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In this paper, extending the idea presented by M. Takeuchi in [13], we introduce the notion of partial matched pair $(H,L)$ involving the concepts of partial action and partial coaction between two Hopf algebras $H$ and $L$. Furthermore, we present necessary conditions for the corresponding bismash product $L# H$ to generate a new Hopf algebra and, as illustration, a family of examples is provided.
In this work the notions of partial action of a weak Hopf algebra on a coalgebra and partial action of a groupoid on a coalgebra will be introduced, just as some important properties. An equivalence between these notions will be presented. Finally, a
In this work we study the deformations of a Hopf algebra $H$ by partial actions of $H$ on its base field $Bbbk$, via partial smash product algebras. We introduce the concept of a $lambda$-Hopf algebra as a Hopf algebra obtained as a partial smash p
In this paper we determine all partial actions and partial coactions of Taft and Nichols Hopf algebras on their base fields. Furthermore, we prove that all such partial (co)actions are symmetric.
We study actions of semisimple Hopf algebras H on Weyl algebras A over a field of characteristic zero. We show that the action of H on A must factor through a group algebra; in other words, if H acts inner faithfully on A, then H is cocommutative. Th
Let $U_q(mathfrak{g})$ be a quantum affine algebra with an indeterminate $q$ and let $mathscr{C}_{mathfrak{g}}$ be the category of finite-dimensional integrable $U_q(mathfrak{g})$-modules. We write $mathscr{C}_{mathfrak{g}}^0$ for the monoidal subcat