Braided Hom-Lie bialgebras

TL;DR

This paper introduces braided Hom-Lie bialgebras, generalizing existing structures, and develops a unified product construction, solving extension problems and exploring special cases within this new framework.

Contribution

It presents the concept of braided Hom-Lie bialgebras and a unified product construction, extending the theory of Hom-Lie bialgebras and solving related extension problems.

Findings

Introduced braided Hom-Lie bialgebras as a new generalization.

Developed a unified product construction for Hom-Lie bialgebras.

Solved the Agore-Militaru extension problem for Hom-Lie bialgebras.

Abstract

We introduce the new concept of braided Hom-Lie bialgebras which is a generalization of Sommerh\"{a}user-Majid's braided Lie bialgebras and Yau's Hom-Lie bialgebras. Using this concept we give the unified product construction for Hom-Lie bialgebras which can be seen as a Hom-Lie version of Bespalov-Drabant's cocycle cross product bialgebras. Some special cases of unified products such as crossed product and matched pair of braided Hom-Lie bialgebras are investigated. As an application, we solve the Agore-Militaru extending problem for Hom-Lie bialgebras by using some non-abelian cohomology theory. Furthermore, one dimensional flag extending structures for Hom-Lie bialgebras are also investigated.