The half-twist for $U_q(\mathfrak{g})$ representations

Abstract

We introduce the notion of a half-ribbon Hopf algebra, which is a Hopf algebra $\mathcal{\mathscr{H}}$ along with a distinguished element $t\in \mathcal{\mathscr{H}}$ such that $\left(\mathcal{\mathscr{H}},R,C\right)$ is a ribbon Hopf algebra, where $R=\left(t^{-1}\otimes t^{-1}\right)\Delta \left(t\right)$ and $C=t^{-2}$. The element $t$ is closely related to the topological “half-twist”, which twists a ribbon by 180 degrees. We construct a functor from a topological category of ribbons with half-twists to the category of representations of any half-ribbon Hopf algebra. We show that $U_q\left(\mathfrak{g}\right)$ is a (topological) half-ribbon Hopf algebra, but that $t^{-2}$ is not the standard ribbon element. For $U_q\left({\mathfrak{s}\mathfrak{l}}_2\right)$, we show that there is no half-ribbon element $t$ such that $t^{-2}$ is the standard ribbon element. We then discuss how ribbon elements can be modified, and some consequences of these modifications.