Rovi, Ana
(2015)
LieRinehart algebras, Hopf algebroids with and without an antipode.
PhD thesis, University of Glasgow.
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Abstract
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.
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