Nijenhuis deformations of Poisson algebras and $F$-manifold algebras
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910512311173120 |
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| author | Baishya, Anusuiya Das, Apurba |
| author_facet | Baishya, Anusuiya Das, Apurba |
| contents | The notion of pre-Poisson algebras was introduced by Aguiar in his study of zinbiel algebras and pre-Lie algebras. In this paper, we first introduce NS-Poisson algebras as a generalization of both Poisson algebras and pre-Poisson algebras. An NS-Poisson algebra has an associated sub-adjacent Poisson algebra. We show that a Nijenhuis operator and a twisted Rota-Baxter operator on a Poisson algebra deforms the structure into an NS-Poisson algebra. The semi-classical limit of an NS-algebra deformation and a suitable filtration of an NS-algebra produce NS-Poisson algebras. On the other hand, F-manifold algebras were introduced by Dotsenko as the underlying algebraic structure of F-manifolds. We also introduce NS-F-manifold algebras as a simultaneous generalization of NS-Poisson algebras, F-manifold algebras and pre-F-manifold algebras. In the end, we show that Nijenhuis deformations of F-manifold algebras and the semi-classical limits of NS-pre-Lie algebra deformations have NS-F-manifold algebra structures. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_18496 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Nijenhuis deformations of Poisson algebras and $F$-manifold algebras Baishya, Anusuiya Das, Apurba Rings and Algebras Mathematical Physics Quantum Algebra 17B63, 17B40, 16S80 The notion of pre-Poisson algebras was introduced by Aguiar in his study of zinbiel algebras and pre-Lie algebras. In this paper, we first introduce NS-Poisson algebras as a generalization of both Poisson algebras and pre-Poisson algebras. An NS-Poisson algebra has an associated sub-adjacent Poisson algebra. We show that a Nijenhuis operator and a twisted Rota-Baxter operator on a Poisson algebra deforms the structure into an NS-Poisson algebra. The semi-classical limit of an NS-algebra deformation and a suitable filtration of an NS-algebra produce NS-Poisson algebras. On the other hand, F-manifold algebras were introduced by Dotsenko as the underlying algebraic structure of F-manifolds. We also introduce NS-F-manifold algebras as a simultaneous generalization of NS-Poisson algebras, F-manifold algebras and pre-F-manifold algebras. In the end, we show that Nijenhuis deformations of F-manifold algebras and the semi-classical limits of NS-pre-Lie algebra deformations have NS-F-manifold algebra structures. |
| title | Nijenhuis deformations of Poisson algebras and $F$-manifold algebras |
| topic | Rings and Algebras Mathematical Physics Quantum Algebra 17B63, 17B40, 16S80 |
| url | https://arxiv.org/abs/2403.18496 |