Cluster algebras and their symmetries: an introduction

Cluster algebras were introduced in 2002 by S. Fomin and
A. Zelevinsky. These algebras are combinatorial structures which appear
in different contexts, from the theory of Lie groups to Teichmüller theory.
Ideal triangulations of boarded surfaces with punctures give rise to some
cluster algebras. We introduce and study the notion of a cluster
automorphism. In the case of cluster algebras arising from triangulations
of surfaces, we relate the group of cluster automorphisms to the mapping
class group of the corresponding surface. (This is a joint work with
I. Assem and R. Schiffer.) I will start with an introduction to the theory
of cluster algebras.