On the formal ribbon extension of a quasitriangular Hopf algebra

TL;DR

This paper explores the formal extension of finite-dimensional quasitriangular Hopf algebras to ribbon Hopf algebras, analyzing their representations and connections to pivotalization, with specific examples like doubled Nichols Hopf algebras.

Contribution

It provides a detailed study of the formal ribbon extension process and its effects on module categories, including a comparison with existing pivotalization methods.

Findings

Every indecomposable module admits two compatible extended actions.

The extension preserves properties of simple, projective, and central modules.

In the semisimple case, the construction aligns with known pivotalization techniques.

Abstract

Any finite-dimensional quasitriangular Hopf algebra $H$ can be formally extended to a ribbon Hopf algebra $\tilde H$ of twice the dimension. We investigate this extension and its representations. We show that every indecomposable $H$-module has precisely two compatible $\tilde H$-actions. We investigate the behavior of simple, projective, and M\"uger central $\tilde H$-modules in terms of these $\tilde H$-actions. We also observe that, in the semisimple case, this construction agrees with the pivotalization/sphericalization construction introduced by Etingof, Nikshych, and Ostrik (2003). As an example, we investigate the formal ribbon extension of odd-index doubled Nichols Hopf algebras $D\mathcal K_n$.