Cluster algebras: Notes for the CDM-03 conference

Abstract

Abstract. This is an expanded version of the notes of our lectures given at the conference Current Developments in Mathematics 2003 held at Harvard University on November 21-22, 2003. We present an overview of the main definitions, results and applications of the theory of cluster algebras.

Maslov dequantization, idempotent and tropical mathematics: A brief introduction

“…Interesting applications of tropical semirings to the traditional representation theory are presented in [11,12,87]. Tropical mathematics is closely related to the very attractive and popular theory of cluster algebras founded by S. Fomin and A. Zelevinsky, see their survey paper [57]. In both cases there are relations with the traditional theory of representations of Lie groups and related topics.…”

Cluster-tilted algebras

“…There are interesting connections to their theory in many directions (see [FZ2]), amongst them to tilting modules over hereditary algebras [MRZ]. Motivated by [MRZ], tilting theory in a particular factor category of the bounded derived category D b (mod H) of the finitely generated modules over a finite dimensional hereditary algebra H over a field k was investigated in [BMRRT], along with the relationship to tilting theory for hereditary algebras.…”

The Maslov Dequantization, idempotent and tropical mathematics: a very brief introduction

“…The so-called tropical combinatorics is discussed in a large survey paper [57] of A. N. Kirillov, see also [11,137]. Tropical mathematics is closely related to the very attractive and popular theory of cluster algebras founded by S. Fomin and A. Zelevinsky, see their survey paper [39]. In both cases there are relations with the traditional theory of representations of Lie groups and related topics.…”