Faster access than browser!
78 relations: ADE classification, Adjoint representation, Affine root system, Algebraic group, Bijection, Cartan matrix, Cartan subalgebra, Characteristic polynomial, Combinatorics, Compact group, Complex number, Congruence (geometry), Coordinate system, Coxeter group, Coxeter–Dynkin diagram, Cross-polytope, Cubic crystal system, Cuboctahedron, Cyclic permutation, Dot product, Dynkin diagram, E6 (mathematics), E7 (mathematics), E8 (mathematics), E8 lattice, Euclidean space, Eugene Dynkin, F4 (mathematics), Field (mathematics), G2 (mathematics), Graded poset, Graph (discrete mathematics), Group (mathematics), Group action, Half-integer, Hexagonal lattice, Hexagram, Hurwitz quaternion, Hyperplane, Integer, Isometry, Isomorphism, John Horton Conway, Killing form, Lattice (discrete subgroup), Lie algebra, Lie group, Linear span, Linear subspace, Mathematics, ..., Mathematische Annalen, Neil Sloane, Parity (mathematics), Partially ordered set, Permutation, Permutation group, Perpendicular, Plant, Reflection (mathematics), Root datum, Root system of a semi-simple Lie algebra, Semisimple Lie algebra, Simple Lie group, Simply connected space, Singularity theory, Spectral graph theory, Square lattice, Standard basis, Triality, Vector space, Weight (representation theory), Weyl group, Wilhelm Killing, Zome, 1 22 polytope, 2 31 polytope, 24-cell, 4 21 polytope. Expand index (28 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.
New!!: Root system and ADE classification · See more »
Adjoint representation
In mathematics, the adjoint representation (or adjoint action) of a Lie group G is a way of representing the elements of the group as linear transformations of the group's Lie algebra, considered as a vector space.
New!!: Root system and Adjoint representation · See more »
Affine root system
In mathematics, an affine root system is a root system of affine-linear functions on a Euclidean space.
New!!: Root system and Affine root system · See more »
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.
New!!: Root system and Algebraic group · See more »
Bijection
In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other set, and each element of the other set is paired with exactly one element of the first set.
New!!: Root system and Bijection · See more »
Cartan matrix
In mathematics, the term Cartan matrix has three meanings.
New!!: Root system and Cartan matrix · See more »
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).
New!!: Root system and Cartan subalgebra · See more »
Characteristic polynomial
In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots.
New!!: Root system and Characteristic polynomial · See more »
Combinatorics
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures.
New!!: Root system and Combinatorics · See more »
Compact group
In mathematics, a compact (topological) group is a topological group whose topology is compact.
New!!: Root system and Compact group · See more »
Complex number
A complex number is a number that can be expressed in the form, where and are real numbers, and is a solution of the equation.
New!!: Root system and Complex number · See more »
Congruence (geometry)
In geometry, two figures or objects are congruent if they have the same shape and size, or if one has the same shape and size as the mirror image of the other.
New!!: Root system and Congruence (geometry) · See more »
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of the points or other geometric elements on a manifold such as Euclidean space.
New!!: Root system and Coordinate system · See more »
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).
New!!: Root system and Coxeter group · See more »
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).
New!!: Root system and Coxeter–Dynkin diagram · See more »
Cross-polytope
In geometry, a cross-polytope, orthoplex, hyperoctahedron, or cocube is a regular, convex polytope that exists in n-dimensions.
New!!: Root system and Cross-polytope · See more »
Cubic crystal system
In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube.
New!!: Root system and Cubic crystal system · See more »
Cuboctahedron
In geometry, a cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces.
New!!: Root system and Cuboctahedron · See more »
Cyclic permutation
In mathematics, and in particular in group theory, a cyclic permutation (or cycle) is a permutation of the elements of some set X which maps the elements of some subset S of X to each other in a cyclic fashion, while fixing (that is, mapping to themselves) all other elements of X. If S has k elements, the cycle is called a k-cycle.
New!!: Root system and Cyclic permutation · See more »
Dot product
In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.
New!!: Root system and Dot product · See more »
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).
New!!: Root system and Dynkin diagram · See more »
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.
