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Infinite-dimensional Lie Algebra
In mathematics, especially in Lie theory, E''n'' is the Kac–Moody algebra whose Dynkin diagram is a bifurcating graph with three branches of length 1, 2 and ''k'', with . In some older books and papers, ''E''2 and ''E''4 are used as names for ''G''2 and ''F''4. Finite-dimensional Lie algebras The E''n'' group is similar to the A''n'' 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 E''n'' is . *E3 is another name for the Lie algebra ''A''1''A''2 of dimension 11, with Cartan determinant 6. *:\left \begin 2 & -1 & 0 \\ -1 & 2 & 0 \\ 0 & 0 & 2 \end\right /math> *E4 is another name for the Lie algebra ''A''4 of dimension 24, with Cartan determinant 5. *:\left \begin 2 & -1 & 0 & 0 \\ -1 & 2 & -1& 0 \\ 0 & -1 & 2 & -1 \\ 0 & 0 & -1 & 2 \end\right /math> *E5 is ano ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unimodular Lattice
In geometry and mathematical group theory, a unimodular lattice is an integral Lattice (group), lattice of Lattice (group)#Dividing space according to a lattice, determinant 1 or −1. For a lattice in ''n''-dimensional Euclidean space, this is equivalent to requiring that the volume of any fundamental domain for the lattice be 1. The E8 lattice, ''E''8 lattice and the Leech lattice are two famous examples. Definitions * A lattice is a free abelian group of finite free abelian group, rank with a symmetric bilinear form (·, ·). * The lattice is integral if (·,·) takes integer values. * The dimension of a lattice is the same as its free module, rank (as a Z-module (mathematics), module). * The norm of a lattice element ''a'' is (''a'', ''a''). * A lattice is positive definite if the norm of all nonzero elements is positive. * The determinant of a lattice is the determinant of the Gram matrix, a matrix (mathematics), matrix with entries (''ai'', ''aj'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 1 K2 Polytope
In geometry, 1k2 polytope is a uniform polytope in n-dimensions (n = k+4) constructed from the En Coxeter group. The family was named by their Coxeter symbol 1k2 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 1-node sequence. It can be named by an extended Schläfli symbol . Family members The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5- demicube (demipenteract) in 5-dimensions, and the 4-simplex (5-cell) in 4-dimensions. Each polytope is constructed from 1k-1,2 and (n-1)- demicube facets. Each has a vertex figure of a ' polytope is a birectified n-simplex, ''t2''. The sequence ends with k=6 (n=10), as an infinite tessellation of 9-dimensional hyperbolic space. The complete family of 1k2 polytope polytopes are: # 5-cell: 102, (5 tetrahedral cells) # 112 polytope, (16 5-cell, and 10 16-cell facets) # 122 polytope, (54 demipenteract facets) # 132 polytope, (56 122 and 126 demihexeract facets) # ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniform 2 K1 Polytope
In geometry, 2k1 polytope is a uniform polytope in ''n'' dimensions (''n'' = ''k''+4) constructed from the En (Lie algebra), En Coxeter group. The family was named by their Coxeter symbol as 2k1 by its bifurcating Coxeter-Dynkin diagram, with a single ring on the end of the 2-node sequence. It can be named by an extended Schläfli symbol . Family members The family starts uniquely as 6-polytopes, but can be extended backwards to include the 5-orthoplex (pentacross) in 5-dimensions, and the 4-simplex (5-cell) in 4-dimensions. Each polytope is constructed from (n-1)-simplex and 2k-1,1 (n-1)-polytope facets, each has a vertex figure as an (n-1)-demihypercube, demicube, '. The sequence ends with k=6 (n=10), as an infinite hyperbolic tessellation of 9-space. The complete family of 2k1 polytope polytopes are: # 5-cell: 201, (5 Tetrahedron, tetrahedra cells) # Pentacross: 211, (32 5-cell (201) facets) # 2 21 polytope, 221, (72 5-simplex and 27 5-orthoplex (211) facets) # 2 31 polytop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiregular K 21 Polytope
In geometry, a uniform ''k''21 polytope is a polytope in ''k'' + 4 dimensions constructed from the ''E''''n'' Coxeter group, and having only regular polytope facets. The family was named by their Coxeter symbol ''k''21 by its bifurcating Coxeter–Dynkin diagram, with a single ring on the end of the ''k''-node sequence. Thorold Gosset discovered this family as a part of his 1900 enumeration of the regular and semiregular polytopes, and so they are sometimes called Gosset's semiregular figures. Gosset named them by their dimension from 5 to 9, for example the ''5-ic semiregular figure''. Family members The sequence as identified by Gosset ends as an infinite tessellation (space-filling honeycomb) in 8-space, called the E8 lattice. (A final form was not discovered by Gosset and is called the E9 lattice: 621. It is a tessellation of hyperbolic 9-space constructed of ∞ 9-simplex and ∞ 9-orthoplex facets with all vertices at infinity.) The family starts uniquely as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nilradical Of A Lie Algebra
