$\text{F-manifolds}$, F$_\text{man}$-algebras and $\text{Poisson-algebra}$ Distributions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914300158803968 |
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| author | Castañeda-Montoya, Santiago Torres-Gomez, Alexander |
| author_facet | Castañeda-Montoya, Santiago Torres-Gomez, Alexander |
| contents | This paper investigates the geometric and algebraic interplay between F-manifolds and a newly defined class of structures termed F$_\text{man}$-algebras. We specialize our study to the category of F-Lie groups, characterized by a Lie group whose associated commutative and associative product of vector fields is left-invariant. We construct a canonical connection on Lie groups uniquely determined by the F$_\text{man}$-algebraic data, and subsequently characterize its curvature tensor and holonomy Lie algebra. A central feature of our investigation is the introduction of the Poisson-algebra distribution, arising from a canonical Poisson subalgebra within the F$_\text{man}$-algebra. We establish the integrability of this distribution, which induces a foliation of the F-Lie group and facilitates a local splitting theorem. The theoretical framework is illustrated through an in-depth analysis of the Heisenberg Lie algebra. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_22795 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\text{F-manifolds}$, F$_\text{man}$-algebras and $\text{Poisson-algebra}$ Distributions Castañeda-Montoya, Santiago Torres-Gomez, Alexander Differential Geometry Symplectic Geometry This paper investigates the geometric and algebraic interplay between F-manifolds and a newly defined class of structures termed F$_\text{man}$-algebras. We specialize our study to the category of F-Lie groups, characterized by a Lie group whose associated commutative and associative product of vector fields is left-invariant. We construct a canonical connection on Lie groups uniquely determined by the F$_\text{man}$-algebraic data, and subsequently characterize its curvature tensor and holonomy Lie algebra. A central feature of our investigation is the introduction of the Poisson-algebra distribution, arising from a canonical Poisson subalgebra within the F$_\text{man}$-algebra. We establish the integrability of this distribution, which induces a foliation of the F-Lie group and facilitates a local splitting theorem. The theoretical framework is illustrated through an in-depth analysis of the Heisenberg Lie algebra. |
| title | $\text{F-manifolds}$, F$_\text{man}$-algebras and $\text{Poisson-algebra}$ Distributions |
| topic | Differential Geometry Symplectic Geometry |
| url | https://arxiv.org/abs/2511.22795 |