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Algebroid (other)
In mathematics, algebroid may refer to several distinct notions, which nevertheless all arise from generalising certain aspects of the theory of algebras or Lie algebras. * Algebroid branch, a formal power series branch of an algebraic curve * Algebroid cohomology * Algebroid multifunction *Courant algebroid, an object generalising Lie bialgebroids *Lie algebroid, the infinitesimal counterpart of Lie groupoids **Atiyah algebroid, a fundamental example of a Lie algebroid associated to a principal bundle *R-algebroid, a categorical construction associated to groupoids In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fun ... {{disambig Mathematics disambiguation pages ... [...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] |
Algebra Over A Field
In mathematics, an algebra over a field (often simply called an algebra) is a vector space equipped with a bilinear map, bilinear product (mathematics), product. Thus, an algebra is an algebraic structure consisting of a set (mathematics), set together with operations of multiplication and addition and scalar multiplication by elements of a field (mathematics), field and satisfying the axioms implied by "vector space" and "bilinear". The multiplication operation in an algebra may or may not be associative, leading to the notions of associative algebras where associativity of multiplication is assumed, and non-associative algebras, where associativity is not assumed (but not excluded, either). Given an integer ''n'', the ring (mathematics), ring of real matrix, real square matrix, square matrices of order ''n'' is an example of an associative algebra over the field of real numbers under matrix addition and matrix multiplication since matrix multiplication is associative. Three-dime ... [...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] |
Algebroid Branch
In mathematics, algebroid may refer to several distinct notions, which nevertheless all arise from generalising certain aspects of the theory of algebras or Lie algebras. * Algebroid branch, a formal power series branch of an algebraic curve * Algebroid cohomology * Algebroid multifunction *Courant algebroid, an object generalising Lie bialgebroids *Lie algebroid, the infinitesimal counterpart of Lie groupoids **Atiyah algebroid, a fundamental example of a Lie algebroid associated to a principal bundle * R-algebroid, a categorical construction associated to groupoids In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fun ... {{disambig Mathematics disambiguation pages ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Algebroid Cohomology
In mathematics, algebroid may refer to several distinct notions, which nevertheless all arise from generalising certain aspects of the theory of algebras or Lie algebras. *Algebroid branch, a formal power series branch of an algebraic curve * Algebroid cohomology * Algebroid multifunction *Courant algebroid, an object generalising Lie bialgebroids *Lie algebroid, the infinitesimal counterpart of Lie groupoids **Atiyah algebroid, a fundamental example of a Lie algebroid associated to a principal bundle *R-algebroid, a categorical construction associated to groupoids In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: * '' Group'' with a partial fun ... {{disambig Mathematics disambiguation pages ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Courant Algebroid
In differential geometry, a field of mathematics, a Courant algebroid is a vector bundle together with an inner product and a compatible bracket more general than that of a Lie algebroid. It is named after Theodore James Courant, Theodore Courant, who had implicitly devised in 1990 the standard prototype of Courant algebroid through his discovery of a skew-symmetric bracket on TM\oplus T^*M, called Courant bracket today, which fails to satisfy the Jacobi identity. The general notion of Courant algebroid was introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids in 1997. Definition A Courant algebroid consists of the data a vector bundle E\to M with a bracket [\cdot,\cdot]:\Gamma E \times \Gamma E \to \Gamma E, a non degenerate fiber-wise inner product \langle \cdot, \cdot \rangle: E\times E\to M\times\R, and a bundle map \rho:E\to TM (called anchor) subject to the following axioms: # Jacobi identity: [\phi, [\chi, \psi = \phi, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Bialgebroid
