Algebra and Representation Theory Seminar
Introduction to cluster algebras
Alyssa Leone, University of Oklahoma
Friday, November 18, 2011
A cluster algebra is a certain subalgebra of the field of rational functions in n variables. Generators are constructed by a series of exchange relations, which in turn induce all relations satisfied by the generators. Defined in 2000 by Fomin and Zelevinsky, the notion of a cluster algebra was motivated by the study of total positivity of matrices. Since then they have been discovered and applied in various areas, such as Poisson geometry, PDE (KP solutions), statistical physics, and Teichmuller theory. The main current areas of application include representation theory of quivers, combinatorics of root systems and Coxeter groups, and triangulations of compact surfaces. This summer I attended a two-week workshop on cluster algebras and cluster combinatorics at MSRI. In this talk, I will look at the basics and construction of cluster algebras, then focus on cluster algebras from triangulations of compact surfaces. In particular, we will see how cluster algebras can be obtained from polygons, and then generalize the polygon case to obtain cluster algebras from surfaces.