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CoxeterDynkin 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 ). It describes a kaleidoscopic construction: each graph "node" represents a mirror (domain facet ) and the label attached to a branch encodes the dihedral angle order between two mirrors (on a domain ridge ). An unlabeled branch implicitly represents order3. Each diagram represents a Coxeter group Coxeter group , and Coxeter groups are classified by their associated diagrams. Dynkin diagrams are closely related objects, which differ from Coxeter diagrams in two respects: firstly, branches labeled "4" or greater are directed , while Coxeter diagrams are undirected ; secondly, Dynkin diagrams must satisfy an additional (crystallographic ) restriction, namely that the only allowed branch labels are 2, 3, 4, and 6 [...More...]  "CoxeterDynkin Diagram" on: Wikipedia Yahoo 

Simple Lie Group In group theory , a SIMPLE LIE GROUP is a connected nonabelian Lie group G which does not have nontrivial connected normal subgroups . A SIMPLE LIE ALGEBRA is a nonabelian Lie algebra Lie algebra whose only ideals are 0 and itself (or equivalently, a Lie algebra Lie algebra of dimension 2 or more, whose only ideals are 0 and itself). Simple Lie groups are a class of Lie groups which play a role in Lie group theory similar to that of simple groups in the theory of discrete groups. Essentially, simple Lie groups are connected Lie groups which cannot be decomposed as an extension of smaller connected Lie groups, and which are not commutative [...More...]  "Simple Lie Group" on: Wikipedia Yahoo 

Exceptional Lie Group In group theory , a SIMPLE LIE GROUP is a connected nonabelian Lie group G which does not have nontrivial connected normal subgroups . A SIMPLE LIE ALGEBRA is a nonabelian Lie algebra Lie algebra whose only ideals are 0 and itself (or equivalently, a Lie algebra Lie algebra of dimension 2 or more, whose only ideals are 0 and itself). Simple Lie groups are a class of Lie groups which play a role in Lie group theory similar to that of simple groups in the theory of discrete groups. Essentially, simple Lie groups are connected Lie groups which cannot be decomposed as an extension of smaller connected Lie groups, and which are not commutative [...More...]  "Exceptional Lie Group" on: Wikipedia Yahoo 

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=n4. In some older books and papers, E2 and E4 are used as names for G2 and F4 . CONTENTS * 1 Finitedimensional Lie algebras * 2 Infinitedimensional Lie algebras * 3 Root lattice * 4 E7½ * 5 See also * 6 References * 7 Further reading FINITEDIMENSIONAL LIE ALGEBRASThe En group is similar to the An group, except the nth node is connected to the 3rd node. So the Cartan matrix appears similar, 1 above and below the diagonal, except for the last row and column, have 1 in the third row and column. The determinant of the Cartan matrix for En is 9n. * E3 is another name for the Lie algebra Lie algebra A1A2 of dimension 11, with Cartan determinant 6 [...More...]  "En (Lie Algebra)" on: Wikipedia Yahoo 

F4 (mathematics) In mathematics , F4 is the name of a Lie group Lie group and also its Lie algebra F4. It is one of the five exceptional simple Lie groups . F4 has rank 4 and dimension 52. The compact form is simply connected and its outer automorphism group is the trivial group . Its fundamental representation is 26dimensional. The compact real form of F4 is the isometry group of a 16dimensional Riemannian manifold known as the octonionic projective plane OP2. This can be seen systematically using a construction known as the magic square , due to Hans Freudenthal and Jacques Tits Jacques Tits . There are 3 real forms : a compact one, a split one, and a third one. They are the isometry groups of the three real Albert algebras [...More...]  "F4 (mathematics)" on: Wikipedia Yahoo 

