In mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems to infinitesimal ones.
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 antiautomorphism 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 mathematics, a commutativity constraint on a monoidal category is a choice of isomorphism for each pair of objects A and B which form a "natural family." In particular, to have a commutativity constraint, one must have for all pairs of objects .
In mathematics, Tannaka–Krein duality theory concerns the interaction of a compact topological group and its category of linear representations. It is a natural extension of Pontryagin duality, between compact and discrete commutative topological groups, to groups that are compact but noncommutative. The theory is named for two men, the Soviet mathematician Mark Grigorievich Krein, and the Japanese Tadao Tannaka. In contrast to the case of commutative groups considered by Lev Pontryagin, the notion dual to a noncommutative compact group is not a group, but a category of representations Π(G) with some additional structure, formed by the finite-dimensional representations of G.
In mathematics, higher category theory is the part of category theory at a higher order, which means that some equalities are replaced by explicit arrows in order to be able to explicitly study the structure behind those equalities. Higher category theory is often applied in algebraic topology, where one studies algebraic invariants of spaces, such as their fundamental weak ∞-groupoid.
André Joyal is a professor of mathematics at the Université du Québec à Montréal who works on category theory. He was a member of the School of Mathematics at the Institute for Advanced Study in 2013, where he was invited to join the Special Year on Univalent Foundations of Mathematics.
In mathematics, the field with one element is a suggestive name for an object that should behave similarly to a finite field with a single element, if such a field could exist. This object is denoted F1, or, in a French–English pun, Fun. The name "field with one element" and the notation F1 are only suggestive, as there is no field with one element in classical abstract algebra. Instead, F1 refers to the idea that there should be a way to replace sets and operations, the traditional building blocks for abstract algebra, with other, more flexible objects. While there is still no field with a single element in these theories, there is a field-like object whose characteristic is one.
In mathematics, especially (higher) category theory, higher-dimensional algebra is the study of categorified structures. It has applications in nonabelian algebraic topology, and generalizes abstract algebra.
Shahn Majid is an English pure mathematician and theoretical physicist, trained at Cambridge University and Harvard University and, since 2001, a Professor of Mathematics at the School of Mathematical Sciences, Queen Mary, University of London.
In mathematics, R-algebroids are constructed starting from groupoids. These are more abstract concepts than the Lie algebroids that play a similar role in the theory of Lie groupoids to that of Lie algebras in the theory of Lie groups..
Ronald Brown is an English mathematician. Emeritus Professor in the School of Computer Science at Bangor University, he has authored many books and more than 160 journal articles.
In mathematics, the concept of groupoid algebra generalizes the notion of group algebra.
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, the Atiyah algebroid, or Atiyah sequence, of a principal G-bundleP 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:
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.
Ross Howard Street is an Australian mathematician specialising in category theory. He did his undergraduate and postgraduate study at the University of Sydney, where his dissertation advisor was Max Kelly. He is an emeritus professor of mathematics at Macquarie University, a fellow of the Australian Mathematical Society (1995), and was elected Fellow of the Australian Academy of Science in 1989. He was awarded the Edgeworth David Medal of the Royal Society of New South Wales in 1977, and the Australian Mathematical Society's George Szekeres Medal in 2012.
Janusz Roman Grabowski Polish mathematician working in differential geometry and mathematical methods in classical and quantum physics.
In mathematics, an associative L-bialgebroid where L is an associative algebra over some ground field k is another associative k-algebra H together with a number of additional structure maps involving H, L and various tensor products of those. Associative bialgebroids are a generalization of a k-bialgebra where a ground ring k is replaced by a possibly noncommutative k-algebra L. Hopf algebroids are associative bialgebroids with an additional antipode map which is an antiautomorphism of H satisfying additional axioms.
Ralph Martin Kaufmann is a German mathematician working in the United States.
arXiv is a repository of electronic preprints approved for posting after moderation, but not full peer review. It consists of scientific papers in the fields of mathematics, physics, astronomy, electrical engineering, computer science, quantitative biology, statistics, mathematical finance and economics, which can be accessed online. In many fields of mathematics and physics, almost all scientific papers are self-archived on the arXiv repository. Begun on August 14, 1991, arXiv.org passed the half-million-article milestone on October 3, 2008, and had hit a million by the end of 2014. By October 2016 the submission rate had grown to more than 10,000 per month.
Mathematical Reviews is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also publishes an associated online bibliographic database called MathSciNet which contains an electronic version of Mathematical Reviews and additionally contains citation information for over 3.5 million items as of 2018.
In computing, a Digital Object Identifier or DOI is a persistent identifier or handle used to identify objects uniquely, standardized by the International Organization for Standardization (ISO). An implementation of the Handle System, DOIs are in wide use mainly to identify academic, professional, and government information, such as journal articles, research reports and data sets, and official publications though they also have been used to identify other types of information resources, such as commercial videos.
