Representation Crossed Category of Group-cograded Multiplier Hopf Algebras

TL;DR

This paper introduces the concept of crossed left G-modules for multiplier Hopf T-coalgebras, establishing their monoidal category structure and characterizing quasitriangular structures via braided monoidal categories.

Contribution

It generalizes the framework of multiplier Hopf algebras by defining crossed modules and linking quasitriangular structures to braided monoidal categories.

Findings

Crossed left G-modules form a monoidal category.

Quasitriangular structures correspond to braided monoidal categories.

Generalization of previous results to multiplier Hopf T-coalgebras.

Abstract

Let $A=\bigoplus_{p\in G}A_{p}$ be a multiplier Hopf $T$-coalgebra over a group $G$, in this paper we give the definition of the crossed left $A$-$G$-modules and show that the category of crossed left $A$-$G$-modules is a monoidal category. Finally we show that a family of multipliers $R = \{R_{p, q} \in M(A_{p}\otimes A_{q})\}_{p, q\in G}$ is a quasitriangular structure of a multiplier $T$-coalgebra $A$ if and only if the crossed left $A$-$G$-module category over $A$ is a braided monoidal category with the braiding $c$ defined by $R$, generalizing the main results in \cite{ZCL11} to the more general framework of multiplier Hopf algebras.