In algebra, the nilradical of a Lie algebra is a nilpotent ideal, which is as large as possible. The nilradical \mathfrak(\mathfrak g) of a finite-dimensional Lie algebra \mathfrak is its maximal nilpotent ideal, which exists because the sum of any two nilpotent ideals is nilpotent. It is an ideal in the radical \mathfrak(\mathfrak) of the Lie algebra \mathfrak. The quotient of a Lie algebra by its nilradical is a reductive Lie algebra \mathfrak^. However, the corresponding short exact sequence In mathematics, an exact sequence is a sequence of morphisms between objects (for example, Group (mathematics), groups, Ring (mathematics), rings, Module (mathematics), modules, and, more generally, objects of an abelian category) such that the Im ... : 0 \to \mathfrak(\mathfrak g)\to \mathfrak g\to \mathfrak^\to 0 does not split in general (i.e., there isn't always a ''subalgebra'' complementary to \mathfrak(\mathfrak g) in \mathfrak). This is in contrast to the Levi decomposition: th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heisenberg Algebra
In mathematics, the Heisenberg group H, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form : \begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end under the operation of matrix multiplication. Elements ''a, b'' and ''c'' can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group"). The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to ''n''-dimensional systems, and most generally, to any symplectic vector space. Three-dimensional case In the three-dimensional case, the product of two Heisenberg matrices is given by : \begin 1 & a & c\\ 0 & 1 & b\\ 0 & 0 & 1\\ \end \begin 1 & a' & c'\\ 0 & 1 & b'\\ 0 & 0 & 1\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
M-theory
In physics, M-theory is a theory that unifies all Consistency, consistent versions of superstring theory. Edward Witten first conjectured the existence of such a theory at a string theory conference at the University of Southern California in 1995. Witten's announcement initiated a flurry of research activity known as the second superstring revolution. Prior to Witten's announcement, string theorists had identified five versions of superstring theory. Although these theories initially appeared to be very different, work by many physicists showed that the theories were related in intricate and nontrivial ways. Physicists found that apparently distinct theories could be unified by mathematical transformations called S-duality and T-duality. Witten's conjecture was based in part on the existence of these dualities and in part on the relationship of the string theories to a field theory (physics), field theory called eleven-dimensional supergravity. Although a complete formulation o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentzian Algebra , spelled without the 't'
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Lorentzian may refer to * Cauchy distribution, also known as the Lorentz distribution, Lorentzian function, or Cauchy–Lorentz distribution * Lorentz lineshape (spectroscopy) * Lorentz transformation * Lorentzian manifold See also * Lorentz (other) *Lorenz (other) Lorenz is a German name. Lorenz may also refer to: Mathematics and science * Lorenz system, a system of equations notable for having chaotic solutions * Lorenz gauge condition, in electromagnetism * Lorenz curve, an income distribution curve * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
E8 Lattice
In mathematics, the E lattice is a special lattice in R. It can be characterized as the unique positive-definite, even, unimodular lattice of rank 8. The name derives from the fact that it is the root lattice of the E root system. The normIn this article, the ''norm'' of a vector refers to its length squared (the square of the ordinary norm). of the E lattice (divided by 2) is a positive definite even unimodular quadratic form in 8 variables, and conversely such a quadratic form can be used to construct a positive-definite, even, unimodular lattice of rank 8. The existence of such a form was first shown by H. J. S. Smith in 1867, and the first explicit construction of this quadratic form was given by Korkin and Zolotarev in 1873. The E lattice is also called the Gosset lattice after Thorold Gosset who was one of the first to study the geometry of the lattice itself around 1900. Lattice points The E lattice is a discrete subgroup of R of full rank (i.e. it spans all of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebra
In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity. In other words, a Lie algebra is an algebra over a field for which the multiplication operation (called the Lie bracket) is alternating and satisfies the Jacobi identity. The Lie bracket of two vectors x and y is denoted ,y/math>. A Lie algebra is typically a non-associative algebra. However, every associative algebra gives rise to a Lie algebra, consisting of the same vector space with the commutator Lie bracket, ,y= xy - yx . Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: every Lie group gives rise to a Lie algebra, which is the tangent space at the identity. (In this case, the Lie bracket measures the failure of commutativity for the Lie group.) Conversely, to any finite-di ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
