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Associative algebra together with a Lie bracket that satisfies Leibniz's law
In mathematics, a Poisson algebra is an associative algebra together with a Lie bracket that also satisfies Leibniz's law; that is, the bracket is also
Poisson_algebra
Operation in Hamiltonian mechanics
more general sense, the Poisson bracket is used to define a Poisson algebra, of which the algebra of functions on a Poisson manifold is a special case
Poisson_bracket
Poisson manifold that is also a Lie group
mathematics, a Poisson–Lie group is a Poisson manifold that is also a Lie group, with the group multiplication being compatible with the Poisson algebra structure
Poisson–Lie_group
Mathematical structure in differential geometry
{\displaystyle M} , making it into a Lie algebra subject to a Leibniz rule (also known as a Poisson algebra). Poisson structures on manifolds were introduced
Poisson_manifold
to finding a (quantum) algebra whose classical limit is a given (classical) algebra such as a Lie algebra or a Poisson algebra. Intuitively, a deformation
Deformation_quantization
deformation quantization of the corresponding Poisson algebra. It is due to Maxim Kontsevich. Given a Poisson algebra (A, {⋅, ⋅}), a deformation quantization
Kontsevich quantization formula
Kontsevich_quantization_formula
properties of the non-commutative algebra pass over to corresponding properties of the Poisson algebra. The Poisson bracket must satisfy the identities
Poisson_ring
Formulation of classical mechanics using momenta
Poisson bracket without resorting to differential equations, see Lie algebra; a Poisson bracket is the name for the Lie bracket in a Poisson algebra.
Hamiltonian_mechanics
Concept in differential geometry
In differential geometry a Poisson supermanifold is a differential supermanifold M such that the supercommutative algebra of smooth functions over it (to
Poisson_supermanifold
algebra Octonion algebra Pre-Lie algebra Poisson algebra Process algebra Quadratic algebra Quadric geometric algebra Quaternion algebra Rees algebra Relation
List_of_algebras
theory as the algebra of generalized Poisson brackets defined on differential forms. A Gerstenhaber algebra is a graded-commutative algebra with a Lie bracket
Gerstenhaber_algebra
Branch of functional analysis
In functional analysis, a branch of mathematics, an operator algebra is an algebra of continuous linear operators on a topological vector space, with
Operator_algebra
bialgebra on a coboundary. They are also called Poisson-Hopf algebras, and are the Lie algebra of a Poisson–Lie group. Lie bialgebras occur naturally in
Lie_bialgebra
Algebra based on a vector space with a quadratic form
mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra with the additional structure
Clifford_algebra
About polynomials in several variables
conjectures are equivalent to the Poisson conjecture, that every endomorphism of the n-th complex Poisson algebra is an automorphism. In consequence
Jacobian_conjecture
Application of mathematical methods to other fields
as a collection of mathematical methods such as real analysis, linear algebra, mathematical modelling, optimisation, combinatorics, probability and statistics
Applied_mathematics
Systematic procedure of turning a classical theory into a quantum one
quotient algebra is converted into a Poisson algebra by introducing a Poisson bracket derivable from the action, called the Peierls bracket. This Poisson algebra
Quantization_(physics)
Concept in mathematics
enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal
Universal_enveloping_algebra
Collection of random variables
and the Poisson process. Louis Bachelier used the Wiener process to model price changes on the Paris Bourse, while A. K. Erlang used the Poisson process
Stochastic_process
Branch of mathematics concerning probability
distributions are the discrete uniform, Bernoulli, binomial, negative binomial, Poisson and geometric distributions. Important continuous distributions include
Probability_theory
Software used in mathematical applications
mathematical suites are computer algebra systems that use symbolic mathematics. They are designed to solve classical algebra equations and problems in human
Mathematical_software
Area of mathematics
algorithm design, computational complexity, numerical methods and computer algebra. Computational mathematics refers also to the use of computers for mathematics
Computational_mathematics
Ring that is also a vector space or a module
differential graded algebra. A Poisson algebra is a commutative associative algebra over a field together with a structure of a Lie algebra so that the Lie
Associative_algebra
Z2-graded generalization of a Poisson algebra
In mathematics, a Poisson superalgebra is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded associative unital algebra A = A 0 ⊕ A 1 {\displaystyle A=A_{0}\oplus
Poisson_superalgebra
Study of discrete mathematical structures
function fields. Algebraic structures occur as both discrete examples and continuous examples. Discrete algebras include: Boolean algebra used in logic gates
Discrete_mathematics
Study of abstract machines and automata
