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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
Algebraic structure used in theoretical physics
there may also be an "ordinary" product, thus giving rise to the Poisson superalgebra and the Gerstenhaber algebra. Such gradings are also observed in
Lie_superalgebra
Associative algebra together with a Lie bracket that satisfies Leibniz's law
Poisson superalgebra and the Gerstenhaber algebra. The difference between the two is in the grading of the product itself. For the Poisson superalgebra, the
Poisson_algebra
Operation in Hamiltonian mechanics
bracket Moyal bracket Peierls bracket Phase space Poisson algebra Poisson ring Poisson superalgebra Poisson superbracket f ( p i , q i , t ) {\displaystyle
Poisson_bracket
Algebraic structure used in theoretical physics
In mathematics and theoretical physics, a superalgebra is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebra. That is, it is an algebra over a commutative
Superalgebra
Concept in differential geometry
the Poisson superbracket turning it into a Poisson superalgebra. Every symplectic supermanifold is a Poisson supermanifold but not vice versa. Poisson manifold
Poisson_supermanifold
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
bracket of degree −1 satisfying the Poisson identity. Everything is understood to satisfy the usual superalgebra sign conventions. More precisely, the
Gerstenhaber_algebra
Concept in mathematics
elements. In this case, the (anti-)commutator of the superalgebra lifts to an (anti-)commuting Poisson bracket. Another possibility is to use something other
Universal_enveloping_algebra
algebra Lie algebra Lie superalgebra Malcev algebra Matrix algebra Non-associative algebra Octonion algebra Pre-Lie algebra Poisson algebra Process algebra
List_of_algebras
Russian mathematician (1936–2006)
Kantor–Koecher–Tits construction, and the Kantor double, a Jordan superalgebra constructed from a Poisson algebra. Kantor, I. L.; Solodovnikov, A. S. (1989) [1973]
Isaiah_Kantor
In mathematics, the Kantor double is a Jordan superalgebra structure on the sum of two copies of a Poisson algebra. It is named after Isaiah Kantor, who
Kantor_double
Formulation of classical mechanics using momenta
has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics
Hamiltonian_mechanics
Writing Lie algebra sets as matrices
structure), then it is a Poisson algebra. The analogous observation for Lie superalgebras gives the notion of a Poisson superalgebra. Representation of a
Lie_algebra_representation
Study of the properties of codes and their fitness
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Coding_theory
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
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
Study of discrete mathematical structures
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Discrete_mathematics
Mathematical approach to quantum physics
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Algebra based on a vector space with a quadratic form
structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras. Let V be a vector space over a
Clifford_algebra
Quantum mechanics with supersymmetry
supersymmetric quantum mechanics, an application of the supersymmetry superalgebra to quantum mechanics as opposed to quantum field theory. It was hoped
Supersymmetric quantum mechanics
Supersymmetric_quantum_mechanics
Branch of applied probability theory
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Decision_theory
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
Set of objects whose state must satisfy limits
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Constraint satisfaction problem
Constraint_satisfaction_problem
Sequence of operations for a task
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Algorithm
Symmetry between bosons and fermions
and the fermions are the odd elements. Such an algebra is called a Lie superalgebra.[citation needed] The simplest supersymmetric extension of the Poincaré
Supersymmetry
Application of mathematical methods to other fields
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Applied_mathematics
Generalization of the BRST formalism
for multi-vector fields is an example of an antibracket. If L is a Lie superalgebra, and Π is the operator exchanging the even and odd parts of a super space
Batalin–Vilkovisky_formalism
Lie algebroid Lie bialgebra Lie coalgebra Lie conformal algebra Lie superalgebra Abelian Lie algebra Affine Lie algebra Anyonic Lie algebra Compact Lie
List of things named after Sophus Lie
List_of_things_named_after_Sophus_Lie
Study of abstract machines and automata
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Automata_theory
Formulation of classical mechanics
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Lagrangian_mechanics
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
Calculus on stochastic processes
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Stochastic_calculus
Area of mathematics
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Computational_mathematics
Branch of mathematics concerning probability
