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Generalized Kac–Moody Algebra
In mathematics, a generalized Kac–Moody algebra is a Lie algebra that is similar to a Kac–Moody algebra, except that it is allowed to have imaginary simple roots. Generalized Kac–Moody algebras are also sometimes called GKM algebras, Borcherds–Kac–Moody algebras, BKM algebras, or Borcherds algebras. The best known example is the monster Lie algebra. Motivation Finite-dimensional semisimple Lie algebras have the following properties: * They have a nondegenerate symmetric invariant bilinear form (,). * They have a grading such that the degree zero piece (the Cartan subalgebra) is abelian. * They have a (Cartan) involution ''w''. * (''a'', ''w(a)'') is positive if ''a'' is nonzero. For example, for the algebras of ''n'' by ''n'' matrices of trace zero, the bilinear form is (''a'', ''b'') = Trace(''ab''), the Cartan involution is given by minus the transpose, and the grading can be given by "distance from the diagonal" so that the Cartan subalgebra is the diagonal elements ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Outer Derivation
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Sporadic Simple Group
In mathematics, a sporadic group is one of the 26 exceptional groups found in the classification of finite simple groups. A simple group is a group ''G'' that does not have any normal subgroups except for the trivial group and ''G'' itself. The classification theorem states that the list of finite simple groups consists of 18 countably infinite plus 26 exceptions that do not follow such a systematic pattern. These 26 exceptions are the sporadic groups. They are also known as the sporadic simple groups, or the sporadic finite groups. Because it is not strictly a group of Lie type, the Tits group is sometimes regarded as a sporadic group, in which case there would be 27 sporadic groups. The monster group is the largest of the sporadic groups, and all but six of the other sporadic groups are subquotients of it. Names Five of the sporadic groups were discovered by Mathieu in the 1860s and the other 21 were found between 1965 and 1975. Several of these groups were predicted to exis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fake Monster Lie Algebra
Fake may refer to: * Deception, an act or a statement intended to deceive ** Charlatan, a person who practices deception to obtain money or other advantages ** Counterfeit, a reproduction of an item, intended to deceive ** Cover-up, an attempt to conceal wrongdoing, error, or other embarrassing information ** Decoy, a person, device, or event meant as a distraction ** Fake news, a form of news consisting of deliberate disinformation or hoaxes spread via traditional news media or online social media. ** Falsity, the deceitfulness of one party culminating in damage to another party ** Feint, a maneuver designed to distract or mislead, often in fencing or military tactics ** Forgery, the falsification of a legal instrument with the intent to defraud ** Hoax, a falsehood deliberately crafted to deceive ** Illusion, a distortion of the senses, which can reveal how the human brain normally organizes and interprets sensory stimulation. * False (other) * Replica, a reproduction of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monstrous Moonshine
In mathematics, monstrous moonshine, or moonshine theory, is the unexpected connection between the monster group ''M'' and modular functions, in particular, the ''j'' function. The term was coined by John Conway and Simon P. Norton in 1979. The monstrous moonshine is now known to be underlain by a vertex operator algebra called the moonshine module (or monster vertex algebra) constructed by Igor Frenkel, James Lepowsky, and Arne Meurman in 1988, which has the monster group as its group of symmetries. This vertex operator algebra is commonly interpreted as a structure underlying a two-dimensional conformal field theory, allowing physics to form a bridge between two mathematical areas. The conjectures made by Conway and Norton were proven by Richard Borcherds for the moonshine module in 1992 using the no-ghost theorem from string theory and the theory of vertex operator algebras and generalized Kac–Moody algebras. History In 1978, John McKay found that the first few ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Monster Group
In the area of abstract algebra known as group theory, the monster group M (also known as the Fischer–Griess monster, or the friendly giant) is the largest sporadic simple group, having order 2463205976112133171923293141475971 = 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000 ≈ 8. The finite simple groups have been completely classified. Every such group belongs to one of 18 countably infinite families, or is one of 26 sporadic groups that do not follow such a systematic pattern. The monster group contains 20 sporadic groups (including itself) as subquotients. Robert Griess, who proved the existence of the monster in 1982, has called those 20 groups the ''happy family'', and the remaining six exceptions ''pariahs''. It is difficult to give a good constructive definition of the monster because of its complexity. Martin Gardner wrote a popular account of the monster group in his June 1980 Mathematical Games column in ''Scientific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Automorphic Form
