Bialgebras, Manin triples of Malcev-Poisson algebras and post-Malcev-Poisson algebras
Fattoum Harrathi, Sami Mabrouk, Nasser Nawel, Sergei Silvestrov
TL;DR
The work extends Malcev-Poisson theory by formulating Malcev-Poisson bialgebras and establishing their equivalence with matched pairs and Manin triples, thereby enriching the duality and compatibility framework for nonassociative Poisson-like structures. It introduces post-Malcev-Poisson algebras as the underlying objects for weighted relative Rota-Baxter operators, linking operator theory to bialgebraic and cohomological perspectives. The paper further develops operator forms of the MPYBE, showing how $r\in A\otimes A$ solutions generate and relate to $\,\mathcal{O}$-operators, and demonstrates the bidirectional bridge between MPYBE solutions and weighted Rota-Baxter constructions. Overall, it provides a cohesive algebraic panorama connecting Malcev-Poisson algebras, their bialgebraic extensions, and post-operator algebraic structures with potential applications in integrable systems and nonassociative geometry.
Abstract
A Malcev-Poisson algebra is a Malcev algebra together with a commutative associative algebra structure related by a Leibniz rule. In this paper, we introduce the notion of Malcev-Poisson bialgebra as an analogue of a Malcev bialgebra and establish the equivalence between matched pairs, Manin triples and Malcev-Poisson bialgebras. Moreover, we introduce a new algebraic structure, called post-Malcev-Poisson algebras. Post-Malcev-Poisson algebras can be viewed as the underlying algebraic structures of weighted relative Rota-Baxter operators on Malcev-Poisson algebras.