$\text{F-manifolds}$ and $\text{Poisson-algebra}$ Distributions
Santiago Castañeda-Montoya, Alexander Torres-Gomez
TL;DR
The work extends Poisson geometry to F-manifolds by introducing Poisson-algebra distributions and proving a Weinstein-like splitting theorem, then specializes to F-Lie groups to construct a canonical left-invariant affine connection. The curvature of this connection decomposes into a Lie-bracket component and a Leibnizator component, yielding precise holonomy descriptions and clarifying how the F-manifold structure augments the underlying Lie-group geometry. The paper also provides a detailed Heisenberg algebra example, classifying four F_man-algebra realizations and analyzing their Poisson, curvature, and holonomy properties. This framework offers a geometrically rich bridge between F-manifolds, Poisson-like distributions, and group-theoretic holonomy, with explicit structure in important low-dimensional cases.
Abstract
This paper introduces the notion of $\text{Poisson-algebra}$ distributions on $\text{F-manifolds}$ and establishes a corresponding splitting theorem for these structures. We then specialize to the case of $\text{F-Lie}$ groups. In this specialized setting, we construct a canonical connection induced by the underlying $\text{F}_\text{man}-\text{algebra}$ structure and proceed to study the associated holonomy Lie algebra.