  The goal of this paper is to give a new method of constructing
finite-dimensional semisimple triangular Hopf algebras, including minimal ones
which are non-trivial (i.e. not group algebras). The paper shows that such Hopf
algebras are quite abundant. It also discovers an unexpected connection of such
Hopf algebras with bijective 1-cocycles on finite groups and set-theoretical
solutions of the quantum Yang-Baxter equation defined by Drinfeld.
