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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
Algebraic structure used in theoretical physics
Lie superalgebra is a generalisation of a Lie algebra to include a Z / 2 Z {\displaystyle \mathbb {Z} /2\mathbb {Z} } ‑grading. Lie superalgebras are
Lie_superalgebra
supergraded Lie superalgebra is a further generalization of this notion to the category of superalgebras in which a graded Lie superalgebra is endowed with
Graded_Lie_algebra
Not-necessarily-associative commutative algebra satisfying (xy)(xx) = x(y(xx))
Victor G (1977), "Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras", Communications in Algebra, 5 (13): 1375–1400, doi:10
Jordan_algebra
Algebra used in 2D conformal field theories and string theory
then V is called a vertex operator superalgebra. One of the simplest examples is the vertex operator superalgebra generated by a single free fermion ψ
Vertex_operator_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
Type of associative algebra that "almost commutes"
supercommutative (associative) algebra (sometimes termed a commutative superalgebra) is a superalgebra (i.e. a Z2-graded algebra) such that for any two homogeneous
Supercommutative_algebra
Supergravity in eleven dimensions
Superstring theory Super vector space Supergeometry Supermathematics Superalgebra Lie superalgebra Super-Poincaré algebra Superconformal algebra Supersymmetry
Eleven-dimensional supergravity
Eleven-dimensional_supergravity
Symmetry between bosons and fermions
applications. The mathematical structure of supersymmetry (graded Lie superalgebras) has subsequently been applied successfully to other topics of physics
Supersymmetry
matrix with entries in a superalgebra (or superring). The most important examples are those with entries in a commutative superalgebra (such as a Grassmann
Supermatrix
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
Graded vector space with applications to theoretical physics
spaces. This leads to a treatment of "superobjects" such as superalgebras, Lie superalgebras, supergroups, etc. that is completely analogous to their ungraded
Super_vector_space
Type of 2D conformal field theory
affine Lie algebra built from the corresponding Lie algebra (or Lie superalgebra). By extension, the name WZW model is sometimes used for any conformal
Wess–Zumino–Witten_model
Semigroup action
of representation theory, a representation of a Lie superalgebra is an action of Lie superalgebra L on a Z2-graded vector space V, such that if A and
Representation of a Lie superalgebra
Representation_of_a_Lie_superalgebra
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
Supersymmetric generalization of the Poincaré algebra
algebras (without central charges or internal symmetries), and are Lie superalgebras. Thus a super-Poincaré algebra is a Z2-graded vector space with a graded
Super-Poincaré_algebra
Lie algebra with imaginary simple roots
interesting examples. It is also possible to extend the definition to superalgebras. A generalized Kac–Moody algebra can be graded by giving ei degree 1
Generalized_Kac–Moody_algebra
Compact astronomical body
supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup Holography Holographic principle AdS/CFT correspondence
Black_hole
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
Index of articles associated with the same name
Differential graded algebra). A superalgebra is a Z 2 {\displaystyle \mathbb {Z} _{2}} -graded algebra. A graded-commutative superalgebra satisfies the "supercommutative"
Graded_structure
Theorem in theoretical physics
The theorem is a generalization of the Coleman–Mandula theorem to Lie superalgebras. It was proved in 1975 by Rudolf Haag, Jan Łopuszański, and Martin Sohnius
Haag–Łopuszański–Sohnius theorem
Haag–Łopuszański–Sohnius_theorem
Sequence of operations for a task
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Algorithm
Natural number
Grigorievich (1977). "Classification of Simple Z-Graded Lie Superalgebras and Simple Jordan Superalgebras". Communications in Algebra. 5 (13). Taylor & Francis:
27_(number)
Canonical commutation or anticommutation relations
elements f , g {\displaystyle f,~g} in V {\displaystyle V} is the obvious superalgebra generalization which unifies CCRs with CARs: if all pure elements are
CCR_and_CAR_algebras
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
Ten-dimensional supergravity
RR fields. The fermionic fields are meanwhile in the NSR sector. The superalgebra for N = ( 1 , 1 ) {\displaystyle {\mathcal {N}}=(1,1)} supersymmetry
Type_IIA_supergravity
Algebraic structure
