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Associative Bialgebroid
In mathematics, if L is an associative algebra over some ground field ''k'', then a left associative L-bialgebroid is another associative ''k''-algebra H together with the following additional maps: an algebra map \alpha:L\to H called the source map, an algebra map \beta:L^\to H called the target map, so that the elements of the images of \alpha and \beta commute in H, therefore inducing an L-bimodule structure on H via the rule a.h.b = \alpha(a)\beta(b) h for a,b\in L, h\in H; an L-bimodule morphism \Delta:H\to H\otimes_L H which is required to be a counital coassociative comultiplication on H in the monoidal category of L-bimodules with monoidal product \otimes_L. The corresponding counit \varepsilon:H\to L is required to be a left character (equivalently, the map H\otimes L\ni h\otimes \ell\mapsto \varepsilon(h\alpha(\ell))\in L must be a left action extending the multiplication L\otimes L\to L along \alpha\otimes\mathrm_L). Furthermore, a compatibility between the comultiplic ... [...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] |
Associative Algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplication operations together give ''A'' the structure of a ring; the addition and scalar multiplication operations together give ''A'' the structure of a vector space over ''K''. In this article we will also use the term ''K''-algebra to mean an associative algebra over the field ''K''. A standard first example of a ''K''-algebra is a ring of square matrices over a field ''K'', with the usual matrix multiplication. A commutative algebra is an associative algebra that has a commutative multiplication, or, equivalently, an associative algebra that is also a commutative ring. In this article associative algebras are assumed to have a multiplicative identity, denoted 1; they are sometimes called unital associative algebras for clarification. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ground Field
In mathematics, a ground field is a field ''K'' fixed at the beginning of the discussion. Use It is used in various areas of algebra: In linear algebra In linear algebra, the concept of a vector space may be developed over any field. In algebraic geometry In algebraic geometry, in the foundational developments of André Weil the use of fields other than the complex numbers was essential to expand the definitions to include the idea of abstract algebraic variety over ''K'', and generic point relative to ''K''. In Lie theory Reference to a ground field may be common in the theory of Lie algebras (''qua'' vector spaces) and algebraic groups (''qua'' algebraic varieties). In Galois theory In Galois theory, given a field extension ''L''/''K'', the field ''K'' that is being extended may be considered the ground field for an argument or discussion. Within algebraic geometry, from the point of view of scheme theory, the spectrum ''Spec''(''K'') of the ground field ''K'' plays the role ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Corestriction
In mathematics, a corestriction of a function is a notion analogous to the notion of a restriction of a function. The duality prefix co- here denotes that while the restriction changes the domain to a subset, the corestriction changes the codomain to a subset. However, the notions are not categorically dual. Given any subset S\subset A, we can consider the corresponding inclusion of sets i_S:S\hookrightarrow A as a function. Then for any function f:A\to B, the restriction f, _S:S\to B of a function f onto S can be defined as the composition f, _S = f\circ i_S. Analogously, for an inclusion i_T:T\hookrightarrow B the corestriction f, ^T:A\to T of f onto T is the unique function f, ^T such that there is a decomposition f = i_T\circ f, ^T. The corestriction exists if and only if T contains the image of f. In particular, the corestriction onto the image always exists and it is sometimes simply called the corestriction of f. More generally, one can consider corestriction of a morphi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bialgebra
In mathematics, a bialgebra over a field ''K'' is a vector space over ''K'' which is both a unital associative algebra and a counital coassociative coalgebra. The algebraic and coalgebraic structures are made compatible with a few more axioms. Specifically, the comultiplication and the counit are both unital algebra homomorphisms, or equivalently, the multiplication and the unit of the algebra both are coalgebra morphisms. (These statements are equivalent since they are expressed by the same commutative diagrams.) Similar bialgebras are related by bialgebra homomorphisms. A bialgebra homomorphism is a linear map that is both an algebra and a coalgebra homomorphism. As reflected in the symmetry of the commutative diagrams, the definition of bialgebra is self-dual, so if one can define a dual of ''B'' (which is always possible if ''B'' is finite-dimensional), then it is automatically a bialgebra. Formal definition (''B'', ∇, η, Δ, ε) is a bialgebra over ''K'' if it h ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ''ring'' is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series. Formally, a ''ring'' is an abelian group whose operation is called ''addition'', with a second binary operation called ''multiplication'' that is associative, is distributive over the addition operation, and has a multiplicative identity element. (Some authors use the term " " with a missing i to refer to the more general structure that omits this last requirement; see .) Whether a ring is commutative (that is, whether the order in which two elements are multiplied might change the result) has ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hopf Algebroid
In mathematics, in the theory of Hopf algebras, a Hopf algebroid is a generalisation of weak Hopf algebras, certain skew Hopf algebras and commutative Hopf ''k''-algebroids. If ''k'' is a field, a commutative ''k''-algebroid is a cogroupoid object in the category of ''k''-algebras; the category of such is hence dual to the category of groupoid ''k''-schemes. This commutative version has been used in 1970-s in algebraic geometry and stable homotopy theory. The generalization of Hopf algebroids and its main part of the structure, associative bialgebroids, to the noncommutative base algebra was introduced by J.-H. Lu in 1996 as a result on work on groupoids in Poisson geometry (later shown equivalent in nontrivial way to a construction of Takeuchi from the 1970s and another by Xu around the year 2000). They may be loosely thought of as Hopf algebras over a noncommutative base ring, where weak Hopf algebras become Hopf algebras over a separable algebra. It is a theorem that a Hopf a ... [...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] |
Internal Bialgebroid
In mathematics, an internal bialgebroid is a structure which generalizes the notion of an associative bialgebroid to the setup where the ambient symmetric monoidal category of vector spaces is replaced by any abstract symmetric monoidal category (''C'', \otimes, ''I'',''s'') admitting coequalizers commuting with the monoidal product \otimes. It consists of two monoids in the monoidal category (''C'', \otimes, ''I''), namely the base monoid A and the total monoid H, and several structure morphisms involving A and H as first axiomatized by G. Böhm.Gabriella Böhm, Internal bialgebroids, entwining structures and corings, in: Algebraic structures and their representations, 207–226, Contemp. Math. 376, Amer. Math. Soc. 2005Cornell University Library, retrieved 11 September, 2017/ref> The coequalizers are needed to introduce the tensor product \otimes_A of (internal) bimodules over the base monoid; this tensor product is consequently (a part of) a monoidal structure on the category ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