In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of the topological group. Automorphic forms are a generalization of the idea of periodic functions in Euclidean space to general topological groups. Modular forms are holomorphic automorphic forms defined over the groups SL(2, R) or PSL(2, R) with the discrete subgroup being the modular group, or one of its congruence subgroups; in this sense the theory of automorphic forms is an extension of the theory of modular forms. More generally, one can use the adelic approach as a way of dealing with the whole family of congruence subgroups at once. From this point of view, an automorphic form over the group ''G''(A''F''), for an algebraic group ''G'' and an algebraic number field ''F'', is a complex-valued function on ''G''(A''F'') that is left ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentzian Algebra
Lorentzian may refer to * Cauchy distribution, also known as the Lorentz distribution, Lorentzian function, or Cauchy–Lorentz distribution * Lorentz transformation * Lorentzian manifold See also *Lorentz (other) Lorentz is a surname and a given name. Lorentz may also refer to: Things named for Hendrik Lorentz * Lorentz factor, Doppler effect *The Lorentz-Lorenz law, the law regarding the refractive index of a substance discovered independently by Hendri ... * Lorenz (other), spelled without the 't' {{Disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Kac–Moody 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. Given an affine Lie algebra, one can also form the associated affine Kac-Moody algebra, as described below. From a purely mathematical point of view, affine Lie algebras are interesting because their representation theory, like representation theory of finite-dimensional semisimple Lie algebras, is much better understood than that of general Kac–Moody algebras. As observed by Victor Kac, the character formula for representations of affine Lie algebras implies certain combinatorial identities, the Macdonald identities. Affine Lie algebras play an important role in string theory and two-dimensional conformal field theory due to the way they are constructed: starting from a simple Lie algebra \mathfrak, one considers the loop algebra, L\mathfrak, formed by the \mathfrak-valued functions on a circle (interpreted as the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl–Kac 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. It was proved by . There is a closely related formula for the character of an irreducible representation of a semisimple Lie algebra. In Weyl's approach to the representation theory of connected compact Lie groups, the proof of the character formula is a key step in proving that every dominant integral element actually arises as the highest weight of some irreducible representation. Important consequences of the character formula are the Weyl dimension formula and the Kostant multiplicity formula. By definition, the character \chi of a representation \pi of ''G'' is the trace of \pi(g), as a function of a group element g\in G. The irreducible representations in this case are all finite-dimensional (this is part of the Peter–Weyl theorem); so the notion of trace is the usual one from linear algebra. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highest Weight Module
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. The importance of the concept, however, stems from its application to representations of Lie algebras and hence also to representations of algebraic and Lie groups. In this context, a weight of a representation is a generalization of the notion of an eigenvalue, and the corresponding eigenspace is called a weight space. Motivation and general concept Given a set ''S'' of n\times n matrices over the same field, each of which is diagonalizable, and any two of which commute, it is always possible to simultaneously diagonalize all of the elements of ''S''.In fact, given a set of commuting matrices over an algebraically closed field, they are simultaneously triangularizable, without needing to assume that they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Bilinear Form
In mathematics, a symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map. In other words, it is a bilinear function B that maps every pair (u,v) of elements of the vector space V to the underlying field such that B(u,v)=B(v,u) for every u and v in V. They are also referred to more briefly as just symmetric forms when "bilinear" is understood. Symmetric bilinear forms on finite-dimensional vector spaces precisely correspond to symmetric matrices given a basis for ''V''. Among bilinear forms, the symmetric ones are important because they are the ones for which the vector space admits a particularly simple kind of basis known as an orthogonal basis (at least when the characteristic of the field is not 2). Given a symmetric bilinear form ''B'', the function is the associated quadratic form on the vector space. Moreover, if the characteristic of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
