Hom-\(\mathcal{O}\)-operators and Hom-Yang-Baxter equations (Q499811)

Summary: In Hom-Lie set, we introduce the concept of Hom-\(\mathcal O\)-operators and study its relation with classical Hom-Yang-Baxter equation, as well as left-symmetric Hom-algebras. We construct the corresponding relation between left-symmetric Hom-algebras and Hom-1-cocycles, which are both related to classical Hom-Yang-Baxter equation. Moreover, in Hom-algebra setting, we establish the equivalent relationship between AHYBE (associative Hom-Yang-Baxter equations) and \(\mathcal O\)-operators on Frobenius monoidal Hom-algebras.