Poly-Jacobi Manifolds: the Dimensioned Approach to Jacobi Geometry

TL;DR

This paper introduces poly-Jacobi manifolds, a new algebraic framework for Jacobi geometry using dimensioned Poisson algebras, linking geometric mechanics with dimensional analysis.

Contribution

It develops a formalism based on dimensioned algebra to algebraically interpret Jacobi manifolds and generalizes to poly-Jacobi manifolds, connecting geometry with algebraic and physical concepts.

Findings

Dimensioned Poisson algebras capture Jacobi manifold structures.

Coisotropic reduction corresponds to algebraic reduction in this framework.

Poly-Jacobi manifolds relate to physical quantities in dimensional analysis.

Abstract

The standard formulation of Jacobi manifolds in terms of differential operators on line bundles, although effective at capturing most of the relevant geometric features, lacks a clear algebraic interpretation similar to how Poisson algebras are understood to be the algebraic counterpart of Poisson manifolds. We propose a formalism, based on the dimensioned algebra technology recently developed by the author, to capture the algebraic counterparts of Jacobi manifolds as dimensioned Poisson algebras. Particularly, we give a generalisation of the functor $\text{C}^\infty:\textsf{Smooth}\to \textsf{Ring}$ for line bundles and Jacobi manifolds, we show that coisotropic reduction of Jacobi manifolds provides an example of algebraic reduction of dimensioned Poisson algebras and we discuss the relation between products of Jacobi manifolds and the tensor products of their associated dimensioned Poisson algebras. These results motivate the definition of the category of poly-line bundles, defined as collections of line bundles over the same base manifold, and the corresponding generalisation of Jacobi structures into this context, the so-called poly-Jacobi manifolds. We present some preliminary results about these generalisations and discuss potential future lines of research. An interesting outcome of this research is a somewhat surprising connection between geometric mechanics and dimensional analysis. This is suggested by the fact that if one assumes a phase space to be a poly-Jacobi manifold, the observables -- which are the dimensioned analogue of real-valued functions -- carry a natural partial addition and total multiplication structure identical to the usual algebra of physical quantities in dimensional analysis.