Hyperbolic Plane In mathematics , HYPERBOLIC GEOMETRY (also called BOLYAI –LOBACHEVSKIAN GEOMETRY or LOBACHEVSKIAN GEOMETRY) is a non Euclidean geometry Euclidean geometry . The parallel postulate of Euclidean geometry is replaced with: For any given line R and point P not on R, in the plane containing both line R and point P there are at least two distinct lines through P that do not intersect R. (compare this with Playfair\'s axiom , the modern version of Euclid Euclid 's parallel postulate ) Hyperbolic plane Hyperbolic plane geometry is also the geometry of saddle surface or pseudospherical surfaces , surfaces with a constant negative Gaussian curvature . A modern use of hyperbolic geometry is in the theory of special relativity , particularly Minkowski spacetime Minkowski spacetime and gyrovector space [...More...]  "Hyperbolic Plane" on: Wikipedia Yahoo 

Fundamental Domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A FUNDAMENTAL DOMAIN is a subset of the space which contains exactly one point from each of these orbits. It serves as a geometric realization for the abstract set of representatives of the orbits. There are many ways to choose a fundamental domain. Typically, a fundamental domain is required to be a connected subset with some restrictions on its boundary, for example, smooth or polyhedral. The images of a chosen fundamental domain under the group action then tile the space. One general construction of fundamental domains uses Voronoi cells . CONTENTS * 1 Hints at general definition * 2 Examples * 3 Fundamental domain Fundamental domain for the modular group * 4 See also * 5 External links HINTS AT GENERAL DEFINITION A lattice in the complex plane and its fundamental domain, with quotient a torus [...More...]  "Fundamental Domain" on: Wikipedia Yahoo 

Geometry GEOMETRY (from the Ancient Greek : γεωμετρία; geo "earth", metron "measurement") is a branch of mathematics concerned with questions of shape, size, relative position of figures, and the properties of space. A mathematician who works in the field of geometry is called a geometer . Geometry Geometry arose independently in a number of early cultures as a practical way for dealing with lengths , areas , and volumes . Geometry Geometry began to see elements of formal mathematical science emerging in the West as early as the 6th century BC. By the 3rd century BC, geometry was put into an axiomatic form by Euclid Euclid , whose treatment, Euclid\'s Elements , set a standard for many centuries to follow. Geometry Geometry arose independently in India, with texts providing rules for geometric constructions appearing as early as the 3rd century BC [...More...]  "Geometry" on: Wikipedia Yahoo 

Greatest Common Divisor In mathematics , the GREATEST COMMON DIVISOR (GCD) of two or more integers , which are not all zero, is the largest positive integer that divides each of the integers. For example, the gcd of 8 and 12 is 4. The greatest common divisor is also known as the GREATEST COMMON FACTOR (GCF), HIGHEST COMMON FACTOR (HCF), GREATEST COMMON MEASURE (GCM), or HIGHEST COMMON DIVISOR. This notion can be extended to polynomials (see Polynomial greatest common divisor ) and other commutative rings (see below ) [...More...]  "Greatest Common Divisor" on: Wikipedia Yahoo 

Group Order In group theory , a branch of mathematics , the term order is used in two unrelated senses: * The ORDER of a group is its cardinality , i.e., the number of elements in its set . Also, the ORDER, sometimes PERIOD, of an element a of a group is the smallest positive integer m such that am = e (where e denotes the identity element of the group, and am denotes the product of m copies of a). If no such m exists, a is said to have infinite order. * The ordering relation of a partially or totally ordered group .This article is about the first sense of order. The order of a group G is denoted by ord(G) or G and the order of an element a is denoted by ord(a) or a. CONTENTS * 1 Example * 2 Order and structure * 3 Counting by order of elements * 4 In relation to homomorphisms * 5 Class equation * 6 Open questions * 7 See also * 8 References EXAMPLEEXAMPLE. The symmetric group S3 has the following multiplication table [...More...]  "Group Order" on: Wikipedia Yahoo 

