Hopf algebras arising from partial (co)actions

TL;DR

This paper introduces the concept of partial matched pairs of Hopf algebras, extending previous ideas, and explores conditions under which their bismash product forms a new Hopf algebra, supported by examples.

Contribution

It defines partial matched pairs of Hopf algebras and establishes conditions for their bismash products to be Hopf algebras, extending the theory of partial actions.

Findings

Necessary conditions for bismash products to be Hopf algebras.

Introduction of partial matched pairs involving partial actions and coactions.

Examples illustrating the new constructions.

Abstract

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.