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

TL;DR

This paper explores the relationship between F-manifolds, F$_ ext{man}$-algebras, and Poisson-algebra distributions, introducing new structures and analyzing their geometric and algebraic properties within Lie groups.

Contribution

It defines F$_ ext{man}$-algebras, constructs a canonical connection on F-Lie groups, and introduces Poisson-algebra distributions with proven integrability and foliation properties.

Findings

Canonical connection characterized by F$_ ext{man}$-algebra data

Poisson-algebra distribution is integrable and induces a foliation

Analysis of the Heisenberg Lie algebra exemplifies the framework

Abstract

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.