisson bracket | Hamiltonian mechanics | Poisson bracket | Nambu mechanics | supermanifold | Filippov-Jacobi identity | PHYSICS, MATHEMATICAL

Nambu-Poisson bracket | Hamiltonian mechanics | Poisson bracket | Nambu mechanics | supermanifold | Filippov-Jacobi identity | PHYSICS, MATHEMATICAL

Journal Article

Modern Physics Letters A, ISSN 0217-7323, 06/2017, Volume 32, Issue 19, p. 1750100

A geometrical approach to the covariant formulation of the dynamics of relativistic systems is introduced. A realization of Peierls brackets by means of a...

Peierls brackets | covariant formalism | field theory | PHYSICS, NUCLEAR | POISSON BRACKETS | PHYSICS, MATHEMATICAL | PHYSICS, PARTICLES & FIELDS

Peierls brackets | covariant formalism | field theory | PHYSICS, NUCLEAR | POISSON BRACKETS | PHYSICS, MATHEMATICAL | PHYSICS, PARTICLES & FIELDS

Journal Article

Journal of Nonlinear Mathematical Physics, ISSN 1402-9251, 04/2019, Volume 26, Issue 2, pp. 202 - 213

A method for the construction of classes of examples of multi-dimensional, multi-component Dubrovin-Novikov brackets of hydrodynamic type is given. This is...

Hamiltonian structures | integrable systems | MATHEMATICS, APPLIED | CLASSIFICATION | POISSON BRACKETS | PHYSICS, MATHEMATICAL

Hamiltonian structures | integrable systems | MATHEMATICS, APPLIED | CLASSIFICATION | POISSON BRACKETS | PHYSICS, MATHEMATICAL

Journal Article

Physics of Plasmas, ISSN 1070-664X, 06/2016, Volume 23, Issue 6, p. 62101

There are several plasma models intermediate in complexity between ideal magnetohydrodynamics (MHD) and two-fluid theory, with Hall and Extended MHD being two...

FLUIDS | PLASMA DYNAMICS | VARIATIONAL PRINCIPLE | WAVES | VLASOV EQUATION | PHYSICS, FLUIDS & PLASMAS | HYDRODYNAMICS | POISSON BRACKETS | ACTION PRINCIPLE FORMULATIONS

FLUIDS | PLASMA DYNAMICS | VARIATIONAL PRINCIPLE | WAVES | VLASOV EQUATION | PHYSICS, FLUIDS & PLASMAS | HYDRODYNAMICS | POISSON BRACKETS | ACTION PRINCIPLE FORMULATIONS

Journal Article

Regular and Chaotic Dynamics, ISSN 1560-3547, 11/2018, Volume 23, Issue 6, pp. 720 - 734

We construct a symplectic realization and a bi-Hamiltonian formulation of a 3-dimensional system whose solution are the Jacobi elliptic functions. We...

Poisson structures | 37J35 | Mathematics | Dynamical Systems and Ergodic Theory | Plücker relations | 53D17 | MATHEMATICS, APPLIED | MECHANICS | Plucker relations | PHYSICS, MATHEMATICAL | DYNAMICAL-SYSTEMS | Research | Functions, Elliptic | Mathematical research | Mappings (Mathematics)

Poisson structures | 37J35 | Mathematics | Dynamical Systems and Ergodic Theory | Plücker relations | 53D17 | MATHEMATICS, APPLIED | MECHANICS | Plucker relations | PHYSICS, MATHEMATICAL | DYNAMICAL-SYSTEMS | Research | Functions, Elliptic | Mathematical research | Mappings (Mathematics)

Journal Article

Indagationes Mathematicae, ISSN 0019-3577, 10/2014, Volume 25, Issue 5, pp. 846 - 871

We introduce the notion of for Poisson structures on with coefficients in Laurent polynomials. To such a Poisson structure we associate a polyhedral cone and a...

Poisson Geometry | Gelfand–Zeitlin integrable system | Gelfand-Zeitlin integrable system | Poisson geometry | MATHEMATICS | LIE-GROUPS | Mathematics - Symplectic Geometry

Poisson Geometry | Gelfand–Zeitlin integrable system | Gelfand-Zeitlin integrable system | Poisson geometry | MATHEMATICS | LIE-GROUPS | Mathematics - Symplectic Geometry

Journal Article

15.
Full Text
Elliptic singularities on log symplectic manifolds and Feigin-Odesskii Poisson brackets

Compositio Mathematica, ISSN 0010-437X, 04/2017, Volume 153, Issue 4, pp. 717 - 744

A log symplectic manifold is a complex manifold equipped with a complex symplectic form that has simple poles on a hypersurface. The possible singularities of...

logarithmic differential form | elliptic curve | Fano manifold | Poisson structure | hypersurface singularity | MATHEMATICS | HILBERT SCHEMES | FOLIATIONS | DEFORMATIONS | GEOMETRY

logarithmic differential form | elliptic curve | Fano manifold | Poisson structure | hypersurface singularity | MATHEMATICS | HILBERT SCHEMES | FOLIATIONS | DEFORMATIONS | GEOMETRY

Journal Article

Regular and Chaotic Dynamics, ISSN 1560-3547, 11/2016, Volume 21, Issue 6, pp. 682 - 696

In this paper, we present Poisson brackets of certain classes of mappings obtained as general periodic reductions of integrable lattice equations. The Poisson...

