Weak Hopf algebras arising from weak matched pairs

TL;DR

This paper generalizes the concept of matched pairs to weak bialgebras and weak Hopf algebras, providing new theoretical frameworks, conditions, and examples in the field of algebraic structures.

Contribution

It introduces the notion of weak matched pairs and establishes conditions for subspaces of smash products to form weak bialgebras or Hopf algebras.

Findings

Defined weak matched pairs in weak bialgebra context

Established conditions for subspaces to be weak bialgebras/Hopf algebras

Presented new examples of (co)actions and integral elements

Abstract

This work extends the idea of matched pairs presented by Majid in \cite{Majid} and Takeuchi in \cite{Takeuchi} for the context of weak bialgebras and weak Hopf algebras. We introduce, also inspired by partial matched pairs \cite{matchedpair}, the notion of weak matched pairs and establish conditions for a subspace of the smash product be a weak bialgebra/Hopf algebra. Further, some new examples of (co)actions of weak bialgebras over algebras and some results about integral elements are presented.