#### Vol. 3, No. 7, 2009

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

### Noah Snyder and Peter Tingley

Vol. 3 (2009), No. 7, 809–834
##### Abstract

We introduce the notion of a half-ribbon Hopf algebra, which is a Hopf algebra $\mathsc{ℋ}$ along with a distinguished element $t\in \mathsc{ℋ}$ such that $\left(\mathsc{ℋ},R,C\right)$ 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.

##### Keywords
quantum group, Hopf algebra, ribbon category
##### Mathematical Subject Classification 2000
Primary: 17B37
Secondary: 57T05, 57M05