Sequential structures in cluster algebras and representation theory

Gellert F (2017)
Bielefeld: Universität Bielefeld.

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Bielefeld Dissertation | English
Abstract
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.
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Gellert F. Sequential structures in cluster algebras and representation theory. Bielefeld: Universität Bielefeld; 2017.
Gellert, F. (2017). Sequential structures in cluster algebras and representation theory. Bielefeld: Universität Bielefeld.
Gellert, F. (2017). Sequential structures in cluster algebras and representation theory. Bielefeld: Universität Bielefeld.
Gellert, F., 2017. Sequential structures in cluster algebras and representation theory, Bielefeld: Universität Bielefeld.
F. Gellert, Sequential structures in cluster algebras and representation theory, Bielefeld: Universität Bielefeld, 2017.
Gellert, F.: Sequential structures in cluster algebras and representation theory. Universität Bielefeld, Bielefeld (2017).
Gellert, Florian. Sequential structures in cluster algebras and representation theory. Bielefeld: Universität Bielefeld, 2017.
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