On the formal ribbon extension of a quasitriangular Hopf algebra

TL;DR

This work formalizes the ribbon extension $ ilde{H}$ of a finite-dimensional quasitriangular Hopf algebra $H$, constructing it as $H ilde{f v}$ with a central ribbon element and deriving the resulting representation theory. It shows that every $H$-module lifts to two nonisomorphic $ ilde{H}$-modules and that the category $ ext{Rep}( ilde{H})$ is braided, with a precise relation to pivotalization in the semisimple setting via an equivalence with $ ilde{ ext{Rep}(H)}$. The paper further analyzes decompositions of $ ilde{H}$ (tensor-product and cocycled crossed product forms) and applies the framework to doubled Nichols Hopf algebras $Dk_n$, highlighting distinct behaviors for even vs. odd $n$ and providing explicit presentations and module structures for $ ilde{Dk_n}$. Overall, it furnishes a robust method to generate (non)semisimple ribbon categories and clarifies how formal ribbon extensions interact with Drinfeld doubles and Nichols algebras, with potential implications for non-semisimple TQFTs and braided category theory.

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 Müger 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 $D\mathcal K_n$.

Theorems & Definitions (58)

• Proposition 2.1.1
• Theorem 2.1.2
• Definition 2.2.1
• Proposition 2.2.1
• proof
• Proposition 2.2.2
• proof
• Remark 2.2.2
• Proposition 2.3.1
• proof