The larger area of my thesis is Algebra; more specifically, my work belongs to the following two major branches of Algebra:
\emph{representation theory} and \emph{invariant theory}.
In brief, the objective of representation theory is to investigate algebraic objects through
their actions on vector spaces;
this allows the well-developed toolkit of linear algebra to be brought to bear on
complex algebraic problems.
The theory has played a crucial role in nearly every subdiscipline of pure mathematics.
Outside of pure mathematics,
representation theory has been successfully used, for instance,
in the study of symmetries of physical systems
and in describing molecular structures in physical chemistry.
Invariant theory is another classical algebraic theme permeating virtually all areas
of pure mathematics and some areas of applied mathematics as well, notably coding theory.
The theory studies actions of algebraic objects, traditionally groups and Lie algebras,
on algebras, that is, vector spaces that are equipped with a multiplication.
\bigskip
The representation theory of (associative) algebras provides a useful setting in which to studymany aspects of the two most classical flavors of representation theory under a common umbrella:
representations of groups and of Lie algebras. However,
it turns out that general algebras fail to capture certain features of group representations
and the same can be said for representations of Lie algebras as well.
The additional structure that is needed in order to access these features is
naturally provided by the important class of \emph{Hopf algebras}.
Besides unifying the representation theories of groups and of Lie algebras, Hopf algebras serve a similar
purpose in invariant theory,
allowing for a simultaneous treatment of group actions (by automorphisms)
and Lie algebras (by derivations) on algebras.
More importantly, actions of Hopf algebras have the potential of capturing additional aspects
of the structure of algebras they act on, uncovering features that cannot be
accessed by ordinary
groups or Lie algebras.
\bigskip
Presently, the theory of Hopf algebras is still nowhere near thelevel that has been achieved for groups and for Lie algebras over the course of the past century
and earlier.
This thesis aims to make a contribution to the representation and invariant theories of Hopf algebras,
focusing for the most part on Hopf algebras that are not necessarily
finite dimensional.
Specifically, the contributions presented here can be grouped under two headings:
\smallskip
\noindent\qquad(i) \textbf{ Invariant Theory:} Hopf algebra actions and prime spectra, and\smallskip
\noindent\qquad(ii)\textbf{ Representation Theory:} the adjoint representation of a Hopf algebra.
\smallskip
In the work done under the heading (i), we were able to use the action of cocommutative Hopf algebras on other algebras to "stratify" the prime spectrum of the algebra being acted upon, and then express each stratum in terms of the spectrum of a commutative domain. Additionally, we studied the transfer of properties between an ideal in the algebra being acted upon, and the largest sub-ideal of that ideal, stable under the action. We were able to achieve results for various families of acting Hopf algebras, namely \emph{cocommutative} and \emph{connected} Hopf algebras.\\The main results concerning heading (ii) concerned the subalgebra of locally finite elements of a Hopf algebra, often called the finite part of the Hopf algebra. This is a subalgebra containing the center that was used successfully to study the ring theoretic properties of group algebras, Lie algebras, and other classical structures.
We prove that the finite is not only a subalgebra, but a coideal subalgebra in general, and in the case of (almost) cocommuative Hopf algebra, it is indeed a Hopf subalgebra.
The results in this thesis generalize earlier theorems that were proved for the prototypical special classes of Hopf algebras: group algebras and enveloping algebras of Lie algebras. / Mathematics
| Identifer | oai:union.ndltd.org:TEMPLE/oai:scholarshare.temple.edu:20.500.12613/6439 |
| Date | January 2021 |
| Creators | Yammine, Ramy |
| Contributors | Lorenz, Martin, 1951-, Dolgushev, Vasily, Letzter, E. S. (Edward S.), 1958-, Riseborough, Peter |
| Publisher | Temple University. Libraries |
| Source Sets | Temple University |
| Language | English |
| Detected Language | English |
| Type | Thesis/Dissertation, Text |
| Format | 70 pages |
| Rights | IN COPYRIGHT- This Rights Statement can be used for an Item that is in copyright. Using this statement implies that the organization making this Item available has determined that the Item is in copyright and either is the rights-holder, has obtained permission from the rights-holder(s) to make their Work(s) available, or makes the Item available under an exception or limitation to copyright (including Fair Use) that entitles it to make the Item available., http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | http://dx.doi.org/10.34944/dspace/6421, Theses and Dissertations |
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