#### Date of Award

2011

#### Degree Type

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematics

#### Advisor(s)

Dan Zacharia

#### Keywords

cluster tilted algebra, complexity, finite-dimensional algebra, tilting theory, trivial extension

#### Subject Categories

Mathematics

#### Abstract

In this thesis we study two types of complexity of modules over finite-dimensional algebras.

In the first part, we examine the Ω-complexity of a family of self-injective k-algebras where k is an algebraically closed field and Ω is the syzygy operator. More precisely, let T be the trivial extension of an iterated tilted algebra of type H. We prove that modules over the trivial extension T all have complexities either 0, 1, 2 or infinity, depending on the representation type of the hereditary algebra H. As part of the proof, we show that a stable equivalence between self-injective algebras preserves the complexity of modules.

In the second part, we study the τ-complexity of modules over cluster tilted algebras where τ is the Auslander-Reiten translate. We prove that modules over the cluster tilted algebra of type H all have complexities either 0, 1, 2 or infinity, depending on the representation type of the hereditary algebra H.

#### Access

Open Access

#### Recommended Citation

Purin, Marju, "Complexity over Finite-Dimensional Algebras" (2011). *Mathematics - Dissertations*. 62.

https://surface.syr.edu/mat_etd/62