Cyclic Lie-Rinehart algebras

TL;DR

This paper introduces cyclic Lie-Rinehart algebras, a new class of algebraic structures linked to differential operators in physics, constructed via duality pairings and cyclic submodules.

Contribution

It defines cyclic Lie-Rinehart algebras and explores their connection to differential operators in mathematical physics.

Findings

Constructed examples of cyclic Lie-Rinehart algebras.

Linked these structures to differential operators in physics.

Provided a framework for duality-based algebraic structures.

Abstract

We study Lie-Rinehart algebra structures in the framework provided by a duality pairing of modules over a unital commutative associative algebra. Thus, we construct examples of Lie brackets corresponding to a fixed anchor map whose image is a cyclic submodule of the derivation module, and therefore we call them cyclic Lie-Rinehart algebras. In a very special case of our construction, these brackets turn out to be related to certain differential operators that occur in mathematical physics.