Universal central extensions of Lie–Rinehart algebras

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.2010 Mathematics Subject Classification. 17B55.

Hom-Lie-Rinehart algebras

“…The above structure gives the categorical product in the category hLR φ A . Note that cartesian product is not the product in this category as expected from the case of Lie-Rinehart algebras [3].…”

On split regular Hom-Leibniz-Rinehart algebras

“…The concept of a Hom-Lie-Rinehart algebra has a geometric analogue which is nowadays called a Hom-Lie algebroid in [2] and [15]. See also [6,[21][22][23]27] for other works on Hom-Lie-Rinehart algebras.…”

On 3-Hom-Lie-Rinehart algebras

“…It presented the notion of extensions of Hom-Lie-Rinehart algebras and deduced a characterisation of low dimensional cohomology spaces in terms of the group of automorphisms of certain abelian extensions and the equivalence classes of those abelian extensions in the category of Hom-Lie-Rinehart algebras in [18]. Later, as a generalization of [4], they introduced a non-abelian tensor product in the category of Hom-Lie-Rinehart algebras and interpreted universal central extensions (and universal α-central extensions) in terms of this non-abelian tensor product in [17]. They also explored a relationship between Hom-Lie-Rinehart algebras and Hom-Batalin-Vilkovisky algebras in [16].…”