nondeterministic finite automata. In the 1960s, a body of algebraic results known as "structure theory" or "algebraic decomposition theory" emerged, which dealt with
Automata_theory
Branch of applied mathematics
some parts of the mathematical fields of linear algebra, the spectral theory of operators, operator algebras and, more broadly, functional analysis. Nonrelativistic
Mathematical_physics
Methods of mathematical approximation
theory was investigated by the classical scholars – Laplace, Siméon Denis Poisson, Carl Friedrich Gauss – as a result of which the computations could be
Perturbation_theory
1960 article by Eugene Wigner
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences
computer algebra system (CAS) is a software product designed for manipulation of mathematical formulae. The principal objective of a computer algebra system
List of open-source software for mathematics
List_of_open-source_software_for_mathematics
Screened Poisson equation Optics Poisson's spot Elasticity Poisson's ratio Dirichlet–Poisson problem Poisson algebra Poisson superalgebra Poisson boundary
List of things named after Siméon Denis Poisson
List_of_things_named_after_Siméon_Denis_Poisson
Field of mathematics
Numerical linear algebra, sometimes called applied linear algebra, is the study of how matrix operations can be used to create computer algorithms which
Numerical_linear_algebra
Physical theory with fields invariant under the action of local "gauge" Lie groups
the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators. For each group generator there necessarily arises
Gauge_theory
Theory of subatomic structure
called algebraic varieties which are defined by the vanishing of polynomials. For example, the Clebsch cubic illustrated on the right is an algebraic variety
String_theory
Sequence of operations for a task
beyond specific numerical solutions to introduce general procedures for algebraic reduction and balancing. This transformed mathematics into a 'mechanical'
Algorithm
Study of the properties of codes and their fitness
needed] The term algebraic coding theory denotes the sub-field of coding theory where the properties of codes are expressed in algebraic terms and then
Coding_theory
Set of objects whose state must satisfy limits
algebra. It turned out that questions about the complexity of CSPs translate into important universal-algebraic questions about underlying algebras.
Constraint satisfaction problem
Constraint_satisfaction_problem
Physical quantities taking values at each point in space and time
these RWEs can deal with complicated mathematical objects with exotic algebraic properties (e.g. spinors are not tensors, so may need calculus for spinor
Field_(physics)
French mathematician and physicist (1781–1840)
Baron Siméon Denis Poisson (/pwɑːˈsɒ̃/, US also /ˈpwɑːsɒn/; French: [si.me.ɔ̃ də.ni pwa.sɔ̃]; 21 June 1781 – 25 April 1840) was a French mathematician
Siméon_Denis_Poisson
Calculus on stochastic processes
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
Stochastic_calculus
Branch of mathematics
the Cauchy sequence, and started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and
Mathematical_analysis
Calculus of vector-valued functions
generalize to higher dimensions, but the alternative approach of geometric algebra, which uses the exterior product, does (see § Generalizations below for
Vector_calculus
Algebraic structure used in analysis
symmetric Lie algebra Poisson algebra Pre-Lie algebra Quantum groups Moyal algebra Quasi-Frobenius Lie algebra Quasi-Lie algebra Restricted Lie algebra Serre
Lie_algebra
Invariant of vertex algebra
b\}=a_{0}b\mod C_{2}(V)} is a Poisson bracket on R V {\displaystyle R_{V}} which gives the C2-algebra the structure of a Poisson algebra. (Zhu's C2-cofiniteness
Zhu_algebra
Branch of computer science
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
Computational_geometry
Algebra of 4D spacetime
name "algebra of physical space" (APS) originally stems from the use of the biquaternions via its definition as the real Clifford or geometric algebra Cl3
Algebra_of_physical_space
Branch of applied probability theory
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
Decision_theory
Harmonic functions as solutions to Laplace's equation
satisfy Poisson's equation—or in the vacuum, Laplace's equation. There is considerable overlap between potential theory and the theory of Poisson's equation
Potential_theory
supersymmetry algebra (or SUSY algebra) is a mathematical formalism for describing the relation between bosons and fermions. The supersymmetry algebra contains
Supersymmetry_algebra
Physical theory describing classical fields
be derived from scalar potentials which satisfied Laplace's equation. Poisson addressed the question of the stability of the planetary orbits, which
Classical_field_theory
Unix-like systems. KCalc, Linux based scientific calculator Maxima: a computer algebra system which bignum integers are directly inherited from its implementation
List of arbitrary-precision arithmetic software
List_of_arbitrary-precision_arithmetic_software
Theory of getting acceptably close inexact mathematical calculations