In differential geometry, a field in mathematics, a Lie bialgebroid consists of two compatible Lie algebroids defined on dual vector bundles. Lie bialgebroids are the vector bundle version of Lie bialgebras. Definition Preliminary notions A Lie algebroid consists of a bilinear skew-symmetric operation cdot,\cdot/math> on the sections \Gamma(A) of a vector bundle ''A \to M'' over a smooth manifold ''M'', together with a vector bundle morphism ''\rho: A \to TM'' subject to the Leibniz rule : phi,f\cdot\psi= \rho(\phi) cdot\psi +f\cdot phi,\psi and Jacobi identity : psi_1,\psi_2.html" ;"title="phi,[\psi_1,\psi_2">phi,[\psi_1,\psi_2 = \phi,\psi_1\psi_2] +[\psi_1,[\phi,\psi_2 where ''\phi,\psi_k'' are sections of ''A'' and ''f'' is a smooth function on ''M''. The Lie bracket '' cdot,\cdotA'' can be extended to polyvector field, multivector fields ''\Gamma(\wedge A)'' graded symmetric via the Leibniz rule : Phi\wedge\Psi,\ChiA = \Phi\wedge Psi,\ChiA +(-1)^ Phi,\ChiA\wedge\P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Algebroid
In mathematics, a Lie algebroid is a vector bundle A \rightarrow M together with a Lie bracket on its space of sections \Gamma(A) and a vector bundle morphism \rho: A \rightarrow TM, satisfying a Leibniz rule. A Lie algebroid can thus be thought of as a "many-object generalisation" of a Lie algebra. Lie algebroids play a similar same role in the theory of Lie groupoids that Lie algebras play in the theory of Lie groups: reducing global problems to infinitesimal ones. Indeed, any Lie groupoid gives rise to a Lie algebroid, which is the vertical bundle of the source map restricted at the units. However, unlike Lie algebras, not every Lie algebroid arises from a Lie groupoid. Lie algebroids were introduced in 1967 by Jean Pradines. Definition and basic concepts A Lie algebroid is a triple (A, cdot,\cdot \rho) consisting of * a vector bundle A over a manifold M * a Lie bracket cdot,\cdot/math> on its space of sections \Gamma (A) * a morphism of vector bundles \rho: A\rightarrow ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Groupoid
In mathematics, a Lie groupoid is a groupoid where the set \operatorname of objects and the set \operatorname of morphisms are both manifolds, all the category operations (source and target, composition, identity-assigning map and inversion) are smooth, and the source and target operations :s,t : \operatorname \to \operatorname are submersions. A Lie groupoid can thus be thought of as a "many-object generalization" of a Lie group, just as a groupoid is a many-object generalization of a group. Accordingly, while Lie groups provide a natural model for (classical) continuous symmetries, Lie groupoids are often used as model for (and arise from) generalised, point-dependent symmetries. Extending the correspondence between Lie groups and Lie algebras, Lie groupoids are the global counterparts of Lie algebroids. Lie groupoids were introduced by Charles Ehresmann under the name ''differentiable groupoids''. Definition and basic concepts A Lie groupoid consists of * two smooth m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Atiyah Algebroid
In mathematics, the Atiyah algebroid, or Atiyah sequence, of a principal '' G''-bundle '' P'' over a manifold '' M'', where '' G'' is a Lie group, is the Lie algebroid of the gauge groupoid of '' P''. Explicitly, it is given by the following short exact sequence of vector bundles over '' M'': : 0 \to P\times_G \mathfrak g\to TP/G \to TM\to 0. It is named after Michael Atiyah, who introduced the construction to study the existence theory of complex analytic connections. It plays a crucial example in the integrability of (transitive) Lie algebroids, and it has applications in gauge theory and geometric mechanics. Definitions As a sequence For any fiber bundle P over a manifold M, the differential d\pi of the projection \pi: P \to M defines a short exact sequence: : 0 \to VP \to TP \xrightarrow \pi^* TM\to 0 of vector bundles over P, where the vertical bundle VP is the kernel of d\pi. If '' P'' is a principal '' G''-bundle, then the group '' G'' acts on the vector bun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Principal Bundle
In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product X \times G of a space X with a group G. In the same way as with the Cartesian product, a principal bundle P is equipped with # An action of G on P, analogous to (x, g)h = (x, gh) for a product space (where (x, g) is an element of P and h is the group element from G; the group action is conventionally a right action). # A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) \mapsto x. Unless it is the product space X \times G, a principal bundle lacks a preferred choice of identity cross-section; it has no preferred analog of x \mapsto (x,e). Likewise, there is not generally a projection onto G generalizing the projection onto the second factor, X \times G \to G that exists for the Cartesian product. They may also have a complicated topology that prevents them from being realized as a product space even if a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
