$\text{F-manifolds}$, F${}_{\text{man}}$-algebras and $\text{Poisson-algebra}$ Distributions

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.