$D$-bialgebras, dendrification and embeddings into AWB of almost Poisson algebras
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866910136250925056 |
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| author | Mabrouk, Sami |
| author_facet | Mabrouk, Sami |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_15346 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | $D$-bialgebras, dendrification and embeddings into AWB of almost Poisson algebras Mabrouk, Sami Representation Theory Mathematical Physics Quantum Algebra Rings and Algebras 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. |
| title | $D$-bialgebras, dendrification and embeddings into AWB of almost Poisson algebras |
| topic | Representation Theory Mathematical Physics Quantum Algebra Rings and Algebras |
| url | https://arxiv.org/abs/2604.15346 |