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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.
These objects were introduced by Böhm, Nill and Szlachányi. The first motivations for studying them came from quantum field theory and operator algebras.[1] Weak Hopf algebras have quite interesting representation theory; in particular modules over a semisimple finite weak Hopf algebra is a fusion category (which is a monoidal category with extra properties). It was also shown by Etingof, Nikshych and Ostrik that any fusion category is equivalent to a category of modules over a weak Hopf algebra.[2]
In mathematics, a Hopfalgebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative)...
spirit, weakHopfalgebras are weak bialgebras together with a linear map S satisfying specific conditions; they are generalizations of Hopfalgebras. These...
universal algebra, an algebraic structure is called an algebra; this term may be ambiguous, since, in other contexts, an algebra is an algebraic structure...
duality, every fusion category arises as the representations of a weakHopfalgebra. Etingof, Pavel; Nikshych, Dmitri; Ostrik, Viktor (2005). "On Fusion...
direct algebraic proof is known. The fundamental theorem of algebra is a corollary of Hopf's theorem. Dropping the requirement of commutativity, Hopf generalized...
extensions are Galois extensions of algebras being acted upon by groups, Hopfalgebras, weakHopfalgebras or Hopf algebroids; for example, suppose a finite...
Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants...
symmetric algebra) inherits a bialgebra structure, and, indeed, a Hopfalgebra structure, from the tensor algebra. See the article on tensor algebras for a...
In mathematics, especially in abstract algebra, a quasigroup is an algebraic structure resembling a group in the sense that "division" is always possible...
types of algebraic structures are studied. Abstract algebra is primarily the study of specific algebraic structures and their properties. Algebraic structures...
Associated bundle Fibration Hopf bundle Classifying space Cofibration Homotopy groups of spheres Plus construction Whitehead theorem Weak equivalence Hurewicz...
a Hopfalgebra do form a group. A primitive element is an element x that satisfies Δ(x) = x ⊗ 1 + 1 ⊗ x. The primitive elements of a Hopfalgebra form...
Representation theory of Hopfalgebras General Associative property, Associator Heap (mathematics) Magma (algebra) Loop (algebra), Quasigroup Nonassociative...
coproduct and a bilinear form making it into a positive selfadjoint graded Hopfalgebra that is both commutative and cocommutative. The study of symmetric functions...
dual vector space. The Hopf algebras associated to groups have a commutative algebra structure, and so general Hopfalgebras are known as quantum groups...
in the books of Morrey and Smoller, following the original statement of Hopf (1927): Let M be an open subset of Euclidean space ℝn. For each i and j between...
generalizes the notion of a fiber bundle and plays an important role in algebraic topology, a branch of mathematics. Fibrations are used, for example, in...
{\displaystyle (M,g)} be a connected and continuous Riemannian manifold. The Hopf–Rinow theorem, in this setting, says that (Gromov 1999) if the metric space...
Mathematics portal Toric geometry Torus Torus based cryptography Hopfalgebra Milne. "Algebraic Groups: The Theory of Group Schemes of Finite Type" (PDF). Archived...
Noetherian (in particular the ring of integers, polynomial rings, and rings of algebraic integers in number fields), and many general theorems on rings rely heavily...
Higher-dimensional algebra Categorical ring Crane, Louis; Frenkel, Igor B. (1994-10-01). "Four-dimensional topological quantum field theory, Hopf categories,...
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