Poisson superbialgebras

TL;DR

This paper introduces Poisson superbialgebras, extending Lie superbialgebra concepts to Poisson superalgebras, and explores their structures, representations, and related equations, establishing foundational theoretical connections.

Contribution

It defines Poisson superbialgebras, develops their structural theory, and links them to Manin triples, Yang-Baxter equations, and $ ext{O}$-operators, advancing the algebraic framework.

Findings

Established the equivalence between Manin triples and matched pairs of Poisson superalgebras.

Connected Poisson superbialgebras with classical and associative Yang-Baxter equations.

Introduced $ ext{O}$-operators and post-Poisson superalgebras, revealing their relationships.

Abstract

We introduce the notion of Poisson superbialgebra as an analogue of Drinfeld's Lie superbialgebras. We extend various known constructions dealing with representations on Lie superbialgebras to Poisson superbialgebras. We introduce the notions of Manin triple of Poisson superalgebras and Poisson superbialgebras and show the equivalence between them in terms of matched pairs of Poisson superalgebras. A combination of the classical Yang-Baxter equation and the associative Yang-Baxter equation is discussed in this framework. Moreover, we introduce notions of $\mathcal{O}$-operator of weight $\lambda\in\mathbb{K}$ of a Poisson superalgebra and post-Poisson superalgebra and interpret the close relationships between them and Poisson superbialgebras.