Bialgebra theory and $\mathcal O$-operators of admissible Hom-Poisson algebras
Karima Benali
TL;DR
This work develops a comprehensive framework for admissible Hom-Poisson algebras, introducing bialgebra theory, matched pairs, and Manin triples in the Hom setting. It establishes deep connections between admissible Hom-Poisson algebras and their dual and pre-Poisson structures via representations, semi-direct products, and coboundary constructions. A key contribution is the characterization of admissible Hom-Poisson bialgebras through standard Manin triples and matched pairs, plus the coboundary specialization using $r$-matrices. The paper also links Hom-$\mathcal{O}$-operators to admissible Hom-pre-Poisson algebras, providing a mechanism to transfer Poisson-type structures into pre-Poisson frameworks. Collectively, these results extend classical Poisson-bialgebra theory to the Hom setting and unify several interconnected algebraic structures through duality, representations, and operator theory.$
Abstract
In this paper, we present and explore several key concepts within the framework of Hom-Poisson algebras. Specifically, we introduce the notions of admissible Hom-Poisson algebras, along with the related ideas of matched pairs and Manin triples for such algebras. We then define the concept of a purely admissible Hom-Poisson bialgebra, placing particular emphasis on its compatibility with the Manin triple structure associated with a nondegenerate symmetric bilinear form. This compatibility is crucial for understanding the structural interplay between these algebraic objects. Additionally, we investigate the notion of Hom-$ \mathcal O$-operators acting on admissible Hom-Poisson algebras. We analyze their properties and establish a connection with admissible Hom-pre-Poisson algebras, shedding light on the relationship between these two structures.