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Differential graded algebra
In mathematics, the Cartan model is a differential graded algebra that computes the equivariant cohomology of a space. Stefan Cordes, Gregory Moore, Sanjaye
Cartan_model
French mathematician (1904–2008)
mathematician Élie Cartan, nephew of mathematician Anna Cartan, oldest brother of composer Jean Cartan [fr; de], physicist Louis Cartan [fr] and mathematician
Henri_Cartan
Classical theory of gravitation
In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation, one of
Einstein–Cartan_theory
French mathematician (1869–1951)
Élie Joseph Cartan ForMemRS (French: [kaʁtɑ̃]; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory
Élie_Cartan
Generalization of affine connections
Cartan connections describe the geometry of manifolds modelled on homogeneous spaces. The theory of Cartan connections was developed by Élie Cartan,
Cartan_connection
Algebraic topology theory
situation), then the above cohomology may be computed using the so-called Cartan model (see equivariant differential forms.) The construction should not be
Equivariant_cohomology
Nilpotent subalgebra of a Lie algebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra h {\displaystyle {\mathfrak {h}}} of a Lie algebra g {\displaystyle
Cartan_subalgebra
Cosmological model in which the observable universe is the interior of a black hole
the Einstein–Cartan–Sciama–Kibble theory of gravity, proposed by Élie Cartan, Dennis Sciama, and Thomas Kibble. Torsion, introduced by Cartan, is a geometric
Black_hole_cosmology
In mathematics, the Cartan–Hadamard conjecture is a fundamental problem in Riemannian geometry and geometric measure theory which states that the classical
Cartan–Hadamard_conjecture
Type of non-Euclidean geometry
regions, where they locally resemble the hyperbolic plane. The hyperboloid model of hyperbolic geometry provides a representation of events one temporal
Hyperbolic_geometry
Field theory of a point particle confined to move on a fixed manifold
Maurer–Cartan form. A common variation of the sigma model is to present it on a symmetric space. The prototypical example is the chiral model, which takes
Sigma_model
Geometrical re-formulation of Newtonian gravity
Newton–Cartan theory (or geometrized Newtonian gravitation) is a geometrical re-formulation, as well as a generalization, of Newtonian gravity first introduced
Newton–Cartan_theory
248-dimensional exceptional simple Lie group
corresponding root lattice, which has rank 8. The designation E8 comes from the Cartan–Killing classification of the complex simple Lie algebras, which fall into
E8_(mathematics)
Type of 2D conformal field theory
root, the corresponding negative root and their commutator, which is a Cartan generator. In the case of the noncompact simple Lie group S L ( 2 , R )
Wess–Zumino–Witten_model
representations σχ. If G is a connected Lie group, then, thanks to the work of Cartan, Malcev, Iwasawa and Chevalley, G has a maximal compact subgroup, unique
Zonal_spherical_function
Approach to general relativity
contractions. The significance of the tetradic formalism appear in the Einstein–Cartan formulation of general relativity. The tetradic formalism of the theory
Tetrad_formalism
Model of subatomic particles
radius of the Compton size and the inner radius of the Cartan size (10−27 m) in the Einstein–Cartan theory of gravity. André-Marie Ampère (1823). "Sur la
Toroidal_ring_model
The Cartan–Karlhede algorithm is a procedure for completely classifying and comparing Riemannian manifolds. Given two Riemannian manifolds of the same
Cartan–Karlhede_algorithm
Classical field theory describing gravitation
These gauge fields naturally lead to a spacetime geometry known as Riemann–Cartan manifold, which generalizes Riemannian geometry by incorporating both curvature
Poincaré_gauge_theory
Dirac equation for self-interacting fermions
self-interacting electrons. The nonlinear Dirac equation appears in the Einstein–Cartan–Sciama–Kibble theory of gravity, which extends general relativity to matter
Nonlinear_Dirac_equation
Argentine children's cable TV channel
children, owned by Pramer. It was associated with the former toy importer, Cartan. Although the channel was inaugurated in mid-1989, its debut was in the
The_Big_Channel