is Z / 2 {\displaystyle \mathbb {Z} /2} , it is also known as a Lie superalgebra. A Lie-isotopic algebra is a generalization of Lie algebras proposed
Generalization of a Lie algebra
Generalization_of_a_Lie_algebra
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
American mathematician
mathematics at the University of California, Berkeley who researches superalgebras and their representations. Serganova graduated from Moscow State School
Vera_Serganova
combines the structures of a supercommutative ring and a graded Lie superalgebra. It is used in the Batalin–Vilkovisky formalism. It appears also in the
Gerstenhaber_algebra
Algebra combining both supersymmetry and conformal symmetry
theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions
Superconformal_algebra
American mathematician (1947–2022)
algebras; combinatorics of Lie algebra representations; graded algebras and superalgebras; and quantum groups and related structures. Benkart received her BS
Georgia_Benkart
Branch of mathematics that studies abstract algebraic structures
based on the representation theory of affine Kac–Moody algebras. Lie superalgebras are generalizations of Lie algebras in which the underlying vector space
Representation_theory
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
Branch of mathematics
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Mathematical_analysis
Algebraic structure used in theoretical physics
purely algebraically as the universal enveloping algebra of the Lie superalgebra. In a similar way one can define an affine algebraic supergroup as a
Supergroup_(physics)
Supersymmetric extension to the Virasoro algebra
of the Virasoro algebra (named after Miguel Ángel Virasoro) to a Lie superalgebra. There are two extensions with particular importance in superstring theory:
Super_Virasoro_algebra
Formulation of classical mechanics using momenta
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Hamiltonian_mechanics
Formulation of classical mechanics
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Lagrangian_mechanics
algebra Kac–Moody algebra Kleene algebra Leibniz algebra Lie algebra Lie superalgebra Malcev algebra Matrix algebra Non-associative algebra Octonion algebra
List_of_algebras
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
Study of discrete mathematical structures
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Discrete_mathematics
Concept in mathematics
The above is exactly how the universal enveloping algebra for Lie superalgebras is constructed. One need only to carefully keep track of the sign, when
Universal_enveloping_algebra
Superstring theory Super vector space Supergeometry Supermathematics Superalgebra Lie superalgebra Super-Poincaré algebra Superconformal algebra Supersymmetry
List of quantum field theories
List_of_quantum_field_theories
Study of abstract machines and automata
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Automata_theory
No-go theorem pertaining the triviality of space-time and internal symmetries
algebras, but the theorem can be generalized by instead considering Lie superalgebras. Doing this allows for additional anticommutating generators known as
Coleman–Mandula_theorem
Supersymmetric generalization of Yang–Mills
Superstring theory Super vector space Supergeometry Supermathematics Superalgebra Lie superalgebra Super-Poincaré algebra Superconformal algebra Supersymmetry
N = 1 supersymmetric Yang–Mills theory
N_=_1_supersymmetric_Yang–Mills_theory
Ten-dimensional supergravity
the same chirality, while the dilatino has the opposite chirality. The superalgebra for type I supersymmetry is given by { Q α , Q β } = ( P γ μ C ) α β
Type_I_supergravity
Branch of mathematics concerning probability
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Probability_theory
Minimal supergravity in four dimensions
satisfied in that case. Its action can be constructed by gauging this superalgebra, yielding the supersymmetry transformation rules for the vielbein and
Pure_4D_N_=_1_supergravity
Application of mathematical methods to other fields
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Applied_mathematics
Physical theory with fields invariant under the action of local "gauge" Lie groups
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Gauge_theory
In the theory of superalgebras, if A is a commutative superalgebra, V is a free right A-supermodule and T is an endomorphism from V to itself, then the
Supertrace
Collection of random variables
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Stochastic_process
Graphical representation of supersymmetric algebras
