Document Type

Dissertation

Date of Degree

Summer 2012

Degree Name

PhD (Doctor of Philosophy)

Degree In

Mathematics

First Advisor

Frauke Bleher

Abstract

Deformation theory studies the behavior of mathematical objects, such as representations or modules, under small perturbations. This theory is useful in both pure and applied mathematics and has been used in the proof of many long-standing problems. In particular, in number theory Wiles and Taylor used universal deformation rings of Galois representations in the proof of Fermat's Last Theorem. The main motivation for determining universal deformation rings of modules for finite dimensional algebras is that deep results from representation theory can be used to arrive at a better understanding of deformation rings. In this thesis, I study the universal deformation rings of certain modules for algebras of dihedral type of polynomial growth which have been completely classied by Erdmann and Skowronski using quivers and relations.

More precisely, let κ be an algebraically closed field and let λ be a κ-algebra of dihedral type which is of polynomial growth. In this thesis, first classify all λ-modules whose stable endomorphism ring is isomorphic to κ and which are given combinatorially by strings, and then I determine the universal deformation ring of each of these modules.

Keywords

deformation theory, finite-dimensional algebras, quivers, representation theory, universal deformation rings

Pages

vii, 75 pages

Bibliography

Includes bibliographical references (page 75).

Copyright

Copyright 2012 Shannon Nicole Talbott

Included in

Mathematics Commons

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