connection called the normal Cartan connection. A conformal connection on an n-manifold M is a Cartan geometry modelled on the conformal sphere, where
Conformal_connection
Construct allowing differentiation of tangent vector fields of manifolds
than affine. In the point of view of Cartan connections, however, the affine subspaces of Euclidean space are model surfaces — they are the simplest surfaces
Affine_connection
Proposed model of gravity
(DGP) model is a model of gravity proposed by Gia Dvali, Gregory Gabadadze, and Massimo Porrati in 2000. The model is popular among some model builders
DGP_model
Special quantum field theory
φ which decouples. The sine-Gordon model is the model with the same Cartan matrix but an imaginary β. This Cartan matrix corresponds to the Lie algebra
Toda_field_theory
Idea in quantum gravity
torsion in an Einstein–Cartan spacetime. This allows one to create a theory of gravity with torsion from a world crystal model of spacetime in which the
Induced_gravity
Field theory in physics that aims to unify the fundamental forces and particles
of the known fundamental forces are mediated by fields. In the Standard Model of particle physics, three of these result from the exchange of gauge bosons
Unified_field_theory
Polish-American mathematician (1913–1998)
are now best known. Eilenberg was a member of Bourbaki and, with Henri Cartan, wrote the 1956 book Homological Algebra. Later in life he worked mainly
Samuel_Eilenberg
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
has odd parity under the Cartan involution, while h {\displaystyle {\mathfrak {h}}} has even parity. That is, denoting the Cartan involution at point p ∈
Killing_vector_field
Model of gravity with dilation
physics, Jackiw–Teitelboim gravity, also known as JT gravity or the R=T model, is a theory of gravity with a dilaton in one spatial and one time dimension
Jackiw–Teitelboim_gravity
Homogeneous quotient space of a semisimple Lie group by a parabolic subgroup
also called a parabolic geometry: any geometry that is modeled on such a space by means of a Cartan connection. The projective space Pn is an example. It
Parabolic geometry (differential geometry)
Parabolic_geometry_(differential_geometry)
Model of mesons in the massless quark limit
by the Killing form acting upon the Maurer–Cartan form of SU(N). The internal global symmetry of this model is G L × G R {\displaystyle G_{L}\times G_{R}}
Chiral_model
Research program on the symmetries of geometry
symmetry group related to each other. Later, Élie Cartan generalized Klein's homogeneous model spaces to Cartan connections on certain principal bundles, which
Erlangen_program
Object in differential geometry
relativity theory, such ideas have been implemented in the form of Einstein–Cartan theory. Let M be a manifold with an affine connection on the tangent bundle
Torsion_tensor
mouth". Historically speaking, the Séminaire Cartan of the late 1940s and early 1950s, around Henri Cartan, was one of the most influential. Publication
French_mathematical_seminars
Study of Lie groups, Lie algebras and differential equations
the areas of mathematics, and was worked out by Wilhelm Killing and Élie Cartan. The foundation of Lie theory is the exponential map relating Lie algebras
Lie_theory
Mathematical discipline
of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable lattice models in quantum statistical mechanics
Quantum_affine_algebra
Kinetic theory of gravity
corpuscles) impacting all material objects from all directions. According to this model, any two material bodies partially shield each other from the impinging
Le Sage's theory of gravitation
Le_Sage's_theory_of_gravitation
frames Cartan's equivalence method Vierbein, tetrad Cartan connection applications Einstein–Cartan theory connection (vector bundle) connection (principal
List of differential geometry topics
List_of_differential_geometry_topics
Theory of rapid universe expansion
The flatness and horizon problems are naturally solved in the Einstein–Cartan–Sciama–Kibble theory of gravity, without needing an exotic form of matter
Cosmic_inflation
Theory of gravitation as curved spacetime
From this, one can deduce that spacetime is curved. The resulting Newton–Cartan theory is a geometric formulation of Newtonian gravity using only covariant
General_relativity
Attraction of masses and energy
