The aim of this note is to communicate a simple example of a Lie-Rinehart algebra whose enveloping algebra is not a Hopf algebroid, neither in the sense of Böhm and Szlachányi, nor in the sense of Lu.

We give examples of Lie-Rinehart algebras whose enveloping algebra is not a full Hopf algebroid in the sense of Bohm and Szlachanyi. We construct these examples as quotients of a canonical Lie-Rinehart algebra over a Jacobi algebra which does admit an antipode

No abstract

Abstract. We describe Lie-Rinehart algebras in the tensor category LM of linear maps in the sense of Loday and Pirashvili and construct a functor from Lie-Rinehart algebras in LM to Leibniz algebroids.