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105 relations: Affine Lie algebra, An Exceptionally Simple Theory of Everything, Antony Garrett Lisi, Automorphisms of the symmetric and alternating groups, Élie Cartan, Bimonster group, Capelli's identity, Cartan connection, Cartan matrix, Catastrophe theory, Cayley plane, Chevalley basis, Classical group, Cluster algebra, Compact group, Coxeter group, Coxeter–Dynkin diagram, Cuboctahedron, David Vogan, Dynkin diagram, E6 (mathematics), E7 (mathematics), E8 (mathematics), Eugene Dynkin, Exceptional isomorphism, Exceptional object, F4 (mathematics), First class constraint, Freudenthal magic square, G2 (mathematics), G2 manifold, Georgi–Glashow model, Glossary of semisimple groups, Grand unification energy, Grand Unified Theory, Group of Lie type, Heptellated 8-simplexes, Hexagon, Hexicated 7-simplexes, Hurwitz's automorphisms theorem, Kac–Moody algebra, Kazhdan's property (T), Killing form, Lace (disambiguation), Lattice (discrete subgroup), Lattice (group), Lie algebra, Lie algebra extension, Lie group, Linear algebraic group, ..., List of group theory topics, List of Lie groups topics, List of long mathematical proofs, List of simple Lie groups, List of string theory topics, List of things named after Sophus Lie, Mostow rigidity theorem, Pathological (mathematics), Pentellated 6-simplexes, Petrie polygon, Pierre Deligne, Quadratic Lie algebra, Quaternion-Kähler symmetric space, Rectified 10-orthoplexes, Rectified 5-orthoplexes, Rectified 6-orthoplexes, Rectified 7-orthoplexes, Rectified 8-orthoplexes, Rectified 9-orthoplexes, Reductive group, Representation theory of SL2(R), Root system, Runcinated 5-cell, Semisimple Lie algebra, Serre's conjecture II (algebra), Simple group, Skip Garibaldi, SL2(R), Smooth projective plane, SO(5), SO(8), Special unitary group, Spin group, Stericated 5-simplexes, Structure constants, Symmetric space, Symplectic group, Symplectomorphism, Thorold Gosset, Topological geometry, Trace (linear algebra), Transversality (mathematics), Triality, Uniform polytope, Vacuum angle, Vladimir Drinfeld, Vogel plane, Wess–Zumino–Witten model, Why Beauty Is Truth, Wilhelm Killing, 1 22 polytope, 2 31 polytope, 24-cell, 4 21 polytope, 42 (number). Expand index (55 more) »
Affine Lie algebra
In mathematics, an affine Lie algebra is an infinite-dimensional Lie algebra that is constructed in a canonical fashion out of a finite-dimensional simple Lie algebra.
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An Exceptionally Simple Theory of Everything
"An Exceptionally Simple Theory of Everything" is a physics preprint proposing a basis for a unified field theory, often referred to as "E8 Theory", which attempts to describe all known fundamental interactions in physics and to stand as a possible theory of everything.
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Antony Garrett Lisi
Antony Garrett Lisi (born January 24, 1968), known as Garrett Lisi, is an American theoretical physicist and adventure sports enthusiast.
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Automorphisms of the symmetric and alternating groups
In group theory, a branch of mathematics, the automorphisms and outer automorphisms of the symmetric groups and alternating groups are both standard examples of these automorphisms, and objects of study in their own right, particularly the exceptional outer automorphism of S6, the symmetric group on 6 elements.
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Élie Cartan
Élie Joseph Cartan, ForMemRS (9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups and their geometric applications.
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Bimonster group
In mathematics, the bimonster is a group that is the wreath product of the monster group M with Z2: The Bimonster is also a quotient of the Coxeter group corresponding to the Dynkin diagram Y555, a Y-shaped graph with 16 nodes: John H. Conway conjectured that a presentation of the bimonster could be given by adding a certain extra relation to the presentation defined by the Y555 diagram; this was proved in 1990 by A. A. Ivanov and Simon P. Norton.
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Capelli's identity
In mathematics, Capelli's identity, named after, is an analogue of the formula det(AB).
