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Lie Algebra Homology
In mathematics, Lie algebra cohomology is a cohomology theory for Lie algebras. It was first introduced in 1929 by Élie Cartan to study the topology of Lie groups and homogeneous spaces by relating cohomological methods of Georges de Rham to properties of the Lie algebra. It was later extended by to coefficients in an arbitrary Lie module. Motivation If G is a compact simply connected Lie group, then it is determined by its Lie algebra, so it should be possible to calculate its cohomology from the Lie algebra. This can be done as follows. Its cohomology is the de Rham cohomology of the complex of differential forms on G. Using an averaging process, this complex can be replaced by the complex of left-invariant differential forms. The left-invariant forms, meanwhile, are determined by their values at the identity, so that the space of left-invariant differential forms can be identified with the exterior algebra of the Lie algebra, with a suitable differential. The construction of ... [...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] |
Derived Functors
In mathematics, certain functors may be ''derived'' to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics. Motivation It was noted in various quite different settings that a short exact sequence often gives rise to a "long exact sequence". The concept of derived functors explains and clarifies many of these observations. Suppose we are given a covariant left exact functor ''F'' : A → B between two abelian categories A and B. If 0 → ''A'' → ''B'' → ''C'' → 0 is a short exact sequence in A, then applying ''F'' yields the exact sequence 0 → ''F''(''A'') → ''F''(''B'') → ''F''(''C'') and one could ask how to continue this sequence to the right to form a long exact sequence. Strictly speaking, this question is ill-posed, since there are always numerous different ways to continue a given exact sequence to the right. But it turns out that (if A is "nice" enough) th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 . Extensions arise in several ways. There is the trivial extension obtained by taking a direct sum of two Lie algebras. Other types are the split extension and the central extension. Extensions may arise naturally, for instance, when forming a Lie algebra from projective group representations. Such a Lie algebra will contain central charges. Starting with a polynomial loop algebra over finite-dimensional simple Lie algebra and performing two extensions, a central extension and an extension by a derivation, one obtains a Lie algebra which is isomorphic with an untwisted affine Kac–Moody algebra. Using the centrally extended loop algebra one may construct a current algebra in two spacetime dimensions. The Virasoro algebra is the universal central extension of the Witt algebra. Central extensions are needed in physi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Rham Cohomology
In mathematics, de Rham cohomology (named after Georges de Rham) is a tool belonging both to algebraic topology and to differential topology, capable of expressing basic topological information about smooth manifolds in a form particularly adapted to computation and the concrete representation of cohomology classes. It is a cohomology theory based on the existence of differential forms with prescribed properties. On any smooth manifold, every exact form is closed, but the converse may fail to hold. Roughly speaking, this failure is related to the possible existence of "holes" in the manifold, and the de Rham cohomology groups comprise a set of topological invariants of smooth manifolds that precisely quantify this relationship. Definition The de Rham complex is the cochain complex of differential forms on some smooth manifold , with the exterior derivative as the differential: :0 \to \Omega^0(M)\ \stackrel\ \Omega^1(M)\ \stackrel\ \Omega^2(M)\ \stackrel\ \Omega^3(M) \to \cd ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connection (mathematics)
In geometry, the notion of a connection makes precise the idea of transporting local geometric objects, such as tangent vectors or tensors in the tangent space, along a curve or family of curves in a ''parallel'' and consistent manner. There are various kinds of connections in modern geometry, depending on what sort of data one wants to transport. For instance, an affine connection, the most elementary type of connection, gives a means for parallel transport of tangent vectors on a manifold from one point to another along a curve. An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of vector fields, measuring the deviation of a vector field from being parallel in a given direction. Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point and the local geometry at another point. Differential geometry embraces severa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a product space B \times F is defined using a continuous surjective map, \pi : E \to B, that in small regions of E behaves just like a projection from corresponding regions of B \times F to B. The map \pi, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber. In the ''trivial'' case, E is just B \times F, and the map \pi is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Identity
