Geometric tool kit for higher spin gravity (part II): An introduction to Lie algebroids and their enveloping algebras

TL;DR

This paper introduces Lie algebroids and their enveloping algebras, proposing a framework for higher-spin gauge symmetries that generalizes traditional Lie algebra approaches.

Contribution

It provides a self-contained introduction to Lie algebroids, connecting them to higher-spin gauge symmetries and enveloping algebras, expanding the mathematical tools for gauge theories.

Findings

Lie algebroids can model higher-spin gauge symmetries

Enveloping algebras of Lie algebroids generalize rigid higher-spin algebras

The framework links gauge symmetries to algebraic structures of vector bundles

Abstract

These notes provide a self-contained introduction to Lie algebroids, Lie-Rinehart algebras and their universal envelopes. This review is motivated by the speculation that higher-spin gauge symmetries should admit a natural formulation as enveloping algebras of Lie algebroids since rigid higher-spin algebras are enveloping algebras of Lie algebras. Nevertheless, the material covered here may be of general interest to anyone interested in the description of gauge symmetries, connections and covariant derivatives, in terms of Lie algebroids. In order to be self-contained, a concise introduction to the algebraic characterisation of vector bundles as projective modules over the algebra of functions on the base manifold is provided.