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

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.

Grantor dc:publisher

Wake Forest University

Year dc:date.issued

2009

Author dc:creator

• Allman, Justin

• Hopf algebra

Language dc:language.iso

en_US

Handle dc:identifier.uri

http://hdl.handle.net/10339/14807

OAI identifier oai:identifier

oai:wakespace.lib.wfu.edu:10339/14807