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Cartan connection
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection.
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Cartan matrix
In mathematics, the term Cartan matrix has three meanings.
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Catastrophe theory
In mathematics, catastrophe theory is a branch of bifurcation theory in the study of dynamical systems; it is also a particular special case of more general singularity theory in geometry.
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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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Classical group
In mathematics, the classical groups are defined as the special linear groups over the reals, the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces.
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Cluster algebra
Cluster algebras are a class of commutative rings introduced by.
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Compact group
In mathematics, a compact (topological) group is a topological group whose topology is compact.
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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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Coxeter–Dynkin diagram
In geometry, a Coxeter–Dynkin diagram (or Coxeter diagram, Coxeter graph) is a graph with numerically labeled edges (called branches) representing the spatial relations between a collection of mirrors (or reflecting hyperplanes).
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Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces.
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David Vogan
David Alexander Vogan, Jr. (born September 8, 1954) is a mathematician at the Massachusetts Institute of Technology who works on unitary representations of simple Lie groups.
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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 (mathematics)
In mathematics, E6 is the name of some closely related Lie groups, linear algebraic groups or their Lie algebras \mathfrak_6, all of which have dimension 78; the same notation E6 is used for the corresponding root lattice, which has rank 6.
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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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Eugene Dynkin
Eugene Borisovich Dynkin (Евге́ний Бори́сович Ды́нкин; 11 May 1924 – 14 November 2014) was a Soviet and American mathematician.
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Exceptional isomorphism
In mathematics, an exceptional isomorphism, also called an accidental isomorphism, is an isomorphism between members ai and bj of two families, usually infinite, of mathematical objects, that is not an example of a pattern of such isomorphisms.
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Exceptional object
Many branches of mathematics study objects of a given type and prove a classification theorem.
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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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First class constraint
A first class constraint is a dynamical quantity in a constrained Hamiltonian system whose Poisson bracket with all the other constraints vanishes on the constraint surface in phase space (the surface implicitly defined by the simultaneous vanishing of all the constraints).
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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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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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G2 manifold
In differential geometry, a G2 manifold is a seven-dimensional Riemannian manifold with holonomy group equal to ''G''2.
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Georgi–Glashow model
In particle physics, the Georgi–Glashow model is a particular grand unification theory (GUT) proposed by Howard Georgi and Sheldon Glashow in 1974.
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Glossary of semisimple groups
This is a glossary for the terminology applied in the mathematical theories of semisimple Lie groups.
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Grand unification energy
The grand unification energy \Lambda_, or the GUT scale, is the energy level above which, it is believed, the electromagnetic force, weak force, and strong force become equal in strength and unify to one force governed by a simple Lie group.
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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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Heptellated 8-simplexes
In eight-dimensional geometry, a heptellated 8-simplex is a convex uniform 8-polytope, including 7th-order truncations (heptellation) from the regular 8-simplex.
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Hexagon
In geometry, a hexagon (from Greek ἕξ hex, "six" and γωνία, gonía, "corner, angle") is a six-sided polygon or 6-gon.
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Hexicated 7-simplexes
In seven-dimensional geometry, a hexicated 7-simplex is a convex uniform 7-polytope, including 6th-order truncations (hexication) from the regular 7-simplex.
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Hurwitz's automorphisms theorem
In mathematics, Hurwitz's automorphisms theorem bounds the order of the group of automorphisms, via orientation-preserving conformal mappings, of a compact Riemann surface of genus g > 1, stating that the number of such automorphisms cannot exceed 84(g − 1).
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Kac–Moody algebra
In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently discovered them) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix.
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Kazhdan's property (T)
In mathematics, a locally compact topological group G has property (T) if the trivial representation is an isolated point in its unitary dual equipped with the Fell topology.
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Killing form
In mathematics, the Killing form, named after Wilhelm Killing, is a symmetric bilinear form that plays a basic role in the theories of Lie groups and Lie algebras.
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Lace (disambiguation)
Lace is a lightweight fabric patterned with open holes.
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Lattice (discrete subgroup)
In Lie theory and related areas of mathematics, a lattice in a locally compact group is a discrete subgroup with the property that the quotient space has finite invariant measure.
