Quasitriangular Hopf group-quasialgebras and generalized quantum Yang-Baxter equations

We introduce and study a large class of algebras (possibly nonassociative) with group-coalgebraic structures, Hopf nonassociative group-coalgebras, which provide a unifying framework for classical Hopf algebras, Turaev's Hopf group-coalgebras, and Hopf quasigroups. Then, we introduce and discuss the notion of a quasitriangular Hopf pi-quasialgebra and show that the category Qrep(H)(R) of quasirepresentations over any quasitriangular Hopf pi-quasialgebra (H, R) is a braided pi-category. As an application of our theory, we give a new solution to the generalized quantum Yang-Baxter equation.