$D$-bialgebras, dendrification and embeddings into AWB of almost Poisson algebras

TL;DR

This paper introduces almost Poisson D-bialgebras, explores their structures and equivalences, and shows how almost Poisson algebras can embed into AWB algebras using averaging operators.

Contribution

It defines almost Poisson D-bialgebras, establishes their key properties, and introduces almost tridendriform Poisson algebras, expanding the algebraic framework of Poisson structures.

Findings

Established the equivalence between matched pairs, Manin triples, and almost Poisson D-bialgebras.

Introduced almost tridendriform Poisson algebras as structures linked to Rota-Baxter operators.

Proved that every almost Poisson algebra can embed into an AWB algebra via averaging operators.

Abstract

An algebra with bracket ({\sf AWB} for short) is an associative algebra endowed with a bilinear bracket satisfying a Leibniz-type compatibility condition, as introduced in \cite{casas}. It can be viewed as a noncommutative generalization of an almost Poisson algebra; indeed, when the associative product is commutative and the bracket is skew-symmetric, one recovers the notion of an almost Poisson algebra. In this paper, we introduce the notion of {almost Poisson Drinfel'd bialgebras ($D$-bialgebras)} as an analogue of Poisson $D$-bialgebras, and we establish the equivalence between matched pairs, Manin triples, and almost Poisson $D$-bialgebras. Furthermore, we define a new algebraic structure, called {almost tridendriform Poisson algebras}, which can be regarded as the underlying algebraic structures associated with relative Rota-Baxter operators on almost Poisson algebras. Finally, we show that every almost Poisson algebra can be embedded into an algebra with bracket ({\sf AWB}) via averaging operators, and more generally via relative averaging operators associated to a given representation of the almost Poisson algebra.