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{{short description|Concept in Hopf algebra and weak Hopf algebra}} ...ref>{{Cite journal |last=Doi |first=Yukio |date=1992-12-15 |title=Unifying Hopf modules |journal=Journal of Algebra |language=en |volume=153 |issue=2 |page ...

...ath>\tau_{V,W}</math>, most importantly the modules over [[quasitriangular Hopf algebra]]s and [[Yetter–Drinfeld category|Yetter–Drinfeld modules]] over [[ If <math>V</math> additionally possesses an [[braided Hopf algebra|algebra structure inside the braided category]] ("braided algebra") ...

{{short description|Example of a non-commutative and non-cocommutative Hopf algebra}} ...an example of an infinite-dimensional [[Hopf algebra]], and '''Sweedler's Hopf algebra''' ''H''₄ is a certain 4-dimensional quotient of it tha ...

{{short description|Concept in Hopf algebra}} ...keuchi|first=M.|title=Matched pairs of groups and bismash products of Hopf algebras|journal=Comm. Algebra|volume=9|issue=8|pages=841&ndash;882|year=1981|doi=10 ...

In mathematics, an '''exp algebra''' is a [[Hopf algebra]] Exp(''G'') constructed from an [[abelian group]] ''G'', and is th The ring Exp(''G'') can be made into a commutative and cocommutative [[Hopf algebra]] as follows. The [[coproduct]] of Exp(''G'') is defined so that a ...

...ebra]] NSymm analogous to the [[Hopf algebra of symmetric functions]]. The Hopf algebra NSymm was introduced by [[Israel M. Gelfand]], Daniel Krob, [[Alain ...quasisymmetric function]]. Over the rational numbers it is isomorphic as a Hopf algebra to the [[universal enveloping algebra]] of the free Lie algebra on ...

===Relation to Hopf algebras=== ...\Delta (x) = x \otimes x</math>. From complete Hopf algebras to Malcev Lie algebras one gets by taking the (completion of) [[primitive element (coalgebra)|prim ...

...map]] S satisfying specific conditions; they are generalizations of [[Hopf algebras]]. ...hat any fusion category is equivalent to a category of modules over a weak Hopf algebra.<ref>Etingof, Nikshych and Ostrik, Cor. 2.22</ref> ...

...ink=John Coleman Moore|year=1965}} classifies an important class of [[Hopf algebras]], of the sort that often show up as [[cohomology]] rings in [[algebraic to ...stic]] zero with <math>\dim A_n < \infty</math> for all ''n'', the natural Hopf algebra homomorphism ...

...[[Henri Poincaré]], is a deformation of the [[Poincaré algebra]] into a [[Hopf algebra]]. In the [[bicrossproduct]] basis, introduced by Majid-Ruegg<ref>{ The [[Hopf algebra|antipodes]] and the [[Hopf algebra|counits]]: ...

...'' is affine, then the duality functor is given by the duality of the Hopf algebras of functions. ==Definition using Hopf algebras== ...

...uantum group]], obtained by deformation of the [[Poincaré group]] into a [[Hopf algebra]]. Also the [[Hopf algebra|antipodes]] and the [[Hopf algebra|counits]] are classical, and represent the group inversion law and ...

...of groupoids|eprint=math/0105137}}</ref>^{pg 2}, comodules over a Hopf algebroid are the purely algebraic analogue of this construction, giving a Given a commutative Hopf-algebroid <math>(A,\Gamma)</math> a '''left''' '''comodule''' <math>M</math ...

...-4401-30-2|access-date=2021-01-05}}</ref> from the noncommutative Steenrod algebras called the '''dual Steenrod algebra'''. This dual algebra has a number of s ...ize this duality through dualizing a commutative diagram of the Steenrod's Hopf algebra structure:<blockquote><math>\mathcal{A}_p^* \xrightarrow{\psi^*} ...

....QA|eprint=1311.0278}}</ref> a concept related to [[quantum group]]s and [[Hopf algebra]]s. ...title=Structure of Algebras|page=22|chapter=Chapt. 2: Ideals and Nilpotent Algebras|orig-year=1939|year=2003|series=Colloquium Publications, Col. 24|publisher= ...

==Associative division algebras== ...enius theorem]] states that [[up to]] [[isomorphism]] there are three such algebras: the reals themselves (dimension 1), the field of [[complex number]]s (dime ...

...maps:<ref>{{citation|arxiv=0805.3806 |last1=Böhm |first1=Gabriella |title=Hopf Algebroids |year=2008 }}</ref> ...lgebra introduced by Takeuchi earlier, in 1977.<ref>M. Takeuchi, Groups of algebras over <math>A \times \bar{A}</math>, J. Math. Soc. Jpn. 29, 459–492, 1977</r ...

...n (1999), [{{Google books|plainurl=y|id=2fcC1EGKz08C|page=97|text=groupoid algebras}} p. 97]</ref> * [[Algebra of functions|Algebras of functions]] ...

...ebra is a weak Hopf algebra, and conversely a weak Hopf algebra ''H'' is a Hopf algebroid over its separable subalgebra ''H^L''. The antipode axi ...der the category <math>\text{CRing}_k</math> of commutative <math>k</math>-algebras. ...

...e [[Drinfled quantum double|Drinfeld double]] of a [[finite group]]s and [[Hopf algebra]]s.<ref>{{Cite journal |last=Drinfel'd |first=V. G. |date=1988-04-0 ...odel is characterized by the representations of the [[Bicrossed product of Hopf algebra|quantum double]] of finite group <math>G</math>. The anyon types a ...