A Class of Quasitriangular Group-cograded Multiplier Hopf Algebras

TL;DR

This paper introduces a new class of group-cograded multiplier Hopf algebras derived from pairings, extending classical finite-dimensional Hopf algebra constructions, and explores conditions for quasitriangular structures.

Contribution

It generalizes finite-dimensional Hopf algebra constructions to multiplier Hopf algebras and establishes conditions for quasitriangular structures within this framework.

Findings

Construction of $D(A, B)$ as a group-cograded multiplier Hopf algebra

Existence of quasitriangular structures under certain pairing conditions

Provision of examples and special cases illustrating the theory

Abstract

For a multiplier Hopf algebra pairing $\langle A, B\rangle$, we construct a class of group-cograded multiplier Hopf algebras $D(A, B)$, generalizing the classical construction of finite dimensional Hopf algebras introduced by Panaite and Staic Mihai. Furthermore, if the multiplier Hopf algebra pairing admits a canonical multiplier in $M(B\otimes A)$ we show the existence of quasitriangular structure on $D(A, B)$. As an application, some special cases and examples are provided.