Hom-associative pseudo-Yang-Baxter equations and infinitesimal Hom-$H$-pseudobialgebras

Abstract

The Hom-version of Lie $H$-pseudobialgebra was introduced in \cite{SL}. In this paper, we investigate infinitesimal Hom-$H$-pseudobialgebra, which is the Hom-associative analog of Hom-Lie $H$-pseudobialgebra. We first provide some examples of this new structure and present the construction theorems. We also consider the subclass of coboundary infinitesimal Hom-$H$-pseudobialgebras and the related Hom-associative pseudo-Yang-Baxter equation. Furthermore, the connection between infinitesimal Hom-$H$-pseudobialgebras and Hom-Lie $H$-pseudobialgebras is depicted. Finally, we show that, under suitable conditions, solutions of the Hom-associative pseudo-Yang-Baxter equation give rise to solutions of the Hom-classical pseudo-Yang-Baxter equation.