  A basis B of a finite dimensional Hopf algebra H is said to be positive if
all the structure constants of H relative to B are non-negative. A
quasi-triangular structure $R\in H\otimes H$ is said to be positive with
respect to B if it has non-negative coefficients in the basis $B \otimes B$ of
$H\otimes H$. In our earlier work, we have classified all finite dimensional
Hopf algebras with positive bases. In this paper, we classify positive
quasi-triangular structures on such Hopf algebras. A consequence of this
classification is a new way of constructing set-theoretical solutions of the
Yang-Baxter equation.
