# A Duality Theorem for Weak Multiplier Hopf Algebra Actions

## TL;DR

This paper unifies the theory of actions for various Hopf algebra types into a single framework for weak multiplier Hopf algebras, establishing a duality theorem and constructing smash products.

## Contribution

It introduces a unified approach to actions of weak multiplier Hopf algebras, defines module algebras, and proves a duality theorem for their actions.

## Key findings

- Established a duality theorem for actions on smash products
- Unified action theories for Hopf, weak Hopf, and multiplier Hopf algebras
- Recovered key results for weak Hopf algebras, multiplier Hopf algebras, and groupoids

## Abstract

The main purpose of this paper is to unify the theory of actions of Hopf algebras, weak Hopf algebras and multiplier Hopf algebras to one of actions of weak multiplier Hopf algebras introduced by A. Van Daele and S. H. Wang. Using such developed actions, we will define the notion of a module algebra over weak multiplier Hopf algebras and construct their smash products. The main result is the duality theorem for actions and their dual actions on the smash product of weak multiplier Hopf algebras. As an application, we recover the main results found in the literature for weak Hopf algebras, multiplier Hopf algebras and groupoids.