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Cyclic Lie-Rinehart algebras

Daniel Beltita, Alina Dobrogowska, Grzegorz Jakimowicz

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.

Cyclic Lie-Rinehart algebras

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.
Paper Structure (4 sections, 13 theorems, 80 equations)

This paper contains 4 sections, 13 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. The basic problem
  3. Covariant differential operators
  4. Duality pairings and pre-Lie-Rinehart algebra structures

Key Result

Lemma 2.9

Every cyclic submodule of $T_R$ is a subalgebra of the Lie algebra $T_R$.

Theorems & Definitions (54)

  • Definition 2.1: Ri63
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4
  • Example 2.5: Lie algebroids
  • Example 2.6: abstract tangent bundle
  • Definition 2.8
  • Lemma 2.9
  • proof
  • Remark 2.10
  • ...and 44 more