distributions are the discrete uniform, Bernoulli, binomial, negative binomial, Poisson and geometric distributions. Important continuous distributions include
Probability_theory
Branch of applied mathematics
contributions to mathematical astronomy, potential theory. Siméon Denis Poisson (1781–1840) worked in analytical mechanics and potential theory. In Germany
Mathematical_physics
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
Physical theory with fields invariant under the action of local "gauge" Lie groups
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Gauge_theory
Theory of getting acceptably close inexact mathematical calculations
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Approximation_theory
Physical quantities taking values at each point in space and time
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Field_(physics)
1960 article by Eugene Wigner
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences
called a Lie superalgebra. Just as one can have representations of a Lie algebra, one can also have representations of a Lie superalgebra, called supermultiplets
Supersymmetry_algebra
Topics referred to by the same term
associative bilinear vector product Superalgebra, a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebra Lie algebras, Poisson algebras, and Jordan algebras
Algebra_(disambiguation)
Academic association dedicated to the use of mathematics in industry
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Society for Industrial and Applied Mathematics
Society_for_Industrial_and_Applied_Mathematics
Quantum field theory enjoying conformal symmetry
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Conformal_field_theory
Methods used to find numerical solutions of ordinary differential equations
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Numerical methods for ordinary differential equations
Numerical_methods_for_ordinary_differential_equations
Branch of computer science
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Computational_geometry
Framework to describe phase transitions
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Statistical_field_theory
Setting of relativistic physics in geometric algebra
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Spacetime_algebra
Study of rational collective decision-making
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Social_choice_theory
The Schouten–Nijenhuis bracket makes the multivector fields into a Lie superalgebra if the grading is changed to the one of opposite parity (so that the
Schouten–Nijenhuis_bracket
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
Modern theory of gravitation that combines supersymmetry and general relativity
supersymmetry (SUSY) generators form together with the Poincaré algebra and superalgebra, called the super-Poincaré algebra, supersymmetry as a gauge theory makes
Supergravity
Software for a class of mathematical problems
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Solver
Concept in theoretical physics
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Renormalization_group
Gell-Mann matrices Poisson bracket Noether's theorem Wigner's classification Gauge theory Grand Unified Theory Supergroup Lie superalgebra Twistor theory
List_of_Lie_groups_topics
Interdisciplinary field of research
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Mathematical_sociology
Software used in mathematical applications
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Mathematical_software
Field of mathematics
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Numerical_linear_algebra
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
List of arbitrary-precision arithmetic software
List_of_arbitrary-precision_arithmetic_software
Branch of mathematics
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Global_optimization
Theory of subatomic structure
Gannon, p. 8 Borcherds, Richard (1992). "Monstrous moonshine and Lie superalgebras" (PDF). Inventiones Mathematicae. 109 (1): 405–444. Bibcode:1992InMat
String_theory
Study of vector bundles, principal bundles, and fibre bundles
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Gauge_theory_(mathematics)
Field of higher mathematics
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Geometric_analysis
European mathematical society
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
European Society for Mathematical and Theoretical Biology
European_Society_for_Mathematical_and_Theoretical_Biology
French scientific society aiming at promoting applied mathematics
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Société de Mathématiques Appliquées et Industrielles
Société_de_Mathématiques_Appliquées_et_Industrielles
Field theory involving topological effects in physics
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Topological quantum field theory
Topological_quantum_field_theory
Branch of numerical analysis
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Numerical methods for partial differential equations
Numerical_methods_for_partial_differential_equations
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Deep backward stochastic differential equation method
Deep_backward_stochastic_differential_equation_method
