Generalizations of Cluster Algebras from Triangulated Surfaces

Abstract

Cluster algebras, first defined by Fomin and Zelevinsky have proven to be a fascinating mathematical object, with connections in a wide range of mathematical fields as well as in physics. Many cluster algebras with desirable properties arise from triangulated surfaces. The goal of this thesis is to explore various generalizations of the narrative of cluster algebras from surfaces and to see which nice properties continue to hold. One fruitful direction is working with generalized cluster algebras from triangulated orbifolds. We give a combinatorial proof of positivity for such generalized cluster algebras by generalizing the snake graphs to the orbifold setting. We also show that our snake graphs give good expansions for generalized arcs and closed curves in an orbifold by showing the expansion formula satisfies the skein relations. We then turn to some applications of this snake graph expansion formula. The first is a generalization of the Markov numbers. Markov numbers can be seen as cluster variables from a certain cluster algebra with each initial cluster variable set to 1. We take a natural generalization of this cluster algebra and again specialize the initial variables to 1 to produce this new family of numbers. Since this generalized cluster algebra comes from an orbifold, we can explicitly describe the shape of the snake graphs corresponding to these numbers. The second application of the snake graph expansion formula is algebraic. Caldero and Chapoton gave a map which takes a module over an algebra and produces a Laurent polynomial; in some cases, the image of this map will be in a cluster algebra related to the original algebra. Using the snake graph expansion formula, we study the result of this map on a class of algebras determined by certain orbifolds. We also look to a different generalization by studying friezes and frieze patterns on surfaces with dissections, following work by Holm and Jorgensen. Caldero and Chapoton showed a connection between frieze patterns and cluster algebras. We study the space of all frieze patterns from dissections and give an algorithm to determine when an arbitrary frieze pattern comes from a dissection. We also give a combinatorial interpretation of the entries.