Dual representations and integrable structures in differential Lie bialgebras

This paper introduces the concept of differential Lie bialgebras, emphasizing the fundamental role of representations of differential Lie algebras on their dual spaces. It is established that such structures can be completely characterized through matched pairs and Manin triples of differential Lie algebras. In the coboundary framework, we further derive the notion of admissible classical Yang-Baxter equations adapted to the differential setting, where skew-symmetric solutions naturally generate differential Lie bialgebras. Moreover, by defining 𝒪-operators on differential Lie algebras, we provide a constructive method for obtaining skew-symmetric solutions to these admissible Yang-Baxter equations.