Universal central extensions of Lie-Rinehart algebras

TL;DR

This paper investigates the universal central extension of Lie-Rinehart algebras, providing a description, exploring automorphism and derivation liftings, and defining a related non-abelian tensor product.

Contribution

It introduces a new non-abelian tensor product for Lie-Rinehart algebras and relates it to universal central extensions, expanding the algebraic framework.

Findings

Describes the universal central extension explicitly.

Establishes conditions for lifting automorphisms and derivations.

Defines a non-abelian tensor product and connects it to central extensions.

Abstract

In this paper we study the universal central extension of a Lie--Rinehart algebra and we give a description of it. Then we study the lifting of automorphisms and derivations to central extensions. We also give a definition of a non-abelian tensor product in Lie--Rinehart algebras based on the construction of Ellis of non-abelian tensor product of Lie algebras. We relate this non-abelian tensor product to the universal central extension.