On Set-Theoretical Solutions to Quantum Yang-Baxter Equation**The project supported in part by National Natural Science Foundation of China, Tianyuan Mathematics Foundation, K.C. Wong Education Foundation, and the Scientific Research Foundation for Young Mainstay Teachers and Returned Overseas Chinese Scholars of the Ministry of Education of China

Abstract

The problem on the set-theoretical solutions to the quantum Yang–Baxter equation was presented by Drinfel'd as a main unsolved problem in quantum group theory. The set-theoretical solutions are a natural extension of the usual (linear) solutions. In this paper, we not only give a further study on some known set-theoretical solutions (the Venkov's solutions), but also find a new kind of set-theoretical solutions which have a geometric interpretation. Moreover, the new solutions lead to the metahomomorphisms in group theory.