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 π-quasialgebra and show that the category Qrep( H) R of quasirepresentations over any quasitriangular Hopf π-quasialgebra ( H, R) is a braided π-category. As an application of our theory, we give a new solution to the generalized quantum Yang–Baxter equation.

我们引入并研究了一大类具有群余代数结构的代数（可能是非结合的），即霍普夫非结合群余代数，它为经典霍普夫代数、图拉耶夫的霍普夫群余代数以及霍普夫拟群提供了一个统一的框架。然后，我们引入并讨论了拟三角霍普夫π - 拟代数的概念，并表明在任何拟三角霍普夫π - 拟代数$(H, R)$上的拟表示范畴$Qrep(H)_R$是一个辫子π - 范畴。作为我们理论的一个应用，我们给出了广义量子杨 - 巴克斯特方程的一个新解。