Solutions of the $\mathrm{BiHom}$-Yang-Baxter equation

On the BiHom-Type Nonlinear Equations

“…This class of algebras was introduced from a categorical approach in [10] which can be viewed as an extension of the class of Hom-algebras. Further research on BiHom-type algebras could be found in [11][12][13][14][15] and so on.…”

“…Long dimodules are the building stones of the Brauer-Long group [18]. The discussion of solutions of BiHom-type Yang-Baxter equation can be seen in [11,19]. The natural consideration is to ask: does there exist algebraic solutions of BiHom-Hopf equation, BiHom-pentagon equation, and BiHom-type D-equation?…”

Hom-Yang-Baxter equations and Hom-Yang-Baxter systems

“…In [29,31] Yau proposed the definition of quasitriangular Hom-Hopf algebras and showed that each quasitriangular Hom-Hopf algebra yields a solution of the Hom-Yang-Baxter equation. Meanwhile, several classes of solutions of the Hom-Yang-Baxter equation were constructed from different respects, including those associated to Hom-Lie algebras [5,25,29,30], Drinfeld (co)doubles [2,34,35] and Hom-Yetter-Drinfeld modules [3,10,13,14,18,26,33].…”

The Hom-Long dimodule category and nonlinear equations

“…The Hom-Yang-Baxter equation reduces to the usual Yang-Baxter equation when the twist map is trivial. Several classes of solutions of the Hom-Yang-Baxter equation were constructed from different respects, including those associated to Hom-Lie algebras [6,28,31,32], Drinfelds (co)doubles [3,37,38], and Hom-Yetter-Drinfeld modules [4,13,17,18,22,29,34].…”