Title:

LieRinehart algebras, Hopf algebroids with and without an antipode

Our main objects of study are Lie{Rinehart algebras, their enveloping algebras and their relation with other structures (Gerstenhaber algebras, Hopf algebroids, Leibniz algebras and algebroids). In particular we focus on two aspects: 1. In the same way that the universal enveloping algebra of a Lie algebra carries a Hopf algebra structure, the universal enveloping algebra of a LieRinehart algebra is one of the richest class of examples of Hopf algebroids (a generalisation of Hopf algebras). We prove that, unlike in the classical Lie algebra case, the universal enveloping algebra of LieRinehart algebras may or may not admit an antipode. We use the characterisation due to Kowalzig and Posthuma [KP11] of the antipode on the Hopf algebroid structure on the enveloping algebra of a LieRinehart algebra in terms of left (and right) modules over its enveloping algebra [Hue98] and give examples of LieRinehart algebras that do not admit these right modules structures and hence no antipode on the universal enveloping algebra of a LieRinehart algebra. Moreover, we prove that some LieRinehart algebras admit a structure weaker than right modules over its enveloping algebra which yields a generator of the corresponding Gerstenhaber algebra while not a squarezero one, hence not a differential. Our examples of these algebras arise when considering Jacobi algebras [Kir76, Lic78], a certain generalisation of Poisson algebras. 2. Following the work of Loday and Pirashvili [LP98] in which they analyse the functorial relation between Lie algebras in the category LM of linear maps (which they define) and Leibniz algebras, we study the relation between LieRinehart algebras and Leibniz algebroids [IdLMP99]: After describing LieRinehart algebra objects in the category LM of linear maps, we construct a functor from LieRinehart algebra objects in LM to Leibniz algebroids.
