Properties, deformation quantizations and bialgebras for dual pre-Poisson algebras
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866918418705285120 |
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| author | Lu, Dilei |
| author_facet | Lu, Dilei |
| contents | A dual pre-Poisson algebra is an algebraic structure that integrates a permutative algebra and a Leibniz algebra under certain compatibility conditions. As the Koszul dual notion of the pre-Poisson algebra, this structure serves as a natural generalization of the Poisson algebra. In this paper, we commence a study on dual pre-Poisson algebras from the algebraic point of view and establish a bialgebra theory for dual pre-Poisson algebras. We begin by investigating the fundamental properties of dual pre-Poisson algebras and provide several explicit constructions. In particular, we prove that the operad of dual pre-Poisson algebras is Koszul, and we compute its Hilbert-Poincaré series and codimension. Furthermore, we introduce the notion of diassociative formal deformations of permutative algebras and show that dual pre-Poisson algebras are the corresponding semi-classical limits. Moreover, we introduce dual pre-Poisson bialgebras, which are characterized both by Manin triples and by matched pairs of dual pre-Poisson algebras, thus developing a bialgebra theory for dual pre-Poisson algebras. Then our study leads to the permutative-Leibniz Yang-Baxter equation (PLYBE) that is composed of the permutative and the classical Leibniz Yang-Baxter equation. We conclude by introducing $\mathcal{O}$-operators and pre-dual pre-Poisson algebras, which provide a systematic method for constructing symmetric solutions to the PLYBE and, consequently, dual pre-Poisson bialgebras. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_11017 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Properties, deformation quantizations and bialgebras for dual pre-Poisson algebras Lu, Dilei Rings and Algebras 16S37, 16T10, 17A30, 17A32, 17B38, 17B63, 18M70 A dual pre-Poisson algebra is an algebraic structure that integrates a permutative algebra and a Leibniz algebra under certain compatibility conditions. As the Koszul dual notion of the pre-Poisson algebra, this structure serves as a natural generalization of the Poisson algebra. In this paper, we commence a study on dual pre-Poisson algebras from the algebraic point of view and establish a bialgebra theory for dual pre-Poisson algebras. We begin by investigating the fundamental properties of dual pre-Poisson algebras and provide several explicit constructions. In particular, we prove that the operad of dual pre-Poisson algebras is Koszul, and we compute its Hilbert-Poincaré series and codimension. Furthermore, we introduce the notion of diassociative formal deformations of permutative algebras and show that dual pre-Poisson algebras are the corresponding semi-classical limits. Moreover, we introduce dual pre-Poisson bialgebras, which are characterized both by Manin triples and by matched pairs of dual pre-Poisson algebras, thus developing a bialgebra theory for dual pre-Poisson algebras. Then our study leads to the permutative-Leibniz Yang-Baxter equation (PLYBE) that is composed of the permutative and the classical Leibniz Yang-Baxter equation. We conclude by introducing $\mathcal{O}$-operators and pre-dual pre-Poisson algebras, which provide a systematic method for constructing symmetric solutions to the PLYBE and, consequently, dual pre-Poisson bialgebras. |
| title | Properties, deformation quantizations and bialgebras for dual pre-Poisson algebras |
| topic | Rings and Algebras 16S37, 16T10, 17A30, 17A32, 17B38, 17B63, 18M70 |
| url | https://arxiv.org/abs/2601.11017 |