supersymmetry generators, i.e., to ( 1 | N ) {\displaystyle (1|N)} superalgebras. In that case, the defining algebraic relationship among the supersymmetry
Adinkra_symbols_(physics)
Physical quantities taking values at each point in space and time
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Field_(physics)
Base space for supersymmetric theories
{\displaystyle (t,\Theta ,\Theta ^{*})} . The coordinates form a Lie superalgebra, in which the gradation degree of t {\displaystyle t} is even and that
Superspace
Superconformal Yang–Mills theory
(in particular the Heisenberg spin chain), but based on a larger Lie superalgebra rather than s u ( 2 ) {\displaystyle {\mathfrak {su}}(2)} for ordinary
N = 4 supersymmetric Yang–Mills theory
N_=_4_supersymmetric_Yang–Mills_theory
Russian mathematician (1936–2006)
the Kantor–Koecher–Tits construction, and the Kantor double, a Jordan superalgebra constructed from a Poisson algebra. Kantor, I. L.; Solodovnikov, A. S
Isaiah_Kantor
Branch of mathematics
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Global_optimization
Supersymmetric generalization of quantum chromodynamics
Superstring theory Super vector space Supergeometry Supermathematics Superalgebra Lie superalgebra Super-Poincaré algebra Superconformal algebra Supersymmetry
Super_QCD
Methods of mathematical approximation
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Perturbation_theory
Ten-dimensional supergravity
Majorana–Weyl fermions. The theory does admit a cosmological constant. The superalgebra for ten-dimensional N = ( 2 , 0 ) {\displaystyle {\mathcal {N}}=(2,0)}
Type_IIB_supergravity
Branch of mathematical physics
branch of mathematical physics which applies the mathematics of Lie superalgebras to the behaviour of bosons and fermions. The driving force in its formation
Supermathematics
Principle in theoretical physics
supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup Holography Holographic principle AdS/CFT correspondence
Holographic_principle
Monster and modular connection
MR 0996026 Borcherds, Richard (1992), "Monstrous Moonshine and Monstrous Lie Superalgebras" (PDF), Invent. Math., vol. 109, pp. 405–444, Bibcode:1992InMat.109
Monstrous_moonshine
Duality between theories of gravity on anti-de Sitter space and conformal field theories
supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup Holography Holographic principle AdS/CFT correspondence
AdS/CFT_correspondence
measurement based Quantum Computing timeline of quantum computing Lie superalgebra supergroup (physics) supercharge supermultiplet supergravity theory of
List of mathematical topics in quantum theory
List_of_mathematical_topics_in_quantum_theory
Writing Lie algebra sets as matrices
Poisson algebra. The analogous observation for Lie superalgebras gives the notion of a Poisson superalgebra. Representation of a Lie group Weight (representation
Lie_algebra_representation
Mathematical approach to quantum physics
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Perturbation theory (quantum mechanics)
Perturbation_theory_(quantum_mechanics)
Branch of applied probability theory
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Decision_theory
Victor G (1977), "Classification of simple Z-graded Lie superalgebras and simple Jordan superalgebras", Communications in Algebra, 5 (13): 1375–1400, doi:10
Kantor–Koecher–Tits construction
Kantor–Koecher–Tits_construction
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
List of arbitrary-precision arithmetic software
List_of_arbitrary-precision_arithmetic_software
Concept in mathematics
theory of SL2(R) Representations of the Lorentz group Stone–von Neumann theorem Unitary representation of a star Lie superalgebra Zonal spherical function
Unitary_representation
Riemannian manifold with SU(n) holonomy
supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup Holography Holographic principle AdS/CFT correspondence
Calabi–Yau_manifold
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
General relativity in M-theory
is generated by the Super-Poincaré algebra, which is a Lie superalgebra. A Lie superalgebra is a Z 2 {\displaystyle \mathbf {Z} _{2}} graded algebra in
Higher-dimensional supergravity
Higher-dimensional_supergravity
Gauge theory with supersymmetry
Superstring theory Super vector space Supergeometry Supermathematics Superalgebra Lie superalgebra Super-Poincaré algebra Superconformal algebra Supersymmetry
Supersymmetric_gauge_theory
Super vector space forming base superspace for supersymmetric field theories
supergroup of nilpotent length 2. This supergroup has the following Lie superalgebra. Suppose that M {\displaystyle M} is Minkowski space (of dimension d
