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# Actions of finite dimensional non-commutative, non-cocommutative Hopf algebras on rings

## Electronic Theses and Dissertations

### Item Details

abstract
In 1954, Shephard and Todd showed that if $A$ is a polynomial ring and $G$ is a finite group acting as automorphisms on $A$, then the ring of invariants $A^G=\{a\in A : g\cdot a = a, \forall g\in G\}$ is again a polynomial ring exactly when $G$ is generated by reflections. The major goal of this thesis is the computation of several examples en route to a conjecture for an analogous result regarding the ring of invariants for some class of "nice" algebras under finite dimensional Hopf algebra actions. We begin with an introduction to the general study of Hopf algebras and their basic properties, then explain why they are a natural choice to generalize the action of finite groups on rings. We then show that in order to generalize existing theories, we must consider actions of "nontrivial" Hopf algebras, in particular, those that are not isomorphic to group rings or their duals. We compute several examples of such actions, and in particular, we prove that there are no actions of nontrivial semisimple Hopf algebras with dimension less than or equal to 15 on polynomial algebras.
subject
Hopf algebra
invariant theory
reflection group
contributor
Kirkman, Ellen (committee chair)
Howards, Hugh (committee member)
Robinson, Stephen (committee member)
date
2009-05-08T17:30:36Z (accessioned)
2010-06-18T18:58:54Z (accessioned)
2009-05-08T17:30:36Z (available)
2010-06-18T18:58:54Z (available)
2009-05-08T17:30:36Z (issued)
degree
Mathematics (discipline)
identifier
http://hdl.handle.net/10339/14807 (uri)
language
en_US (iso)
publisher
Wake Forest University
rights
Release the entire work immediately for access worldwide. (accessRights)
title
Actions of finite dimensional non-commutative, non-cocommutative Hopf algebras on rings
type
Thesis