craft. The physical models of gravity, like all physical models, are expressed mathematically. Physicists use several different models, depending on the
Gravity
Models spontaneously breaking Lorentz symmetry
free of ghosts and tachyons, remains an open problem. Standard-Model Extension Riemann–Cartan geometry Antimatter tests of Lorentz violation Lorentz-violating
Bumblebee_models
Lie algebra, usually infinite-dimensional
that can be defined by generators and relations through a generalized Cartan matrix. These algebras form a generalization of finite-dimensional semisimple
Kac–Moody_algebra
Type of Kac–Moody algebras
{g}}} and ⟨ ⋅ | ⋅ ⟩ {\displaystyle \langle \cdot |\cdot \rangle } is the Cartan-Killing form on g . {\displaystyle {\mathfrak {g}}.} The affine Lie algebra
Affine_Lie_algebra
introducing the vector space V over Q generated by the weight lattice of a Cartan subalgebra; on the vector space of piecewise-linear paths in V connecting
Littelmann_path_model
Two geometries based on axioms closely related to those specifying Euclidean geometry
appropriate curvature to model a portion of hyperbolic space and in a second paper in the same year, defined the Klein model, which models the entirety of hyperbolic
Non-Euclidean_geometry
Proposed theories of gravity
sketch of Cartan's theory as restated by Trautman. Cartan suggested a simple generalization of Einstein's theory of gravitation. He proposed a model of space
Alternatives to general relativity
Alternatives_to_general_relativity
Hypothetical physical concept
the Standard Model of physics, a unification of all forces except gravity. The lone fundamental force not built into the Standard Model is gravity. General
Theory_of_everything
Fringe theory of physics
are eigenvalues of the Cartan subalgebra generators, and are called roots or weights of a representation. In the Standard Model of particle physics, each
An Exceptionally Simple Theory of Everything
An_Exceptionally_Simple_Theory_of_Everything
Gravity theories that are invariant under Weyl transformations
potentials around a gravitational force in this model. By adding a suitable gravitational term to the Standard Model action in curved spacetime, the theory develops
Conformal_gravity
Concept in mathematics
for example). The modern theory of pseudogroups was developed by Élie Cartan in the early 1900s. A pseudogroup imposes several conditions on sets of
Pseudogroup
Type of transport in differential geometry
Projective connections are modeled on the geometry of projective space. In modern terms, they may be described as Cartan connections modeled on projective space;
Projective_connection
Toy model of general relativity
The Callan–Giddings–Harvey–Strominger (CGHS) model is a toy model of general relativity in 1 spatial and 1 time dimension. It is named after Curtis Callan
CGHS_model
Special subset of a partially ordered set
techniques in mathematical logic. Filters on sets were introduced by Henri Cartan in 1937. Nicolas Bourbaki, in their book Topologie Générale, popularized
Filter_(mathematics)
Simple Lie group; the automorphism group of the octonions
which we now call g 2 {\displaystyle {\mathfrak {g}}_{2}} . In 1893, Élie Cartan published a note describing an open set in C 5 {\displaystyle \mathbb {C}
G2_(mathematics)
Early attempts to explain gravity
such models are no longer regarded as viable theories within the mainstream scientific community because general relativity is now the standard model to
Mechanical explanations of gravitation
Mechanical_explanations_of_gravitation
Theory of quantum gravity
by the neutron-star merger GW170817 contravene predictions made by this model of gravity. Some have revised the theory to account for this. Hořava originally
Hořava–Lifshitz_gravity
Concept in mathematics
{\displaystyle [h,f]=-2f} , and [ h , e ] = 2 e {\displaystyle [h,e]=2e} . This is a Cartan-Weyl basis for s l 2 C {\displaystyle {\mathfrak {sl}}_{2}\mathbb {C} }
Special_linear_Lie_algebra
Study of angle-preserving transformations of a geometric space
conformal connection, which can be handled either as a type of Cartan connection modelled on the associated Möbius geometry, or as a Weyl connection. This
Conformal_geometry
German-born theoretical physicist (1879–1955)
antisymmetric part, called the torsion. This modification was made by Einstein and Cartan in the 1920s. In general relativity, gravitational force is reimagined as
