#### Document Type

Dissertation

#### Date of Degree

Summer 2015

#### Degree Name

PhD (Doctor of Philosophy)

#### Degree In

Mathematics

#### First Advisor

Frauke Bleher

#### Abstract

This thesis applies methods from the representation theory of finite dimensional algebras, specifically Brauer tree algebras, to the study of versal deformation rings of modules for these algebras. The main motivation for studying Brauer tree algebras is that they generalize *p*-modular blocks of group rings with cyclic defect groups.

We consider the case when Λ is a Brauer tree algebra over an algebraically closed field *K* and determine the structure of the versal deformation ring *R(Λ,V)* of every indecomposable Λ-module *V* when the Brauer tree is a star whose exceptional vertex is at the center. The ring *R(Λ,V)* is a complete local commutative Noetherian *K*-algebra with residue field *K*, which can be expressed as a quotient ring of a power series algebra over *K* in finitely many commuting variables. The defining property of *R(Λ,V)* is that the isomorphism class of every lift of *V* over a complete local commutative Noetherian *K*-algebra *R* with residue field *K* arises from a local ring homomorphism α *: R(Λ, V )→R* and that α is unique if *R* is the ring of dual numbers *k[t]/(t ^{2})*. In the case where Λ is a star Brauer tree algebra and

*V*is an indecomposable Λ-module such that the

*K*-dimension of Ext

^{1}

_{Λ}(V,V) is equal to

*R*, we explicitly determine generators of an ideal

*J*of

*k*[[t

_{1},....,t

_{r}]] such that

*R(Λ,V)≅*.

*k*[[t_{1},....,t_{r}]]/*J*#### Public Abstract

This thesis applies methods from the representation theory of finite dimensional algebras, specifically Brauer tree algebras, to the study of versal and universal deformation rings of representations. A Brauer tree algebra Λ over an algebraically closed field *k* can be described by a directed graph together with relations. A representation *V* of Λ is given by assigning a finite dimensional vector space to each vertex and a linear map to each arrow of the directed graph such that the relations of the algebra are satisfied. The first step is to lift such a representation *V* to various rings *R* that are quotient rings of power series rings over *k* in finitely many commuting variables. The goal is then to use these lifts to find what is called the versal deformation ring *R*(Λ, *V*) of *V*. This ring has the property that it can be used to describe the isomorphism class of every lift of *V* over any ring *R* as above. In this thesis, we determine the versal deformation ring of every indecomposable representation of a Brauer tree algebra in the case where the Brauer tree is a star whose exceptional vertex is at the center.

#### Keywords

publicabstract, algebra, representation theory

#### Pages

ix, 110 pages

#### Bibliography

Includes bibliographical references (pages 109-110).

#### Copyright

Copyright 2015 Daniel Joseph Wackwitz