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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] |
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 poin ... [...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 product. Thus, an algebra is an algebraic structure consisting of a set together with operations of multiplication and addition and scalar multiplication by elements of a 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 and non-associative algebras. Given an integer ''n'', the ring of real 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-dimensional Euclidean space with multiplication given by the vector cross product is an example of a nonassociative algebra over the field of real numbers since the vector cross product is nonassociative, satisfying the Jacobi ... [...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. The Lie bracket of two vectors x and y is denoted ,y/math>. The vector space \mathfrak g together with this operation is a non-associative algebra, meaning that the Lie bracket is not necessarily associative. Lie algebras are closely related to Lie groups, which are groups that are also smooth manifolds: any Lie group gives rise to a Lie algebra, which is its tangent space at the identity. Conversely, to any finite-dimensional Lie algebra over real or complex numbers, there is a corresponding connected Lie group unique up to finite coverings ( Lie's third theorem). This correspondence allows one to study the structure and classification of Lie groups in terms of Lie algebras. In physics, Lie groups appear as symmetry grou ... [...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 a field of mathematics known as differential geometry, a Courant geometry was originally introduced by Zhang-Ju Liu, Alan Weinstein and Ping Xu in their investigation of doubles of Lie bialgebroids in 1997. Liu, Weinstein and Xu named it after Courant, who had implicitly devised earlier 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. Both this standard example and the double of a Lie bialgebra are special instances of Courant algebroids. Definition A Courant algebroid consists of the data a vector bundle E\to M with a bracket ,.\Gamma E \times \Gamma E \to \Gamma E, a non degenerate fiber-wise inner product \langle.,.\rangle: E\times E\to M\times\R, and a bundle map \rho:E\to TM subject to the following axioms, : chi,_\psi.html" ;"title="phi, [\chi, \psi">phi, [\chi, \psi = \phi, \chi \psi] + [\chi, [\phi, \psi :[\phi, f\psi] = \r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Bialgebroid
A Lie bialgebroid is a mathematical structure in the area of non-Riemannian differential geometry. In brief a Lie bialgebroid are two compatible Lie algebroids defined on dual vector bundles. They form the vector bundle version of a Lie bialgebra. Definition Preliminary notions Remember that a ''Lie algebroid'' is defined as a skew-symmetric operation ,.on the sections Γ(''A'') of a vector bundle ''A→M'' over a smooth manifold ''M'' together with a vector bundle morphism ''ρ: A→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 ''Φ'', ''ψ''k are sections of ''A'' and ''f'' is a smooth function on ''M''. The Lie bracket ,.sub>''A'' can be extended to polyvector field, multivector fields Γ(⋀''A'') graded symmetric via the Leibniz rule : Phi\wedge\Psi,\ChiA = \Phi\wedge Psi,\ChiA +(-1)^ Ph ... [...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 TM ... [...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 ma ... [...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 bund ... [...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. # A projection onto X. For a product space, this is just the projection onto the first factor, (x,g) \mapsto x. Unlike a product space, principal bundles lack a preferred choice of identity cross-section; they have no preferred analog of (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 number of arbitrary choices are made to try to define such a structure by defining it on smaller pieces of the space. A common example of a princi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
