Representations and actions of Hopf algebras

Abstract

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.