Cluster algebras: notes for 2004 IMCC (Chonju, Korea, August 2004)

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.

This is a brief and informal introduction to cluster algebras. It roughly follows the historical path of their discovery, made jointly with A.Zelevinsky. Total positivity serves as the main motivation.

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

This is an expanded version of the notes of my three lectures at a NATO Advanced Study Institute ``Symmetric functions 2001: surveys of developments and perspectives" (Isaac Newton Institute for Mathematical Sciences, Cambridge, UK; June 25-July 6, 2001). Lecture I presents a unified expression due to A. Berenstein and the author for generalized Littlewood-Richardson coefficients (= tensor product multiplicities) for any complex semisimple Lie algebra. Lecture II outlines a p...

These are notes for a series of lectures presented at the ASIDE conference 2016. The definition of a cluster algebra is motivated through several examples, namely Markov triples, the Grassmannians $Gr_2(\mathbb{C})$, and the appearance of double Bruhat cells in the theory of total positivity. Once the definition of cluster algebras is introduced in several stages of increasing generality, proofs of fundamental results are sketched in the rank 2 case. From these foundations we...

This is an extended abstract of my talk at the Oberwolfach-Workshop "Representation Theory of Finite-Dimensional Algebras" (February 6 - 12, 2005). It gives self-contained and simplified definitions of quantum cluster algebras introduced and studied in a joint work with A.Berenstein (math.QA/0404446).

Sergey Fomin and Andrei Zelevinsky have invented cluster algebras at the beginning of this decade with the aim of creating an algebraic framework for the study of canonical bases in quantum groups and total positivity in algebraic groups. It soon turned out that the combinatorics of cluster algebras also appear in many other subjects and notably in the representation theory of quivers and finite-dimensional algebras. In this talk, we give a concise introduction to cluster alg...

We use the representation theory of preprojective algebras to construct and study certain cluster algebras related to semisimple algebraic groups.

This paper continues the study of cluster algebras initiated in math.RT/0104151. Its main result is the complete classification of the cluster algebras of finite type, i.e., those with finitely many clusters. This classification turns out to be identical to the Cartan-Killing classification of semisimple Lie algebras and finite root systems, which is intriguing since in most cases, the symmetry exhibited by the Cartan-Killing type of a cluster algebra is not at all apparent f...

We survey some recent constructions of cluster algebra structures on coordinate rings of unipotent subgroups and unipotent cells of Kac-Moody groups. We also review a quantized version of these results.