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59 relations: ADE classification, Algebraic group, ATLAS of Finite Groups, Automorphism, Cartan subalgebra, Cartan subgroup, Cayley plane, Chevalley basis, Classification of finite simple groups, Compact space, Complex dimension, Convex hull, Coxeter group, Cubic surface, Cyclic group, Dimensional reduction, Dual representation, Dynkin diagram, E6 polytope, E7 (mathematics), E8 (mathematics), En (Lie algebra), F4 (mathematics), Finite field, Freudenthal magic square, Fundamental representation, G2 (mathematics), Galois cohomology, Gauge theory, Graduate Texts in Mathematics, Grand Unified Theory, Group of Lie type, Index of a subgroup, Isometry group, Jordan algebra, Lang's theorem, Lie algebra, Lie group, Linear span, Mathematics, Outer automorphism group, Particle physics, Perfect field, Riemannian manifold, Root system, Schur multiplier, Simple group, Simple Lie group, Singlet state, Springer Science+Business Media, ..., Standard Model, Supergravity, Symmetry breaking, Symmetry group, University of Chicago Press, Vector space, Weyl character formula, Weyl equation, Weyl group. Expand index (9 more) »
ADE classification
In mathematics, the ADE classification (originally A-D-E classifications) is a situation where certain kinds of objects are in correspondence with simply laced Dynkin diagrams.
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Algebraic group
In algebraic geometry, an algebraic group (or group variety) is a group that is an algebraic variety, such that the multiplication and inversion operations are given by regular maps on the variety.
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ATLAS of Finite Groups
The ATLAS of Finite Groups, often simply known as the ATLAS, is a group theory book by John Horton Conway, Robert Turner Curtis, Simon Phillips Norton, Richard Alan Parker and Robert Arnott Wilson (with computational assistance from J. G. Thackray), published in December 1985 by Oxford University Press and reprinted with corrections in 2003.
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Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself.
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Cartan subalgebra
In mathematics, a Cartan subalgebra, often abbreviated as CSA, is a nilpotent subalgebra \mathfrak of a Lie algebra \mathfrak that is self-normalising (if \in \mathfrak for all X \in \mathfrak, then Y \in \mathfrak).
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Cartan subgroup
In mathematics, a Cartan subgroup of a Lie group or algebraic group G is one of the subgroups whose Lie algebra is a Cartan subalgebra.
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Cayley plane
In mathematics, the Cayley plane (or octonionic projective plane) P2(O) is a projective plane over the octonions.
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Chevalley basis
In mathematics, a Chevalley basis for a simple complex Lie algebra is a basis constructed by Claude Chevalley with the property that all structure constants are integers.
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Classification of finite simple groups
In mathematics, the classification of the finite simple groups is a theorem stating that every finite simple group belongs to one of four broad classes described below.
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Compact space
In mathematics, and more specifically in general topology, compactness is a property that generalizes the notion of a subset of Euclidean space being closed (that is, containing all its limit points) and bounded (that is, having all its points lie within some fixed distance of each other).
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Complex dimension
In mathematics, complex dimension usually refers to the dimension of a complex manifold M, or a complex algebraic variety V. If the complex dimension is d, the real dimension will be 2d.
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Convex hull
In mathematics, the convex hull or convex envelope or convex closure of a set X of points in the Euclidean plane or in a Euclidean space (or, more generally, in an affine space over the reals) is the smallest convex set that contains X. For instance, when X is a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around X., p. 3.
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Coxeter group
In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors).
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Cubic surface
A cubic surface is a projective variety studied in algebraic geometry.
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Cyclic group
In algebra, a cyclic group or monogenous group is a group that is generated by a single element.
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Dimensional reduction
Dimensional reduction is the limit of a compactified theory where the size of the compact dimension goes to zero.
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Dual representation
In mathematics, if is a group and is a linear representation of it on the vector space, then the dual representation is defined over the dual vector space as follows: The dual representation is also known as the contragredient representation.
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Dynkin diagram
In the mathematical field of Lie theory, a Dynkin diagram, named for Eugene Dynkin, is a type of graph with some edges doubled or tripled (drawn as a double or triple line).
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E6 polytope
In 6-dimensional geometry, there are 39 uniform polytopes with E6 symmetry.
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E7 (mathematics)
In mathematics, E7 is the name of several closely related Lie groups, linear algebraic groups or their Lie algebras e7, all of which have dimension 133; the same notation E7 is used for the corresponding root lattice, which has rank 7.
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E8 (mathematics)
In mathematics, E8 is any of several closely related exceptional simple Lie groups, linear algebraic groups or Lie algebras of dimension 248; the same notation is used for the corresponding root lattice, which has rank 8.
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En (Lie algebra)
In mathematics, especially in Lie theory, En is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1,2, and k, with k.
