The Ribbon Elements of Drinfeld Double of Radford Hopf Algebra

Let m, n be two positive integers, be an algebraically closed field with char() mn. Radford constructed an mn²-dimensional Hopf algebra Rₘₙ(q) such that its Jacobson radical is not a Hopf ideal. We show that the Drinfeld double D(Rₘₙ(q)) of Radford Hopf algebra Rₘₙ(q) has ribbon elements if and only if n is odd. Moreover, if m is even and n is odd, then D(Rₘₙ(q)) has two ribbon elements, if both m and n are odd, then D(Rₘₙ(q)) has only one ribbon element. Finally, we compute explicitly all ribbon elements of D(Rₘₙ(q)).