The crossed structures on the solutions of some nonlinear equations

In this paper, we mainly discuss crossed Long dimodules of Turaev group coalgebras and of Turaev group algebras. We describe solutions to the Long equation and the quantum Yang–Baxter equation derived from crossed Long dimodules. Additionally, we introduce generalized crossed Long dimodules, which is different from the crossed Long dimodules defined above. Finally, we demonstrate that generalized Long dimodules can provide solutions to the crossed Yang–Baxter equation and yield a new crossed ribbon category.