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

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.