New!!: Root system and E6 (mathematics) · See more »
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.
New!!: Root system and E7 (mathematics) · See more »
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.
New!!: Root system and E8 (mathematics) · See more »
E8 lattice
In mathematics, the E8 lattice is a special lattice in R8.
New!!: Root system and E8 lattice · See more »
Euclidean space
In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.
New!!: Root system and Euclidean space · See more »
Eugene Dynkin
Eugene Borisovich Dynkin (Евге́ний Бори́сович Ды́нкин; 11 May 1924 – 14 November 2014) was a Soviet and American mathematician.
New!!: Root system and Eugene Dynkin · See more »
F4 (mathematics)
In mathematics, F4 is the name of a Lie group and also its Lie algebra f4.
New!!: Root system and F4 (mathematics) · See more »
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined, and behave as when they are applied to rational and real numbers.
New!!: Root system and Field (mathematics) · See more »
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.
New!!: Root system and G2 (mathematics) · See more »
Graded poset
In mathematics, in the branch of combinatorics, a graded poset is a partially ordered set (poset) P equipped with a rank function ρ from P to N satisfying the following two properties.
New!!: Root system and Graded poset · See more »
Graph (discrete mathematics)
In mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related".
New!!: Root system and Graph (discrete mathematics) · See more »
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set of elements equipped with an operation that combines any two elements to form a third element and that satisfies four conditions called the group axioms, namely closure, associativity, identity and invertibility.
New!!: Root system and Group (mathematics) · See more »
Group action
In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.
New!!: Root system and Group action · See more »
Half-integer
In mathematics, a half-integer is a number of the form where n is an integer.
New!!: Root system and Half-integer · See more »
Hexagonal lattice
The hexagonal lattice or triangular lattice is one of the five 2D lattice types.
New!!: Root system and Hexagonal lattice · See more »
Hexagram
A hexagram (Greek) or sexagram (Latin) is a six-pointed geometric star figure with the Schläfli symbol, 2, or.
New!!: Root system and Hexagram · See more »
Hurwitz quaternion
In mathematics, a Hurwitz quaternion (or Hurwitz integer) is a quaternion whose components are either all integers or all half-integers (halves of an odd integer; a mixture of integers and half-integers is excluded).
New!!: Root system and Hurwitz quaternion · See more »
Hyperplane
In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.
New!!: Root system and Hyperplane · See more »
Integer
An integer (from the Latin ''integer'' meaning "whole")Integer 's first literal meaning in Latin is "untouched", from in ("not") plus tangere ("to touch").
New!!: Root system and Integer · See more »
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective.
New!!: Root system and Isometry · See more »
Isomorphism
In mathematics, an isomorphism (from the Ancient Greek: ἴσος isos "equal", and μορφή morphe "form" or "shape") is a homomorphism or morphism (i.e. a mathematical mapping) that can be reversed by an inverse morphism.
New!!: Root system and Isomorphism · See more »
John Horton Conway
John Horton Conway FRS (born 26 December 1937) is an English mathematician active in the theory of finite groups, knot theory, number theory, combinatorial game theory and coding theory.
New!!: Root system and John Horton Conway · See more »
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.
New!!: Root system and Killing form · See more »
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.
New!!: Root system and Lattice (discrete subgroup) · See more »
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.
New!!: Root system and Lie algebra · See more »
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.
New!!: Root system and Lie group · See more »
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.
New!!: Root system and Linear span · See more »
Linear subspace
In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space.
New!!: Root system and Linear subspace · See more »
Mathematics
Mathematics (from Greek μάθημα máthēma, "knowledge, study, learning") is the study of such topics as quantity, structure, space, and change.
New!!: Root system and Mathematics · See more »
Mathematische Annalen
Mathematische Annalen (abbreviated as Math. Ann. or, formerly, Math. Annal.) is a German mathematical research journal founded in 1868 by Alfred Clebsch and Carl Neumann.
New!!: Root system and Mathematische Annalen · See more »
Neil Sloane
Neil James Alexander Sloane (born October 10, 1939) is a British-American mathematician.
New!!: Root system and Neil Sloane · See more »
Parity (mathematics)
In mathematics, parity is the property of an integer's inclusion in one of two categories: even or odd.
