Representation Crossed Category of Group-cograded Multiplier Hopf Algebras

Abstract

Let $A={⨁}_{p\in G}{A}_p$ 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.

12 pages