39A20 | periodic reduction | Lagrangian | 39A14 | 70H06 | Poisson bracket | Mathematics | Dynamical Systems and Ergodic Theory | lattice equation | 70H15 | MATHEMATICS, APPLIED | INTEGRABLE MAPPINGS | MECHANICS | HIROTA | SYSTEMS | PHYSICS, MATHEMATICAL | Lattice theory | Research | Mathematical research | Mappings (Mathematics) | Physics - Exactly Solvable and Integrable Systems

39A20 | periodic reduction | Lagrangian | 39A14 | 70H06 | Poisson bracket | Mathematics | Dynamical Systems and Ergodic Theory | lattice equation | 70H15 | MATHEMATICS, APPLIED | INTEGRABLE MAPPINGS | MECHANICS | HIROTA | SYSTEMS | PHYSICS, MATHEMATICAL | Lattice theory | Research | Mathematical research | Mappings (Mathematics) | Physics - Exactly Solvable and Integrable Systems

Journal Article

Journal of Mathematical Physics, ISSN 0022-2488, 06/2015, Volume 56, Issue 6, p. 1

We find that in presence of noncommutative Poisson brackets, the relation between Lagrangian and Hamiltonian is modified. We discuss this property by using...

Integrals | Poisson distribution | Oscillators | Physics

Integrals | Poisson distribution | Oscillators | Physics

Journal Article

Classical and Quantum Gravity, ISSN 0264-9381, 08/2018, Volume 35, Issue 18, p. 185012

A hypersurface composed of two null sheets, or 'light fronts', swept out by the two congruences of future null normal geodesics emerging from a spacelike...

canonical general relativity | Poisson bracket | observables | free | null initial data | Peierls bracket | PHYSICS, MULTIDISCIPLINARY | ASTRONOMY & ASTROPHYSICS | SURFACE | GEOMETRY | PHYSICS, PARTICLES & FIELDS | Physics - General Relativity and Quantum Cosmology

canonical general relativity | Poisson bracket | observables | free | null initial data | Peierls bracket | PHYSICS, MULTIDISCIPLINARY | ASTRONOMY & ASTROPHYSICS | SURFACE | GEOMETRY | PHYSICS, PARTICLES & FIELDS | Physics - General Relativity and Quantum Cosmology

Journal Article

Theoretical and Mathematical Physics, ISSN 0040-5779, 8/2018, Volume 196, Issue 2, pp. 1129 - 1149

The theory of multidimensional Poisson vertex algebras provides a completely algebraic formalism for studying the Hamiltonian structure of partial differential...

Poisson vertex algebra | Theoretical, Mathematical and Computational Physics | hydrodynamic Poisson bracket | Applications of Mathematics | Physics | Hamiltonian operator | Poisson cohomology | PHYSICS, MULTIDISCIPLINARY | DUBROVIN-NOVIKOV BRACKETS | EQUATIONS | HAMILTONIAN OPERATORS | PHYSICS, MATHEMATICAL | Algebra | Differential equations

Poisson vertex algebra | Theoretical, Mathematical and Computational Physics | hydrodynamic Poisson bracket | Applications of Mathematics | Physics | Hamiltonian operator | Poisson cohomology | PHYSICS, MULTIDISCIPLINARY | DUBROVIN-NOVIKOV BRACKETS | EQUATIONS | HAMILTONIAN OPERATORS | PHYSICS, MATHEMATICAL | Algebra | Differential equations

Journal Article

Journal of Noncommutative Geometry, ISSN 1661-6952, 2018, Volume 12, Issue 2, pp. 577 - 636

For any graded bialgebras A and B, we define a commutative graded algebra A(B) representing the functor of B-representations of A. When A is a cocommutative...

Gerstenhaber algebra | Representation algebra | Hopf algebra | Poisson algebra | Quasi-poisson algebra | Double Poisson algebra | MATHEMATICS, APPLIED | representation algebra | POISSON STRUCTURES | PHYSICS, MATHEMATICAL | MODULI SPACES | MATHEMATICS | double Poisson algebra | quasi-Poisson algebra | MANIFOLDS | SURFACES

Gerstenhaber algebra | Representation algebra | Hopf algebra | Poisson algebra | Quasi-poisson algebra | Double Poisson algebra | MATHEMATICS, APPLIED | representation algebra | POISSON STRUCTURES | PHYSICS, MATHEMATICAL | MODULI SPACES | MATHEMATICS | double Poisson algebra | quasi-Poisson algebra | MANIFOLDS | SURFACES

Journal Article

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