On the formal ribbon extension of a quasitriangular Hopf algebra

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 $D{\mathcal{K}}_n$.

22 pages. Minor edits to correct typos