BiHom-structures And Hom-Yang-Baxter Equations

Motivated by recent works on BiHom-algebras,we introduce two new solutions of a Hom-Yang-Baxter equation by using tools of BiHom-structures.On the one hand,we first give a definition for a Yetter-Drinfeld module over a BiHom-bialgebra H and study the structure of HHBYD of the category of Yetter-Drinfeld modules over H.Then we prove that one of its subcategories HHBYDb is a quasi-braided tensor category,by which we can get a new solution of a Hom-Yang-Baxter equation.On the other hand,we first define a quasi-triangular BiHom-bialgebra and discuss some properties of it.Then we make a connection between a Yetter-Drinfeld module and a quasi-triangular BiHom-bialgebra.At last,we prove that the universal R-matrix R of a quasi-triangular BiHom-bialgebra satisfies quantum BiHom-Yang-Baxter equations,by which we can get another new solution of a Hom-Yang-Baxter equation.