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Lattice (group)
In geometry and group theory, a lattice in \mathbbR^n is a subgroup of the additive group \mathbb^n which is isomorphic to the additive group \mathbbZ^n, and which spans the real vector space \mathbb^n.
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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 algebra extension
In the theory of Lie groups, Lie algebras and their representation theory, a Lie algebra extension is an enlargement of a given Lie algebra by another Lie algebra.
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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 algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible n×n matrices (under matrix multiplication) that is defined by polynomial equations.
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List of group theory topics
No description.
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List of Lie groups topics
This is a list of Lie group topics, by Wikipedia page.
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List of long mathematical proofs
This is a list of unusually long mathematical proofs.
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List of simple Lie groups
In mathematics, the simple Lie groups were first classified by Wilhelm Killing and later perfected by Élie Cartan.
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List of string theory topics
This is a list of string theory topics.
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List of things named after Sophus Lie
This is a list of things named after Sophus Lie.
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Mostow rigidity theorem
In mathematics, Mostow's rigidity theorem, or strong rigidity theorem, or Mostow–Prasad rigidity theorem, essentially states that the geometry of a complete, finite-volume hyperbolic manifold of dimension greater than two is determined by the fundamental group and hence unique.
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Pathological (mathematics)
In mathematics, a pathological phenomenon is one whose properties are considered atypically bad or counterintuitive; the opposite is well-behaved.
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Pentellated 6-simplexes
In six-dimensional geometry, a pentellated 6-simplex is a convex uniform 6-polytope with 5th order truncations of the regular 6-simplex.
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Petrie polygon
In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every (n – 1) consecutive sides (but no n) belongs to one of the facets.
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Pierre Deligne
Pierre René, Viscount Deligne (born 3 October 1944) is a Belgian mathematician.
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Quadratic Lie algebra
A quadratic Lie algebra is a Lie algebra together with a compatible symmetric bilinear form.
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Quaternion-Kähler symmetric space
In differential geometry, a quaternion-Kähler symmetric space or Wolf space is a quaternion-Kähler manifold which, as a Riemannian manifold, is a Riemannian symmetric space.
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Rectified 10-orthoplexes
In ten-dimensional geometry, a rectified 10-orthoplex is a convex uniform 10-polytope, being a rectification of the regular 10-orthoplex.
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Rectified 5-orthoplexes
In five-dimensional geometry, a rectified 5-orthoplex is a convex uniform 5-polytope, being a rectification of the regular 5-orthoplex.
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Rectified 6-orthoplexes
In six-dimensional geometry, a rectified 6-orthoplex is a convex uniform 6-polytope, being a rectification of the regular 6-orthoplex.
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Rectified 7-orthoplexes
In seven-dimensional geometry, a rectified 7-orthoplex is a convex uniform 7-polytope, being a rectification of the regular 7-orthoplex.
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Rectified 8-orthoplexes
In eight-dimensional geometry, a rectified 8-orthoplex is a convex uniform 8-polytope, being a rectification of the regular 8-orthoplex.
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Rectified 9-orthoplexes
In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-orthoplex.
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Reductive group
In mathematics, a reductive group is a type of linear algebraic group over a field.
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Representation theory of SL2(R)
In mathematics, the main results concerning irreducible unitary representations of the Lie group SL(2,'''R''') are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952).
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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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Runcinated 5-cell
In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination (a 3rd order truncation, up to face-planing) of the regular 5-cell.
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Semisimple Lie algebra
In mathematics, a Lie algebra is semisimple if it is a direct sum of simple Lie algebras, i.e., non-abelian Lie algebras \mathfrak g whose only ideals are and \mathfrak g itself.
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Serre's conjecture II (algebra)
In mathematics, Jean-Pierre Serre conjectured the following statement regarding the Galois cohomology of a simply connected semisimple algebraic group.
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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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Skip Garibaldi
Skip Garibaldi is an American mathematician doing research on algebraic groups and especially exceptional groups.
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SL2(R)
In mathematics, the special linear group SL(2,R) or SL2(R) is the group of 2 × 2 real matrices with determinant one: a & b \\ c & d \end \right): a,b,c,d\in\mathbf\mboxad-bc.