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
Approximation_theory
Type of approximation to an underlying physical theory
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
Effective_field_theory
Generalization of the BRST formalism
graded algebra (DGA) with differential Δ. A BV 1-algebra has vanishing antibracket. Let there be given an (n|n) supermanifold with an odd Poisson bi-vector
Batalin–Vilkovisky_formalism
Study of rational collective decision-making
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
Social_choice_theory
Formulation of classical mechanics
Gannon, Terry (2006). Moonshine beyond the monster: the bridge connecting algebra, modular forms and physics. Cambridge University Press. p. 267. ISBN 0-521-83531-3
Lagrangian_mechanics
Field theory involving topological effects in physics
theory, the theory of four-manifolds, and algebraic topology, and to the theory of moduli spaces in algebraic geometry. Donaldson, Jones, Witten, and Kontsevich
Topological quantum field theory
Topological_quantum_field_theory
Mathematical approach to quantum physics
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Physics-mathematics connection
properties of elementary particles to the structure of Lie groups and Lie algebras. According to this connection, the different quantum states of an elementary
Particle physics and representation theory
Particle_physics_and_representation_theory
Branch of numerical analysis
integration of ordinary differential equations (ODEs) and differential algebraic equations (DAEs), to be used. A large number of integration routines have
Numerical methods for partial differential equations
Numerical_methods_for_partial_differential_equations
Organization
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
International Council for Industrial and Applied Mathematics
International_Council_for_Industrial_and_Applied_Mathematics
Software for a class of mathematical problems
problems Systems of ordinary differential equations Systems of differential algebraic equations Boolean satisfiability problems, including SAT solvers Quantified
Solver
Methods used to find numerical solutions of ordinary differential equations
Wanner, Solving ordinary differential equations II: Stiff and differential-algebraic problems, second edition, Springer Verlag, Berlin, 1996. ISBN 3-540-60452-9
Numerical methods for ordinary differential equations
Numerical_methods_for_ordinary_differential_equations
Generalization of Hamiltonian mechanics involving multiple Hamiltonians
Nambu dynamics. Hamiltonian mechanics Symplectic manifold Poisson manifold Poisson algebra Integrable system Conserved quantity Hamiltonian Fluid Mechanics
Nambu_mechanics
Typically linear operator defined in terms of differentiation of functions
appears, for instance, in an associative algebra structure on a deformation quantization of a Poisson algebra. A microdifferential operator is a type of
Differential_operator
Calculus of functions of several variables
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
Multivariable_calculus
Setting of relativistic physics in geometric algebra
spacetime algebra (STA) is the application of Clifford algebra Cl1,3(R), or equivalently the geometric algebra G(M4) of physics. Spacetime algebra provides
Spacetime_algebra
Concept in theoretical physics
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
Renormalization_group
Mathematical concept
In mathematics, and specifically in potential theory, the Poisson kernel is an integral kernel, used for solving the two-dimensional Laplace equation
Poisson_kernel
Field of higher mathematics
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
Geometric_analysis
Property of certain dynamical systems
the Poisson algebra consists only of constants), it must have even dimension 2 n , {\displaystyle 2n,} and the maximal number of independent Poisson commuting
Integrable_system
Quantum field theory enjoying conformal symmetry
conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes
Conformal_field_theory
well as in algebraic geometry (over the complex numbers for definiteness). The glossary also includes notions from Hamiltonian geometry, Poisson geometry
Glossary of symplectic geometry
Glossary_of_symplectic_geometry
Topics referred to by the same term
_{2}} -graded algebra Lie algebras, Poisson algebras, and Jordan algebras, important examples of (potentially) nonassociative algebras In functional analysis:
Algebra_(disambiguation)
Writing Lie algebra sets as matrices
Lie algebra on itself is a representation on an algebra (i.e., acts by derivations on the associative algebra structure), then it is a Poisson algebra. The
Lie_algebra_representation
Symmetry between bosons and fermions
algebra requires the introduction of a Z2-grading under which the bosons are the even elements and the fermions are the odd elements. Such an algebra
Supersymmetry
Interdisciplinary field of research
kinship. The linkage of mathematics and sociology here involved abstract algebra, in particular, group theory. This, in turn, led to a focus on a data-analytical
Mathematical_sociology
Branch of mathematics
