Lie-Rinehart algebras, Hopf algebroids with and without an antipode

Rovi, Ana (2015) Lie-Rinehart 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 Lie-Rinehart 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 Lie-Rinehart 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 Lie-Rinehart algebra in terms of left (and right) modules over its enveloping algebra [Hue98] and give examples of Lie-Rinehart algebras that do not admit these right modules structures and 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 than right modules over its enveloping algebra which yields a generator of the corresponding Gerstenhaber algebra while not a square-zero 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 Lie-Rinehart algebras and Leibniz algebroids [IdLMP99]: After describing Lie-Rinehart algebra objects in the category LM of linear maps, we construct a functor from Lie-Rinehart algebra objects in LM to Leibniz algebroids.

Item Type: Thesis (PhD)
Qualification Level: Doctoral
Keywords: Lie algebras, Hopf algebroids, Leibniz algebras, Jacobi algebras
Subjects: Q Science > QA Mathematics
Colleges/Schools: College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Funder's Name: UNSPECIFIED
Supervisor's Name: Kraehmer, Dr. Ulrich
Date of Award: 2015
Depositing User: Ms Ana Rovi
Unique ID: glathesis:2015-6510
Copyright: Copyright of this thesis is held by the author.
Date Deposited: 25 Jun 2015 10:10
Last Modified: 25 Jun 2015 10:25
URI: http://theses.gla.ac.uk/id/eprint/6510

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