The Hom-Yang-Baxter equation, Hom-Lie algebras, and quasi-triangular bialgebras

We study a twisted version of the Yang-Baxter equation, called the Hom-Yang-Baxter equation (HYBE), which is motivated by Hom-Lie algebras. Three classes of solutions of the HYBE are constructed, one from Hom-Lie algebras and the others from Drinfeld's (dual) quasi-triangular bialgebras. Each solution of the HYBE can be extended to operators that satisfy the braid relations. Assuming an invertibility condition, these operators give a representation of the braid group