Class of mathematical software
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Tensor_software
Branch of functional analysis
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Operator_algebra
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Clifford_analysis
Algebra of 4D spacetime
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Algebra_of_physical_space
Supergeometric generalization of a manifold
well as in purely mathematical subjects including the theory of Lie superalgebras and supergroups. Alongside the standard locally ringed-space formulation
Supermanifold
Theory of stochastic partial differential equations
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Supersymmetric theory of stochastic dynamics
Supersymmetric_theory_of_stochastic_dynamics
Organization
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
International Council for Industrial and Applied Mathematics
International_Council_for_Industrial_and_Applied_Mathematics
Japanese counterpart of the Society for Industrial and Applied Mathematics
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Japan Society for Industrial and Applied Mathematics
Japan_Society_for_Industrial_and_Applied_Mathematics
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
List of open-source software for mathematics
List_of_open-source_software_for_mathematics
Type of approximation to an underlying physical theory
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Effective_field_theory
Calculus of vector-valued functions
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Vector_calculus
Physics-mathematics connection
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Particle physics and representation theory
Particle_physics_and_representation_theory
Theory in theoretical physics
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Topological_string_theory
Infinitesimal calculus on functions defined on a geometric algebra
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Geometric_calculus
Compact astronomical body
Archived from the original on 3 September 2019. Retrieved 28 June 2020. Poisson, Eric; Israel, Werner (1990). "Internal Structure of Black Holes". Physical
Black_hole
Operation measuring the failure of two entities to commute
ISBN 0471010901 Lavrov, P.M. (2014), "Jacobi -type identities in algebras and superalgebras", Theoretical and Mathematical Physics, 179 (2): 550–558, arXiv:1304
Commutator
Application of Clifford algebra
colloquially as the "sandwich product". Since geometric algebras are superalgebras, the result should be negated in the (relatively rare) case that A {\textstyle
Plane-based_geometric_algebra
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Validated_numerics
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
List of interactive geometry software
List_of_interactive_geometry_software
Kazakh mathematician and physicist (born 1956)
3. – P. 509–517. Dzhumadildaev A.S., Odd central extensions of Lie superalgebras // Functional Analysis and its Applications. – 1995. – V.29, No.3. –
Askar_Dzhumadildayev
Properties underlying modern physics
transformation. Electroweak symmetry Electroweak symmetry breaking A Lie superalgebra is an algebra in which (suitable) basis elements either have a commutation
Symmetry_in_quantum_mechanics
Hypothetical elementary particle that mediates gravity
supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup Holography Holographic principle AdS/CFT correspondence
Graviton
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
List of finite element software packages
List_of_finite_element_software_packages
Differential algebra
structure of a *-algebra, and can be unified as even and odd terms of a superalgebra, as discussed in CCR and CAR algebras. Weyl algebras also generalize
Weyl_algebra
Algebraic structure used in analysis
example, a graded Lie algebra is a Lie algebra (or more generally a Lie superalgebra) with a compatible grading. A differential graded Lie algebra also comes
Lie_algebra
Calculus of functions of several variables
representation theory Feynman integral Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences
Multivariable_calculus
Property of a differential manifold that includes complex structures
at the point, much like Weinstein's theorem for the local structure of Poisson manifolds. The remaining question of the local structure is: what does
Generalized_complex_structure
Unified field theory
supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup Holography Holographic principle AdS/CFT correspondence
Kaluza–Klein_theory
Graduate-level textbooks in mathematics
Commutative Algebra Viviana Ene, Jürgen Herzog 2011 978-0-8218-7287-1 131 Lie Superalgebras and Enveloping Algebras Ian M. Musson 2012 978-0-8218-6867-6 132 Topics
Graduate Studies in Mathematics
Graduate_Studies_in_Mathematics
Hypothetical particle
supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup Holography Holographic principle AdS/CFT correspondence
Dilaton
POISSON SUPERALGEBRA
POISSON SUPERALGEBRA
POISSON SUPERALGEBRA
POISSON SUPERALGEBRA
POISSON SUPERALGEBRA
POISSON SUPERALGEBRA
POISSON SUPERALGEBRA
POISSON SUPERALGEBRA
POISSON SUPERALGEBRA