Affine Dynkin Diagram In the mathematical field of Lie theory , a DYNKIN DIAGRAM, named for Eugene Dynkin Eugene Dynkin , is a type of graph with some edges doubled or tripled (drawn as a double or triple line). The multiple edges are, within certain constraints, directed . The main interest in Dynkin diagrams are as a means to classify semisimple Lie algebras over algebraically closed fields . This gives rise to Weyl groups , i.e. to many (although not all) finite reflection groups . Dynkin diagrams may also arise in other contexts. The term "Dynkin diagram" can be ambiguous [...More...]  "Affine Dynkin Diagram" on: Wikipedia Yahoo 

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 g 2 {displaystyle {mathfrak {g}}_{2}} , as well as some algebraic groups . They are the smallest of the five exceptional simple Lie groups . G2 has rank 2 and dimension 14. It has two fundamental representations , with dimension 7 and 14. The compact form of G2 can be described as the automorphism group of the octonion algebra or, equivalently, as the subgroup of SO(7) that preserves any chosen particular vector in its 8dimensional real spinor representation [...More...]  "G2 (mathematics)" on: Wikipedia Yahoo 

Rectangle In Euclidean plane geometry , a RECTANGLE is a quadrilateral with four right angles . It can also be defined as an equiangular quadrilateral, since equiangular means that all of its angles are equal (360°/4 = 90°). It can also be defined as a parallelogram containing a right angle. A rectangle with four sides of equal length is a square . The term OBLONG is occasionally used to refer to a nonsquare rectangle. A rectangle with vertices ABCD would be denoted as ABCD. The word rectangle comes from the Latin Latin rectangulus, which is a combination of rectus (as an adjective, right, proper) and angulus (angle ). A CROSSED RECTANGLE is a crossed (selfintersecting) quadrilateral which consists of two opposite sides of a rectangle along with the two diagonals. It is a special case of an antiparallelogram , and its angles are not right angles [...More...]  "Rectangle" on: Wikipedia Yahoo 

Perpendicular In elementary geometry , the property of being PERPENDICULAR (PERPENDICULARITY) is the relationship between two lines which meet at a right angle (90 degrees ). The property extends to other related geometric objects . A line is said to be perpendicular to another line if the two lines intersect at a right angle. Explicitly, a first line is perpendicular to a second line if (1) the two lines meet; and (2) at the point of intersection the straight angle on one side of the first line is cut by the second line into two congruent angles . Perpendicularity can be shown to be symmetric , meaning if a first line is perpendicular to a second line, then the second line is also perpendicular to the first. For this reason, we may speak of two lines as being perpendicular (to each other) without specifying an order. Perpendicularity easily extends to segments and rays [...More...]  "Perpendicular" on: Wikipedia Yahoo 

Square (geometry) In geometry , a SQUARE is a regular quadrilateral , which means that it has four equal sides and four equal angles (90degree angles, or right angles ). It can also be defined as a rectangle in which two adjacent sides have equal length. A square with vertices ABCD would be denoted {displaystyle square } ABCD [...More...]  "Square (geometry)" on: Wikipedia Yahoo 

Polygon In elementary geometry , a POLYGON (/ˈpɒlɪɡɒn/ ) is a plane figure that is bounded by a finite chain of straight line segments closing in a loop to form a closed polygonal chain or circuit. These segments are called its edges or sides, and the points where two edges meet are the polygon's vertices (singular: vertex) or corners. The interior of the polygon is sometimes called its body. An NGON is a polygon with n sides; for example, a triangle is a 3gon. A polygon is a 2dimensional example of the more general polytope in any number of dimensions. The basic geometrical notion of a polygon has been adapted in various ways to suit particular purposes. Mathematicians are often concerned only with the bounding closed polygonal chain and with simple polygons which do not selfintersect, and they often define a polygon accordingly. A polygonal boundary may be allowed to intersect itself, creating star polygons and other selfintersecting polygons [...More...]  "Polygon" on: Wikipedia Yahoo 