Super_Minkowski_space
Comprehensive physical model
including Lie 3-algebras and Lie superalgebras. Neither of these fit with Yang–Mills theory. In particular Lie superalgebras would introduce bosons with
Grand_Unified_Theory
Theory in theoretical physics
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Topological_string_theory
Software for a class of mathematical problems
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Solver
Theory of stochastic partial differential equations
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Supersymmetric theory of stochastic dynamics
Supersymmetric_theory_of_stochastic_dynamics
Type of Riemannian manifold
supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup Holography Holographic principle AdS/CFT correspondence
Hyperkähler_manifold
Calculus of vector-valued functions
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Vector_calculus
mathematics, a supermodule is a Z2-graded module over a superring or superalgebra. Supermodules arise in super linear algebra which is a mathematical framework
Supermodule
1960 article by Eugene Wigner
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences
Eight-dimensional Riemannian manifold
supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup Holography Holographic principle AdS/CFT correspondence
Spin(7)-manifold
2D supersymmetric generalization to the conformal algebra
the 2D N = 2 superconformal algebra is an infinite-dimensional Lie superalgebra, related to supersymmetry, that occurs in string theory and two-dimensional
N_=_2_superconformal_algebra
Japanese counterpart of the Society for Industrial and Applied Mathematics
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Japan Society for Industrial and Applied Mathematics
Japan_Society_for_Industrial_and_Applied_Mathematics
Branch of applied mathematics
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Mathematical_physics
Mathematical concept
supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup Holography Holographic principle AdS/CFT correspondence
Worldsheet
Differential geometry of supermanifolds
Superstring theory Super vector space Supergeometry Supermathematics Superalgebra Lie superalgebra Super-Poincaré algebra Superconformal algebra Supersymmetry
Supergeometry
Secondary characteristic classes of 3-manifolds
supergravity Type IIA supergravity Type IIB supergravity Superspace Lie superalgebra Lie supergroup Holography Holographic principle AdS/CFT correspondence
Chern–Simons_form
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
List of interactive geometry software
List_of_interactive_geometry_software
Infinite product identities associated to affine root systems
analogs of the Weyl denominator formula for affine Kac–Moody algebras and superalgebras. Demazure, Michel (1977), "Identités de Macdonald", Séminaire Bourbaki
Macdonald_identities
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
Field theory involving topological effects in physics
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Topological quantum field theory
Topological_quantum_field_theory
Algebra of 4D spacetime
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Algebra_of_physical_space
Poisson algebra Quantum group Renormalization group Spacetime algebra Superalgebra Supersymmetry algebra Decision sciences Game theory Operations research
Clifford_analysis
SUPERALGEBRA
SUPERALGEBRA
SUPERALGEBRA
SUPERALGEBRA
Boy/Male
Hindu
Lord of all beings
Boy/Male
Indian, Sanskrit, Tamil
Curious
Boy/Male
Tamil
Vishruth | விஷà¯à®°à¯à®¤
Celebrated or renowned, Much heard of, Famous, Pleased, Delighted, Happy, Son of Vasudeva (Brahma purana, Lord Vishnu
Boy/Male
African, American, Anglo, British, English, Jamaican
Watchman's Hill; Wear Valley
Girl/Female
Indian
Pretty, Beautiful
Boy/Male
Norse
A man freed by Skallagrim.
Boy/Male
Arabic, Basque
Ocean; Seas
Surname or Lastname
English
English : metonymic occupational name for a maker and seller of gloves or a nickname for a wearer of particularly fine gloves, from Middle English cuffe ‘glove’ (of uncertain origin; attested in this sense from the 14th century, with the modern meaning first in the 16th century).Irish : Anglicized form of Gaelic Mac Dhuibh, a variant of Mac Duibh ‘son of the black one’ (see Duff).Irish : approximate translation of Gaelic Ó DoirnÃn (see Dornan).Cornish : nickname from Cornish cuf ‘dear’, ‘kind’.
Girl/Female
Arabic, Muslim
Miracle; Astonishment
Surname or Lastname
English
English : from a reduced form of the personal name Sebastian.French : from a diminutive of Bast.
SUPERALGEBRA
SUPERALGEBRA
SUPERALGEBRA
SUPERALGEBRA
SUPERALGEBRA