Geometry & Cluster Algebras: Finite Type Classification & Double Bruhat Cells

Abstract

In 2000, Fomin and Zelevinsky introduced a new language called cluster algebras for describing rings with certain combinatorial structures. Cluster algebras enjoy a variety of nice properties such as well-established collection of classification results and interesting geometric properties with the upper cluster algebra. We approach cluster algebras rst from the perspective of operations on quivers, then reacquaint ourselves with a more general definition. We then present the classification of cluster algebras of nite type and explore cluster algebra structures on the ring of regular functions of double Bruhat cells. Show more