equations and optimization problems. Real algebra is the part of algebra which is relevant to real algebraic (and semialgebraic) geometry. It is mostly
Global_optimization
Overview of mechanics based on the least action principle
A(q, p, t) and B(q, p, t) are two scalar valued dynamical variables, the Poisson bracket is defined by the generalized coordinates and momentums: { A ,
Analytical_mechanics
Academic association dedicated to the use of mathematics in industry
Engineering Geometric Design Geosciences Imaging Science Life Sciences Linear Algebra Mathematical Aspects of Materials Science Mathematics of Planet Earth Nonlinear
Society for Industrial and Applied Mathematics
Society_for_Industrial_and_Applied_Mathematics
1017/s0004972711002954. Kosmann-Schwarzbach, Yvette (1996). "From Poisson algebras to Gerstenhaber algebras". Annales de l'Institut Fourier. 46 (5): 1243–1274. doi:10
Leibniz_algebra
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
List of finite element software packages
List_of_finite_element_software_packages
Infinitesimal calculus on functions defined on a geometric algebra
In mathematics, geometric calculus extends geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to
Geometric_calculus
software (licence CC BY-ND). GeoGebra is software that combines geometry, algebra and calculus for mathematics education in schools and universities. It
List of interactive geometry software
List_of_interactive_geometry_software
involution with respect to Poisson–Lie brackets, introduced by Mishchenko and Fomenko (1978). They used it to prove that the Poisson algebra of a finite-dimensional
Argument_shift_method
American mathematician
representations of the corresponding curves on the surfaces generate a Poisson algebra, whose Lie bracket has a topological description in terms of the intersections
William Goldman (mathematician)
William_Goldman_(mathematician)
Class of mathematical software
executable, C/C++ library, and Octave/MATLAB API. Cadabra is a computer algebra system (CAS) designed specifically for the solution of problems encountered
Tensor_software
Russian mathematician (1936–2006)
construction, and the Kantor double, a Jordan superalgebra constructed from a Poisson algebra. Kantor, I. L.; Solodovnikov, A. S. (1989) [1973], Hypercomplex numbers
Isaiah_Kantor
Theory in theoretical physics
other topics. The operators in topological string theory represent the algebra of operators in the full string theory that preserve a certain amount[clarification
Topological_string_theory
Theorem of dynamical systems
springs is expressed mathematically using an algebraic construct known as the Poisson algebra, which is an algebra on all (smooth) functions of position and
Liouville–Arnold_theorem
Journal of Linear Algebra, Volume 34, Pages 137-151, March 2018 Shinya Miyajima, Fast verified computation for solutions of algebraic Riccati equations
Validated_numerics
European mathematical society
Biology. Retrieved 6 July 2026. Feliu, Elisenda (2020). "On the role of algebra in models in molecular biology". Journal of Mathematical Biology. 80 (4):
European Society for Mathematical and Theoretical Biology
European_Society_for_Mathematical_and_Theoretical_Biology
Study of vector bundles, principal bundles, and fibre bundles
as well as giving alternative descriptions of important structures in algebraic geometry such as moduli spaces of vector bundles and coherent sheaves
Gauge_theory_(mathematics)
Algebraic structure used in theoretical physics
be an "ordinary" product, thus giving rise to the Poisson superalgebra and the Gerstenhaber algebra. Such gradings are also observed in deformation theory
Lie_superalgebra
Algebraic structure used in theoretical physics
superalgebra is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into
Superalgebra
Framework to describe phase transitions
stochastic dynamics Algebraic structures Algebra of physical space Particle physics and representation theory Feynman integral Poisson algebra Quantum group
Statistical_field_theory
Theory of quantum gravity merging quantum mechanics and general relativity
known as the master equation. That the master constraint Poisson algebra is an honest Lie algebra opens the possibility of using a method, known as group
Loop_quantum_gravity
Associative algebra generalizing the Virasoro algebra
theory and representation theory, a W-algebra is an associative algebra that generalizes the Virasoro algebra. W-algebras were introduced by Alexander Zamolodchikov
W-algebra
Clifford analysis, using Clifford algebras named after William Kingdon Clifford, is the study of Dirac operators, and Dirac type operators in analysis
Clifford_analysis
Process in quantum mechanical theories
consider a Poisson manifold instead of a symplectic space for the classical theory and perform an ħ-deformation of the corresponding Poisson algebra or even
Canonical_quantization
POISSON ALGEBRA
POISSON ALGEBRA
Boy/Male
Australian, British, English
Son of Adam
Boy/Male
Indian, Sanskrit
Poison Spewing
Surname or Lastname
English
English : patronymic from Middle English prest ‘priest’, i.e. ‘son of the priest’.French : occupational name for a presser of wine or oil, from a derivative of presser ‘to press’.