Albert_Einstein
Theory in modern physics that describes gravity as an entropic force
between gravity and entropy. In 2009, Erik Verlinde proposed a conceptual model that describes gravity as an entropic force. He argues (similar to Jacobson's
Entropic_gravity
78-dimensional exceptional simple Lie group
The designation E6 comes from the Cartan–Killing classification of the complex simple Lie algebras (see Élie Cartan § Work). This classifies Lie algebras
E6_(mathematics)
Pseudonym of a group of mathematicians
use dated texts. While teaching at the University of Strasbourg, Henri Cartan complained to his colleague André Weil of the inadequacy of available course
Nicolas_Bourbaki
Quantum gravity Theory of everything Classical Poincaré gauge theory Einstein–Cartan Teleparallelism Bimetric theories Gauge theory gravity Composite gravity
Gravity_filtration
Type of metric space in mathematics
{CAT} (k)} was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov (although
CAT(k)_space
Generalization of a complex manifold that allows the use of singularities
S2CID 122113902. Cartan, H.; Bruhat, F.; Cerf, Jean; Dolbeault, P.; Frenkel, Jean; Hervé, Michel; Malatian; Serre, J-P. "Séminaire Henri Cartan, Tome 4 (1951-1952)"
Complex_analytic_variety
mathematical physics, nonlinear realization of a Lie group G possessing a Cartan subgroup H is a particular induced representation of G. In fact, it is a
Nonlinear_realization
Family of subsets representing "large" sets
neighborhoods of a point in a topological space. Filters were introduced by Henri Cartan in 1937 in the context of general topological spaces and were subsequently
Filter_on_a_set
American actress
ISSN 0362-4331. Retrieved May 4, 2021. Amber Gray, Jennifer Laura Thompson, Ryan McCartan, More Featured in Industry Reading of Into the Wild Musical Theatre World
Amber_Gray
Theoretical model of gravity
model for emergent or induced gravity in an Einstein–Cartan theory of gravitation (which embraces Einstein's theory of General Relativity). The model
World_crystal
French painter (1886–1970)
Hermine Lionette Cartan David (19 April 1886 – 1 December 1970) was a French painter. Hermine David was born in Paris in 1886. She was born out of wedlock;
Hermine_David
Geometric model of the physical space
space" when the context is clear). In classical physics, it serves as a model of the physical universe, in which all known matter exists. When relativity
Three-dimensional_space
Country in Eastern Europe and North Asia
Archived from the original on 19 March 2022. Retrieved 13 January 2022. McCartan, E. F. (1963). "The Long Voyages-Early Russian Circumnavigation". The Russian
Russia
Theory of gravity in which the graviton has nonzero mass
Despite the fact that the OP model coincides with the ghost-free massive gravity models rediscovered in dRGT, the OP model has been almost unknown among
Massive_gravity
Topics referred to by the same term
bundle along a vector field Cartan connection, achieved by identifying tangent spaces with the tangent space of a certain model Klein geometry Ehresmann
Connection
Theory of quantum gravity merging quantum mechanics and general relativity
categoryPages displaying short descriptions of redirect targets Einstein–Cartan theory – Classical theory of gravitation Rovelli 2008. Ashtekar, Abhay (3
Loop_quantum_gravity
Theory extending Einstein gravity
1016/j.physletb.2020.135773. Salvio, Alberto (1 July 2019). "Quasi-Conformal Models and the Early Universe". The European Physical Journal C. 79 (9) 750. arXiv:1907
Quadratic_gravity
Isomorphism of commutative rings constructed in the theory of Lie algebras
of the symmetric algebra S ( h ) {\displaystyle S({\mathfrak {h}})} of a Cartan subalgebra h {\displaystyle {\mathfrak {h}}} that are invariant under the
Harish-Chandra_isomorphism
Unified field theory
the Standard Model, SU(3) × SU(2) × U(1). However, an attempt to convert this interesting geometrical construction into a bona-fide model of reality flounders
Kaluza–Klein_theory
Aspect of astrophysics history
mathematician Élie Cartan formulated Einstein's theory in the language of bundles and connections, a generalization of Riemannian geometry to which Cartan made important
History of loop quantum gravity
History_of_loop_quantum_gravity
