In mathematics, a Hopf algebra, named after Heinz Hopf, is a structure that is simultaneously an (unital associative) algebra and a (counital coassociative)...

associative algebra, the coalgebra must be co-associative, and, if the algebra is unital, then the co-algebra must be co-unital as well. A Hopf algebra is a...

noncommutative algebras with additional structure. These include Drinfeld–Jimbo type quantum groups (which are quasitriangular Hopf algebras), compact matrix...

be extended by giving an antipode to create a Hopf algebra structure. Note: In this article, all algebras are assumed to be unital and associative. The...

In mathematics, a Hopf algebra, H, is quasitriangular if there exists an invertible element, R, of H ⊗ H {\displaystyle H\otimes H} such that R   Δ ( x...

as a group algebra, for it is indeed an algebra over the given ring. A group algebra over a field has a further structure of a Hopf algebra; in this case...

abstract algebra, a representation of a Hopf algebra is a representation of its underlying associative algebra. That is, a representation of a Hopf algebra H...

the group Hopf algebra of a given group is a certain construct related to the symmetries of group actions. Deformations of group Hopf algebras are foundational...

both algebras and coalgebras but for which the compatibility conditions between the two structures have been "weakened". In the same spirit, weak Hopf algebras...

In quantum group, Hopf algebra and weak Hopf algebra, the Doi-Hopf module is a crucial construction that has many applications. It's named after Japanese...