Girl/Female
Arabic, Farsi, Iranian
Poison
Male
English
Variant spelling of English unisex Addison, ADISSON means "son of Adam."
Boy/Male
Hindu
Poison
Surname or Lastname
English
English : patronymic from Middle English Pole or Poul, vernacular forms of Paul.Americanized spelling of Scandinavian Poulsen.
Girl/Female
Tamil
Poison
Boy/Male
Hindu, Indian
Poison
Girl/Female
Indian, Telugu
Poison
Boy/Male
Indian
Poison
Girl/Female
Gujarati, Hindu, Indian
Poison; Earth
Boy/Male
Tamil
Poison
Surname or Lastname
English
English : patronymic from Phil, a short form of the personal name Philip.
Surname or Lastname
English
English : variant of Grissom.
Surname or Lastname
English
English : topographic name for someone who lived by a postern gate, from Old French posterne; in some cases it would have been a metonymic occupational name for a gatekeeper.English : habitational name from Poston in Herefordshire or Poston in Shropshire, which is named with an Old English personal name Possa + þorn ‘thorn tree’.
Surname or Lastname
English and French
English and French : from Old French pinson ‘finch’, perhaps a nickname applied to a bright and cheerful person.English and French : metonymic occupational name for someone who made pincers or forceps or who used them in their work, from Old French pinson ‘pincers’ (a derivative of pincier ‘to pinch’).
Surname or Lastname
English (Midlands)
English (Midlands) : habitational name from Pointon in Lincolnshire, Poynton in Cheshire, or Poynton Green in Shropshire. The first is named from Old English Pohhingtūn ‘settlement (Old English tūn) associated with Pohha’, a byname apparently meaning ‘bag’; the others have as the first element the Old English personal names Pofa and Pēofa respectively.
Surname or Lastname
English
English : variant spelling of Pierson.
Girl/Female
Biblical
Poison, tricks.
POISSON ALGEBRA
POISSON ALGEBRA
POISSON ALGEBRA
POISSON ALGEBRA
POISSON ALGEBRA
POISSON ALGEBRA
POISSON ALGEBRA
n.
Venom; poison.
v. t.
To poison; to drug.
p. pr. & vb. n.
of Poison
n.
That which taints or destroys moral purity or health; as, the poison of evil example; the poison of sin.
n.
Poison spittle; poison ejected from the mouth.
imp. & p. p.
of Poison
v. t.
To poison; to infect with poison.
v. t.
To imprison; to shut up in, or as in, a prison; to confine; to restrain from liberty.
n.
Poison.
n.
Rat poison; white arsenic.
pl.
of Cornet-a-piston
v. i.
To act as, or convey, a poison.
n.
Poison; venom.
n.
A kind of antidote for poisons; a counter poison formerly in vogue.
n.
Any agent which, when introduced into the animal organism, is capable of producing a morbid, noxious, or deadly effect upon it; as, morphine is a deadly poison; the poison of pestilential diseases.
n.
A four-wheeled carriage for conveying ammunition, consisting of two parts, a body and a limber. In light field batteries there is one caisson to each piece, having two ammunition boxes on the body, and one on the limber.
n.
To injure or kill by poison; to administer poison to.
n.
To put poison upon or into; to infect with poison; as, to poison an arrow; to poison food or drink.
n.
To taint; to corrupt; to vitiate; as, vice poisons happiness; slander poisoned his mind.
n.
The California poison oak (Rhus diversiloba). See under Poison, a.