The Ribbon Elements of Drinfeld Double of Radford Hopf Algebra

Let $m$, $n$ be two positive integers, $\mathbb{k}$ be an algebraically closed field with char($\mathbb{k})\nmid mn$. Radford constructed an $m{n}^2$-dimensional Hopf algebra $R_{mn}(q)$ such that its Jacobson radical is not a Hopf ideal. We show that the Drinfeld double $D(R_{mn}(q))$ of Radford Hopf algebra $R_{mn}(q)$ has ribbon elements if and only if $n$ is odd. Moreover, if $m$ is even and $n$ is odd, then $D(R_{mn}(q))$ has two ribbon elements, if both $m$ and $n$ are odd, then $D(R_{mn}(q))$ has only one ribbon element. Finally, we compute explicitly all ribbon elements of $D(R_{mn}(q))$.