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F4 (mathematics)
In mathematics, F4 is the name of a Lie group and also its Lie algebra f4.
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Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements.
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Freudenthal magic square
In mathematics, the Freudenthal magic square (or Freudenthal–Tits magic square) is a construction relating several Lie algebras (and their associated Lie groups).
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Fundamental representation
In representation theory of Lie groups and Lie algebras, a fundamental representation is an irreducible finite-dimensional representation of a semisimple Lie group or Lie algebra whose highest weight is a fundamental weight.
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G2 (mathematics)
In mathematics, G2 is the name of three simple Lie groups (a complex form, a compact real form and a split real form), their Lie algebras \mathfrak_2, as well as some algebraic groups.
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Galois cohomology
In mathematics, Galois cohomology is the study of the group cohomology of Galois modules, that is, the application of homological algebra to modules for Galois groups.
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Gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under certain Lie groups of local transformations.
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Graduate Texts in Mathematics
Graduate Texts in Mathematics (GTM) (ISSN 0072-5285) is a series of graduate-level textbooks in mathematics published by Springer-Verlag.
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Grand Unified Theory
A Grand Unified Theory (GUT) is a model in particle physics in which, at high energy, the three gauge interactions of the Standard Model which define the electromagnetic, weak, and strong interactions, or forces, are merged into one single force.
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Group of Lie type
In mathematics, specifically in group theory, the phrase group of Lie type usually refers to finite groups that are closely related to the group of rational points of a reductive linear algebraic group with values in a finite field.
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Index of a subgroup
In mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" (cosets) of H that fill up G. For example, if H has index 2 in G, then intuitively half of the elements of G lie in H. The index of H in G is usually denoted |G: H| or or (G:H).
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Isometry group
In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation.
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Jordan algebra
In abstract algebra, a Jordan algebra is an nonassociative algebra over a field whose multiplication satisfies the following axioms.
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Lang's theorem
In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if G is a connected smooth algebraic group over a finite field \mathbf_q, then, writing \sigma: G \to G, \, x \mapsto x^q for the Frobenius, the morphism of varieties is surjective.
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Lie algebra
In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.
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Lie group
In mathematics, a Lie group (pronounced "Lee") is a group that is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure.
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Linear span
In linear algebra, the linear span (also called the linear hull or just span) of a set of vectors in a vector space is the intersection of all subspaces containing that set.
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Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
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Outer automorphism group
In mathematics, the outer automorphism group of a group,, is the quotient,, where is the automorphism group of and) is the subgroup consisting of inner automorphisms.
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Particle physics
Particle physics (also high energy physics) is the branch of physics that studies the nature of the particles that constitute matter and radiation.
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Perfect field
In algebra, a field k is said to be perfect if any one of the following equivalent conditions holds.
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Riemannian manifold
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.
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Root system
In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties.
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Schur multiplier
In mathematical group theory, the Schur multiplier or Schur multiplicator is the second homology group H2(G, Z) of a group G. It was introduced by in his work on projective representations.
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Simple group
In mathematics, a simple group is a nontrivial group whose only normal subgroups are the trivial group and the group itself.
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Simple Lie group
In group theory, a simple Lie group is a connected non-abelian Lie group G which does not have nontrivial connected normal subgroups.
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Singlet state
In quantum mechanics, a singlet state usually refers to a system in which all electrons are paired.
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Springer Science+Business Media
Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.
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Standard Model
The Standard Model of particle physics is the theory describing three of the four known fundamental forces (the electromagnetic, weak, and strong interactions, and not including the gravitational force) in the universe, as well as classifying all known elementary particles.
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Supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity where supersymmetry obeys locality; in contrast to non-gravitational supersymmetric theories such as the Minimal Supersymmetric Standard Model.
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Symmetry breaking
In physics, symmetry breaking is a phenomenon in which (infinitesimally) small fluctuations acting on a system crossing a critical point decide the system's fate, by determining which branch of a bifurcation is taken.
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Symmetry group
In group theory, the symmetry group of an object (image, signal, etc.) is the group of all transformations under which the object is invariant with composition as the group operation.
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University of Chicago Press
The University of Chicago Press is the largest and one of the oldest university presses in the United States.
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Vector space
A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.
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Weyl character formula
In mathematics, the Weyl character formula in representation theory describes the characters of irreducible representations of compact Lie groups in terms of their highest weights.
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Weyl equation
In physics, particularly quantum field theory, the Weyl equation is a relativistic wave equation for describing massless spin-1/2 particles called Weyl fermions.
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Weyl group
In mathematics, in particular the theory of Lie algebras, the Weyl group of a root system Φ is a subgroup of the isometry group of the root system.
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E6 (Mathematics), E6 (math), E₆.
References
[1] https://en.wikipedia.org/wiki/E6_(mathematics)