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Smooth projective plane
In geometry, smooth projective planes are special projective planes.
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SO(5)
In mathematics, SO(5), also denoted SO5(R) or SO(5,R), is the special orthogonal group of degree 5 over the field R of real numbers, i.e. (isomorphic to) the group of orthogonal 5×5 matrices of determinant 1.
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SO(8)
In mathematics, SO(8) is the special orthogonal group acting on eight-dimensional Euclidean space.
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Special unitary group
In mathematics, the special unitary group of degree, denoted, is the Lie group of unitary matrices with determinant 1.
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Spin group
In mathematics the spin group Spin(n) is the double cover of the special orthogonal group, such that there exists a short exact sequence of Lie groups (with) As a Lie group, Spin(n) therefore shares its dimension,, and its Lie algebra with the special orthogonal group.
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Stericated 5-simplexes
In five-dimensional geometry, a stericated 5-simplex is a convex uniform 5-polytope with fourth-order truncations (sterication) of the regular 5-simplex.
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Structure constants
In mathematics, the structure constants or structure coefficients of an algebra over a field are used to explicitly specify the product of two basis vectors in the algebra as a linear combination.
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Symmetric space
In differential geometry, representation theory and harmonic analysis, a symmetric space is a pseudo-Riemannian manifold whose group of symmetries contains an inversion symmetry about every point.
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Symplectic group
In mathematics, the name symplectic group can refer to two different, but closely related, collections of mathematical groups, denoted and, the latter is called the compact symplectic group.
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Symplectomorphism
In mathematics, a symplectomorphism or symplectic map is an isomorphism in the category of symplectic manifolds.
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Thorold Gosset
John Herbert de Paz Thorold Gosset (16 October 1869 – December 1962) was an English lawyer and an amateur mathematician.
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Topological geometry
Topological geometry deals with incidence structures consisting of a point set P and a family \mathfrak of subsets of P called lines or circles etc.
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Trace (linear algebra)
In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis.
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Transversality (mathematics)
In mathematics, transversality is a notion that describes how spaces can intersect; transversality can be seen as the "opposite" of tangency, and plays a role in general position.
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Triality
In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces.
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Uniform polytope
A uniform polytope of dimension three or higher is a vertex-transitive polytope bounded by uniform facets.
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Vacuum angle
In quantum gauge theories, in the Hamiltonian formulation (Hamiltonian system), the wave function is a functional of the gauge connection \,A and matter fields \,\phi.
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Vladimir Drinfeld
Vladimir Gershonovich Drinfeld (Володи́мир Ге́ршонович Дрінфельд; Влади́мир Ге́ршонович Дри́нфельд; born February 14, 1954), surname also romanized as Drinfel'd, is a Ukrainian-American mathematician, currently working at the University of Chicago.
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Vogel plane
In mathematics, the Vogel plane is a method of parameterizing simple Lie algebras by eigenvalues α, β, γ of the Casimir operator on the symmetric square of the Lie algebra, which gives a point (α: β: γ) of P2/S3, the projective plane P2 divided out by the symmetric group S3 of permutations of coordinates.
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Wess–Zumino–Witten model
In theoretical physics and mathematics, a Wess–Zumino–Witten (WZW) model, also called a Wess–Zumino–Novikov–Witten model, is a type of two-dimensional conformal field theory named after Julius Wess, Bruno Zumino, Sergei Novikov and Edward Witten.
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Why Beauty Is Truth
Why Beauty Is Truth: A History of Symmetry is a 2007 book by Ian Stewart.
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Wilhelm Killing
Wilhelm Karl Joseph Killing (10 May 1847 – 11 February 1923) was a German mathematician who made important contributions to the theories of Lie algebras, Lie groups, and non-Euclidean geometry.
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1 22 polytope
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.
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2 31 polytope
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.
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24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
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4 21 polytope
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group.
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42 (number)
42 (forty-two) is the natural number that succeeds 41 and precedes 43.
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References
[1] https://en.wikipedia.org/wiki/Simple_Lie_group
