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.

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
• 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: https://theses.gla.ac.uk/id/eprint/6510