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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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
F-coalgebra
In mathematics, specifically in category theory, an F-coalgebra is a structure defined according to a functor F, with specific properties as defined below. For both algebras and coalgebras, a functor is a convenient and general way of organizing a signature. This has applications in computer science: examples of coalgebras include lazy, infinite data structures, such as streams, and also transition systems. F-coalgebras are dual to F-algebras. Just as the class of all algebras for a given signature and equational theory form a 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 : \mathcal\longrightarrow \mathcal be an endofunctor on a category \mathcal. An F-coalgebra is an object A of \mathcal together with a morphism :\alpha : A \longrightarrow FA of \mathcal, usually written as (A, \alpha). An F-coalgebra homomorphism from (A, \alpha) to another F-coalgebra (B, ... [...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 art ... [...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 clarificatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Representation Theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essence, a representation makes an abstract algebraic object more concrete by describing its elements by matrices and their algebraic operations (for example, matrix addition, matrix multiplication). The theory of matrices and linear operators is well-understood, so representations of more abstract objects in terms of familiar linear algebra objects helps glean properties and sometimes simplify calculations on more abstract theories. The algebraic objects amenable to such a description include groups, associative algebras and Lie algebras. The most prominent of these (and historically the first) is the representation theory of groups, in which elements of a group are represented by invertible matrices in such a way that the group operatio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Universal Enveloping Algebra
In mathematics, the universal enveloping algebra of a Lie algebra is the unital associative algebra whose representations correspond precisely to the representations of that Lie algebra. Universal enveloping algebras are used in the representation theory of Lie groups and Lie algebras. For example, Verma modules can be constructed as quotients of the universal enveloping algebra. In addition, the enveloping algebra gives a precise definition for the Casimir operators. Because Casimir operators commute with all elements of a Lie algebra, they can be used to classify representations. The precise definition also allows the importation of Casimir operators into other areas of mathematics, specifically, those that have a differential algebra. They also play a central role in some recent developments in mathematics. In particular, their dual provides a commutative example of the objects studied in non-commutative geometry, the quantum groups. This dual can be shown, by the Gelfand ... [...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] |
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 the fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Clebsch–Gordan Coefficients
In physics, the Clebsch–Gordan (CG) coefficients are numbers that arise in angular momentum coupling in quantum mechanics. They appear as the expansion coefficients of total angular momentum eigenstates in an uncoupled tensor product basis. In more mathematical terms, the CG coefficients are used in representation theory, particularly of compact Lie groups, to perform the explicit direct sum decomposition of the tensor product of two irreducible representations (i.e., a reducible representation into irreducible representations, in cases where the numbers and types of irreducible components are already known abstractly). The name derives from the German mathematicians Alfred Clebsch and Paul Gordan, who encountered an equivalent problem in invariant theory. From a vector calculus perspective, the CG coefficients associated with the SO(3) group can be defined simply in terms of integrals of products of spherical harmonics and their complex conjugates. The addition of spins in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Free Object
In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between elements of the free object are those that follow from the defining axioms of the algebraic structure. Examples include free groups, tensor algebras, or free lattices. The concept is a part of universal algebra, in the sense that it relates to all types of algebraic structure (with finitary operations). It also has a formulation in terms of category theory, although this is in yet more abstract terms. Definition Free objects are the direct generalization to categories of the notion of basis in a vector space. A linear function between vector spaces is entirely determined by its values on a basis of the vector space The following definition translates this to any category. A concrete category is a category that is equipped with a f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lift (mathematics)
In category theory, a branch of mathematics, given a morphism ''f'': ''X'' → ''Y'' and a morphism ''g'': ''Z'' → ''Y'', a lift or lifting of ''f'' to ''Z'' is a morphism ''h'': ''X'' → ''Z'' such that . We say that ''f'' factors through ''h''. A basic example in topology is lifting a path in one topological space to a path in a covering space. For example, consider mapping opposite points on a sphere to the same point, a continuous map from the sphere covering the projective plane. A path in the projective plane is a continuous map from the unit interval ,1 We can lift such a path to the sphere by choosing one of the two sphere points mapping to the first point on the path, then maintain continuity. In this case, each of the two starting points forces a unique path on the sphere, the lift of the path in the projective plane. Thus in the category of topological spaces with continuous maps as morphisms, we have :\begin f\colon\, & ,1\to \mathbb^2 &&\ \text \\ g\colon\ ... [...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 t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