In mathematics, the Jacobi identity is a property of a binary operation that describes how the order of evaluation, the placement of parentheses in a multiple product, affects the result of the operation. By contrast, for operations with the associative property, any order of evaluation gives the same result (parentheses in a multiple product are not needed). The identity is named after the German mathematician Carl Gustav Jacob Jacobi. The cross product a\times b and the Lie bracket operation ,b/math> both satisfy the Jacobi identity. In analytical mechanics, the Jacobi identity is satisfied by the Poisson brackets. In quantum mechanics, it is satisfied by operator commutators on a Hilbert space and equivalently in the phase space formulation of quantum mechanics by the Moyal bracket. Definition Let + and \times be two binary operations, and let 0 be the neutral element for +. The is :x \times (y \times z) \ +\ y \times (z \times x) \ +\ z \times (x \times y)\ =\ 0. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Graded Algebra
In mathematics, in particular abstract algebra and topology, a differential graded algebra is a graded associative algebra with an added chain complex structure that respects the algebra structure. __TOC__ Definition A differential graded algebra (or DG-algebra for short) ''A'' is a graded algebra equipped with a map d\colon A \to A which has either degree 1 (cochain complex convention) or degree −1 (chain complex convention) that satisfies two conditions: A more succinct way to state the same definition is to say that a DG-algebra is a monoid object in the monoidal category of chain complexes. A DG morphism between DG-algebras is a graded algebra homomorphism which respects the differential ''d''. A differential graded augmented algebra (also called a DGA-algebra, an augmented DG-algebra or simply a DGA) is a DG-algebra equipped with a DG morphism to the ground ring (the terminology is due to Henri Cartan). ''Warning:'' some sources use the term ''DGA'' for a DG-alge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transpose Of A Linear Map
In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces. The transpose or algebraic adjoint of a linear map is often used to study the original linear map. This concept is generalised by adjoint functors. Definition Let X^ denote the algebraic dual space of a vector space X. Let X and Y be vector spaces over the same field \mathcal. If u : X \to Y is a linear map, then its algebraic adjoint or dual, is the map ^ u : Y^ \to X^ defined by f \mapsto f \circ u. The resulting functional ^ u(f) := f \circ u is called the pullback of f by u. The continuous dual space of a topological vector space (TVS) X is denoted by X^. If X and Y are TVSs then a linear map u : X \to Y is weakly continuous if and only if ^ u\left(Y^\right) \subseteq X^, in which case we let ^t u : Y^ \to X^ denote the restriction of ^ u to Y^. The map ^t u is called the transpose or algebraic ad ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Levi Decomposition
In Lie theory and representation theory, the Levi decomposition, conjectured by Wilhelm Killing and Élie Cartan and proved by , states that any finite-dimensional real Lie algebra ''g'' is the semidirect product of a solvable ideal and a semisimple subalgebra. One is its radical, a maximal solvable ideal, and the other is a semisimple subalgebra, called a Levi subalgebra. The Levi decomposition implies that any finite-dimensional Lie algebra is a semidirect product of a solvable Lie algebra and a semisimple Lie algebra. When viewed as a factor-algebra of ''g'', this semisimple Lie algebra is also called the Levi factor of ''g''. To a certain extent, the decomposition can be used to reduce problems about finite-dimensional Lie algebras and Lie groups to separate problems about Lie algebras in these two special classes, solvable and semisimple. Moreover, Malcev (1942) showed that any two Levi subalgebras are conjugate by an (inner) automorphism of the form :\exp(\mathrm(z ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl's Theorem On Complete Reducibility
In algebra, Weyl's theorem on complete reducibility is a fundamental result in the theory of Lie algebra representations (specifically in the representation theory of semisimple Lie algebras). Let \mathfrak be a semisimple Lie algebra over a field of characteristic zero. The theorem states that every finite-dimensional module over \mathfrak is semisimple as a module (i.e., a direct sum of simple modules.) The enveloping algebra is semisimple Weyl's theorem implies (in fact is equivalent to) that the enveloping algebra of a finite-dimensional representation is a semisimple ring in the following way. Given a finite-dimensional Lie algebra representation \pi: \mathfrak \to \mathfrak(V), let A \subset \operatorname(V) be the associative subalgebra of the endomorphism algebra of ''V'' generated by \pi(\mathfrak g). The ring ''A'' is called the enveloping algebra of \pi. If \pi is semisimple, then ''A'' is semisimple. (Proof: Since ''A'' is a finite-dimensional algebra, it is an Artini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