New!!: Root system and Parity (mathematics) · See more »
Partially ordered set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.
New!!: Root system and Partially ordered set · See more »
Permutation
In mathematics, the notion of permutation relates to the act of arranging all the members of a set into some sequence or order, or if the set is already ordered, rearranging (reordering) its elements, a process called permuting.
New!!: Root system and Permutation · See more »
Permutation group
In mathematics, a permutation group is a group G whose elements are permutations of a given set M and whose group operation is the composition of permutations in G (which are thought of as bijective functions from the set M to itself).
New!!: Root system and Permutation group · See more »
Perpendicular
In elementary geometry, the property of being perpendicular (perpendicularity) is the relationship between two lines which meet at a right angle (90 degrees).
New!!: Root system and Perpendicular · See more »
Plant
Plants are mainly multicellular, predominantly photosynthetic eukaryotes of the kingdom Plantae.
New!!: Root system and Plant · See more »
Reflection (mathematics)
In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.
New!!: Root system and Reflection (mathematics) · See more »
Root datum
In mathematical group theory, the root datum of a connected split reductive algebraic group over a field is a generalization of a root system that determines the group up to isomorphism.
New!!: Root system and Root datum · See more »
Root system of a semi-simple Lie algebra
In mathematics, there is a one-to-one correspondence between reduced crystallographic root systems and semisimple Lie algebras.
New!!: Root system and Root system of a semi-simple Lie algebra · See more »
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.
New!!: Root system and Semisimple Lie algebra · See more »
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.
New!!: Root system and Simple Lie group · See more »
Simply connected space
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the space) into any other such path while preserving the two endpoints in question.
New!!: Root system and Simply connected space · See more »
Singularity theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite.
New!!: Root system and Singularity theory · See more »
Spectral graph theory
In mathematics, spectral graph theory is the study of the properties of a graph in relationship to the characteristic polynomial, eigenvalues, and eigenvectors of matrices associated with the graph, such as its adjacency matrix or Laplacian matrix.
New!!: Root system and Spectral graph theory · See more »
Square lattice
In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space.
New!!: Root system and Square lattice · See more »
Standard basis
In mathematics, the standard basis (also called natural basis) for a Euclidean space is the set of unit vectors pointing in the direction of the axes of a Cartesian coordinate system.
New!!: Root system and Standard basis · See more »
Triality
In mathematics, triality is a relationship among three vector spaces, analogous to the duality relation between dual vector spaces.
New!!: Root system and Triality · See more »
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.
New!!: Root system and Vector space · See more »
Weight (representation theory)
In the mathematical field of representation theory, a weight of an algebra A over a field F is an algebra homomorphism from A to F, or equivalently, a one-dimensional representation of A over F. It is the algebra analogue of a multiplicative character of a group.
New!!: Root system and Weight (representation theory) · See more »
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.
New!!: Root system and Weyl group · See more »
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.
New!!: Root system and Wilhelm Killing · See more »
Zome
The term zome is used in several related senses.
New!!: Root system and Zome · See more »
1 22 polytope
In 6-dimensional geometry, the 122 polytope is a uniform polytope, constructed from the E6 group.
New!!: Root system and 1 22 polytope · See more »
2 31 polytope
In 7-dimensional geometry, 231 is a uniform polytope, constructed from the E7 group.
New!!: Root system and 2 31 polytope · See more »
24-cell
In geometry, the 24-cell is the convex regular 4-polytope (four-dimensional analogue of a Platonic solid) with Schläfli symbol.
New!!: Root system and 24-cell · See more »
4 21 polytope
In 8-dimensional geometry, the 421 is a semiregular uniform 8-polytope, constructed within the symmetry of the E8 group.
New!!: Root system and 4 21 polytope · See more »
Redirects here:
An (mathematics), Bn (mathematics), Cn (mathematics), Co-root, Coroot, Crystallographic root system, D4 (root system), Dn (mathematics), Negative root, Negative roots, Positive root, Positive roots, Root Space, Root diagram, Root lattice, Root reflection, Root space, Root systems, Root vector, Simple root (root system).
References
[1] https://en.wikipedia.org/wiki/Root_system
