Universal enveloping algebras of Lie-Rinehart algebras as a left adjoint functor

TL;DR

This paper demonstrates that the universal enveloping algebra constructions for Lie-Rinehart and anchored Lie algebras are naturally left adjoint functors, providing a categorical perspective that clarifies their universal properties and related structures.

Contribution

It establishes the universal enveloping algebra constructions as left adjoint functors, offering a new categorical understanding of Lie-Rinehart algebra structures.

Findings

Universal enveloping algebra constructions are left adjoint functors.

Provides new categorical insights into Lie-Rinehart algebra morphisms.

Clarifies the definitions of modules and gauge algebras in this context.

Abstract

We prove how the universal enveloping algebra constructions for Lie-Rinehart algebras and anchored Lie algebras are naturally left adjoint functors. This provides a conceptual motivation for the universal properties these constructions satisfy. As a supplement, the categorical approach offers new insights into the definitions of Lie-Rinehart algebra morphisms, of modules over Lie-Rinehart algebras and of the infinitesimal gauge algebra of a module.