Increase in distance between parts of the universe
predicted by Newtonian gravity formulated in the geometrical language of Cartan. This avoids fundamental problems of Newtonian gravity in an infinite Euclidean
Expansion_of_the_universe
Model for the origin of the universe
showed that a nonsingular Big Bounce appears naturally in the Einstein–Cartan–Sciama–Kibble theory of gravity. This theory extends general relativity
Big_Bounce
Conformal anomaly free CGHS model
The Russo–Susskind–Thorlacius model or RST model in short is a modification of the CGHS model to take care of conformal anomalies and render it analytically
RST_model
52-dimensional exceptional simple Lie group
1 {\displaystyle C_{2}=1} with three distinct principal curvatures, E. Cartan, 1939). The characters of finite dimensional representations of the real
F4_(mathematics)
Quantum gravity Theory of everything Classical Poincaré gauge theory Einstein–Cartan Teleparallelism Bimetric theories Gauge theory gravity Composite gravity
Bi-scalar tensor vector gravity
Bi-scalar_tensor_vector_gravity
Concept in physics
Quantum gravity Theory of everything Classical Poincaré gauge theory Einstein–Cartan Teleparallelism Bimetric theories Gauge theory gravity Composite gravity
Nonsymmetric gravitational theory
Nonsymmetric_gravitational_theory
Attempt to describe spacetime and associated phenomena in terms of geometry
curvature and the Bianchi identities via topological ghosts. In such a graded Cartan formalism, the nilpotency of the ghost operators is on par with the Poincaré
Geometrodynamics
characteristic zero can be described by Maurer–Cartan elements of an appropriate differential graded Lie algebra. A Maurer-Cartan element is a degree −1 element, x
Differential graded Lie algebra
Differential_graded_Lie_algebra
non-Euclidean geometry Henry Frederick Baker (1866–1956) – algebraic geometry Élie Cartan (1869–1951) Dmitri Egorov (1869–1931) – differential geometry Veniamin Kagan
List_of_geometers
Cosmological fine-tuning problem
also to avoid it. The flatness problem is naturally solved by the Einstein–Cartan–Sciama–Kibble theory of gravity, without an exotic form of matter required
Flatness_problem
Restatement of Newton's law of universal gravitation
Quantum gravity Theory of everything Classical Poincaré gauge theory Einstein–Cartan Teleparallelism Bimetric theories Gauge theory gravity Composite gravity
Gauss's_law_for_gravity
Geometric algebra approach to gravity
principle. A spin tensor can also be supported in a manner similar to Einstein–Cartan–Sciama–Kibble theory. GTG was first proposed by Lasenby, Doran, and Gull
Gauge_theory_gravity
Topics referred to by the same term
a German whitelist for bulk email senders Chinese Islamic Association Cartan subalgebra Central simple algebra Client SMTP Authorization Common Scrambling
CSA
comparison theorem (differential equations) Cartan–Kähler theorem (partial differential equations) Cartan–Kuranishi prolongation theorem (partial differential
List_of_theorems
Geometric model of the planar projection of the physical universe
Ahmes Alhazen Apollonius Archimedes Atiyah Baudhayana Bolyai Brahmagupta Cartan Chern Coxeter Descartes Euclid Euler Gauss Gromov Hilbert Huygens Jyeṣṭhadeva
Euclidean_plane
Nonpertubative field theoretic approach to quantum gravity
particle physics, astrophysics and cosmology, for instance. The Standard Model in combination with asymptotic safety might be valid up to arbitrarily high
Physics applications of asymptotically safe gravity
Physics_applications_of_asymptotically_safe_gravity
Line along which a quadratic form applied to any two points' displacement is zero
collection of all such isotropic lines forms the light cone at the origin. Élie Cartan expanded the concept of isotropic lines to multivectors in his book on spinors
Isotropic_line
General relativity in 2+1 dimensions
for a positive one. This theory can be exactly solved, making it a toy model for quantum gravity. In this formulation the action can be written using
(2+1)-dimensional topological gravity
(2+1)-dimensional_topological_gravity
Non-tensorial representation of the spin group
quadratically from a spinor. Spinors were introduced in geometry by Élie Cartan in 1913. In the 1920s physicists discovered that spinors are essential to
Spinor
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