Versal deformation rings of modules over Brauer tree algebras

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]/(t²). In the case where Λ is a star Brauer tree algebra and V is an indecomposable Λ-module such that the K-dimension of Ext¹_Λ(V,V) is equal to R, we explicitly determine generators of an ideal J of k[[t₁,....,t_r]] such that R(Λ,V)≅k[[t₁,....,t_r]]/J.