Graduate Course on Cluster Algebras

SEMESTER 2, 2007/8. Lecturer: Robert Marsh

This course will take place on Wednesdays, 1-3pm in room MALL 2 in the School of Mathematics. The first lecture will be on Wednesday 23 January 2008 (note that the second hour will be in the MAGIC room for the first week only).

Abstract

A cluster algebra, in its simplest form, is a subring of the field of quotients of polynomials in a finite number of indeterminates. The algebra is defined using initial data (a sign-skew-symmetric matrix and a free generating set of the field), known as a seed. Via a combinatorial process known as mutation, a subset of the field is produced; this is then a generating set for the cluster algebra.

Cluster algebras possess an interesting combinatorial structure. For example, those whose generating set above is finite are classified by the Dynkin diagrams. The cluster algebra corresponding to a Dynkin diagram is closely connected to the corresponding root system. The number of generators of such cluster algebras can be regarded as generalised Catalan numbers; indeed there is a polytope associated to each finite type cluster algebra which can be regarded as a generalisation of the classical associahedron, or Stasheff polytope, of binary trees.

Cluster algebras were introduced by Fomin and Zelevinsky [FZ02] in order to study the dual canonical basis of the quantum group of a simple Lie algebra over the complex numbers. The canonical basis was the culmination of much effort to understand the representation theory of simple Lie algebras, and describing and understanding it is still a big open problem in Lie theory. There are many connections between cluster algebras and other fields (including the representation theory of algebras and discrete integrable systems, Y-systems).

The course will be an introduction to cluster algebras from the beginning.

Some references to get started:

[FR05] Root systems and generalized associahedra, Sergey Fomin and Nathan Reading, Preprint arXiv:math/0505518v2, 2005 - notes from a summer school.
[FZ02] Cluster algebras. I. Foundations, Sergey Fomin and Andrei Zelevinsky, J. Amer. Math. Soc. 15 (2002), no. 2, 497--529 - the paper first defining and studying cluster algebras.
[FZ03] Cluster algebras. II. Finite type classification, Sergey Fomin and Andrei Zelevinsky, Invent. Math. 154 (2003), no. 1, 63--121 - classification of cluster algebras of finite type via root systems.
The Cluster Algebras Portal contains much information about cluster algebras, related papers, conferences, etc.