TY  - THES
AB  - 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: <br /><br />

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?<br /><br />

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.
DA  - 2017
KW  - (Quantum) Cluster Algebra
KW  - Cluster Category
KW  - (Maximal) Green Sequences
KW  - Representation Theory of Quivers
LA  - eng
PY  - 2017
TI  - Sequential structures in cluster algebras and representation theory
UR  - https://nbn-resolving.org/urn:nbn:de:0070-pub-29121975
Y2  - 2024-11-24T19:51:38
ER  -