Our main objects of study are Lie{Rinehart algebras, their enveloping algebras and theirrelation with other structures (Gerstenhaber algebras, Hopf algebroids, Leibniz algebrasand algebroids). In particular we focus on two aspects:1. In the same way that the universal enveloping algebra of a Lie algebra carries aHopf algebra structure, the universal enveloping algebra of a Lie-Rinehart algebrais one of the richest class of examples of Hopf algebroids (a generalisation of Hopfalgebras). We prove that, unlike in the classical Lie algebra case, the universalenveloping algebra of Lie-Rinehart algebras may or may not admit an antipode.We use the characterisation due to Kowalzig and Posthuma [KP11] of the antipodeon the Hopf algebroid structure on the enveloping algebra of a Lie-Rinehart algebrain terms of left (and right) modules over its enveloping algebra [Hue98] and giveexamples of Lie-Rinehart algebras that do not admit these right modules structuresand hence no antipode on the universal enveloping algebra of a Lie-Rinehart algebra.Moreover, we prove that some Lie-Rinehart algebras admit a structure weaker thanright modules over its enveloping algebra which yields a generator of the correspondingGerstenhaber algebra while not a square-zero one, hence not a differential. Ourexamples of these algebras arise when considering Jacobi algebras [Kir76, Lic78], acertain generalisation of Poisson algebras.2. Following the work of Loday and Pirashvili [LP98] in which they analyse the functorialrelation between Lie algebras in the category LM of linear maps (which theydefine) and Leibniz algebras, we study the relation between Lie-Rinehart algebrasand Leibniz algebroids [IdLMP99]: After describing Lie-Rinehart algebra objects inthe category LM of linear maps, we construct a functor from Lie-Rinehart algebraobjects in LM to Leibniz algebroids.
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Lie-Rinehart algebras, Hopf algebroids with and without an antipode
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