Sequential structures in cluster algebras and representation theory

Gellert F. Sequential structures in cluster algebras and representation theory. Bielefeld: Universität Bielefeld; 2017.The thesis deals with a range of questions in cluster algebras and the representation theory of quivers. In particular, we provide solutions to the following problems: 1. Does a cluster algebra admit a quantisation and if it does, how unique is it? 2. What is the smallest simply-laced quiver without loops and 2-cycles whose principal extension does not admit a maximal green sequence? 3. Considering the poset of quiver representations of certain orientations of type A diagrams induced by inclusion, what is the width of such a poset? In particular, for a given cluster algebra we construct a basis of those matrices which provide a quantisation. Leading to the smallest simply-laced quiver as proposed above, we prove several combinatorial lemmas for particular quivers with up to four mutable vertices. Furthermore, we introduce a new kind of periodicity in the oriented exchange graph of principally extended cluster algebras. This periodicity we study in more detail for a particularextended Dynkin quiver of exceptional type A and show that it yields an infinite sequence of cluster tilting objects inside the preinjective component of the associated cluster category