In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are known as adjoint functors, one being the left adjoint and the other the right adjoint. Pairs of adjoint functors are ubiquitous in mathematics and often arise from constructions of "optimal solutions" to certain problems, such as the construction of a free group on a set in algebra, or the construction of the Stone–Čech compactification of a topological space in topology.
In mathematics, the distributive property of binary operations is a generalization of the distributive law, which asserts that the equality
In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an algebra and a coalgebra, with these structures' compatibility making it a bialgebra, and that moreover is equipped with an antihomomorphism satisfying a certain property. The representation theory of a Hopf algebra is particularly nice, since the existence of compatible comultiplication, counit, and antipode allows for the construction of tensor products of representations, trivial representations, and dual representations.
In category theory, a branch of mathematics, a monad is a monoid in the category of endofunctors of some fixed category. An endofunctor is a functor mapping a category to itself, and a monad is an endofunctor together with two natural transformations required to fulfill certain coherence conditions. Monads are used in the theory of pairs of adjoint functors, and they generalize closure operators on partially ordered sets to arbitrary categories. Monads are also useful in the theory of datatypes, the denotational semantics of imperative programming languages, and in functional programming languages, allowing languages without mutable state to do things such as simulate for-loops; see Monad.
In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space.
In mathematical physics, the conformal symmetry of spacetime is expressed by an extension of the Poincaré group, known as the conformal group. The extension includes special conformal transformations and dilations. In three spatial plus one time dimensions, conformal symmetry has 15 degrees of freedom: ten for the Poincaré group, four for special conformal transformations, and one for a dilation.
In mathematics, categorification is the process of replacing set-theoretic theorems with category-theoretic analogues. Categorification, when done successfully, replaces sets with categories, functions with functors, and equations with natural isomorphisms of functors satisfying additional properties. The term was coined by Louis Crane.
In mathematics, especially in the fields of representation theory and module theory, a Frobenius algebra is a finite-dimensional unital associative algebra with a special kind of bilinear form which gives the algebras particularly nice duality theories. Frobenius algebras began to be studied in the 1930s by Richard Brauer and Cecil Nesbitt and were named after Georg Frobenius. Tadashi Nakayama discovered the beginnings of a rich duality theory, . Jean Dieudonné used this to characterize Frobenius algebras. Frobenius algebras were generalized to quasi-Frobenius rings, those Noetherian rings whose right regular representation is injective. In recent times, interest has been renewed in Frobenius algebras due to connections to topological quantum field theory.
In theoretical physics, the superconformal algebra is a graded Lie algebra or superalgebra that combines the conformal algebra and supersymmetry. In two dimensions, the superconformal algebra is infinite-dimensional. In higher dimensions, superconformal algebras are finite-dimensional and generate the superconformal group.
In theoretical physics, a primary field, also called a primary operator, or simply a primary, is a local operator in a conformal field theory which is annihilated by the part of the conformal algebra consisting of the lowering generators. From the representation theory point of view, a primary is the lowest dimension operator in a given representation of the conformal algebra. All other operators in a representation are called descendants; they can be obtained by acting on the primary with the raising generators.
In mathematics, specifically in symplectic geometry, the momentum map is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including symplectic (Marsden–Weinstein) quotients, discussed below, and symplectic cuts and sums.
In category theory, a Kleisli category is a category naturally associated to any monad T. It is equivalent to the category of free T-algebras. The Kleisli category is one of two extremal solutions to the question Does every monad arise from an adjunction? The other extremal solution is the Eilenberg–Moore category. Kleisli categories are named for the mathematician Heinrich Kleisli.
In category theory, a strong monad over a monoidal category (C, ⊗, I) is a monad (T, η, μ) together with a natural transformation tA,B : A ⊗ TB → T(A ⊗ B), called (tensorial) strength, such that the diagrams
In category theory, a monoidal monad is a monad on a monoidal category such that the functor is a lax monoidal functor and the natural transformations and are monoidal natural transformations. In other words, is equipped with coherence maps and satisfying certain properties, and the unit and multiplication are monoidal natural transformations. By monoidality of , the morphisms and are necessarily equal.
In mathematics, a Rota–Baxter algebra is an associative algebra, together with a particular linear map R which satisfies the Rota–Baxter identity. It appeared first in the work of the American mathematician Glen E. Baxter in the realm of probability theory. Baxter's work was further explored from different angles by Gian-Carlo Rota, Pierre Cartier, and Frederic V. Atkinson, among others. Baxter’s derivation of this identity that later bore his name emanated from some of the fundamental results of the famous probabilist Frank Spitzer in random walk theory.
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. 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 algebroid satisfying a finite projectivity condition over a separable algebra is a weak Hopf algebra, and conversely a weak Hopf algebra H is a Hopf algebroid over its separable subalgebra HL. The antipode axioms have been changed by G. Böhm and K. Szlachányi in 2004 for tensor categorical reasons and to accommodate examples associated to depth two Frobenius algebra extensions.
In mathematics, weak bialgebras are a generalization of bialgebras that are both algebras and coalgebras but for which the compatibility conditions between the two structures have been "weakened". In the same spirit, weak Hopf algebras are weak bialgebras together with a linear map S satisfying specific conditions; they are generalizations of Hopf algebras.
In mathematics, a Verlinde algebra is a finite-dimensional associative algebra introduced by Erik Verlinde, with a basis of elements φλ corresponding to primary fields of a rational two-dimensional conformal field theory, whose structure constants Nν
λμ describe fusion of primary fields.
In differential geometry, the Kirwan map, introduced by British mathematician Frances Kirwan, is the homomorphism
Alexander Nikolaevich Varchenko is a Soviet and Russian mathematician working in geometry, topology, combinatorics and mathematical physics.
