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Coalgebra
In mathematics, coalgebras or cogebras are structures that are dual (in the category-theoretic sense of reversing arrows) to unital associative algebras. The axioms of unital associative algebras can be formulated in terms of commutative diagrams. Turning all arrows around, one obtains the axioms of coalgebras. Every coalgebra, by (vector space) duality, gives rise to an algebra, but not in general the other way. In finite dimensions, this duality goes in both directions ( see below). Coalgebras occur naturally in a number of contexts (for example, representation theory, universal enveloping algebras and group schemes). There are also F-coalgebras, with important applications in computer science. Informal discussion One frequently recurring example of coalgebras occurs in representation theory, and in particular, in the representation theory of the rotation group. A primary task, of practical use in physics, is to obtain combinations of systems with different states of angu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F-coalgebra
In mathematics, specifically in category theory, an F-coalgebra is a Mathematical structure, structure defined according to a functor F, with specific properties as defined below. For both algebraic structure, algebras and coalgebras, a functor is a convenient and general way of organizing a Signature (logic), signature. This has applications in computer science: examples of coalgebras include Lazy evaluation, lazy, Recursion_(computer_science)#Recursive_data_structures_.28structural_recursion.29, infinite data structures, such as Stream (computing), streams, and also State transition system, transition systems. F-coalgebras are Dual (category theory), dual to F-algebra, F-algebras. Just as the class of all algebraic structure, algebras for a given signature and equational theory form a Variety (universal algebra), variety, so does the class of all F-coalgebras satisfying a given equational theory form a covariety, where the signature is given by F. Definition Let :F : \mathca ... [...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. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing ''V'', in the sense of the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of ''T''(''V''). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras. The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure. ''Note'': In this article, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Algebra
In mathematics, the tensor algebra of a vector space ''V'', denoted ''T''(''V'') or ''T''(''V''), is the algebra of tensors on ''V'' (of any rank) with multiplication being the tensor product. It is the free algebra on ''V'', in the sense of being left adjoint to the forgetful functor from algebras to vector spaces: it is the "most general" algebra containing ''V'', in the sense of the corresponding universal property (see below). The tensor algebra is important because many other algebras arise as quotient algebras of ''T''(''V''). These include the exterior algebra, the symmetric algebra, Clifford algebras, the Weyl algebra and universal enveloping algebras. The tensor algebra also has two coalgebra structures; one simple one, which does not make it a bialgebra, but does lead to the concept of a cofree coalgebra, and a more complicated one, which yields a bialgebra, and can be extended by giving an antipode to create a Hopf algebra structure. ''Note'': In this article, ... [...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 mathematical analysis, 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 mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spin (physics)
Spin is a conserved quantity carried by elementary particles, and thus by composite particles (hadrons) and atomic nuclei. Spin is one of two types of angular momentum in quantum mechanics, the other being ''orbital angular momentum''. The orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution and appears when there is periodic structure to its wavefunction as the angle varies. For photons, spin is the quantum-mechanical counterpart of the polarization of light; for electrons, the spin has no classical counterpart. The existence of electron spin angular momentum is inferred from experiments, such as the Stern–Gerlach experiment, in which silver atoms were observed to possess two possible discrete angular momenta despite having no orbital angular momentum. The existence of the electron spin can also be inferred theoretically from the spin–statistics theorem and from the Pauli exclusion principle—and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product Of Modules
In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps (e.g. multiplication) to be carried out in terms of linear maps. The module construction is analogous to the construction of the tensor product of vector spaces, but can be carried out for a pair of modules over a commutative ring resulting in a third module, and also for a pair of a right-module and a left-module over any ring, with result an abelian group. Tensor products are important in areas of abstract algebra, homological algebra, algebraic topology, algebraic geometry, operator algebras and noncommutative geometry. The universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. It allows the study of bilinear or multilinear operations via linear operations. The tensor product of an algebra and a module can be used for extension of scalars. For a commutative ring, the tensor product of modules can be iterated to f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multilinear Map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function :f\colon V_1 \times \cdots \times V_n \to W\text where V_1,\ldots,V_n and W are vector spaces (or modules over a commutative ring), with the following property: for each i, if all of the variables but v_i are held constant, then f(v_1, \ldots, v_i, \ldots, v_n) is a linear function of v_i. A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, a multilinear map of ''k'' variables is called a ''k''-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra. If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating ''k''-linear maps. The latter coincide if the underlying rin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers. Many other fields, such as fields of rational functions, algebraic function fields, algebraic number fields, and ''p''-adic fields are commonly used and studied in mathematics, particularly in number theory and algebraic geometry. Most cryptographic protocols rely on finite fields, i.e., fields with finitely many elements. The relation of two fields is expressed by the notion of a field extension. Galois theory, initiated by Évariste Galois in the 1830s, is devoted to understanding the symmetries of field extensions. Among other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Littlewood–Richardson Rule
In mathematics, the Littlewood–Richardson rule is a combinatorial description of the coefficients that arise when decomposing a product of two Schur functions as a linear combination of other Schur functions. These coefficients are natural numbers, which the Littlewood–Richardson rule describes as counting certain skew tableaux. They occur in many other mathematical contexts, for instance as multiplicity in the decomposition of tensor products of finite-dimensional representations of general linear groups, or in the decomposition of certain induced representations in the representation theory of the symmetric group, or in the area of algebraic combinatorics dealing with Young tableaux and symmetric polynomials. Littlewood–Richardson coefficients depend on three partitions, say \lambda,\mu,\nu, of which \lambda and \mu describe the Schur functions being multiplied, and \nu gives the Schur function of which this is the coefficient in the linear combination; in other words ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shuffle Product
In mathematics, a shuffle algebra is a Hopf algebra with a basis corresponding to words on some set, whose product is given by the shuffle product ''X'' ⧢ ''Y'' of two words ''X'', ''Y'': the sum of all ways of interlacing them. The interlacing is given by the riffle shuffle permutation. The shuffle algebra on a finite set is the graded dual of the universal enveloping algebra of the free Lie algebra on the set. Over the rational numbers, the shuffle algebra is isomorphic to the polynomial algebra in the Lyndon words. The shuffle product occurs in generic settings in non-commutative algebras; this is because it is able to preserve the relative order of factors being multiplied together - the riffle shuffle permutation. This can be held in contrast to the divided power structure, which becomes appropriate when factors are commutative. Shuffle product The shuffle product of words of lengths ''m'' and ''n'' is a sum over the ways of interleaving the two words, as shown in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Algebra
In mathematics, especially in the area of abstract algebra known as ring theory, a free algebra is the noncommutative analogue of a polynomial ring since its elements may be described as "polynomials" with non-commuting variables. Likewise, the polynomial ring may be regarded as a free commutative algebra. Definition For ''R'' a commutative ring, the free (associative, unital) algebra on ''n'' indeterminates is the free ''R''-module with a basis consisting of all words over the alphabet (including the empty word, which is the unit of the free algebra). This ''R''-module becomes an ''R''-algebra by defining a multiplication as follows: the product of two basis elements is the concatenation of the corresponding words: :\left(X_X_ \cdots X_\right) \cdot \left(X_X_ \cdots X_\right) = X_X_ \cdots X_X_X_ \cdots X_, and the product of two arbitrary ''R''-module elements is thus uniquely determined (because the multiplication in an ''R''-algebra must be ''R''-bilinear). This ''R''- ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
