Splitting Property of quasitriangular Hopf algebras

TL;DR

This paper explores the splitting property of quasitriangular Hopf algebras, establishing conditions under which it holds and providing new obstructions for their structure, with implications for finite-dimensional cases.

Contribution

It introduces new criteria for the splitting property in quasitriangular Hopf algebras using quotient structures and coinvariants, advancing understanding of their algebraic properties.

Findings

Splitting property linked to factorizable quotient Hopf algebras.

Conditions involving coinvariants determine the splitting property.

Provides new obstructions for quasitriangular structures in finite-dimensional Hopf algebras.

Abstract

We investigate the splitting property of quasitriangular Hopf algebras through the lens of twisted tensor products. Specifically, we demonstrate that an infinite-dimensional quasitriangular Hopf algebra possesses the splitting property if it admits a factorizable quotient Hopf algebra. We also establish that the splitting property holds if there exists a full rank quotient Hopf algebra where the left and right coinvariants coincide. As a consequence, we obtain a new obstruction for quasitriangular structure in finite-dimensional Hopf algebras.